U.S. patent application number 09/974623 was filed with the patent office on 2002-05-02 for visible-object determination for interactive visualization.
Invention is credited to Sowizral, Henry, Zikan, Karel.
Application Number | 20020050990 09/974623 |
Document ID | / |
Family ID | 22122149 |
Filed Date | 2002-05-02 |
United States Patent
Application |
20020050990 |
Kind Code |
A1 |
Sowizral, Henry ; et
al. |
May 2, 2002 |
Visible-object determination for interactive visualization
Abstract
A system and method for performing visible object determination
based upon a dual search of a cone hierarchy and a bounding (e.g.
hull) hierarchy. Visualization software running on a host processor
represents space with a hierarchy of cones constructed by recursive
refinement, and represents a collection of objects with a hierarchy
of bounding hulls. The visualization software searches the cone and
hull hierarchies starting with the root cone and the root hull.
Before exploring a given cone-hull pair, a cone-restricted minimum
distance between the cone and the hull is measured and compared to
the visibility distance value of the cone. Only when the former is
smaller than the latter will the cone be searched against the
hull.
Inventors: |
Sowizral, Henry; (Los Altos,
CA) ; Zikan, Karel; (Seattle, WA) |
Correspondence
Address: |
Jeffrey C. Hood
Conley, Rose, & Tayon, P.C.
P.O. Box 398
Austin
TX
78767
US
|
Family ID: |
22122149 |
Appl. No.: |
09/974623 |
Filed: |
October 9, 2001 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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09974623 |
Oct 9, 2001 |
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09247466 |
Feb 9, 1999 |
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6300965 |
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60074868 |
Feb 17, 1998 |
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Current U.S.
Class: |
345/421 |
Current CPC
Class: |
G06T 15/30 20130101;
G06T 15/10 20130101 |
Class at
Publication: |
345/421 |
International
Class: |
G06T 015/40 |
Claims
1. A method for displaying visible objects on a display device, the
method comprising: searching (a) a bounding hierarchy generated
from a collection of objects and (b) a cone hierarchy, to determine
nearest objects for a subset of cones of the cone hierarchy; and
displaying the nearest objects for the subset of cones of the cone
hierarchy.
2. The method of claim 1, wherein said subset of cones comprises
the leaf cones of the cone hierarchy.
3. The method of claim 1, wherein, for a first cone in the cone
hierarchy and a first bound in the bounding hierarchy, said
searching includes: determining a first measurement value of
separation between the first cone and the first bound; determining
whether the first measurement value satisfies an inequality
condition with respect to a measurement value associated with the
first cone; searching the first bound with respect to the first
cone in response to an affirmative determination that the first
measurement value satisfies the inequality condition with respect
to the measurement value associated with the first cone.
4. The method of claim 3 further comprising constructing the
bounding hierarchy from the collection of objects by: clustering
said objects to form clusters; bounding each object and each
cluster with a corresponding bound; allocating a node in the
bounding hierarchy for each object and each cluster; associating
parameters with each node, wherein the parameters associated with
the node describe the corresponding bound; organizing the nodes so
that node relationships represent cluster membership.
5. The method of claim 4, wherein the bound for each object and
each cluster is a hull.
6. The method of claim 3, wherein said determining the first
measurement of separation between the first cone and the first
bound comprises computing a penalty of separation between the first
cone and the first bound.
7. The method of claim 6, wherein said computing the penalty of
separation between the first cone and the first bound comprises
performing computations which minimize an increasing function of
separation distance between the vertex of the first cone and points
in the intersection of the first cone and the first bound.
8. The method of claim 7, wherein said determining whether the
first measurement value satisfies the inequality condition
comprises determining whether the first measurement value is
smaller than the measurement value associated with the first
cone.
9. The method of claim 7, wherein the increasing function of
separation distance comprises a distance measure given by
.parallel.s.parallel., wherein s is a displacement vector
representing the separation between the vertex of the first cone
and a point in the intersection of the first cone and the first
bound, wherein .parallel..multidot..parallel. denotes a vector
norm.
10. The method of claim 3, wherein said determining the first
measurement of separation between the first cone and the first
bound comprises computing a merit of separation between the first
cone and the first bound.
11. The method of claim 10, wherein said computing the merit of
separation between the first cone and the first bound comprises
performing computations which maximize a decreasing function of
separation distance between the vertex of the first cone and points
in the intersection of the first cone and the first bound.
12. The method of claim 11, wherein said determining whether the
first measurement value satisfies the inequality condition
comprises determining whether the first measurement value is larger
than the measurement value associated with the first cone.
13. The method of claim 3, wherein said determining the first
measurement value of separation between the first cone and the
first bound comprises solving a nonlinear programming problem using
a first set of constraints defining the first cone and a second set
of constraints defining the first bound in order to determine an
extremal distance between the vertex of the first cone and points
in the intersection of the first cone and the first bound.
14. The method of claim 13, wherein said solving a nonlinear
programming problem comprises solving a linear programming
problem.
15. The method of claim 13, wherein said solving a nonlinear
programming problem comprises solving a quadratic programming
problem.
16. The method of claim 3, wherein said cone hierarchy is generated
by refining an initial cone.
17. The method of claim 16, wherein said initial cone contains a
view frustum.
18. The method of claim 17, wherein said initial cone is a view
frustrum.
19. The method of claim 16, wherein said initial cone is the entire
space.
20. The method of claim 3, wherein said searching the cone
hierarchy and the bounding hierarchy comprises performing a level
order search of the cone hierarchy.
21. The method of claim 3, wherein said searching the cone
hierarchy and the bounding hierarchy comprises performing a level
order search of the bounding hierarchy.
22. The method of claim 3, wherein said searching the cone
hierarchy and the bounding hierarchy comprises performing an
iterative search of said cone hierarchy.
23. The method of claim 3, wherein said searching the cone
hierarchy and the bounding hierarchy comprises performing an
iterative search of the bounding hierarchy.
24. The method of claim 3, wherein said searching the cone
hierarchy and the bounding hierarchy comprises performing a
recursive search of said cone hierarchy.
25. The method of claim 3, wherein said searching the cone
hierarchy and the bounding hierarchy comprises performing a
recursive search of the bounding hierarchy.
26. The method of claim 3, further comprising initializing each
cone of the cone hierarchy with a measurement value which
corresponds to un-occluded visibility.
27. The method of claim 3, wherein said searching the first bound
with respect to the first cone comprises: determining whether or
not said first cone is a leaf of the cone hierarchy and said first
bound is a leaf of the bounding hierarchy; setting the measurement
value associated with the first cone equal to the first measurement
value of separation between the first bound and the first cone;
setting a visible object attribute associated with the first cone
equal to the first bound; wherein said setting of the measurement
value associated with the first cone and said setting of the
visible object attribute associated with the first cone are
performed in response to an affirmative determination that the
first cone is a leaf of the cone hierarchy and the first bound is a
leaf of the hull hierarchy.
28. The method of claim 3, wherein said searching the first bound
with respect to the first cone comprises: determining whether said
first cone is a leaf of the cone hierarchy and said first bound is
not a leaf of the bounding hierarchy; conditionally exploring
sub-bounds of said first bound with respect to said first cone in
response to an affirmative determination that said first cone is a
leaf of the cone hierarchy and said first bound is not a leaf of
the bounding hierarchy.
29. The method of claim 28, wherein said conditionally exploring
sub-bounds of said first bound includes: computing a cone-bound
separation value for each of the sub-bounds of the first bound with
respect to said first cone; conditionally searching said sub-bounds
of said first bound with respect to said first cone in ascending
order of separation from the first cone.
30. The method of claim 29, wherein said conditionally searching
said sub-bounds of said first bound includes: determining whether
the cone-bound separation value of a first subbound among said
subbounds satisfies the inequality condition with respect to the
measurement value associated with the first cone; searching said
first sub-bound with respect to said first cone in response to an
affirmative determination that said cone-bound separation value of
said first sub-bound satisfies the inequality condition with
respect to the measurement value associated with the first
cone.
31. The method of claim 3, wherein said searching of the first
bound with respect to the first cone includes: determining if the
first bound is a leaf of the bounding hierarchy and the first cone
is not a leaf of the cone hierarchy; conditionally searching
subcones of the first cone with respect to the first bound in
response to an affirmative determination that said first bound is a
leaf of the bounding hierarchy and said first cone is not a leaf of
the cone hierarchy.
32. The method of claim 31, wherein said conditionally searching
subcones of the first cone with respect to the first bound
includes: computing a cone-bound separation value for the first
bound with respect to a first subcone of the first cone;
determining whether the cone-bound separation value satisfies the
inequality condition with respect to a measurement value associated
with the first subcone; searching said first subcone with respect
to said first bound in response to an affirmative determination
that said cone-bound separation value satisfies the inequality
condition with respect to the measurement value associated with the
first subcone.
33. The method of claim 32, wherein said searching the first bound
with respect to the first cone further comprises setting the
measurement value associated with the first cone equal to an
extremum of the measurement values associated with said subcones
after said conditionally searching said subcones with respect to
the first bound.
34. The method of claim 3, wherein said searching the first bound
with respect to the first cone includes: computing a first
cone-bound separation value for the first bound with respect to a
first subcone of the first cone; determining whether the first
cone-bound separation value satisfies the inequality condition with
respect to a measurement value associated with said first subcone;
conditionally exploring subbounds of said first bound with respect
to said first subcone in response to an affirmative determination
that said first cone-bound separation value satisfies the
inequality condition with respect to the measurement value
associated with the first subcone.
35. The method of claim 34, wherein said conditionally exploring
subbounds of said first bound with respect to said first subcone
includes: computing a second cone-bound separation value for each
of said subbounds with respect to said first subcone; conditionally
searching said subbounds of said first bound with respect to said
first subcone in ascending order of their separation from the first
subcone.
36. The method of claim 35, wherein said conditionally searching
said subbounds of said first bound with respect to said first
subcone includes: determining whether the second cone-hull
separation value for a first subbound among said subbounds
satisfies the inequality condition with respect to the measurement
value associated with the first subcone; searching said first
subbound with respect to said first subcone in response to an
affirmative determination that said second cone-hull separation
value for the first subbound satisfies the inequality condition
with respect to the measurement value associated with the first
subcone.
37. The method of claim 36, wherein said searching the first bound
with respect to the first cone further comprises setting the
measurement value associated with the first cone equal to an
extremum of the measurement values associated with subcones of the
first cone including said first subcone.
38. The method of claim 2, wherein each of the leaf cones of the
cone hierarchy subtends an area of a screen on the display
device.
39. The method of claim 38, wherein said area contains one or more
pixels.
40. The method of claim 38, wherein said area corresponds to a
portion of a pixel.
41. A computer system for displaying visible objects, the computer
system comprising: a display device; a memory for storing a
visibility search program; a processor coupled to the memory and
configured to execute the visibility search program, wherein, in
response to execution of the visibility search program, the
processor is configured to search (a) a bounding hierarchy
generated from a collection of objects and (b) a cone hierarchy, to
determine nearest objects for a subset of cones of the cone
hierarchy; wherein the display device is operable to display the
nearest objects for the subset of cones in the cone hierarchy.
42. The computer system of claim 41, wherein the subset of cones
comprises the leaf cones of the cone hierarchy.
43. The computer system of claim 41, wherein, in response to
execution of said visibility search program, said processor is
operable to search the cone hierarchy and the bounding hierarchy
by: determining a first measurement value of separation between a
first cone of the cone hierarchy and a first bound of the bounding
hierarchy; determining whether the first measurement value
satisfies an inequality condition with respect to a measurement
value associated with the first cone; searching the first bound
with respect to the first cone in response to an affirmative
determination that the first measurement value satisfies the
inequality condition with respect to the measurement value
associated with the first cone.
44. The computer system of claim 43 further comprising a graphics
accelerator coupled to the processor, wherein said processor is
configured to transmit visibility information including a
specification of the nearest object for each leaf cone of the cone
hierarchy to the graphics accelerator, wherein the graphics
accelerator is configured to render an image on the display screen
based on the nearest object for each leaf cone of the cone
hierarchy.
45. The computer system of claim 43, wherein, in response to
execution of said visibility search program, said processor is
operable to construct the bounding hierarchy by: clustering said
objects to form clusters; bounding each object and each cluster
with a corresponding bound; allocating a node in the bounding
hierarchy for each object and each cluster; associating parameters
with each node, wherein the parameters associated with each node
describe the corresponding bound; and organizing the nodes so that
node relationships represent cluster membership.
46. The computer system of claim 45, wherein the bound for each
object and each cluster is a hull.
47. The computer system of claim 43, wherein, in order to determine
the first measurement of separation between the first cone and the
first bound, the processor computes a penalty of separation between
the first cone and the first bound.
48. The computer system of claim 47, wherein, in computing the
penalty of separation between the first cone and the first bound,
the processor performs computations which minimize an increasing
function of separation distance between the vertex of the first
cone and points in the intersection of the first cone and the first
bound.
49. The computer system of claim 48, wherein, in determining
whether the first measurement value satisfies the inequality
condition, the processor determines whether the first measurement
value is smaller than the measurement value associated with the
first cone.
50. The computer system of claim 48, wherein the increasing
function of separation distance comprises a distance measure given
by .parallel.s.parallel., wherein s is a displacement vector
representing the separation between the vertex of the first cone
and a point in the intersection of the first cone and the first
bound, wherein .parallel..multidot..parallel. denotes a vector
norm.
51. The computer system of claim 43, in determining the first
measurement of separation between the first cone and the first
bound, the processor computes a merit of separation between the
first cone and the first bound.
52. The computer system of claim 51, wherein, in computing the
merit of separation between the first cone and the first bound, the
processor performs computations which maximize a decreasing
function of separation distance between the vertex of the first
cone and points in the intersection of the first cone and the first
bound.
53. The computer system of claim 52, wherein, in determining
whether the first measurement value satisfies the inequality
condition, the processor determines whether the first measurement
value is larger than the measurement value associated with the
first cone.
54. The computer system of claim 43, wherein, in determining the
first measurement value of separation between the first cone and
the first bound, the processor solves a nonlinear programming
problem using a first set of constraints defining the first cone
and a second set of constraints defining the first bound in order
to determine an extremal distance between the vertex of the first
cone and points in the intersection of the first cone and the first
bound.
55. The computer system of claim 54, wherein the nonlinear
programming problem comprises a linear programming problem.
56. The computer system of claim 54, wherein the nonlinear
programming problem comprises a quadratic programming problem.
57. The computer system of claim 43, wherein the processor is
further operable to construct the cone hierarchy by refining an
initial cone.
58. The computer system of claim 57, wherein said initial cone
contains a view frustum.
59. The computer system of claim 58, wherein said initial cone is a
view frustum.
60. The computer system of claim 57, wherein said initial cone is
the entire space.
61. The computer system of claim 43, wherein, in searching the cone
hierarchy and the bounding hierarchy, the processor performs a
level order search of the cone hierarchy.
