U.S. patent application number 11/074309 was filed with the patent office on 2005-09-15 for system and method for morphable model design space definition.
Invention is credited to Pawlicki, Richard R., Smith, Randall C., Warn, David Robert.
Application Number | 20050200623 11/074309 |
Document ID | / |
Family ID | 34994189 |
Filed Date | 2005-09-15 |
United States Patent
Application |
20050200623 |
Kind Code |
A1 |
Smith, Randall C. ; et
al. |
September 15, 2005 |
System and method for morphable model design space definition
Abstract
A system and method for designing, on a display device, a new
design. The method includes receiving a plurality of at least two
predefined models defined as parameters in a common coordinate
system, extracting patterns and relationships from the predefined
models and capturing the patterns and relationships in a processor
memory to form a statistical model providing plural, selectable
vehicle shapes, and generating a new design based on a selectable
shape of the statistical model.
Inventors: |
Smith, Randall C.;
(Rochester Hills, MI) ; Pawlicki, Richard R.;
(Sterling Heights, MI) ; Warn, David Robert;
(Royal Oak, MI) |
Correspondence
Address: |
KATHRYN A MARRA
General Motors Corporation
Legal Staff, Mail Code 482-C23-B21
P.O. Box 300
Detroit
MI
48265-3000
US
|
Family ID: |
34994189 |
Appl. No.: |
11/074309 |
Filed: |
March 7, 2005 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
60598290 |
Aug 3, 2004 |
|
|
|
60552975 |
Mar 12, 2004 |
|
|
|
Current U.S.
Class: |
345/419 ;
345/646 |
Current CPC
Class: |
Y10S 715/964 20130101;
G06F 2111/08 20200101; G06T 2210/44 20130101; G06T 19/20 20130101;
G06T 17/20 20130101; G06F 30/15 20200101; G06F 2111/04 20200101;
G06T 2219/2021 20130101 |
Class at
Publication: |
345/419 ;
345/646 |
International
Class: |
G06F 017/50; G06T
015/00; G09G 005/00 |
Claims
1. A method for generating a geometric design, comprising:
receiving input of first components of a geometric design of a high
dimensional space; a first mapping of the first components of a
high dimensional space into first components of a low dimensional
space of principal components; applying the first components of a
low dimensional space of principal components to an optimization
computation to generate second components of a low dimensional
space of principal components; a second mapping of the second
components of a low dimensional space of principal components to
second components of a new geometric design of a high dimensional
space.
2. A method as recited in claim 1, comprising: on a display,
displaying a representation of the first components of a geometric
design of a high dimension; on a display, displaying a
representations of the second components of a geometric design of
high dimension.
3. A method as recited in claim 1, further comprising: on a display
screen, a user manipulating a representation of the first
components of a geometric design of high dimension; in sequence the
first mapping occurs; in sequence, the optimization computation
applies constraints to the first components of a low dimensional
space of principal components to generate second components of a
low dimensional space of principal components; in sequence the
second mapping occurs; on the display screen, displaying a
representation of the second components of a geometric design of
high dimension.
4. A method as recited in claim 1, wherein the first components of
a geometric design of a high dimensional space is a statistic model
generated from a set of exemplar models.
5. A method as recited in claim 1 wherein the optimization
computation comprises applying constraints to principal
components.
6. A method as recited in claim 3 wherein the mapping and
optimization computation steps are transparent to the user.
7. A method as recited in claim 1 wherein the first components of a
geometric design of a high dimensional space represent an
automobile.
8. A system for generating a geometric design, comprising: an input
module for receiving of first components of a geometric design of a
high dimensional space; a first mapping module for mapping the
first components of a high dimensional space into first components
of a low dimensional space of principal components; an optimization
computation module for applying the first components of a low
dimensional space of principal components to an optimization
computation to generate second components of a low dimensional
space of principal components; a second mapping module for mapping
the second components of a low dimensional space of principal
components to second components of a new geometric design of a high
dimensional space.
9. A system as recited in claim 8, comprising: a display module for
displaying, on a display screen, a representation of the first
components of a geometric design of a high dimension and on the
display, displaying a representations of the second components of a
geometric design of high dimension.
10. A system as recited in claim 9, further comprising: a
manipulation module for a user manipulating on a display screen, a
representation of a first component of the first components of a
geometric design of high dimension.