62. The computer system of claim 43, wherein, in searching the cone
hierarchy and the bounding hierarchy, the processor performs a
level order search of the bounding hierarchy.
63. The computer system of claim 43, wherein, in searching the cone
hierarchy and the bounding hierarchy, the processor performs an
iterative search of said cone hierarchy.
64. The computer system of claim 43, wherein, in searching the cone
hierarchy and the bounding hierarchy, the processor performs an
iterative search of the bounding hierarchy.
65. The computer system of claim 43, wherein, in searching the cone
hierarchy and the bounding hierarchy, the processor performs a
recursive search of said cone hierarchy.
66. The computer system of claim 43, wherein, in searching the cone
hierarchy and the bounding hierarchy, the processor performs a
recursive search of the bounding hierarchy.
67. The computer system of claim 43 wherein said processor
initializes each cone of the cone hierarchy with a measurement
value which corresponds to un-occluded visibility.
68. The computer system of claim 43, wherein, in searching the
first bound with respect to the first cone, the processor is
configured to: determine whether or not said first cone is a leaf
of the cone hierarchy and said first bound is a leaf of the
bounding hierarchy; set the measurement value associated with the
first cone equal to the first measurement value of separation
between the first bound and the first cone; set a visible object
attribute associated with the first cone equal to the first bound;
wherein said setting of the measurement value associated with the
first cone and said setting of the visible object attribute
associated with the first cone are performed in response to an
affirmative determination that the first cone is a leaf of the cone
hierarchy and the first bound is a leaf of the hull hierarchy.
69. The computer system of claim 43, wherein, in searching the
first bound with respect to the first cone, the processor is
configured to: determine whether said first cone is a leaf of the
cone hierarchy and said first bound is not a leaf of the bounding
hierarchy; conditionally explore sub-bounds of said first bound
with respect to said first cone in response to an affirmative
determination that said first cone is a leaf of the cone hierarchy
and said first bound is not a leaf of the bounding hierarchy.
70. The computer system of claim 69, wherein, in conditionally
exploring sub-bounds of said first bound, the processor is
configured to: compute a cone-bound separation value for each of
the sub-bounds of the first bound with respect to said first cone;
conditionally search said sub-bounds of said first bound with
respect to said first cone in ascending order of separation from
the first cone.
71. The computer system of claim 70, wherein, in conditionally
searching said sub-bounds of said first bound, the processor is
configured to: determine whether the cone-bound separation value of
a first subbound among said subbounds satisfies the inequality
condition with respect to the measurement value associated with the
first cone; search said first sub-bound with respect to said first
cone in response to an affirmative determination that said
cone-bound separation value of said first sub-bound satisfies the
inequality condition with respect to the measurement value
associated with the first cone.
72. The computer system of claim 43, wherein, in searching the
first bound with respect to the first cone, the processor is
configured to: determine if the first bound is a leaf of the
bounding hierarchy and the first cone is not a leaf of the cone
hierarchy; conditionally search subcones of the first cone with
respect to the first bound in response to an affirmative
determination that said first bound is a leaf of the bounding
hierarchy and said first cone is not a leaf of the cone
hierarchy.
73. The computer system of claim 72, wherein, in conditionally
searching subcones of the first cone with respect to the first
bound, the processor is configured to: computing a cone-bound
separation value for the first bound with respect to a first
subcone of the first cone; determining whether the cone-bound
separation value satisfies the inequality condition with respect to
a measurement value associated with the first subcone; searching
said first subcone with respect to said first bound in response to
an affirmative determination that said cone-bound separation value
satisfies the inequality condition with respect to the measurement
value associated with the first subcone.
74. The computer system of claim 73, wherein, in searching the
first bound with respect to the first cone, the processor sets the
measurement value associated with the first cone equal to an
extremum of the measurement values associated with said subcones
after said conditionally searching said subcones with respect to
the first bound.
75. The computer system of claim 43, wherein, in searching the
first bound with respect to the first cone, the processor is
configured to: compute a first cone-bound separation value for the
first bound with respect to a first subcone of the first cone;
determine whether the first cone-bound separation value satisfies
the inequality condition with respect to a measurement value
associated with said first subcone; conditionally explore subbounds
of said first bound with respect to said first subcone in response
to an affirmative determination that said first cone-bound
separation value satisfies the inequality condition with respect to
the measurement value associated with the first subcone.
76. The computer system of claim 75, wherein, in conditionally
exploring subbounds of said first bound with respect to said first
subcone, the processor is configured to: compute a second
cone-bound separation value for each of said subbounds with respect
to said first subcone; conditionally search said subbounds of said
first bound with respect to said first subcone in ascending order
of their separation from the first subcone.
77. The computer system of claim 76, wherein, in conditionally
searching said subbounds of said first bound with respect to said
first subcone, the processor is configured to: determine whether
the second cone-hull separation value for a first subbound among
said subbounds satisfies the inequality condition with respect to
the measurement value associated with the first subcone; search
said first subbound with respect to said first subcone in response
to an affirmative determination that said second cone-hull
separation value for the first subbound satisfies the inequality
condition with respect to the measurement value associated with the
first subcone.
78. The computer system of claim 77, wherein, in order to search
the first bound with respect to the first cone, the processor sets
the measurement value associated with the first cone equal to an
extremum of the measurement values associated with subcones of the
first cone including said first subcone.
79. The computer system of claim 42, wherein each of the leaf cones
of the cone hierarchy subtends an area of a screen on the display
device.
80. The computer system of claim 79, wherein said area contains one
or more pixels.
81. The computer system of claim 79, wherein said area corresponds
to a portion of a pixel.
82. A memory media which stores program instructions for
determining visible objects for display on a display device,
wherein the program instructions are executable by a processor to
implement: searching (a) a bounding hierarchy generated from a
collection of objects and (b) a cone hierarchy, to determine
nearest objects for a subset of cones of the cone hierarchy;
displaying the nearest objects for the subset of cones of the cone
hierarchy.
83. The memory media of claim 82, wherein the subset of cones
comprises the leaf cones of the cone hierarchy.
84. The memory media of claim 82, wherein, for a first cone in the
cone hierarchy and a first bound in the bounding hierarchy, said
searching includes: determining a first measurement value of
separation between the first cone and the first bound; determining
whether the first measurement value satisfies an inequality
condition with respect to a measurement value associated with the
first cone; searching the first bound with respect to the first
cone in response to an affirmative determination that the first
measurement value satisfies the inequality condition with respect
to the measurement value associated with the first cone.
85. The memory media of claim 84, wherein the program instructions
are executable by the processor to control a construction of the
bounding hierarchy by: clustering said objects to form clusters;
bounding each object and each cluster with a corresponding bound;
allocating a node in the bounding hierarchy for each object and
each cluster; associating parameters with each node, wherein the
parameters associated with the node describe the corresponding
bound; organizing the nodes so that node relationships represent
cluster membership.
86. The memory media of claim 85, wherein the bound for each object
and each cluster is a hull.
87. The memory media of claim 84, wherein said determining the
first measurement of separation between the first cone and the
first bound comprises computing a penalty of separation between the
first cone and the first bound.
88. The memory media of claim 87, wherein said computing the
penalty of separation between the first cone and the first bound
comprises performing computations which minimize an increasing
function of separation distance between the vertex of the first
cone and points in the intersection of the first cone and the first
bound.
89. The memory media of claim 88, wherein said determining whether
the first measurement value satisfies the inequality condition
comprises determining whether the first measurement value is
smaller than the measurement value associated with the first
cone.
90. The memory media of claim 88, wherein the increasing function
of separation distance comprises a distance measure given by
.parallel.s.parallel., wherein s is a displacement vector
representing the separation between the vertex of the first cone
and a point in the intersection of the first cone and the first
bound, wherein .parallel..multidot..parallel. denotes a vector
norm.
91. The memory media of claim 84, wherein said determining the
first measurement of separation between the first cone and the
first bound comprises computing a merit of separation between the
first cone and the first bound.
92. The memory media of claim 91, wherein said computing the merit
of separation between the first cone and the first bound comprises
performing computations which maximize a decreasing function of
separation distance between the vertex of the first cone and points
in the intersection of the first cone and the first bound.
93. The memory media of claim 92, wherein said determining whether
the first measurement value satisfies the inequality condition
comprises determining whether the first measurement value is larger
than the measurement value associated with the first cone.
94. The memory media of claim 84, wherein said determining the
first measurement value of separation between the first cone and
the first bound comprises solving a nonlinear programming problem
using a first set of constraints defining the first cone and a
second set of constraints defining the first bound in order to
determine an extremal distance between the vertex of the first cone
and points in the intersection of the first cone and the first
bound.
95. The memory media of claim 94, wherein said solving a nonlinear
programming problem comprises solving a linear programming
problem.
96. The memory media of claim 94, wherein said solving a nonlinear
programming problem comprises solving a quadratic programming
problem.
97. The memory media of claim 84, wherein said cone hierarchy is
generated by refining an initial cone.
98. The memory media of claim 97, wherein said initial cone
contains a view frustum.
99. The memory media of claim 98, wherein said initial cone is a
view frustum.
100. The memory media of claim 97, wherein said initial cone is the
entire space.
101. The memory media of claim 84, wherein said searching the cone
hierarchy and the bounding hierarchy comprises performing a level
order search of the cone hierarchy.
102. The memory media of claim 84, wherein said searching the cone
hierarchy and the bounding hierarchy comprises performing a level
order search of the bounding hierarchy.
103. The memory media of claim 84, wherein said searching the cone
hierarchy and the bounding hierarchy comprises performing an
iterative search of said cone hierarchy.
104. The memory media of claim 84, wherein said searching the cone
hierarchy and the bounding hierarchy comprises performing an
iterative search of the bounding hierarchy.
105. The memory media of claim 84, wherein said searching the cone
hierarchy and the bounding hierarchy comprises performing a
recursive search of said cone hierarchy.
106. The memory media of claim 84, wherein said searching the cone
hierarchy and the bounding hierarchy comprises performing a
recursive search of the bounding hierarchy.
107. The memory media of claim 84, further comprising initializing
each cone of the cone hierarchy with a measurement value which
corresponds to un-occluded visibility.
108. The memory media of claim 84, wherein said searching the first
bound with respect to the first cone comprises: determining whether
or not said first cone is a leaf of the cone hierarchy and said
first bound is a leaf of the bounding hierarchy; setting the
measurement value associated with the first cone equal to the first
measurement value of separation between the first bound and the
first cone; setting a visible object attribute associated with the
first cone equal to the first bound; wherein said setting of the
measurement value associated with the first cone and said setting
of the visible object attribute associated with the first cone are
performed in response to an affirmative determination that the
first cone is a leaf of the cone hierarchy and the first bound is a
leaf of the hull hierarchy.
109. The memory media of claim 84, wherein said searching the first
bound with respect to the first cone comprises: determining whether
said first cone is a leaf of the cone hierarchy and said first
bound is not a leaf of the bounding hierarchy; conditionally
exploring sub-bounds of said first bound with respect to said first
cone in response to an affirmative determination that said first
cone is a leaf of the cone hierarchy and said first bound is not a
leaf of the bounding hierarchy.
110. The memory media of claim 109, wherein said conditionally
exploring sub-bounds of said first bound includes: computing a
cone-bound separation value for each of the sub-bounds of the first
bound with respect to said first cone; conditionally searching said
sub-bounds of said first bound with respect to said first cone in
ascending order of separation from the first cone.
111. The memory media of claim 110, wherein said conditionally
searching said sub-bounds of said first bound includes: determining
whether the cone-bound separation value of a first subbound among
said subbounds satisfies the inequality condition with respect to
the measurement value associated with the first cone; searching
said first sub-bound with respect to said first cone in response to
an affirmative determination that said cone-bound separation value
of said first sub-bound satisfies the inequality condition with
respect to the measurement value associated with the first
cone.
112. The memory media of claim 84, wherein said searching of the
first bound with respect to the first cone includes: determining if
the first bound is a leaf of the bounding hierarchy and the first
cone is not a leaf of the cone hierarchy; conditionally searching
subcones of the first cone with respect to the first bound in
response to an affirmative determination that said first bound is a
leaf of the bounding hierarchy and said first cone is not a leaf of
the cone hierarchy.
113. The memory media of claim 112, wherein said conditionally
searching subcones of the first cone with respect to the first
bound includes: computing a cone-bound separation value for the
first bound with respect to a first subcone of the first cone; p1
determining whether the cone-bound separation value satisfies the
inequality condition with respect to a measurement value associated
with the first subcone; searching said first subcone with respect
to said first bound in response to an affirmative determination
that said cone-bound separation value satisfies the inequality
condition with respect to the measurement value associated with the
first subcone.
114. The memory media of claim 113, wherein said searching the
first bound with respect to the first cone further comprises
setting the measurement value associated with the first cone equal
to an extremum of the measurement values associated with said
subcones after said conditionally searching said subcones with
respect to the first bound.
115. The memory media of claim 84, wherein said searching the first
bound with respect to the first cone includes: computing a first
cone-bound separation value for the first bound with respect to a
first subcone of the first cone; determining whether the first
cone-bound separation value satisfies the inequality condition with
respect to a measurement value associated with said first subcone;
conditionally exploring subbounds of said first bound with respect
to said first subcone in response to an affirmative determination
that said first cone-bound separation value satisfies the
inequality condition with respect to the measurement value
associated with the first subcone.
116. The memory media of claim 115, wherein said conditionally
exploring subbounds of said first bound with respect to said first
subcone includes: computing a second cone-bound separation value
for each of said subbounds with respect to said first subcone;
conditionally searching said subbounds of said first bound with
respect to said first subcone in ascending order of their
separation from the first subcone.
117. The memory media of claim 116, wherein said conditionally
searching said subbounds of said first bound with respect to said
first subcone includes: determining whether the second cone-hull
separation value for a first subbound among said subbounds
satisfies the inequality condition with respect to the measurement
value associated with the first subcone; searching said first
subbound with respect to said first subcone in response to an
affirmative determination that said second cone-hull separation
value for the first subbound satisfies the inequality condition
with respect to the measurement value associated with the first
subcone.
118. The memory media of claim 117, wherein said searching the
first bound with respect to the first cone further comprises
setting the measurement value associated with the first cone equal
to an extremum of the measurement values associated with subcones
of the first cone including said first subcone.
119. The memory media of claim 83, wherein each of the leaf cones
of the cone hierarchy subtends an area of a screen on the display
device.
120. The memory media of claim 119, wherein said area contains one
or more pixels.
121. The memory media of claim 119, wherein said area corresponds
to a portion of a pixel.
Description
CONTINUATION DATA
[0001] This application is a continuation of U.S. patent
application Ser. No. 09/247,466 filed on Feb. 09, 1999, U.S. Pat.