12. A system as recited in claim 8, wherein the first components of
a geometric design of a high dimensional space are a statistic
model generated from a set of exemplar models.
13. A system as recited in claim 8 wherein the optimization
computation comprises: a constraint module for applying constraints
to principal components.
14. A method as recited in claim 8 wherein operations of the first
mapping, second mapping and optimization modules are transparent to
the user.
15. A method as recited in claim 8 wherein the first components of
a geometric design of a high dimensional space represent an
automobile.
16. A method for design, comprising: defining a first space for a
first design to occupy; defining a second space for a second design
to occupy; manipulating the second design within the first
space.
17. A method as recited in claim 16, wherein defining the first
space comprises: receiving a plurality of at least two predefined
models defined as parameters in a common coordinate system;
calculating means from correspondences between parameters of the
predefined models; calculating a covariance based on the means.
18. A method as recited in claim 16 wherein defining the second
space comprises: receiving a plurality of at least two predefined
models defined as parameters in a common coordinate system;
calculating means from correspondences between parameters of the
predefined models; calculating a covariance based on the means.
19. A method as recited in claim 16 wherein the manipulating step
comprises: receiving the second design of a high dimensional space;
a first mapping of the first components of the high dimensional
space into first components of a low dimensional space of principal
components; applying the first components of a low dimensional
space of principal components to an optimization computation to
generate second components of a low dimensional space of principal
components; a second mapping of the second components of a low
dimensional space of principal components to second components of a
new geometric design of a high dimensional space.
20. A method as recited in claim 16, further comprising: on a
display screen, a user manipulating the second design of high
dimensional space.
Description
CROSS REFERENCE TO RELATED APPLICATIONS
[0001] This application claims priority to U.S. Provisional
Application Ser. No. 60/598,290, titled, "A SYSTEM AND METHOD FOR
GEOMETRIC SHAPE DESIGN," filed Jul. 30, 2004, which itself claims
priority to U.S. Provisional Application Ser. No. 60/552,975,
titled, "CAPTURING AND MANIPULATING AUTOMOTIVE DESIGN
CHARACTERISTICS IN A STATISTICAL SHAPE MODEL," filed Mar. 12, 2004,
both of which are incorporated by reference herein in their
entirety.
TECHNICAL FIELD
[0002] A design tool for use in manufacture is disclosed. More
particularly, this disclosure relates to computer-aided geometric
shape design tools.
BACKGROUND OF THE INVENTION
[0003] Computer design tools useful for industrial design are most
commonly "computer aided design" (CAD) based. Users of CAD programs
often undergo training and have much experience in the use of CAD
before their proficiency reaches a level high enough for complex
design and engineering.
[0004] In the automotive industry, car body designers typically
sketch their designs. Car body designers are creative artists who
produce styling sketches and typically do not use CAD programs.
From sketches and discussion with the car body designer, a CAD
designer will rework the sketch onto the computer. Accordingly, the
sketch is engineered into three-dimensional detail. There are often
many instances of refinement discussed between the artist and CAD
user. Oftentimes, for example, the designer's sketches are not in
correct perspective and details may need to be added and changed,
and therefore, the process to completion by a CAD user may become
tedious and repetitive. The CAD tool requires construction of a
shape, piece by piece. The overall shape may not emerge until a
significant amount of work has been done.
[0005] It would be advantageous for a designer to have available a
design tool that is simple to operate and can readily replace hand
sketching, and whose resultant design is captured in a computer
file. Therefore, the time consuming, step of refinement between the
artist and a CAD designer may be substantially eliminated.
[0006] In automobile design, a designer most likely must keep the
design within certain style parameters. For example, the task at
hand for the designer may be to design a new Cadillac. In this way,
it may be advantageous for the designer to have available Cadillac
designs and then to change some aspect or another to create a new
look in keeping with the brand character of the Cadillac.
[0007] Alternatively, a designer may want to create a design that
is an intermediate between two designs, or is a blend of three or
more designs. In any of these events, the process currently depends
on the designer's strong familiarity with the various automobiles'
designs. In this way, the ability to use a computer to maintain
data on designs and create automobile designs from any number of
combinations would be particularly advantageous for the design
process.
[0008] Complex design shapes such as automobiles may have
topologies that vary greatly. The list of automobiles, even for
just one automobile manufacturer, is extensive and the styling is
diverse and includes many discrete variations. The functional
categories of automobiles for a single manufacturer may include
coupes, sedans, SUVs, sports cars, and trucks. Secondary categories
may include minivans, wagons and convertibles. A computer based
design tool that would allow a designer to combine any number of
models to form a resultant new style or model would be
advantageous.
[0009] It would be advantageous if the design tool visually offered
to the artist a plurality of automobiles to choose from and
provided the ability to combine them into a combined resultant
automobile design. If the designer desires a sportier car, or, for
example, a Buick to be more Cadillac-like, or to use the grill of
one car on another car, it would be advantageous to provide in a
design tool the flexibility to the user to reach his design goals
or otherwise explore options.
[0010] Once a user has created a resultant combined design by
combining as many models as desired, an additional benefit would
come from the ability to change or morph that resultant design. A
design tool that would be useful for morphing automobile designs is
preferably flexible enough to allow a designer to explore different
combinations and then provide the ability to morph the resultant
design into many possible designs.
[0011] Furthermore, a design tool that provides many options for
morphable features, and allows the addition of constraints on the
features in the morphing process, would be advantageous as
well.
[0012] In this manner it would be also advantageous that a computer
operable design tool provide the output so that after the vehicle
designer's initial design is complete, a CAD programmer would then
be able to work from the output.
SUMMARY OF THE INVENTION
[0013] This invention is a software package, a system, a method,
and an apparatus for designing geometric shapes for automobiles or
any other manufactured objects.
[0014] On a display screen a first set of exemplar
designs-hereinafter referred to as a catalog--is provided for
selecting a second set of exemplars to create a resultant design
space or mixture. A design space includes space defined by features
of a mixture. Once a user has created a resultant combined design,
also available is the ability for the designer to explore the
design space and therefore to change or morph the resultant design.
Automobile designs may be embodied as the exemplars. The term
exemplar includes a model that is a registered model as defined
below.
[0015] Once the resultant combined design is chosen then
manipulated, altered, or morphed according to one or more
statistical models, the result may be a new design. A statistical
model includes a probabilistic object derived from a design space.
The mathematical space and statistical manipulations of the
mathematical space allow the user to explore the space allowing for
stylistic and functional interpretations. A mathematical algorithm
described herein allows the user to select one or more feature
constraints and apply the constraints to the morphable model.
[0016] Accordingly, a CAD user may input the model generated by the
described design tool and begin the process of model making
therefrom or use the model directly.
BRIEF DESCRIPTION OF THE DRAWINGS
[0017] FIG. 1 is a system and apparatus diagram;
[0018] FIG. 2 is a screen display showing a catalog;
[0019] FIG. 3 is a screen display where three exemplars are shown
in the background and a morphable vehicle is shown in the
foreground;
[0020] FIG. 4 is a screen display where three exemplars are shown
in the background and a morphed vehicle is shown in the
foreground;
[0021] FIG. 5 is a flowchart illustrating different actions
available to the user;
[0022] FIG. 6 shows a truck and a coupe;
[0023] FIG. 7 shows steps using information relating to the truck
and coup;
[0024] FIG. 8 is a flowchart of the model updating algorithm;
[0025] FIG. 9 is an illustration showing several user
interfaces;
[0026] FIG. 10 is an illustration showing several user
interfaces;
[0027] FIG. 11 is an illustration showing several user
interfaces;
[0028] FIG. 12 is an illustration showing an additional display and
interface;
[0029] FIG. 13 is an illustration showing another user
interface;
[0030] FIG. 14 is a flowchart of the shifting method described
herein; and
[0031] FIG. 15 is a flowchart of the method incorporating the
shifted output.
DESCRIPTION OF THE PREFERRED EMBODIMENT
[0032] The system and method, apparatus and product includes
central system modules 100 as shown in FIG. 1 including, for
example, an input module 102, a processor module 104 in
communication with software instructions from software 106, a
display module 108 to display operations to a user 110, and an
apparatus 112 for direct input such as a keyboard, touch-screen,
voice recognition, joy stick, a mouse, and/or other apparatus for
user input. The user 110 manipulates data that is processed by
processor 104 according to computer instructions. Data may be shown
on display 108 and/or generated as output to output module 114. The
output of a new vehicle design may be communicated to a new vehicle
manufacture process.