No. 6,300,965 entitled "Visible-Object Determination for
Interactive Visualization" which claims the benefit of U.S.
provisional application Ser. No. 60/074,868 filed on Feb. 17, 1998
entitled "Visible-Object Determination for Interactive
Visualization".
FIELD OF THE INVENTION
[0002] The present invention relates generally to the field of
computer graphics, and more particularly, to the problem of
determining the set of objects/surfaces visible from a defined
viewpoint in a graphics environment.
BACKGROUND OF THE INVENTION
[0003] Visualization software has proven to be very useful in
evaluating three-dimensional designs long before the physical
realization of those designs. In addition, visualization software
has shown its cost effectiveness by allowing engineering companies
to find design problems early in the design cycle, thus saving them
significant amounts of money. Unfortunately, the need to view more
and more complex scenes has outpaced the ability of graphics
hardware systems to display them at reasonable frame rates. As
scene complexity grows, visualization software designers need to
carefully use the rendering resource provided by graphic hardware
pipelines.
[0004] A hardware pipeline wastes rendering bandwidth when it
discards triangle work. Rendering bandwidth waste can be decreased
by not asking the pipeline to draw triangles that it will discard.
Various software methods for reducing pipeline waste have evolved
over time. Each technique reduces waste at a different point within
the pipeline. As examples, software frustum culling can
significantly reduce discards in a pipeline's clipping computation
while software backface culling can reduce discards in a pipeline's
lighting computation.
[0005] The z-buffer is the final part of the graphics pipeline that
discards work. In essence, the z-buffer retains visible surfaces
and discards those not visible. As scene complexity increases,
especially in walk through and CAD environments, the number of
occluded surfaces rises rapidly and as a result the number of
surfaces that the z-buffer discards rises as well. A frame's
average depth complexity determines roughly how much work (and thus
rendering bandwidth) the z-buffer discards. In a frame with a
per-pixel depth complexity of d the pipeline's effectiveness is
1/d. As depth complexity rises, the hardware pipeline thus becomes
proportionally less and less effective.
[0006] Software occlusion culling has been proposed as an
additional tool for improving rendering effectiveness. A
visualization program which performs occlusion culling effectively
increases the graphic hardware's overall rendering bandwidth by not
asking the hardware pipeline to draw occluded objects. Computing a
scene's visible objects is the complementary problem to that of
occlusion culling. Rather than removing occluded objects from the
set of objects in a scene or even a frustum culled scene, a program
instead computes which objects are visible and draws just those. A
simple visualization program can compute the set of visible objects
and draw those objects from the current viewpoint, allowing the
pipeline to remove backfacing polygons and the z-buffer to remove
any non-visible surfaces.
[0007] One technique for computing the visible object set uses ray
casting. RealEyes [Sowizral, H. A., Zikan, K., Esposito, C., Janin,
A., Mizell, D. "RealEyes: A System for Visualizing Very Large
Physical Structures", SIGGRAPH '94, Visual Proceedings, 1994, p.
228], a system that implemented the ray casting technique, was
demonstrated in SIGGRAPH 1994's BOOM room. At interactive rates,
visitors could "walk" around the interior of a Boeing 747 or
explore the structures comprising Space Station Freedom's lab
module.
[0008] The intuition for the use of rays in determining visibility
relies on the properties of light. The first object encountered
along a ray is visible since it alone can reflect light into the
viewer's eye. Also, that object interposes itself between the
viewer and all succeeding objects along the ray making them not
visible. In the discrete world of computer graphics, it is
difficult to propagate a continuum of rays. So a discrete subset of
rays is invariably used. Of course, this implies that visible
objects or segments of objects smaller than the resolution of the
ray sample may be missed and not discovered. This is because rays
guarantee correct determination of visible objects only up to the
density of the ray-sample. FIG. 1 illustrates the ray-based method
of visible object detection. Rays that interact with one or more
objects are marked with a dot at the point of their first contact
with an object. It is this point of first contact that determines
the value of the screen pixel corresponding to the ray. Also
observe that the object denoted A is small enough to be entirely
missed by the given ray sample.
[0009] Visible-object determination has its roots in
visible-surface determination. Foley et al. [Foley, J., van Dam,
A., Feiner, S. and Hughes, J. Computer Graphics: Principles and
Practice, 2nd ed., Addison-Wesley, Chapter 15, pp.649-718, 1996]
divide visible-surface determination approaches into two broad
groups: image-precision and object-precision algorithms. Image
precision algorithms typically operate at the resolution of the
display device and tend to have superior performance
computationally. Object precision approaches operate in object
space-usually performing object to object comparisons.
[0010] A prototypical image-precision visible-surface-determination
algorithm casts rays from the viewpoint through the center of each
display pixel to determine the nearest visible surface along each
ray. The list of applications of visible-surface ray casting (or
ray tracing) is long and distinguished. Appel ["Some Techniques for
Shading Machine Rendering of Solids", SJCC'68, pp. 37-45, 1968]
uses ray casting for shading. Goldstein and Nagel [Mathematical
Applications Group, Inc., "3-D Simulated Graphics Offered by
Service Bureau," Datamation, 13(1), Febuaray 1968, p. 69.; see also
Goldstein, R. A. and Nagel, R. "3-D Visual Simulation", Simulation,
16(1), pp.25-31, 1971] use ray casting for boolean set operations.
Kay et al. [Kay, D. S. and Greenberg, D., "Transparency for
Computer Synthesized Images," SIGGRAPH'79, pp.158-164] and Whitted
["An Improved Illumination Model for Shaded Display", CACM, 23(6).
pp.343-349, 1980] use ray tracing for refraction and specular
reflection computations. Airey et al. [Airey, J. M., Rohlf, J. H.
and Brooks, Jr. F. P. Towards Image Realism with Interactive Update
Rates in Complex Virtual Building Environments. ACM SIGGRAPH
Symposium on Interactive 3D Graphics, 24, 2(1990), pp. 41-50] uses
ray casting for computing the portion of a model visible from a
given cell.
[0011] Another approach to visible-surface determination relies on
sending beams or cones into a database of surfaces [see Dadoun et
al., "Hierarchical approachs to hidden surface intersection
testing." Proceeedings of Graphics Interface '82, Toronto, May
1982, 49-56; see also Dadoun et al., "The geometry of beam
tracing." In Joseph O'Rourke, ed., Proceeedings of the Symposium on
Computational Geometry, pp.55-61, ACM Press, New York, 1985].
Essentially, beams become a replacement for rays. The approach
usually results in compact beams decomposing into a set of possibly
non-connected cone(s) after interacting with an object.
[0012] A variety of spatial subdivision schemes have been used to
impose a spatial structure on the objects in a scene. [The
following four references pertain to spatial subdivision schemes:
(a) Glassner, "Space subdivision for fast ray tracing," IEEE
CG&A, 4(10):15-22, October 1984; (b) Jevans et al., "Adaptive
voxel subdivision for ray tracing," Proceedings Graphics Interface
'89, 164-172, June 1989; (c) Kaplan, M. "The use of spatial
coherence in ray tracing," in Techniques for Computer Graphics . .
. , Rogers, D. and Earnshaw, R. A. (eds), Springer-Verlag, New
York, 1987; and (d) Rubin, S. M. and Whitted, T. "A 3-dimensional
representation for fast rendering of complex scenes," Computer
Graphics, 14(3):110-116, July 1980. ]
[0013] Kay et al. [Kay, T. L. and Kajiya, J. T. "Ray Tracing
Complex Scenes", SIGGRAPH 1986, pp. 269-278,1986], concentrating on
the computational aspect of ray casting, employed a hierarchy of
spatial bounding volumes in conjunction with rays, to determine the
visible objects along each ray. Of course, the spatial hierarchy
needs to be precomputed. However, once in place, such a hierarchy
facilitates a recursive computation for finding objects. If the
environment is stationary, the same data-structure facilitates
finding the visible object along any ray from any origin.
[0014] Teller et al. [Teller, S. and Sequin, C. H. "Visibility
Preprocessing for Interactive Walkthroughs," SIGGRAPH '91,
pp.61-69] use preprocessing to full advantage in visible-object
computation by precomputing cell-to-cell visibility. Their approach
is essentially an object precision approach and they report over 6
hours of preprocessing time to calculate 58 Mbytes of visibility
information for a 250,000 polygon model on a 50 MIP machine
[Teller, S. and Sequin. C. H. "Visibility computations in
polyhedral three-dimensional environments," U.C. Berkeley Report
No. UCB/CSD 92/680, April 1992 ].
[0015] In a different approach to visibility computation, Greene et
al. [Greene, N., Kass, M., and Miller, G. "Hierarchical z-Buffer
Visibility," SIGGRAPH '93, pp.231-238] use a variety of
hierarchical data structures to help exploit the spatial structure
inherent in object space (an octree of objects), the image
structure inherent in pixels (a Z pyramid), and the temporal
structure inherent in frame-by-frame rendering (a list of
previously visible octree nodes). The Z-pyramid permits the rapid
culling of large portions of the model by testing for visibility
using a rapid scan conversion of the cubes in the octree.
[0016] The depth complexity of graphical environments continues to
increase in response to consumer demand for realism and
performance. Thus, the efficiency of an algorithm for visible
object determination has a direct impact on the marketability of a
visualization system. The computational bandwidth required by the
visible object determination algorithm determines the class of
processor required for the visualization system, and thereby
effects overall system cost. Thus, a system or method for improving
the efficiency of visible object determination is greatly
desired.
SUMMARY OF THE PRESENT INVENTION
[0017] The present invention comprises a system and method for
displaying visible objects in a graphics environment. In
particular, a system and method for performing visible object
determination based upon a dual search of a cone hierarchy and a
bounding hierarchy is herein disclosed. The system includes a
processor, a display device, system memory, and optionally a
graphics accelerator. The processor executes visualization software
which provides for visualization of a collection of
three-dimensional objects on the display device. The objects reside
in a three-dimensional space and thus admit the possibility of
occluding one another.
[0018] The visualization software represents space in terms of a
hierarchy of cones emanating from the viewpoint. In one embodiment,
the leaf-cones of the hierarchy, i.e. the cones at the highest
level of refinement, subtend an area which corresponds to a
fraction of a pixel in screen area. For example, two cones may
conveniently fill the area of a pixel. Alternatively, the leaf-cone
may subtend areas which include one or more pixels.
[0019] An initial view frustum or neighborhood of the view frustum
is recursively tessellated (i.e. refined) to generate a cone
hierarchy. Alternatively, the entire space around the viewpoint may
be recursively tessellated to generate the cone hierarchy. In this
case, the cone hierarchy does not need to be recomputed for changes
in the viewpoint and view-direction.
[0020] The visualization software also generates a hierarchy of
bounds from the collection of objects. In particular, the bounding
hierarchy is generated by: (a) recursively grouping clusters
starting with the objects themselves as order-zero clusters, (b)
bounding each object and cluster (of all orders) with a
corresponding bound, e.g. a polytope hull, (c) allocating a node in
the bounding hierarchy for each object and cluster, and (d)
organizing the nodes in the bounding hierarchy to reflect cluster
membership. For example if node A is the parent of node B, the
cluster corresponding to node A contains a subcluster (or object)
corresponding to node B. Each node stores parameters which
characterize the bound of the corresponding cluster or object.
[0021] The visualization software performs a search of the cone and
bounding hierarchies starting with the root cone and the root
bound. Each leaf-cone is assigned a visibility distance value which
represents the distance to the closest known object as perceived
from within the leaf-cone. Each leaf-cone is also assigned an
object attribute which specifies the closest known object within
view of the leaf-cone. Similarly, each non-leaf cone is assigned a
visibility distance value. However, the visibility distance value
of a non-leaf cone is set equal to the maximum of the visibility
distance values for its subcone children. This implies that the
visibility distance value for each non-leaf cone equals the maximum
of the visibility distance values of its leaf-cone descendents.
[0022] The visibility software operates on cone-bound pairs. Before
exploring a given cone-bound pair, the distance between the cone
and the bound is measured. This involves determining the minimum
distance to points residing in both the bound and the cone from the
vertex of the cone. This cone-bound distance is then compared to
the visibility distance value of the cone. If the cone-bound
distance is larger than the visibility distance value of the cone,
all of the leaf-cone descendents of the given cone have known
visible objects closer than the given bound by definition of the
visibility distance value. Thus, no benefit can be gained from
exploring the cone-bound pair. In contrast, if the cone-bound
distance is smaller than the visibility distance value of the cone,
the bound may contain objects which will affect the visibility
distance values of one or more leaf-cone descendents of the given
cone. The cone-bound pair must be searched. According to the
present invention, cone-bound pairs are advantageously searched
only when there is a possibility that the given bound may affect
the visibility of the cone's descendents. Thus, the search
algorithm of the present invention avoids unnecessary cone-bound
explorations and thereby saves considerable computational
bandwidth.
[0023] Supposing that the search condition is satisfied, the bound
is explored with respect to the given cone. If the cone and bound
are both leaves of their respective hierarchies, the bound
specifies an object which is closer than the closest known object
for the leaf-cone. Thus, the visibility distance value of the
leaf-cone is updated with the cone-bound distance between the cone
and bound. Also, the object attribute for the cone is updated to
point to the given bound.
[0024] In the case that the cone is a leaf-cone and the bound is a
non-leaf bound, the search algorithm examines subbounds of the
given bound, and conditionally explores these subbounds in
ascending order of their cone-bound distance from the given cone.
Again, exploration of a subbound is conditioned upon the subbound
achieving a cone-bound distance to the given cone which is smaller
than the cone's visibility distance value.
[0025] In the case that the cone is a non-leaf cone and the bound
is a leaf bound (i.e. one which bounds a single object), the search
algorithm conditionally explores subcones of the given cone with
respect to the given bound. Exploration of a subcone is conditioned
upon the subcone achieving a cone-bound distance to the given bound
which is smaller than the subcone's visibility distance value.
[0026] In the case that the cone is a non-leaf cone and the bound
is a non-leaf bound, the search algorithm conditionally explores
subbounds of the given bound against the subcones of the given
cone. Consider a particular subcone of the given cone for the sake
of discussion. The subbounds of the given bound are conditionally
explored against the subcone in ascending order of their cone-bound
distances from the subcone. Because the closest subbound is
searched first, and potentially decreases the visibility distance
value of the given subcone, succeeding (more distant) subbounds
will have more difficulty passing the search condition, i.e. of
having a cone-bound distance to the given subcone which is less
than the visibility distance value of the subcone. Thus, the
probability is maximized that the fewest number of subbounds will
need to be explored by ordering the conditional explorations
according to cone-bound distance.
[0027] When the search of the two trees is completed, the object
attribute of each leaf-cone points to the object which is visible
to the leaf-cone, and the visibility distance value of the
leaf-cone specifies the distance to the visible object. This
visibility information is provided to the graphics accelerator so
that the graphics accelerator may render the visible objects (or
visible portions of visible object) on the display device.