[0033] This invention may be embodied in the form of any number of
computer-implemented processes and apparatuses for practicing those
processes. Embodiments of the invention may be in the form of
computer program code containing instructions embodied in tangible
media, such as floppy diskettes, CD-ROMs, hard drives, or any other
computer-readable storage medium, wherein, when the computer
program code is loaded into and executed by a computer, the
computer becomes an apparatus for practicing the invention. The
present invention may also be embodied in the form of computer
program code, for example, whether stored in a storage medium,
loaded into and/or executed by a computer, or transmitted over some
transmission medium, such as over electrical wiring or cabling,
through fiber optics, or via electromagnetic radiation, wherein,
when the computer program code is loaded into and executed by a
computer, the computer becomes an apparatus for practicing the
invention. When implemented on a general-purpose microprocessor,
the computer program code segments configure the microprocessor to
create specific logic circuits.
[0034] The user 110 may view on display screen 108 a variety of
models shown in catalog form 118 (see FIG. 2). The user may begin
the method by selecting a mix, or set of designs, from the catalog,
to define and populate a mathematical design space. A model as
referred to herein includes a single geometric object(s). A
reference model includes a template for a class of objects, for
example of vehicles. A catalog includes a collection of exemplars
from one object class. The terms "vehicle" and "object" are used
interchangeably in this disclosure and, where vehicle is used,
object applies.
[0035] Exemplars that correspond to existing models may inherently
incorporate engineering constraints. The term constraint used in
this disclosure is a mathematical term and used in the sense of a
mathematical operation, and is not intended to add limitations to
the meaning of the features with which it is used in conjunction.
Packaging and criteria information exists in the data stored in
datastore 116 or elsewhere of the cataloged vehicles providing
information early in the design process. Existing vehicles used as
input to the statistical model may already have well-coordinated
packaging and aesthetic characteristics.
[0036] The design tool therefore provides the ability to morph
models in many ways. Basic geometric shape features such as points
can be constrained to lie in certain positions. Other features that
are selectable for constraint during morphing may include primary
criteria such as vehicle height, wheelbase, H-point, and steering
wheel position. A feature may include a simple attribute of the
reference model, for example, a point.
[0037] Therefore, the catalog provides the designer exemplars to
incorporate into a resultant object that may meet target design
requirements. Accordingly, certain criteria including engineering
constraints may be built into the resultant average vehicle as
exemplars are existing models and inherently include engineering
constraints. Many engineering criteria are manipulable, changeable
or morphable in accordance with this process, method and apparatus.
Engineering constraints available for application in the morphing
process may be displayed as, for example, options in drop down
menus.
[0038] There may be a style that the designer may want to emulate,
e.g., a Buick--having a particular brand flavor or identity that in
this example should remain in the mix. Alternatively, favorite
models may be mixed together and new design elements may be added.
It will be appreciated that there are many ways to approach design
with the design tool described herein, including such
considerations as aesthetics, brand identity, and function.
Selectable morphable features include features named herein as well
as others that may be recognized by persons skilled in the art of
vehicle design.
[0039] In a first embodiment, the user's interaction 110 with the
system and method is driven by a user interface, the display 108
and manual input 112. Referring now to FIG. 2, on the display 108,
the MIX button 120 brings up the set of vehicles from which the
user may choose. In general, this set of vehicles may be called a
catalog 118. Here we refer to the drawing of the 2D data although
in another embodiment 3D data may also be used.
[0040] Returning again to FIG. 2, a smaller set is selected by the
user from a larger set of exemplar vehicles, in this example
including forty-four exemplars. Here, for example, a plurality of
vehicles 130, 132 and 134, are combined. On one or more screens
there may be any number of exemplar options. Exemplars may be added
to the mix 136 or removed from the mix 138. In this way the user
110 can select particular exemplars to build a morphable model (via
140). Selecting the mix provides a step that sets the initial
probabilities of the statistical model. A morphable model includes
a statistical model derived from a design space or a PCA (Principal
Component Analysis) space.
[0041] A morphable model summarizes the space defined by exemplars.
Any point in the space can be considered a "morph" of the
exemplars, i.e. a weighted linear combination of their features.
However, constraints are also applied to features of the morphable
model to produce updates of the shape based on optimal estimation.
These actions create a new design (a single entity in space), and
the process would not typically be called morphing, although the
result could still be considered a combination of the
exemplars.
[0042] Under the ADD button 136 there may be general categories in
a drop down menu. Options that may be provided are, for example:
processing exemplars, processing only the cars; only the trucks;
only the Chevrolets; or only the GMCs. Similarly, one could also
remove items by the REMOVE button 138. In a similar manner,
vehicles may be chosen and the Cadillacs removed. Accordingly,
everything would then be in a selected set except the
Cadillacs.