[0028] In one embodiment, the visualization software provides for
interactive visualization by reading user inputs to control the
current viewpoint and view-direction in the graphics environment.
Additional software ensures efficient computation through the use
of careful state management and parallelism.
[0029] In one alternative embodiment, the cone hierarchy and
bounding hierarchy are searched iteratively. In a second
alternative embodiment, a level order search is performed on the
cone hierarchy and the bounding hierarchy.
[0030] The present invention contemplates a wide variety of
techniques for measuring the extent of separation or proximity
between a bound and a cone. One set of embodiments focus of
minimizing an increasing function of separation distance between
the vertex of the cone and points in the intersection of the cone
and the bound. Another set of embodiments involve maximizing a
decreasing function of separation distance between the vertex of
the cone and points in the intersection of the cone and the bound.
In general, any wavefront with a boundary that obeys a mild "star
shape" condition may provide the basis for a measurement of
separation between a bound and a cone.
BRIEF DESCRIPTION OF THE FIGURES
[0031] FIG. 1 illustrates the ray-based method of visible object
detection according to the prior art;
[0032] FIG. 2A illustrates a graphical computing system according
to the present invention;
[0033] FIG. 2B is a block diagram illustrating one embodiment of
the graphical computing system of the present invention;
[0034] FIG. 3 illustrates several main phases of visualization
software according to the present invention;
[0035] FIG. 4A illustrates a collection of objects in a graphics
environment;
[0036] FIG. 4B illustrates a first step in the first formation of a
hull hierarchy, i.e. the step of bounding objects with containing
hulls and allocating hull nodes for the containing hulls;
[0037] FIG. 4C illustrates the process of grouping together hulls
to form higher order hulls, and allocating nodes in the hull
hierarchy which correspond to the higher order hulls;
[0038] FIG. 4D illustrates the culmination of the recursive
grouping process wherein all objects are contained in a universal
containing hull which corresponds to the root node of the hull
hierarhcy;
[0039] FIG. 5A illustrates the mathematical expressions which
describe lines and half-planes in two dimensional space;
[0040] FIG. 5B illustrates the description of a rectangular region
as the intersection of four half-planes in a two dimensional
space;
[0041] FIG. 6 illustrates a two-dimensional cone partitioned into a
number of subcones which interact with a collection of objects by
means of wavefronts propagating within each of the subcones;
[0042] FIG. 7 illustrates polyhedral cones with rectangular and
triangular cross-section emanating from the origin;
[0043] FIG. 8A illustrates the mathematical expressions which
describe a line through the origin and a corresponding half-plane
given a normal vector in two-dimensional space;
[0044] FIG. 8B illustrates the specification of a two-dimensional
conic region as the intersection of two half-planes;
[0045] FIGS. 9A-9C illustrate the formation of a cone hierarchy
based on repeated subdivision of an initial cone with rectangular
cross-section;
[0046] FIG. 10 illustrates a visibility search algorithm according
to the present invention;
[0047] FIG. 11 illustrates a method for displaying visible objects
in a graphics environment;
[0048] FIG. 12 illustrates the process of recursively clustering a
collection of objects to form a bounding hierarchy;
[0049] FIG. 13 illustrates the processing steps to be performed
when the visibility search arrives at a terminal cone and a
terminal bound;
[0050] FIG. 14 illustrates the processing steps to be performed
when the visibility search arrives at a terminal cone and a
non-terminal bound;
[0051] FIG. 15 further elaborates step 712 of FIG. 14, i.e. the
step of conditionally exploring subbounds of the first bound with
respect to the first cone;
[0052] FIG. 16 further elaborates step 722 of FIG. 15, i.e. the
step of conditionally searching the subbounds of the first bound
with respect to the first cone, by means of a generic subbound as
an example;
[0053] FIG. 17 illustrates the processing steps to be performed
when the visibility search arrives at a terminal bound and a
non-terminal cone;
[0054] FIG. 18 further elaborates step 742 of FIG. 17, i.e. the
step of conditionally searching subcones of the first cone with
respect to the first bound, by means of a generic subcone
example;
[0055] FIG. 19 illustrates the processing steps to be performed
when the visibility search arrives at a non-terminal bound and a
non-terminal cone;
[0056] FIG. 20 further elaborates step 762 of FIG. 19, i.e. the
step of conditionally exploring subbounds of the first bound with
respect to the first subcone; and
[0057] FIG. 21 further elaborates step 772 of FIG. 20, i.e. the
step of conditionally searching the subbounds of the first bound
with respect to the first subcone, by means of a generic subhull
example.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
[0058] U.S. patent application Ser. No. 09/247,466 filed on Feb. 9,
1999 entitled "Visible-Object Determination for Interactive
Visualization" is hereby incorporated by reference in its
entirety.
[0059] FIG. 2A presents a graphical computing system 80 according
to the preferred embodiment of the present invention. Graphical
computing system 80 comprises a system unit 82, and a display
device 84 coupled to the system unit 82. The display device 84 may
be realized by any various types of video monitors or graphical
displays. Graphics computer system 80 also includes a keyboard 86
and preferably a mouse 88.
[0060] FIG. 2B is a block diagram illustrating one embodiment of
graphical computing system 80. As shown, graphical computing system
80 includes a host central processing unit (CPU) 102 coupled to a
high speed bus or system bus 104. A system memory 106 is also
preferably coupled to system bus 104. The system memory 106 may
include any of various types of memory subsystems including random
access memory, read only memory, and/or mass storage devices. The
host processor 102 executes a visualization software program which
determines the set of visible objects in a scene. The visualization
software program is stored in system memory 106. In an alternative
embodiment, the visualization software program executes on a
processor comprised within graphics accelerator 112.
[0061] An optional 3-D graphics accelerator 112 may be coupled to
system bus 104. If 3D accelerator 112 is not included in graphical
computing system 80, then display device 84 may couple directly to
system bus 104. It is assumed that various other peripheral
devices, or other buses, may be connected to system bus 104, as is
well known in the art. Display device 84 couples to 3-D graphics
accelerator 112. CPU 102 may transfer information to and from the
graphics accelerator 112 according to a programmed input/output
(I/O) protocol over the system bus 104. Furthermore, the graphics
accelerator 112 may access system memory 106 according to a direct
memory access (DMA) protocol or through intelligent bus
mastering.
[0062] 3-D graphics accelerator 112 is a specialized graphics
rendering subsystem which is designed to off-load the 3-D rendering
functions from the host processor, thus providing improved system
performance. In a system with a 3-D graphics accelerator, a
graphics application program executing on the host processor 102
generates three-dimensional geometry data that defines
three-dimensional graphics elements for output on a display device
84. The application program transfers the geometry data to the
graphics accelerator 112. The graphics accelerator 112 receives the
geometry data and renders the corresponding graphics elements on
the display device. A fundamental goal of computer graphics is to
render objects that are realistic to the user. The problem of
visual realism, however, is quite complex. Objects in the "real
world" include an incredible amount of detail, including subtle
variations in color, reflections, and surface irregularities.
Applications which display three-dimensional graphics require a
tremendous amount of processing bandwidth. For example, for a
computer system to generate smooth 3-D motion video, the computer
system is required to maintain a frame rate or update rate of
between 20 to 30 frames per second. This typically requires a 3-D
graphics accelerator capable of processing over a million triangles
per second.
[0063] The computer system 80 includes visualization software
according to the present invention. The visualization software is
preferably executed by the host processor 102, or alternatively may
be executed by a processor comprised on the graphics accelerator
112.
[0064] Visualization Software Architecture
[0065] As illustrated in FIG. 3, the visualization software of the
present invention comprises three main phases. In an initial step
210, the visualization software loads a scene. The scene includes a
plurality of objects. Scene loading includes constructing an object
hierarchy--if it has not already been constructed. In step 220 the
visualization software discovers the set of visible objects in the
scene with respect to a current viewpoint. In the preferred
embodiment, the visualization software is designed to compute
visibility for three-dimensional objects from a view point in a
three-dimensional coordinate space. However, the methodologies
herein described naturally generalize to spaces of arbitrary
dimension.
[0066] In one embodiment of the visualization software, the
viewpoint in the graphical environment may be changed by user
input. For example, by manipulating the mouse 88 and/or depressing
keys on keyboard 86, the user may cause the viewpoint and view
orientation to change. Thus, the visualization software preferably
recomputes the set of visible objects whenever the viewpoint or
view orientation changes.
[0067] In step 230 the visualization software in conjunction with
graphics accelerator 112 displays the visible objects, i.e.
commands the graphics accelerator to 112 to render those objects in
the display screen. By performing visible object determination in
software on the host computer, the graphics accelerator 112 may
operate with greatly increased efficiency. Thereby, the overall
processing capacity of the graphics system 80 is improved.
Discovery and display of the visible-object-set occur repeatedly as
the viewpoint or view direction (i.e. orientation) changes. The
discovery and display steps are preferably performed as concurrent
processes. However, in an alternative embodiment, they are
performed sequentially.
[0068] Unless otherwise stated, it is assumed that all objects in
the model are opaque convex polytopes. A three-dimensional solid is
said to be convex if any two points in the solid (or on the surface
of the solid) may be connected with a line segment which resides
entirely within the solid. Thus a solid cube is convex, while a
donut is not. A polytope is an object with planar sides (e.g. cube,
tetrahedron, etc.). The methodologies described herein for opaque
objects naturally extend to transparent or semi-transparent objects
by not allowing such objects to terminate a cone computation. The
convexity assumption presents more of a problem. However, every
object can be approximated as a union of convex polytopes. It is
helpful to note that the visible-object-set computation does not
require quire an exact computation, but rather a conservative one.
In other words, it is permissible to over-estimate the set of
visible objects.
[0069] Constructing the Object Hierarchy
[0070] Initially, the objects in the scene are organized into a
hierarchy that groups objects spatially. An octree is one
possibility for generating the object hierarchy. However, in the
preferred embodiment, a clustering algorithm is used which groups
nearby objects then recursively clusters pairs of groups into
larger containing spaces. The clustering algorithm employs a simple
distance measure and thresholding operation to achieve the object
clustering. FIGS. 4A-4D illustrate the clustering process for a
collection of four objects J00 through J11. The objects are indexed
in a fashion which anticipates their ultimate position in a binary
tree of object groups. The objects are depicted as polygons
situated in a plane (see FIG. 4A). However, the reader should
imagine these objects as arbitrary three-dimensional objects. In
one embodiment, the objects are three-dimensional polytopes.
[0071] Each object is bounded, i.e. enclosed, by a corresponding
bounding surface referred to herein as a bound. In the preferred
embodiment, the bound for each object is a polytope hull (i.e. a
hull having planar faces) as shown in FIG. 4B. The hulls H00
through H11 are given labels which are consistent with the objects
they bound. For example, hull H00 bounds object J00. The hulls are
illustrated as rectangles with sides parallel to a pair of
coordinate axes. These hulls are intended to represent rectangular
boxes (parallelopipeds) in three dimensions whose sides are normal
to a fixed set of coordinate axes. For each hull a corresponding
node data structure is generated. The node stores parameters which
characterize the corresponding hull.
[0072] Since a hull has a surface which is comprised of a finite
number of planar components, the description of a hull is
intimately connected to the description of a plane in three-space.
In FIG. 5A, a two dimensional example is given from which the
equation of an arbitrary plane may be generalized. A unit vector n
[any vector suffices but a vector of length one is convenient for
discussion] defines a line L through the origin of the two
dimensional space. By taking the dot product of a vector v with the
unit vector n, denoted v.multidot.n, one obtains the length of the
projection of vector v in the direction defined by unit vector n.
Thus, given a real constant c it follows that the equation
x.multidot.n=c, where x is a vector variable, defines a line M
perpendicular to line L and situated at a distance c from the
origin along line L. In the context of three-dimensional space,
this same equation defines a plane perpendicular to the line L,
again displaced distance c from the origin along line L. Observe
that the constant c may be negative, in which case the line (or
plane) M is displaced from the origin distance
.vertline.c.vertline. along line L in the direction opposite to
unit vector n.
[0073] The line x.multidot.n=c divides the plane into two
half-planes. By replacing the equality in the above equation with
an inequality, one obtains the description of one of these
half-planes. The equality x.multidot.n<c defines the half-plane
which contains the negative infinity end of line L. [The unit
vector n defines the positive direction of line L.] In three
dimensions, the plane x.multidot.n=c divides the three-dimensional
space into two half-spaces. The inequality x.multidot.n<c
defines the half-space which contains the negative infinity end of
line L.
[0074] FIG. 5B shows how a rectangular region may be defined as the
intersection of four half-planes. Given four normal vectors n.sub.1
through n.sub.4, and four corresponding constants c.sub.1 through
c.sub.4, a rectangular region is defined as the set of points which
simultaneously satisfy the set of inequalities
x.multidot.n.sup.i<c.su- b.i, where i ranges from one to four.
This system of inequalities may be summarized by the matrix-vector
expression N.multidot.x<c, where the rows are matrix N are the
normal vectors n.sub.1 through n.sub.4, and the components of
vector c are the corresponding constants c.sub.1 through c.sub.4.
If the normal vectors are chosen so as to lie in the positive and
negative axial directions (as shown in FIG. 5B), the resulting
rectangular region has sides parallel to the axes. It is noted that
the rectangular hulls H00 through H11 shown in FIG. 4B all use a
common set of normal vectors. Thus, each hull is characterized by
its unique c vector.
[0075] In three-dimensional space, a rectangular box may be
analogously defined as the intersection of six half-spaces. Given
six normal vectors n.sub.1 through n.sub.6, oriented in each of the
three positive and three negative axial directions, and six
corresponding constants c.sub.1 through c.sub.6, the simultaneous
solution of the inequalities x.multidot.n.sub.i<c.sub.i, where i
runs from one to six, defines a rectangular box with sides parallel
to the coordinate planes. Thus, a rectangular box may be compactly
represented with the same matrix-vector expression Nx<c, where
matrix N now has six rows for the six normal vectors, and vector c
has six elements for the six corresponding constants.
[0076] To construct an object hierarchy, object hulls H00 through
H11 are paired together as shown in FIG. 4C. Each pair of object
hulls is bounded by a containing hull. For example, hulls H00 and
H01 are paired together and bounded by containing hull H0.
Containing-hull H0 efficiently contains the two component hulls H00
and H01. Likewise, object hulls H10 and H11 are paired together and
bounded by containing-hull H1. In addition, two parent nodes are
generated in the object hierarchy, one for each of the
containing-hulls H0 and H1. For simplicity, the parent nodes are
commonly labeled as their corresponding containing-hulls. Thus,
parent node H0 points to its children nodes H00 and H01, while
parent node H1 points to its children nodes H10 and H11. Each
parent node contains the characterizing c vector for the
corresponding containing-hull.