[0043] Exemplars in the catalog have been through a registration
process that may be automatic or manual. Exemplars have been fit so
that there is a substantially one-to-one correspondence between
them. For example, a particular curve, the front piece of a hood on
a car, is numbered 1, and another element is numbered 2. The
registration is possible by a process of segmentation that includes
knowledge of one of ordinary skill in the art of design of
manufactured objects such as vehicles, of how many degrees of
freedom are needed to describe a particular car or other geometric
shape and how one can correlate that same information to all, or
substantially all of the cars in the catalog. The result of
segmentation is a set of features that can be put into
substantially 1-to-1 correspondence across input vehicles. From
these vehicles in correspondence, a statistical model can be
produced that captures the similarities in character, style, shape
and proportions of the mixed set. An input vehicle includes a
particular model, for example, a particular car. Registration
includes matching features of the reference model to an input
model. A registered model is an input model after registration.
[0044] Still referring to FIG. 2, the user 110 uses the interface
driven by the MIX button 120 to select items and then build the
morphable model 140. As shown in FIG. 2, a user can select
individual vehicles, such as a Corvette 134, a Hummer 130 and an
SSR 132. A mixture includes a selected subset of a catalog.
[0045] Initially, by selecting a mix as described above, the
initial probabilities and initial mathematical space are set. An
average or mean of these chosen cars is taken to represent the
morphable model. A morphable model is a statistical model derived
from the cars chosen to populate the design space. A design
includes a model in the design space. The morphable model can be
changed by the user in the manner that the user chooses. The user
is able to drive the design tool's output to satisfy aesthetics,
and impose functional and engineering constraints on the desired
design. Once a morphable model is generated, the user can select a
particular place, space, or feature to modify, for example, a
windshield or window. As an example shown in FIGS. 3 and 4, a rear
window is changed from the initial, average model to that of a
morphed model. As discussed above, selectable features to control
morphing are numerous. In this example of FIGS. 3 and 4, the user
can click a point at the base of the rear window and move it to
modify the shape.
[0046] FIG. 3 shows a display screen 108 having a mix of a Hummer
H2 130, the Chevrolet SSR 132, and the Chevrolet Corvette 134. The
average of the three vehicles 150 is shown, representing the
morphable model. In this way it is possible to combine different
categories such as trucks, SUVs and cars together in a single
representation.
[0047] On the display screen 108, the average model 150 may be
shown in one color for example and the exemplars may be in other
colors behind the average model indicating the mixture. The
designer may choose the colors.
[0048] On the display 108, grayed out, the Hummer and the Corvette,
and the SSR, are displayed; their average may be displayed in
black. The resultant average vehicle 150 is approximately the size
of the SSR.
[0049] In this example the user then may pick a point or a curve on
the average vehicle to pursue changes to the morphable model. For
example, the user may pick a point at the rear window 152a. By
selecting the rear glass to modify the user can move that point up
to make the vehicle more Hummer like and move it down to make it
more Corvette like. It may continue to include some of the
character of the SSR.
[0050] A drop-down menu or any other user interface may provide a
list of points or options for the designer to pick from to alter
the resultant vehicle. Or the user may simply click near or on the
rear window to select the point 152a. In the example, the point
152a represents the rear window. By clicking the point 152a, a
cluster of points 156a, 156b and 156c may be provided on the
display 108. The cluster of points shows the space in which the mix
of cars' rear windows reside as shown in FIGS. 3 and 4. In other
words, the cluster of points gives the user an immediate indication
as to the extent of the design space for that feature from the
exemplars. The cluster may or may not be shown on the display but
if shown may be used as a guide. In this way the user can drag the
point 152a to the position of 152b in FIG. 4, knowing that he is
still within the prescribed design space of the mixture.
[0051] The average vehicle 150 morphs into the morphed vehicle 160.
As the point is dragged the morphing may show on the screen in a
continuous interactive manner. From the registration process,
curves that are related to each other from the registered exemplars
defining the design space may change consistently together.
Manipulating exterior vehicle curves will change the packaging
curves and visa versa. This dynamic interaction of styling and
packaging is available immediately to the designer to aid the
aesthetic creation process. Alternatively, when the point is
dragged, the morphed model 160 may update when the mouse button is
released or at any point therebetween.