[0077] The containing-hulls H0 and H1 may be referred to as first
order containing-hulls since they are the result of a first pairing
operation on the original object hulls. A second pairing operation
is applied to the first-order containing hulls to obtain
second-order containing-hulls. Each second-order containing-hull
efficiently contains two first-order hulls. For each of the
second-order containing-hulls a parent node is generated in the
object hierarchy. The parent node reflects the same parent-child
relationship as the corresponding second-order containing-hull. For
example, in FIG. 4D, second-order containing-hull H efficiently
contains first-order containing-hulls H0 and H1. Thus, parent node
H in the object hierarchy points to children nodes H0 and H1.
Parent node H stores the characterizing vector c for the
containing-hull H. In the example presented in FIGS. 4A-4D, the
object hierarchy is complete after two pairing operations since the
original object collections contained only four objects.
[0078] In general, a succession of pairing operations is performed.
At each stage, a higher-order set of containing-hulls and
corresponding nodes for the object hierarchy are generated. Each
node contains the describing vector c for the corresponding
containing-hull. At the end of the process, the object hierarchy
comprises a binary tree with a single root node. The root node
corresponds to a total containing-hull which contains all sub-hulls
of all orders including all the original object-hulls. The object
hierarchy, because it comprises a hierarchy of bounding hulls, will
also be referred to as the hull hierarchy. In the preferred
embodiment, the pairing operations are based on proximity, i.e.
objects (and hulls of the same order) are paired based on
proximity. Proximity based pairing results in a more efficient
visible object determination algorithm. This tree of containing
hulls provides us with a computationally efficient, hierarchical
representation of the entire scene. For instance, when a cone
completely misses a node's containing-hull, none of the node's
descendents need to be examined.
[0079] Bounding hulls (i.e. containing hulls) serve the purpose of
simplifying and approximating objects. Any hierarchy of containing
hulls works in principle. However, hierarchies of hulls based on a
common set of normal vectors are particularly efficient
computationally. A collection of hulls based on a common set of
normal vectors will be referred to herein as a fixed-direction or
commonly-generated collection. As described above, a polytope hull
is described by a bounding system of linear inequalities
{x:Nx.ltoreq.c}, where the rows of the matrix N are a set of normal
vectors, and the elements of the vector c define the distances to
move along each of the normal vectors to obtain a corresponding
side of the polytope. In a fixed-direction collection of hulls, the
normal matrix N is common to all the hulls in the collection, while
the vector c is unique for each hull in the collection. The problem
of calculating the coefficient vector c for a containing hull given
a collection of subhulls is greatly simplified when a common set of
normal vectors is used. In addition, the nodes of the hull
hierarchy may advantageously consume less memory space since the
normal matrix N need not be stored in the nodes. In the preferred
embodiment of the invention, the hull hierarchy comprises a
fixed-direction collection of hulls.
[0080] In a first embodiment, six normal vectors oriented in the
three positive and three negative axial directions are used to
generate a fixed-direction hierarchy of hulls shaped like
rectangular boxes with sides parallel to the coordinate planes.
These axis-aligned bounding hulls provide a simple representation
that has excellent local computational properties. It is easy to
transform or compare two axis-aligned hulls. However, the
approximation provided by axis-aligned hulls tends to be rather
coarse, often proving costly at more global levels.
[0081] In a second embodiment, eight normal vectors directed
towards the coners of a cube are used to generate a hierarchy of
eight-sided hulls. For example, the eight vectors (.+-.1, .+-.1,
.+-.1) may be used to generate the eight-sided hulls. The
octahedron is a special case of this hull family.
[0082] In a third embodiment, fourteen normal vectors, i.e. the six
normals which generate the rectangular boxes plus the eight normals
which generate the eight-sided boxes, are used to generate a hull
hierarchy with fourteen-sided hulls. These fourteen-sided hulls may
be described as rectangular boxes with coners shaved off. It is
noted that as the number of normal vectors and therefore side
increases, the accuracy of the hull's approximation to the
underlying object increases.
[0083] In a fourth embodiment, twelve more normals are added to the
fourteen normals just described to obtain a set of twenty-six
normal vectors. The twelve additional normals serve to shave off
the twelve edges of the rectangular box in addition to the coners
which have already been shaved off. This results in twenty-six
sided hulls. For example, the twelve normal vectors (.+-.1, .+-.1,
0), (.+-.1, 0, .+-.1), and (0, .+-.1, .+-.1) may be used as the
additional vectors.
[0084] In the examples given above, hulls are recursively grouped
in pairs to generate a binary tree. However, in other embodiments,
hulls are grouped together in groups of size G, where G is larger
than two. In one embodiment, the group size varies from group to
group. This may be particularly advantageous for scenes which have
non-uniform object density. For example, if a large number of small
objects are clustered in a scene, it may be advantageous to include
these in a single group, i.e. bound them with a single containing
hull. Larger objects may be assembled into groups with fewer
members.
[0085] Although the above discussion has focussed on the use of
polytope hulls as bounds for object and clusters, it is noted that
any type of bounding surfaces may be used, thereby generating a
hierarchy of bounds referred to herein as a bounding hierarchy.
Each node of the bounding hierarchy corresponds to an object or
cluster and stores parameters which characterize the corresponding
bound for that object or cluster. For example, polynomial surfaces
such as quadratic surfaces may be used to generate bounds for
objects and/or clusters. Spheres and ellipsoids are examples of
quadratic surfaces.
[0086] Cones in Visible Object Determination
[0087] In addition to the bounding hierarchy (e.g. hull hierarchy)
discussed above, the present invention makes use of a hierarchy of
spatial cones. An initial cone which represents the view frustum is
recursively subdivided into a hierarchy of sub-cones. Then a
simultaneous double recursion is performed through the pair of
trees (the object tree and cone tree) to rapidly determine the set
of visible objects. This cone-based method provides a substantial
computational gain over the prior art method based on ray-casting.
This is partially due to the fact that the ray-casting methods
require multiple unrelated visibility queries to cover a region
equivalent to a cone.
[0088] Cones discretize the spatial continuum differently than
rays. Consider the simultaneous propagation of all possible rays
from a point and the ensuing spherical wavefront. The first object
encountered by each ray is visible. If consideration is restricted
to those rays that form a cone, the same observation still applies.
The first object encountered by the cone's wavefront is visible.
Now, if the view frustum is partitioned into some number of cones,
the objects visible from the viewpoint can be determined up to the
resolution of the cones.
[0089] FIG. 6 illustrates a two-dimensional cone C in a
two-dimensional environment. Cone C is defined by the region
interior to the rays R1 and R2 (and inclusive of those rays). The
cone C is partitioned into a number of sub-cones. The ambient space
is populated with a collection of two-dimensional objects. A
wavefront propagates within each of the subcones. Each wavefront
terminates upon its first interaction with an object. Three of the
subcones are especially highlighted, i.e. subcones C1, C2 and C3.
Subcone C1 interacts with two objects, i.e. an ellipse and a star
shaped object. However, the wavefront propagating within subcone C1
terminates, by definition, upon its first interaction with an
object. Since the wavefront interacts with the ellipse first, the
wavefront terminates on the ellipse, and the star object is defined
to be invisible with respect to this subcone. If higher resolution
is desired, subcone C1 may be further partitioned into a set of
smaller cones. Some of these smaller cones would then terminate on
the ellipse, while others would terminate on the star-shaped
object. Subcone C2 interacts with only one object, i.e. object B.
The wavefront propagating with subcone C2 terminates at the point
of first contact with object B.
[0090] It is noted that the cone-based object visibility query
(modeled on the wavefront propagation concept) is an inherently
spatial computation. Thus, the object visibility query for subcone
C3 detects the small object A even though this object lies entirely
in the interior of subcone C3. The visibility query computation
will be explained in detail later.
[0091] Polyhedral Cones
[0092] The spatial cones used in the preferred embodiment are
polyhedral cones. The generic polyhedral cone has a polygonal
cross-section. FIG. 7 give two examples of polyhedral cones. The
first polyhedral cone PC1 has a rectangular cross-section, while
the second polyhedral cone PC2 has a triangular cross-section. The
view frustum is a cone with rectangular cross-section like cone
PC1. Polyhedral cones are defined by homogeneous linear
inequalities. Given a normal vector n, the equation n.multidot.x=0
involving vector argument x defines a plane passing through the
origin and perpendicular to the normal vector n. This plane divides
space into two half-spaces. The linear inequality n.multidot.x<0
defines the half-space from which the normal vector n points
outward. FIG. 8A gives a two-dimensional example. As shown, the
equation n.multidot.x=0 specifies the set of points (interpreted as
vectors) which are perpendicular to normal n. This perpendicular
line L divides the plane into two half-planes. The half-plane
defined by the inequality n.multidot.x<0 is denoted by shading.
Observe that the normal vector n points out of this half-plane.
[0093] A polyhedral cone is constructed by intersection of multiple
half-spaces. For example, solid cone PC2 of FIG. 7 is the
intersection of three half-spaces. Similarly, solid cone PC1 is the
intersection of four half-spaces. FIG. 8B provides a
two-dimensional example of intersecting half-planes to generate a
conic area. The two normal vectors n.sub.1 and n.sub.2define
perpendicular lines L.sub.1 and L.sub.2 respectively. The
inequality n.sub.1.multidot.x<0 specifies the half-plane which
is south-west (i.e. left and below) of the line L.sub.1. The
inequality n.sub.2.multidot.x<0 defines the half-plane which is
to the right of line L.sub.2. The solution to the simultaneous
system of inequalities n.sub.1.multidot.x<0 and
n.sub.2.multidot.x<0 is the intersection region denoted in
shading. This system of inequalities may be summarized by the
matrix equation Sx<0, where the rows of matrix S are the normal
vectors. From this discussion, it may be observed that solid cone
PC1 of FIG. 7 is determined by four normal vectors. The normal
matrix S would then have four rows (for the four normal vectors)
and three columns corresponding to the dimension of the ambient
space.
[0094] Thus, a polyhedral cone emanating from the origin is defined
as the set of points satisfying a system of linear inequalities
Sx.ltoreq.0. [There is no loss of generality in assuming the origin
to be the viewpoint.] According to this definition, half-spaces,
planes, rays, and the origin itself may be considers as polyhedral
cones. In addition, the entire space may be considered to be a
polyhedral cone, i.e. that cone which is defined by an empty matrix
S.
[0095] Distance Measurement
[0096] In view of the discussion concerning wave propagation, the
distance of a object, hull, or bound from a particular viewpoint is
defined to be the minimum distance to the object, hull, or bound
from the viewpoint. So, assuming a viewpoint at the origin, the
distance of the object, hull, or bound X from the viewpoint is
defined as 1 f ( X ) = min x X ; x r; ,
[0097] where .parallel.x.parallel. is the norm of vector x. When
the object, hull, or bound X is empty, the distance is taken to be
positive infinity.
[0098] Any vector norm may be chosen for the measurement of
distance. In one embodiment, the Euclidean norm is chosen for
distance measurements. The Euclidean norm results in a spherically
shaped wavefront. Any wavefront shape may be used as long as it
satisfies a mild "star-shape" criterion: the entire boundary of the
wavefront must be unobstructed when viewed from the origin. All
convex wavefronts satisfy this condition, and many non-convex ones
do as well. In general, the level curves of a norm are recommended
as the wavefront shapes. From a computational standpoint, the
spherical wavefront shape given by the L.sub.2 norm, and the
piecewise-linear wavefront shapes given by the L.sub.1, and
L.sub..infin. norms provide good choices for visibility detection.
Not surprisingly, piecewise-linear approximations of such norms,
for example the approximation to the L.sub.2 norm also works well:
such an approximation is illustrated in FIG. 6.
[0099] Cones and Visibility
[0100] From a viewpoint located within a large set of objects,
there exists at least one point (on some object) nearest to the
viewpoint. Since that point (or set of points) is closest to the
viewpoint, nothing can occlude the view of that point (or those
points). This implies that the object (or objects) containing the
nearest point (or points) is (are) at least partially visible.
[0101] Now, consider an arbitrary cone K emanating from the origin
as a viewpoint. The unobstructed visibility argument holds even if
all distance measurements are restricted to points that fall within
the cone. Define the distance of an object, hull, or bound X
relative to the cone K as 2 f K ( X ) = min x X K ; x r; .
[0102] If the distance f.sub.K is computed for each object in a
scene, the nearest object, i.e. the object which achieves a minimum
distance value, is visible with respect to cone K. In fact, an
object X is visible if and only if there exists a cone K such that
(a)f.sub.K(X) is finite and (b)f.sub.K(X).ltoreq.f.sub.K(Y) holds
for all objects Y in the set of objects.
[0103] It is noted that rays may be viewed as degenerate cones that
emanate from the viewpoint and pass through the center of each
pixel. The nearest object along each ray is visible and thus
determines the value of the corresponding pixel. Similarly, it is
possible to construct cones which emanate from the viewpoint and
cover each pixel. For example, two or more cones with triangular
cross-section may neatly cover the area of a pixel. The nearest
object within each cone is visible and generates the value of the
corresponding pixel.
[0104] As discussed above, the ray-based methods of the prior art
are able to detect objects only up the resolution of the ray
sample. Small visible objects or small portions of larger objects
may be missed entirely due to insufficient ray density. In
contrast, cones can completely fill space. Thus, the cone-based
method of the present invention may advantageously detect small
visible objects or portions of objects that would be missed by a
ray-based method with equal angular resolution.
[0105] Generalized Separation Measurement
[0106] For the purposes of performing a visibility search
procedure, it is necessary to have a method for measuring the
extent of separation (or conversely proximity) of objects, bounds,
or hulls with respect to cones. There exists a great variety of
such methods in addition to those based on minimizing vector norms
defined above. As alluded to above, a measurement value indicating
the extent of separation between a set X and a cone K may be
obtained by propagating a wavefront internal to the cone from the
vertex of the cone and observing the radius of first interaction of
the internal wavefront with the set X. As mentioned above, the
wavefront must satisfy a mild "star shape" condition: the entire
boundary of the wavefront must be visible from the vertex of the
cone.
[0107] In one embodiment, the measurement value is obtained by
computing a penalty of separation between the set X and the cone K.
The penalty of separation is evaluated by minimizing an increasing
function of separation distance between the vertex of the cone K
and points in the intersection of the cone K and set X. For
example, any positive power of a vector norm gives such an
increasing function.
[0108] In another embodiment, the measurement value is obtained by
computing a merit of proximity between the set X and the cone K.
The merit of separation is evaluated by maximizing a decreasing
function of separation distance between the vertex of the cone K
and points in the intersection of the cone K and set X. For
example, any negative power of a vector norm gives such a
decreasing function.