[0052] The designer may drag point 152a to a point outside of the
space created within 156a, 156b and 156c. However, the model may
break down when manipulated outside the bounds of the design space
as defined by the examples.
[0053] The catalog of designs includes geometric design data, coded
into a representation such that different, but related, designs may
be put into correspondence. The coding may, for example, entail
segmenting a geometric design into points, curves or surfaces. Some
or all designs in the catalog may be represented as an array of
values that characterize the vertices and curves for that design.
Corresponding values in different arrays map to corresponding
vertices and curves in different designs. On the other hand, one or
more catalogs of any number of objects may be included. One or more
catalogs may provide a statistical basis for the class of shapes
that are generated by the method, system and apparatus described
herein.
[0054] To build a morphable model from the selected set of
exemplars, the set of mathematical arrays, one array representing
selected designs 130, 132 and 134, may be manipulated using
statistical methods to extract their relationships and take
advantage of redundancies in the data arrays representing the
geometric designs. For example, a mean or average array may be
calculated, whose elements are calculated by averaging
corresponding elements of the arrays. Similarly, a covariance
matrix may be calculated from the data arrays, once the average
array is calculated.
[0055] Morphing with statistics supports consistency of the
resulting design. In the statistical methods as described in detail
below, the more models used, the more accurately the model class is
described by the statistics. Other statistical models may be used
that operate on less data. Mathematical methods including
statistics are described for their ability to reach results, and
any other methods used to reach the results are considered within
the scope of this disclosure.
[0056] One skilled in the art of vehicle design will recognize that
the vehicle design process typically begins with vehicle criteria.
Some of such criteria include points called "hard points" which in
certain circumstances should not be changed. Hard points may be
categorized according to vehicle architecture class, for example. A
resulting set of dimensions for hard points is therefore developed
and communicated to a designer. The designer may then typically
design the vehicle exterior with constraints given on the location
of at least some of the vehicle hard points.
[0057] The designer input 102 of FIG. 1 is receptive of designer
selections relating to manipulations of the patterns and
relationships contained in the statistical vehicle model. A
modeling module including, for example, a pattern and relationship
extraction module 460 and a constraint module 467 (see FIG. 5), is
adapted to build and modify the statistical model in the design
space based on the designer input.
[0058] The designer may then manipulate the features of the
statistical model corresponding to the hard points to conform with
standard dimensions. The model is drawn on the display 108 of FIG.
1 based on the modified parameters, which in the example are
positions of hard points. The modeling module described in detail
below is adapted to manipulate positions of the visually rendered
vehicle hard points in design space to accomplish designer creation
of a new vehicle shape.
[0059] The process, system and apparatus described herein may be
applied by vehicle designers in additional ways. For example the
vehicle models may be used to design a vehicle interior from a
plurality of vehicle interior exemplars. Also, the interior and
exterior may be designed simultaneously. Additionally, plural
vehicle interior exemplars and plural exterior exemplars may be
used independently to form separate exterior and interior resultant
statistical models. These separate models may then be combined into
a common coordinate system by, for example, causing the hard points
of the models to conform to one another. The designer may then
adjust the interior and exterior models simultaneously with a
single selection or with multiple selections. Cluster points
similar to those as described above in reference to FIGS. 3 and 4
may be used to guide morphing of interior models.
[0060] As described above, the method, system and apparatus
described herein provide the designer the ability to be creative
and to drive the output. For example, the designer may be working
in a space that includes angular exemplars, but may wish to add
some curvature. The designer may add a previously registered round
shape to the mix for that result. In this way, design qualities or
features may be added into the mix. Also they may be weighted--such
as 50 cars and 1 round shape, or weighted as 25 round shapes and 25
cars. The average and the statistics can be changed by the
designer's choice. A slide bar or other similar interface may
provide the user the ability to specify weights.
[0061] As noted with the previous example, exemplars may include
other items than existing vehicles. New designs, not yet built into
production models, may also be included. Different shapes having
other purposes than vehicle bodies may also be used as
exemplars.
[0062] The design space may be large. The designer can use the
mixer to select a smaller set of models appropriate to his design
goal. For example, if only Cadillac vehicles are chosen for the
mixture, the design space will reflect the characteristics of
Cadillacs.
[0063] Referring now to FIG. 5, a flowchart showing an embodiment
of the method described herein is shown at 400. The flowchart is a
high level overview of the options available to the user in the
creative process. During the steps, mathematical algorithms are
used to process data and provide the output along the way.
[0064] A catalog of exemplar designs is provided at 402. Designs of
the catalog may comprise a collection of points and curves (or
other features) conforming to a design topology. Here the design
topology, also referred to herein as a topology, and denoted
.OMEGA., is a finite set P of points p.sub.1, p.sub.2, p.sub.3, . .
. , and a finite set C of curves c.sub.1, c.sub.2, c.sub.3, . . . ,
along with a mapping between the two sets. This mapping may be such
that exactly two points map to a given curve. These two points
constitute the endpoints of the curve, which may be taken to be a
straight line segment. In another embodiment, four points may map
to a given curve, which may be taken to be a Bezier curve. In this
case, two of the points are endpoints of the curve, and the other
two points are the remaining two control vertices for the
particular Bezier curve. In addition, except for exceptional
instances, an endpoint maps to at least two curves. Either of these
specifications of the design topology defines how the curves are
connected one to another by specifying the points that are common
between two or more curves. Further specifications of the design
topology may instead employ NURBS, or polygons, and so on. Designs
used as exemplars for the same design space use the same topology,
and this enables designs to be placed in substantially one-to-one
correspondence, previously discussed in reference to the catalog
shown in FIG. 2.
[0065] As an aside to illustrate the correspondence, FIG. 6 shows
partially detailed outlines of a coupe 450 and a truck 452 that
have features in correspondence. Corresponding curves in the coupe
and truck outline may be different, but the correspondence provides
that they may be represented by the same curves, Curve 1 through
Curve k. For example the coupe 450 and truck 452 may be used as
exemplars chosen from the catalog to create a model design
space.
[0066] It may also be part of the definition of a design topology
that two curves meet at a common endpoint, i.e., no two curves
cross at an interior point of either curve. Compliance with this
part of the definition of a design topology may be checked for
compliance after coordinates have been provided for the points
p.sub.1, p.sub.2, p.sub.3, . . . .
[0067] Note that this definition of design topology is not limited
to points and curves. As one example, this definition of design
topology may be extended to include, for example, a finite set T of
triangles t.sub.1, t.sub.2, t.sub.3, . . . with mappings between P
and T and between C and T, consistent with one another and with the
mapping between P and C, so that three endpoints map to a given
triangle, and three curves map to a given triangle. With this
definition of design topology, the connectedness of the triangles
used in, say, a triangulation of a vehicle body, may be specified
by the points that are common between any two triangles, and the
curves, or edges, that are common between any two triangles. With
this definition, 3D design exemplars may be used. It will be
appreciated that other modifications and extensions to the
definition of design topology may be used.
[0068] In the case where a topology includes a set of triangles T,
it may further be a part of the definition of the topology that two
triangles intersect at a common endpoint or common edge, so that no
two triangles cross in the interior of one or the other. As with
the part of the definition of design topology discussed earlier
that two curves meet at a common endpoint, this further part of the
definition of the topology may be checked for compliance after
coordinates have been provided for the points p.sub.1, p.sub.2,
p.sub.3, . . .
[0069] It is to be noted that neither non-intersection requirement
discussed above is a requirement or restriction on the design
topology itself, but rather a stipulation of what it means for a
design to conform to the topology.
[0070] Returning now to discussion of the method shown in the
flowchart of FIG. 5, exemplars in the catalog have values assigned
for points p.sub.1, p.sub.2, p.sub.3, . . . A point p.sub.i may
have an ordered set of values--e.g., (x.sub.i,y.sub.i) for 2D, or
(x.sub.i,y.sub.i, z.sub.i) for 3D. The set of points P with their
ordered values comprise a feature vector x. For 2D design data, for
example, x may have the form, x=(x.sub.1, y.sub.1, X.sub.2,
y.sub.2, . . . ). Thus, an exemplar has a feature vector x, whose
elements appear in a canonical order prescribed by the topology.
The number d of components of x depends on the type of design data,
e.g., 2D, 3D, and so on. It is also within the scope of this
discussion that x may contain numerical data other than coordinate
values, e.g., engineering analysis data or consumer response
data.
[0071] A selection of exemplar designs from the catalog is chosen
at 404. This selection may be made through a user interface, as
discussed earlier in reference to FIGS. 1 and 2.