[0109] A Cone Hierarchy
[0110] The visibility determination method of the present invention
relies on the use of a hierarchy of cones in addition to the
hierarchy of hulls described above. The class of polyhedral cones
is especially well suited for generating a cone hierarchy:
polyhedral cones naturally decompose into polyhedral subcones by
the insertion of one or more separating planes. The ability to nest
cones into a hierarchical structure allows a very rapid examination
of object visibility. As an example, consider two neighboring cones
that share a common face. By taking the union of these two cones, a
new composite cone is generated. The composite cone neatly contains
its children, and is thus capable of being used in querying exactly
the same space as its two children. In other words, the children
cones share no interior points with each other and they completely
fill the parent without leaving any empty space.
[0111] A typical display and its associated view frustum has a
rectangular cross-section. There are vast array of possibilities
for tessellating this rectangular cross-section to generate a
system of sub-cones. For example, the rectangle naturally
decomposes into four rectangular cross-sections, or two triangular
cross-sections. Although these examples illustrate decompositions
using regular components, irregular components may be used as
well.
[0112] FIGS. 9A-9C illustrate a hierarchical decomposition of an
initial view frustum C. FIG. 9A depicts the rectangular
cross-section of the view frustum and its bisection into two cones
with triangular cross-section, i.e. cones C0 and C1. The view
frustum C corresponds to the root node of a cone tree. Cones and
their corresponding nodes in the cone tree are identically labeled
for simplicity. Each node of the cone tree stores the matrix S of
normal vectors which generates the corresponding cone. The root
node points to two children nodes corresponding to cones C0 and C1.
FIG. 9B illustrates a second decomposition stage. Each of the cones
C0 and C1 is bisected into two sub-cones (again with triangular
cross-section). Cone C0 decomposes into the two sub-cones C00 and
C01. Likewise, cone C1 is bisected into two sub-cones C10 and C11.
Nodes are added to the cone tree to reflect the structure of this
decomposition. The parent-child relation of nodes in the cone tree
reflects the superset-subset relation of the respective cones in
space. FIG. 9C illustrates the pattern of successive cone
bisections according to one embodiment. Each cone in the hierarchy
may be decomposed into two sub-cones by means a bisecting plane.
FIG. 9C illustrate several successive descending bisections which
generate cones C0, C10, C110, and C1110, and so on. The initial
cone C (i.e. the view frustum) may be decomposed to any desired
resolution. In one embodiment, the bisections terminate when the
resultant cones intercept some fraction of a pixel such as, for
example, 1/2 a pixel. The corresponding terminal nodes of the cone
tree are called leaves. Alternate embodiments are contemplated
where the bisections terminate when the resultant leaf-cones
intercept areas which subtend (a) a portion of pixel such as 1/N
where N is a positive integer, or (b) areas including one or more
pixels.
[0113] The triangular hierarchical decomposition shown in FIGS.
9A-9C has a number of useful properties. By decomposing the
original rectangular cone based on recursive bisection, a binary
tree of cones of arbitrary depth is generated. Triangular cones
have the fewest sides making them computational more attractive.
Lastly, triangular cones can also tessellate the unit cube (or unit
sphere). Thus, it is possible to create a hierarchical cone
representation for the entire space surrounding the viewpoint.
[0114] It is noted that any cone decomposition strategy may be
employed to generate a cone hierarchy. In a second embodiment, the
view frustum is decomposed into four similar rectangular cones;
each of these subcones is decomposed into four more rectangular
subcones, and so on. This results in a cone tree with four-fold
branches.
[0115] Discovering the Set of Visible Objects
[0116] Once the hull hierarchy and the cone hierarchy have been
constructed, the set of visible objects is computed from the
current viewpoint. In one embodiment, the visible object set is
repeatedly recomputed for a succession of viewpoints and viewing
directions. The successive viewpoints and viewing directions may be
specified by a user through an input device such as a mouse,
joystick, keyboard, trackball, or any combination thereof. The
visible object determination method of the present invention is
organized as a simultaneous search of the hull tree and the cone
tree. The search process involves recursively performing cone-hull
queries. Given a cone node K and a hull node H, a cone-hull query
on cone K and hull H investigates the visibility of hull H and its
descendent hulls with respect to cone K and its descendent cones.
The search process has a computational complexity of order log M,
where M equals the number of cone nodes times the number of hull
nodes. In addition, many cone-hull queries can occur in parallel
allowing aggressive use of multiple processors in constructing the
visible-object-set.
[0117] Viewing the Scene
[0118] Independently, and also concurrently, the set of visible
objects from the current viewpoint may be rendered on one or more
displays. The rendering can occur concurrently because the
visible-object-set remains fairly constant between frames in a
walkthrough environment. Thus the previous set of visible objects
provides an excellent approximation to the current set of visible
objects.
[0119] Managing the Visible-Object-Set
[0120] The visualization software must manage the
visible-object-set. Over time, as an end-user navigates through a
model, just inserting objects into the visible-object-set would
result in a visible-object-set that contains too many objects. To
ensure good rendering performance, the visualization process must
therefore remove objects from the visible-object-set when those
objects no longer belong to the set-- or soon thereafter. A variety
of solutions to object removal are possible. One solution is based
on object aging. The system removes any object from the
visible-object-set that has not been rediscovered by the cone query
within a specified number of redraw cycles.
[0121] Computing Visibility Using Cones
[0122] Substantial computation leverage is provided by recursively
searching the hierarchical tree of cones in conjunction with the
hierarchical tree of hulls. Whole groups of cones may be tested
against whole groups of hulls in a single query. For example, if a
parent cone does not intersect a parent hull, it is obvious that no
child of the parent cone can intersect any child of the parent
hull. In such a situation, the parent hull and all of its
descendants may be removed from further visibility considerations
with respect to the parent cone.
[0123] Visibility Search Algorithm
[0124] In the preferred embodiment, the visibility search algorithm
of the present invention is realized in a visibility search program
stored in memory 106. Thus, hereinafter the terms visibility search
program and visibility search algorithm will be used
interchangeably. The visibility search program performs a recursive
search of the two trees (the object tree and the cone tree) to
assign visible objects to each of the leaf cones of the cone tree.
The host processor 102 executes the visibility search program. In
an alternate embodiment, the visibility search program executes on
a processor comprised within the graphics accelerator 112.
[0125] The recursive search of the two trees provides a number of
opportunities for aggressive pruning of the search space. Central
to the search is the object-cone distance measure defined above,
i.e. given a cone K and an object (or hull) X, the object-cone
distance is defined as 3 f K ( X ) = min x X K ; x r; .
[0126] It is noted that this minimization is in general a nonlinear
programming problem since the cones and object hulls are defined by
constraint equations, i.e. planes in three-space. If the vector
norm .parallel.x.parallel. is the L.sup.1 norm (i.e. the norm
defined as the sum of absolute values of the components of vector
x), the nonlinear programming problem reduce to a linear
programming problem. If the vector norm .parallel.x.parallel. is
the Euclidean norm, the nonlinear programming problem reduce to a
quadratic programming problem. Given a collection of objects, the
object X which achieves the smallest distance f.sub.K(X) is closest
to the cone's viewpoint, and therefore must be at least partially
visible.
[0127] The recursive search starts with the root H of the hull tree
and the root C of the cone tree (see FIGS. 4 and 9). Remember that
each node of the hull tree specifies a bounding hull which contains
the hulls of all its descendant nodes. Initially the distance
between the root cone and the root hull is computed. If that
distance is infinite, then no cone in the cone hierarchy intersects
any hull in the hull hierarchy and there are no visible objects. If
the distance is finite, then further searching is required. Either
tree may be refined at this point. In the preferred embodiment,
both trees are refined in a predefined order.
[0128] The pruning mechanism is built upon several basic elements.
A distance measurement function computes the distance f.sub.K(X) of
a hull X from the viewpoint of a cone K. In other words, the
function determines the cone-restricted distance to the hull X. In
the preferred embodiment, the minimization associated with
evaluating the distance measurement function is implemented by
solving an associated linear programming problem.
[0129] To facilitate the search process, each leaf-cone, i.e. each
terminal node of the cone tree, is assigned an extent value which
represents its distance to the closest known object-hull. Thus,
this extent value may be referred to as the visibility distance.
The visibility distance of a leaf-cone is non-increasing, i.e. it
decreases as closer objects (i.e. object hulls) are discovered in
the search process. Visibility distances for all leaf-cones are
initialized to positive infinity. [An object-hull is a hull that
directly bounds an object. Thus, object-hulls are terminal nodes of
the hull tree.] In addition to a visibility distance value, each
leaf-cone node is assigned storage for an currently visible object.
This object attribute is initialized to NO_OBJECT or BACKGROUND
depending upon the scene context.
[0130] In addition, each non-leaf cone, i.e. each cone at a
non-final refinement level, is assigned an extent value which
equals the maximum of its sub-cones. Or equivalently, the extent
value for a non-leaf cone equals the maximum of its leaf-cone
descendents. These extent values are also referred to as visibility
distance values. The visibility distance values for all non-leaf
cones are initialized to positive infinity also (consistent with
initialization of the leaf-cones). Suppose a given non-leaf cone K
and a hull H achieve a cone-object distance f.sub.K(X). If this
distance f.sub.K(X) is greater than the visibility distance value
of the cone K, then all of the leaf-cone descendents of cone K
already have known objects closer than the hull H. Therefore, no
benefit can be gained by searching hull H against cone K and its
descendents. In contrast, if a hull H achieves a distance
f.sub.K(X) from cone K which is less than the visibility distance
value of cone K, it is possible that hull H contains objects which
will strictly decrease the visibility distance of some leaf-cone
descendent of cone K. Thus, the hull H and its descendents should
be searched against cone K and its descendents.
[0131] The following code fragment illustrates the beginning of the
search process. The variables hullTree and coneTree point to the
root nodes of the hull tree and cone tree respectively.
1 main(hullTree, coneTree) { cone.extent = infinity; distance =
DIST(hullTree, coneTree); if(distance < infinity) find
Visible(hullTree, coneTree, distance); }
[0132] The DIST function evaluates the distance between the root
hull and the root cone. If this distance is less than positive
infinity, the function findVisible is called with the root hull,
root cone, and their hull-cone distance as arguments. The function
findVisible performs the recursive search of the two trees.
[0133] FIGS. 10A-10C illustrate one embodiment of the findVisible
function (also referred to by the contraction FV). Refer to the
Appendix for a source code embodiment of the findVisible function.
The findVisible function receives several input variables from a
calling routine as illustrated by the following function
prototype:
findVisible ( H, C, d)
[0134] where H is a hull node to be explored against the cone node
C. The value d represents the cone-hull distance between cone C and
hull H. In step 300, the findVisible function determines whether
the cone C and hull H are both leaf nodes, i.e. terminal nodes of
their respective trees. If so, the visibility distance value
d.sub.C for the leaf-cone C is set equal to the current distance to
object hull H. In addition, the object attribute of leaf-cone C is
assigned the value of hull H. Then control is returned to the
calling routine.
[0135] If the hull H and cone C are not both leaves, step 302 is
performed. In step 302, the findVisible function determines if the
cone C is itself a leaf node of the cone tree. If so, steps 304
through 310 are performed. In step 304, the children of hull H are
investigated, i.e. the distances of cone C to the subhulls H0 and
H1 are compared. [The children of hull H will be generically
referred to as H0 and H1.] If the distance to subhull H0 is smaller
than the distance to subhull H1, steps 306 and 308 are performed.
Otherwise, step 310 is performed. Step 304 is performed in order to
determine which subhull will be allowed the opportunity of first
interaction with cone C. In the preferred embodiment of the
invention, the closer subhull is searched first.
[0136] In step 306, the findVisible function determines if the
cone-hull distance of subhull H0 from cone C is less than the
visibility distance value d.sub.C of cone C. If so, the
findVisibility function is called again in order to search subhull
H0 against cone C. The cone-hull distance between cone C and
subhull H0 is provided as the distance argument for the function
call. If the distance to subhull H0 is not less than the current
visibility distance, step 308 is performed.
[0137] In step 308, the findVisible function determines if the
cone-hull distance of subhull H1 from cone C is smaller than the
visibility distance value of cone C. If it is smaller, the
findVisibility function is called again in order to search subhull
H1 against cone C. The cone-hull distance between cone C and
subhull H1 is provided as the distance argument for the function
call. If the distance to subhull H1 is not less than the current
visibility distance, program control returns to the calling
program.
[0138] It is noted that the call to the function findVisible in
step 306 may cause the visibility distance value for cone C to
decrease. Thus, the visibility distance value for cone C may
actually have a smaller value when tested in step 308 than when
tested in step 306. This underscores the importance of the test
performed in step 304 to determine the subhull which is closer to
cone C. The closer hull is searched first and allowed to preempt
the more distant subhull. In other words, after searching the
closer subhull, the more distant subhull may not need to be
searched. Thus, the distance conditioned search order for subhulls
advantageously improves the efficiency of the visibility search
algorithm.
[0139] In step 310, the subhulls are searched in the opposite order
in response to the determination of step 304 that subhull H0
achieves a cone-hull distance from cone C which is not less than
the corresponding distance for subhull H1. Again, the closer
subhull H1 is searched first and allowed to preempt the search of
the farther subhull H0.
[0140] If, in step 302, it is determined that cone C is not a leaf,
step 320 is performed (see FIG. 10B). In step 320, the
findVisibility function determines if the hull node H is a leaf
node, i.e. an object hull, of the hull hierarchy. If so, the
children of cone C are explored with respect to object hull H in
steps 322 through 326. In step 322, the findVisibility function
determines if the cone-hull distance between hull H and child
subcone C0 is less than the visibility distance value of child
subcone C0. If so, the findVisibility function is called again in
order to explore hull H with respect to subcone C0. Thus, hull H,
subcone C0 and the cone-hull distance between these two are
supplied to the findVisibility function as arguments. When the
findVisibility function returns, control passes to step 324. Also,
if the inequality test of step 322 fails, control passes to step
324.
[0141] In step 324, the findVisibility function determines if the
cone-hull distance of hull H from subcone C1 is less than the
visibility distance value for subcone C1. If so, the findVisibility
function is called again in order to explore hull H with respect to
subcone C1. Thus, hull H, subcone C1 and the cone-hull distance
between these two are supplied to the findVisibility function as
arguments. When the findVisibility function returns, control passes
to step 326. Also, if the inequality test of step 324 fails,
control passes to step 326.
[0142] Since the visibility distance values for subcones C0 and C1
may have decreased by the interactions with hull H in steps 322 and
324 respectively, the visibility distance value for cone C must be
updated. Thus, in step 326, the visibility distance value for cone
C is set equal to the maximum of the visibility distance values of
subcones C0 and C1. After step 326, program control returns to the
calling routine.
[0143] If, in step 320, the hull node H is determined not to be a
leaf node, then step 330 is performed (see FIG. 10C). Thus, step
330 and succeeding steps may assume that neither hull H nor cone C
are leaf nodes in their respective hierarchies. Steps 332, 334, 335
and 336 explore the children of hull H with respect to cone C0
conditioned upon the result of testing step 330. Similarly, step
340 explores the children of hull H with respect to subcone C1
conditioned upon the result of testing step 338.
[0144] In step 330, the findVisibility function determines if the
cone-hull distance between hull H and subcone C0 is less than the
current visibility distance value of subcone C0. If so, step 332 is
performed. In step 332, the cone-hull distances of subhulls H0 and
H1 from subcone C0 are compared. The closer subhull is given the
opportunity of first interaction with subcone C0. Thus, if the
cone-hull distance of subhull H0 from subcone C0 is smaller than
the cone-hull distance of subhull H1 from subcone C0, step 334 is
performed. Otherwise step 336 is performed.
[0145] In step 334, the findVisibility function determines if the
cone-hull distance of subhull H0 from subcone C0 is less than the
current visibility distance value of subcone C0. If so, the
findVisibility function is called again in order to explore subhull
H0 against subcone C0. Thus, subhull H0, subcone C0, and the
cone-hull distance between these two are provided as arguments to
the findVisibility function. When this call to the findVisibility
function returns, step 335 is performed.
[0146] If, in step 334, it is determined that the cone-hull
distance of subhull H0 from subcone C0 is not smaller than the
visibility distance value of subcone C0, step 335 is performed. In
step 335, the findVisibility function determines if the cone-hull
distance of subhull H1 from subcone C0 is smaller the visibility
distance value of subcone C0. If so, the findVisibility function is
called again in order to explore subhull H1 with respect to subcone
C0. Thus, subhull H1, subcone C0, and the cone-hull distance
between them are provided as arguments to the findVisibility
function. When this call to the findVisibility function returns,
program control passes to step 338. Additionally, if the inequality
test of step 335 fails, step 338 is performed.
[0147] Since the closer subhull H0 is allowed the first interaction
with cone C0, it may alter (i.e. decrease) the visibility distance
value of subcone C0. Thus, the farther subhull H1 may have more
difficulty in passing the inequality test of step 335 due to the
prior interaction of subhull H0. In other words, the exploration of
subhull H1 with respect to subcone C0 may not need to be performed,
and advantageously skipped, in view of the prior interaction with
the closer subhull H0. Thus, the search efficiency of the
visibility determination method is improved by avoiding unnecessary
subhull explorations.
[0148] Step 336 is performed in response to a determination in step
332 that the cone-hull distance of subhull H0 from subcone C0 is
not less than the cone-hull distance of subhull H1 from subcone C0.
In other words, step 336 is performed when subhull H1 is closer to
subcone C0 than subhull H0. Step 336 involves operations similar to
those performed in steps 334 and 335. However, the conditional
search of subhull H1 to performed before the conditional search of
subhull H0, i.e. the closer subhull is searched first. After step
336 control passes to step 338.
[0149] If, in step 330, it is determined that the cone-hull
distance of hull H from cone C0 is not less than the visibility
distance of subcone C0, control passes to step 338. In step 338,
the findVisibility function determines if the cone-hull distance of
hull H from subcone C1 is less than the visibility distance value
of subcone C1. If so, step 340 is performed. Step 340 performs
conditional exploration of subhulls H0 and H1 with respect to
subcone C1. Thus, step 340 involves operations similar to those
described in steps 332, 334, 335 and 336 with the roles of subcone
C0 and subcone C1 are interchanged. After step 340, step 342 is
performed. In addition, if the inequality test of step 338 fails,
step 342 is performed.
[0150] In step 342, the visibility distance value of cone C is set
equal to the maximum of the visibility distance values of subcones
C0 and C1. After step 342, program control returns to the calling
routine.
[0151] As explained above, the visibility distance value assigned
to each cone in the cone tree equals the maximum of the visibility
distance values assigned to its subcone children. Thus, if a given
hull node achieves a distance to a cone which is larger than the
cone's current visibility distance value, it is immediately
apparent that none of the hulls descendents will be of interest to
any of the cone's descendents. The given hull node may be skipped
so far this cone is concerned.
[0152] A cone's visibility distance value decreases as the
recursion tests more and more object-cone leaf pairs. As nearby
objects are discovered, a cone's visibility distance value
decreases and the probability of skipping unpromising hull nodes
increases. A leaf in the hull tree bounds the volume of the
associated object and also approximates that object's contents.
Thus, cone visibility distance values, set during recursion, are
usually not the real distances to objects but a conservative
approximation of those distances. If the conservative approximation
is inadequate for use in an application, then that application can
invoke an exact-computation for the visibility distance values.
[0153] Throughout the above discussion of the visibility search
algorithm it has been assumed that the DIST function used to
compute cone-hull separation distance is based on minimizing an
increasing function of separation distance between the vertex of
the given cone and points in the intersection of the given cone and
the given bound/hull. However, it is noted that the DIST function
may be programmed to compute a merit of proximity between a given
cone and given bound/hull. The resulting merit value increases with
increasing proximity and decreases with increasing separation,
converse to the typical behavior of a distance function. In this
case, the visibility search algorithm performs a search of
bound/hull H against cone K only if the merit value of separation
between cone K and bound/hull H is greater than the current merit
value associated with cone K. Furthermore, after a search of
subcones of cone K is completed, the merit value associated with
the cone K is updated to equal the minimum of the merit values of
its subcone children.
[0154] In general, the DIST function determines a cone-hull
measurement value of separation by computing the extremum (i.e.
minimum or maximum) of some monotonic (increasing or decreasing)
function of separation between the vertex of the cone K and points
in the intersection of cone K and bound/hull H. The search of cone
K against a bound/hull H is conditioned on the bound/hull H
achieving a cone-hull measurement value with respect to cone K
which satisfies an inequality condition with respect to measurement
value assigned to cone K. The sense of the inequality, i.e. less
than or greater than, depends on the whether the DIST function uses
an increasing or decreasing function of separation.
[0155] While the search of the hull and cone hierarchies described
above assumes a recursive form, it is noted that any search
strategy may be employed. In one alternate embodiment, the hull
and/or cone hierarchies are searched iteratively. Such a brute
force solution may be advantageous when a large array of processors
is available to implement the iterative search. In another
embodiment, a level-order search is performed on the hull and/or
cone hierarchies.
[0156] Method for Displaying Visible Objects
[0157] A method for displaying visible objects in a graphics
environment is described in the flowchart of FIG. 11. A visibility
search algorithm executing on CPU 102 or optionally graphics
accelerator 112 controls the determination of a set of visible
objects for display on display screen 84. In step 501, a collection
of objects is accessed from memory 106. In step 502, a bounding
hierarchy is constructed from the collection of objects. The
construction of the bounding hierarchy will be described in detail
below in conjunction with FIG. 12. In step 503 a cone hierarchy is
constructed by refining an initial cone. Refer to the discussion
above in connection with FIGS. 6-9 for more information on
construction of the cone hierarchy. In one embodiment, the initial
cone is the entire space. In a second embodiment, the initial cone
corresponds to a neighborhood which contains the view frustum and a
neighborhood around the view frustum. Thus, if the viewing
orientation (i.e. direction) changes so that the new view frustum
remains within the neighborhood of the initial cone, the cone
hierarchy may not need to be reconstructed, thereby saving
computational bandwidth. Each cone in the cone hierarchy is
allocated a measurement value which corresponds to totally
un-occluded visibility.
[0158] In step 504, a search of the cone hierarchy and the bounding
hierarchy is performed. The search procedure determines a nearest
object for each of the leaf cones of the cone hierarchy. It is
noted that some leaf cones may not interact with any object in the
given collection. In this case, the nearest object may be
interpreted to be the background.
[0159] The search procedure performs conditional explorations of
cone-bound pairs starting with the root cone of the cone hierarchy
and the root bound of the bounding hierarchy. Given a first cone in
the cone hierarchy and a first bound in the bounding hierarchy, the
search algorithm determines a first measurement value which
indicates an extent of separation between the first cone and the
first bound as shown in step 504A.
[0160] In step 504B, the first measurement value is compared to a
measurement value associated with the first cone. If the first
measurement value satisfies an inequality condition with respect to
the measurement value associated with the first cone, the first
bound is searched with respect to the first cone as shown in step
504C. In general, the measurement value associated with a non-leaf
cone reflects the maximum of the measurement values of its
leaf-cone descendents, and the measurement value associated with a
leaf-cone indicates the extent of separation between the leaf-cone
and its nearest known object. Therefore, no benefit can be gained
by searching the first bound with respect to the first cone if the
comparison of step 504B indicates that the first bound is farther
than the nearest known object for each of the leaf-cone descendents
of the first cone. Steps 504B and 504C are together referred to as
a conditioned search. Thus, the first bound and first cone are
explored only if there is evidence that the first bound is closer
than the nearest known object for at least one leaf descendent of
the first cone.
[0161] In step 506, the nearest objects for a subset of cones of
the cone hierarchy are displayed on display device 84. In the
preferred embodiment, the subset of cones comprises the leaf-cones
of the cone hierarchy, i.e. the nearest object for each leaf-cone
in the cone hierarchy is displayed on display device 84. Tremendous
processing bandwidth is saved by requesting rendering computations
only for the objects which are visible. The nearest object to a
leaf cone is the visible object for that cone. It is noted that the
visibility search performed in step 504 may be performed repeatedly
for different viewpoints and/or viewing directions. Thus, the
display step 506 may advantageously be performed in parallel with
step 504. While, the visibility search 504 is being performed for a
current viewpoint/view-direction, the results from a previous
visibility search may be concurrently displayed in step 506.
[0162] In one embodiment, the first measurement value comprises a
penalty of separation between the first cone and the first bound
which is determined by minimizing an increasing function of
separation distance between the vertex of the first cone and points
in the intersection of the first cone and the first bound. In this
case, step 504B determines whether the first measurement value is
smaller than the measurement value associated with the first cone
since smaller measurement values indicate closer proximity. In
other words, the inequality condition of step 504B is satisfied
when the first measurement value is less than the measurement value
associated with the first cone. The increasing function of
separation distance may be specified by a vector norm. For example,
the expression .parallel.s.parallel. defines an increasing function
of separation distance, where s is a displacement vector
representing the vector difference between the vertex of the first
cone and an arbitrary point in the intersection of the first cone
and the first bound, and .parallel..multidot..parallel. denotes a
vector norm. Examples, of vector norms include the L.sup.1 norm,
the L.sup.2 norm, the L.sup..infin. norm, and in general, any of
the L.sup.p norms where p is any real number in the range from one
to infinity inclusive. The L.sup.1 norm of a vector is defined as
the sum of the absolute values of its components. The L.sup.2 norm
of a vector is defined as the square root of the sum of the squares
of the vector components. The L.sup..infin. norm of a vector is
defined as the maximum of the absolute values of the vector
components.
[0163] In the preferred embodiment, the first measurement value of
separation between the first bound and the first cone is computed
by solving a linear programming problem using the linear
constraints given by normal matrix S for the first cone, and the
linear constraints given by the normal matrix N and the extent
vector c for the first bound. Recall the discussion in connection
with FIGS. 9A-9C.
[0164] In an alternate embodiment, the first measurement value
comprises a merit of proximity (i.e. closeness) between the first
cone and the first bound which is determined by maximizing a
decreasing function of separation distance between the vertex of
the first cone and points in the intersection of the first cone and
the first bound. In this case, step 504B determines whether the
first measurement value is larger than the measurement value of
associated with the first cone since larger measurement values
indicate closer proximity. In other words, the inequality condition
of step 504B is satisfied when the first measurement value is
greater than the measurement value associated with the first
cone.
[0165] FIG. 12 illustrates step 502, i.e. the construction of the
bounding hierarchy from the collection of objects. In step 602, the
objects in the graphics scene are recursively clustered. Objects
are assembled into clusters preferably based on proximity. These
first order clusters are themselves assembled into second order
clusters. Clusters of successively higher order are formed until
all the objects are contained in one universal cluster. Objects may
be considered as order zero clusters. In step 604, each cluster of
all orders is bounded with a corresponding bound. The bounds are
preferably polytope hulls as described above in connection with
FIGS. 4 and 5. However, other types of bounds are contemplated such
as, e.g., quadratic surfaces, generalized polynomial bounds,
etc.
[0166] In step 606, a hierarchical tree of bounds is generated by
allocating a node for each of the objects and clusters. In step
608, each node is assigned parameters which describe (characterize)
the corresponding bound. In one embodiment this parameter
assignment comprises storing the extent vector c which locates the
polytope hull faces as described in connection with FIGS. 5A and
5B. In step 610, the nodes are organized so that node relationships
correspond to cluster membership. For example, if node A is the
parent of node B in the bounding hierarchy, then the cluster
corresponding to node A contains a subcluster corresponding to node
B, and the bound for node A contains the bound for node B.
[0167] Although the construction of the cone hierarchy above has
been described in terms of recursive clustering, it is noted
alternative embodiments are contemplated which use other forms of
clustering such as iterative clustering.
[0168] A Terminal Cone-Bound Pair
[0169] FIG. 13 illustrates a first portion of step 504C, i.e. the
step of searching the first bound with respect to the first cone,
when both the first bound and the first cone are leaves of their
respective hierarchies. In step 702, the visibility search
algorithm determines whether the first cone and the first bound are
leaves of their respective hierarchies. If they are both leaves,
the visibility search algorithm (a) sets the measurement value
associated with the first cone equal to the first measurement value
of separation between the first bound and the first cone as shown
in step 704, and (b) sets a visible object attribute associated
with the first cone equal to the first bound as shown in step 706.
It is noted that steps 704 and 706 may be performed in any order or
in parallel.
[0170] A Terminal Cone with a Non-Terminal Bound
[0171] FIG. 14 illustrates a second portion of step 504C, i.e. the
step of searching the first bound with respect to the first cone,
when the first cone is a leaf of the cone hierarchy, and the first
bound is not a leaf of the bounding hierarchy. In step 710, the
visibility search algorithm determines whether the first cone is
leaf-cone of the cone hierarchy, and the first bound is a non-leaf
bound of the bounding hierarchy, i.e. a non-terminal bound. In
response to an affirmative determination that the first cone is a
leaf cone and the first bound is not a leaf bound, the sub-bounds
of the first bound are conditionally explored with respect to the
first cone.
[0172] FIG. 15 illustrates step 712, i.e. the step of conditionally
exploring the sub-bounds of the first bound with respect to the
first cone. In step 720, the visibility search algorithm computes a
cone-bound separation value for each of the sub-bounds of the first
bound with respect to the first cone. In one embodiment, the
cone-bound separation values are computed in the same way as the
first measurement value. In step 722, the visibility search
algorithm conditionally searches sub-bounds of the first bound with
respect to the first cone in ascending order of their separation
from the first cone. The cone-bound separation values are used to
determine the ascending order. In other words, if sub-bound A is
closer to the first cone than sub-bound B, sub-bound A is
conditionally searched before sub-bound B.
[0173] FIG. 16 illustrates step 722, i.e. the conditional search of
the sub-bounds of the first bound with respect to the first cone by
means of a generic first subbound of the first bound. In step 730,
the visibility search algorithm determines whether the cone-bound
separation value of the first subbound satisfies an inequality
condition with respect to the measurement value associated with the
first cone. The sense of the inequality tested in step 730 may be
the same as the sense of the inequality tested in step 504B.
[0174] In step 732, the visibility search algorithm searches the
first subbound with respect to the first cone in response to an
affirmative determination that the cone-bound separation value of
the first subbound satisfies the inequality condition with respect
to the first cone's measurement value.
[0175] A Terminal Hull with a Non-Terminal Cone
[0176] FIG. 17 illustrates a third portion of step 504C, i.e. the
step of searching the first bound with respect to the first cone
when the first bound is a leaf-bound of the bounding hierarchy, and
the first cone is a non-leaf cone of the cone hierarchy. In step
740, the visibility search algorithm determines if the first bound
is a leaf of the bounding hierarchy and the first cone is a
non-leaf of the cone hierarchy. In step 742, visibility search
algorithm conditionally searches the subcones of the first cone
with respect to the first bound in response to an affirmative
determination that the first bound is a leaf bound and the first
cone is a non-leaf cone.
[0177] FIG. 18 illustrates step 742, i.e. the step of conditionally
searching subcones of the first cone with respect to the first
bound by means of a generic first subcone. The generic
representative is intended to typify the process of conditionally
searching any of the subcones of the first cone. In step 749 the
visibility search algorithm computes a cone-bound separation value
for the first bound with respect to the first subcone. In step 750,
the visibility search algorithm determines whether the cone-bound
separation value satisfies an inequality condition with respect to
a measurement value associated with the first subcone. In one
embodiment, the cone-bound separation value is computed using the
same method as used to compute the first measurement value in step
504A. In this case, the sense of the inequality tested in step 750
may be consistent with the sense of the inequality tested in step
504B.
[0178] In step 752, the visibility search algorithm searches the
first subcone with respect to the first bound in response to an
affirmative determination that the cone-bound separation value
satisfies the inequality condition with respect to the measurement
value associated with the first subcone.
[0179] After the subcones of the first cone have been searched
against the first bound, the measurement value associated with the
first cone is updated. Namely, the measurement value associated
with the first cone may be set equal to an extremum (i.e. maximum
or minimum) of the measurement values associated with the subcones
of the first cone. The choice of the maximum as the extremum is
associated with embodiments which compute cone-bound separation
based on an increasing function of separation. The choice of
minimum as the extremum is associated with embodiments which
compute cone-bound separation based on a decreasing function of
separation.
[0180] A Non-Terminal Cone with a Non-Terminal Bound
[0181] FIG. 19 illustrates a fourth portion of step 504C, i.e. the
step of searching the first bound with respect to the first cone
when neither the first cone nor the first bound are leaf nodes of
their respective hierarchies. In step 759, the visibility search
algorithm computes a first cone-bound separation value for the
first bound with respect to a first subcone of the first cone. In
step 760, the visibility search algorithm determines whether the
first cone-bound separation value satisfies an inequality condition
with respect to a measurement value associated with the first
subcone. In one embodiment, the first cone-bound separation value
is computed using the same method used to compute the first
measurement value in step 504A. In this case, the sense of the
inequality tested in step 760 may be consistent with the sense of
the inequality tested in step 504B. In step 762, the visibility
search algorithm conditionally explores subbounds of the first
bound with respect to the first subcone in response to an
affirmative determination that the first cone-bound separation
value satisifes the inequality condition with respect to the
measurement value associated with the first subcone.
[0182] FIG. 20 illustrates step 762, i.e. the step of conditionally
exploring the subhulls of the first hull with respect to the first
subcone. In step 770, the visibility search algorithm computes a
second cone-bound separation value for each of the subbounds of the
first bound with respect to the first subcone. The second
cone-bound separation values may be computed using the same method
as used to compute the first measurement value in step 504A. In
step 772, the visibility search algorithm conditionally searches
the subbounds of the first bound with respect to the first subcone
in ascending order of their separation from the first subcone, i.e.
subbounds which are closer to the first subcone are searched first.
This ordering of the search advantageously allows closer subbounds
to preempt the search of farther subbounds, and thus unneeded
searches are avoided.
[0183] FIG. 21 illustrates step 772, i.e. the step of conditionally
searching subbounds of the first bound with respect to the first
subcone. A generic subhull of the first hull is used to illustrate
the conditioned search. In step 780, the visibility search
algorithm determines whether the second cone-hull separation value
for a first subbound among the subbounds of the first bound
satisfies an inequality condition with respect to the measurement
value associated with the first subcone. It is noted that the sense
of the inequality tested in step 780 may be similar to the sense of
the inequality tested in step 504B, especially if the method used
to compute the second cone-hull separation value is the same as
that used to compute the first measurement value.
[0184] In step 782, the visibility search algorithm searches the
first subbound with respect to the first subcone in response to an
affirmative determination that the second cone-hull separation
value for the first subbound satisfies the inequality condition
with respect to the measurement value associated with the first
subcone.
[0185] After the subbounds of the first bound have been searched
against the first subcone, a second subcone of the first cone is
tested. In particular, the visibility search algorithm computes a
measure of the separation of the first bound from the second
subcone. If this measure of separation satisfies an inequality
condition with respect to a measurement value associated with the
second subcone, then subbounds of the first bound may be
conditionally explored with respect to the second subcone. The
conditional exploration of the second subcone is similar to the
conditonal exploration of the first subcone described in
conjunction with FIGS. 20 and 21.
[0186] After all subcones of the first cone have been tested and
conditionally searched as described above, the measurement value
associated with the first cone is set equal to an extremum of the
measurement values associated with the subcones. As noted above,
the maximum is chosen as the extremum in embodiments which use an
increasing function of separation to compute cone-bound separation
values, and the minimum is chosen in embodiments which use a
decreasing function of separation to compute the cone-bound
separation values.
[0187] In one alternate embodiment, the leaf-cones subtend angular
sectors larger than one pixel. Thus, after termination of the
visibility search algorithm described above, the leaf-cones may be
further processed by a ray-based exploration method to determine
the values for individual pixels within leaf-cones.
[0188] Although the search of the bounding hierarchy and the cone
hierarchy described above assumes a recursive form, alternate
embodiments are contemplated where a level-order search or
iterative search is performed on one or both of the bounding
hierarchy and cone hierarchy.
[0189] Computing the Cone Restricted Distance Function
[0190] Recall that evaluation of the cone-hull distance f.sub.C(H)
of a hull H from a cone C calls for minimizing
.parallel.x.parallel. subject to the hull constraints Ax.ltoreq.b
and the cone constraints Sx.ltoreq.0. The rows of matrix A comprise
normals for the hull surfaces. The rows of matrix S comprise
normals for the cone surfaces. This minimization may be formulated
as a nonlinear programming problem. For example, the nonlinear
programming problem reduces to a quadratic programming problem when
a Euclidean norm is used, and a linear programming problem when the
L.sup.1 norm is used. The cone-hull distance computation is herein
referred to as a geometric query.
[0191] It is also noted that cone-hull separation may be measured
by maximizing an increasing function separation such as
.parallel.x.parallel..sup.-1 for points x satisfying the bound/hull
constraints and the cone constraints. Thus, in general a cone-hull
separation value may be computed by determining an extremal (i.e.
minimal or maximal) value of the separation function subject to the
cone constraints and the bound/hull constraints.
[0192] The use of a hierarchy of cones instead of a collection of
rays is motivated by the desire for computational efficiency.
Thanks to early candidate pruning that results from the double
recursion illustrated earlier, fewer geometric queries are
performed. These queries however are more expensive than the
queries used in the ray casting method. Therefore, the cone query
calculation must be designed meticulously. A sloppy algorithm could
end up wasting most of the computational advantage provided by
improvements in the dual tree search. For the linear programming
case, a method for achieving a computationally tight query will now
be outlined.
[0193] A piecewise-linear formulation of distance f.sub.C leads to
the following linear program:
min(v.sup.Tx) subject to Ax.ltoreq.b, Sx.ltoreq.0.
[0194] The vector v is some member of the cone that is polar to the
cone C. For instance, V=-S.sup.Te, where e is the vector of all
ones. [It is noted that the matrix S of cone normals S are outward
normals to the cone surfaces. Thus, the negation of the sum of the
normal vectors gives a polar vector.] The condition Ax.ltoreq.b
implies that the point x is within the bounding hull. The condition
Sx.ltoreq.0 implies that the point x is within the cone C. For an
efficient solution method, the linear program problem is restated
in term of its dual:
max(b.sup.Ty) subject to A.sup.Ty+S.sup.TZ=v, 0.ltoreq.y,
0.ltoreq.z.
[0195] The dual objective value, b.sup.Ty is infinite when the cone
and bounding hull do not intersect (the variables y and z are the
Lagrange multipliers of the previous problem's constraints).
[0196] In the preferred embodiment, the bounding hulls have sides
normal to a fixed set of normal vectors. Thus, the matrix A.sup.T
is the same for all hulls. For a given cone, the matrix S.sup.T and
the vector v are also fixed. From this observation, it is apparent
that the multi-dimensional polyhedron
{(y,z): A.sup.Ty+S.sup.Tz=v, 0.ltoreq.y, 0.ltoreq.z}
[0197] is associated with the cone. (In one embodiment, this
polyhedron has seventeen dimensions. Fourteen of those dimensions
come from the type of the fixed-direction bounding hull and an
three additional dimensions come from the cone.) Since the
polyhedron depends only on the cone matrix S, it is feasible to
completely precompute the extremal structure of the polygon for
each cone in the cone hierarchy. By complementary slackness, the
vertices of the polyhedron will have at most three elements. The
edges and extremal rays will have at most four non-zero elements.
An abbreviated, simplex-based, hill-climbing technique can be used
to quickly solve the query in this setting.
[0198] To establish the setting, the orientation of the cone
hierarchy needs to be fixed. This is not feasible if the cone
hierarchy changes orientation with changes in orientation of the
view frustum. Thus, the entire space is tessellated with cones, and
visible objects are detected within the entire space. Only after
this entire space computation is the set of visible of objects
culled to conform to the current view frustum.
[0199] In an alternative embodiment, a less aggressive approach may
be pursued. Namely, by noting which subsets of the cones correspond
to the current orientation of the view frustum, only this subset
may be included in the visible-object-set computation.
[0200] Memory Media
[0201] As described above, the visibility software and visibility
search program of the present invention are preferably stored in
memory 106 for access by processor 102. In addition, the visibility
software and visibility search program may be stored in any desired
memory media such as an installation media (e.g. CD-ROM, floppy
disk, etc.), a non-volatile memory (e.g. hard disk, optical
storage, magnetic tape, zip drive storage, ROM, etc.), various
kinds of volatile memory such as RAM, or any combination thereof.
The present invention contemplates the deposition and storage of
the visibility software and visibility search program on memory
media for distribution to end users and/or customers.
[0202] Multiple Objects Per Cone
[0203] According to the visibility search algorithm, a single
nearest object is identified for each leaf cone (i.e. terminal
cone). If each of the leaf cones have the ultimate resolution, i.e.
the resolution of a pixel, then the strategy of identifying the
nearest object in each leaf cone is guaranteed to detect all
visible objects. However, the visibility search of the fully
resolved cone hierarchy is computationally expensive. The
computational expense may be decreased by having fewer levels of
refinement in the cone hierarchy. But fewer levels of refinement
implies that the size of the leaf cones is larger. As the size of
the leaf cones increases, there is an increasing probability that
two or more objects will be visible to a single leaf cone, i.e.
that the nearest object is not the only object visible to the cone.
Therefore, there is an increased probability of reporting less than
the full set of visible objects as the size of the leaf-cones
increases, or equivalently, when fewer levels of cone refinement
are used in the cone hierarchy.
[0204] In order to increase the probability of capturing the full
set of visible objects, the visibility search algorithm may be
modified to identify the first K nearest objects for each leaf
cone, where K is an integer greater than or equal to two.
Advantageously the integer K may be a function of cone size. Thus,
if the cone hierarchy is close to ultimate resolution K may be
close to one. Conversely, if the cone hierarchy is poorly resolved,
i.e. includes only a few levels of refinement, the integer K may be
larger.
[0205] The present invention contemplates maximizing computational
efficiency along the axis of high-cone-resolution/low-K-value on
the one hand and low-cone-resolution/high-K-value on the other.
[0206] Adaptive Refinement of the Cone Hierarchy
[0207] In the foregoing discussion, the cone hierarchy is described
as being constructed prior to initiation of the search for visible
objects by the visibility search algorithm, and remains static
during the search. Another alternative is to adaptively refine the
cone hierarchy during the search procedure. In this fashion, the
cone hierarchy may not waste storage for cones which will never
interact with any objects. The cone hierarchy may be refined in
response to user inputs. For example, cones which correspond to the
user's current direction of gaze may warrant additional refinement.
A given cone may remain unrefined until the search procedure
discovers a bound which interacts with the given cone, at which
time the cone may be refined. The refinement of a given cone may be
further refined as additional interacting objects/bounds are
discovered in order to more adequately distinguish the objects. In
the context where objects are in motion, the movement of an object
into a given cone's field of view may induce increased refinement
of the given cone. If the user in a virtual environment stops to
look at a given object, the cones defining that object may be
increasingly refined.
[0208] Refinement of the cone hierarchy may be subject to the
availability of computational cycles. According to the paradigm of
successive warming, the initial cone tree may have only one or a
few cones allowing a crude initial estimate of visible object set
to be immediately displayed. As computational cycles become
available the cone hierarchy may be successively refined and
searched in order to provide an increasingly accurate display of
the visible object set.
[0209] In general the cones of the cone hierarchy may be at
differing levels of refinement. Cone refinement is allowed only if
the cone interacts with an object or bound (e.g. hull). Adaptive
refinement of a cone terminates when the cone resolution equals
that of a pixel or when no object occurs in the cone.
[0210] It is noted that a combination of fixed refinement and
adaptive refinement of the cone hierarchy may be used.
[0211] The present invention also contemplates adaptively refining
the cone hierarchy and identifying the K nearest objects/bound for
each cone, where K changes the refinement level changes.
[0212] Non-Occluding Objects
[0213] Non-occluding objects are objects which do not totally
occlude (i.e. block visibility) of other objects. For example, a
transparent, semi-transparent, or translucent object may be a
non-occluder. A screen door, tinted glass, a window with slats may
be classified as non-occluders. Objects behind a non-occluder may
be partially visible. The present invention contemplates certain
modifications to the visibility search algorithm to allow for the
presence of non-occluding objects (NOOs) in the collection of
objects to be searched. In particular, the visibility search
algorithm may be configured to search for the first K nearest
occluding objects and any NOO closer than the K.sup.th occluder in
each leaf cone, where K may be a function of leaf cone size.
* * * * *