[0072] Once a mix of designs has been chosen in step 404, the
design space is defined at 406 which is a module of the extraction
module 460. Since designs in the mix substantially conform to a
particular design topology, a particular point defined in the
topology has a value for its coordinates, for the designs in the
mix. Thus, an element of P, e.g., p.sub.4, may be associated with a
cluster or "cloud" of points, the points of the cloud coming from
the particular values of p.sub.4 for one of the exemplars in the
mix. The collection of the clusters, roughly speaking, defines the
design space.
[0073] More precisely, the design space is defined by the
statistics of the feature vectors x for the set of designs selected
for the mix. This definition may be augmented by associating with
the set of chosen designs a joint Gaussian probability distribution
for the values of the coordinates of points of the design topology,
i.e., a joint distribution for the feature vector components. This
joint distribution is constructed from the set of selected designs
so as to have the same values for the first and second moments as
would be obtained from the statistics of the set of selected
designs.
[0074] It will be appreciated that the use of a Gaussian
distribution is not necessary; another probability distribution may
be more appropriate in particular applications. In fact, it may be
desirable to provide a choice of probability modeling functions
through buttons or menu pick lists on the user interface.
Preferably, a Gaussian distribution may be used as a default
probability modeling function. However, use of a Gaussian
distribution in this discussion is not intended to limit this
disclosure. The basic properties of the design space defined at 406
derive from the statistics of the feature vectors of the set of
designs selected for the mix.
[0075] In particular, these statistics provide for determining the
average value for the coordinates of the points p.sub.1, p.sub.2
p.sub.3, . . . , i.e., the average feature vector. This is shown in
the flowchart at 408. The set of average values for the feature
vector x comprises the average design. This average design may be
output; outputting of the design may include storage in memory or a
more permanent storage medium, rendering to the display, and so on,
at 410. Preferably the average design is at least rendered to the
display. These statistics also provide for determining the
covariances among the feature vector components. The statistical
model is also referred to herein as a morphable model. To
illustrate steps 406, 408 and 410, that are the pattern and
relationship extraction module 460, refer to FIGS. 6 and 7 that
show the data flow from the vector parameters. By combining curves
1 on each vehicle, the average curve 1 results. When done for all
the curves, the average vehicle results. Since the curves 1 through
curve k are substantially registered, the mean of the coupe 450 and
the truck 452 curves may be derived, and then the covariance matrix
may be determined. These algorithmic steps carried out by the
system and apparatus are performed to form the mix of vehicles
suitable for morphing, alteration or manipulation as described
below.
[0076] Again referring to FIG. 5, designer input is provided for in
the step at 412. Here, a user may opt to save the design, as a
completed design, as shown at 414. A user may instead opt to add an
additional exemplar design to the mix, or remove a selected design
from the mix, as shown at 416. Either of these actions by the user
at 416 generally causes a return to the design space definition
step at 406.
[0077] Now that a morphable model is generated, a user may opt at
412 to select a feature constraint, as shown at 418. Such a feature
constraint may include a choice of a point of the design, or a
choice of an engineering constraint, such as H-point or even
aerodynamic drag (using a linear approximation to the drag
coefficient); or a choice of a styling constraint, such as
wheelbase. The constraint choice may be implemented with a matrix
function H as discussed below. The choice of a point may generally
be made through the use of a mouse or other pointing device, as
known in the art. In the example of morphing as shown in FIGS. 3
and 4 where a point on the rear window was moved causing the entire
vehicle to change, the point may have been selected from a list of
morphable features.
[0078] Feature constraints are functions of the model features,
referenced as "H" in the mathematical detail later. For example, a
wheelbase length constraint could be defined as in the following. A
vehicle may be aligned with a coordinate axis--for example, the X
axis--so that the distance between two points along this axis is
just the difference of their x-coordinates. Suppose the features of
the model are kept in a column vector of numbers called `x`, that
includes the points for the centers of the front and rear wheels on
the driver's side. One of the rows of the matrix H--say the
i.sup.th --may be composed of all zeros, except with a `1` in the
same position of the ith row of H (for example H[i,j.sub.1]) that
the x-coordinate of the center of the rear wheel occupies in the
feature vector (x[j.sub.1]), and a `-1` in the position of the ith
row of H (for example H[i,j.sub.2]) that the x-coordinate of the
center of the front wheel occupies in the feature vector
(x[j.sub.2]). The ith element of the product H*x evaluates to a
result that in this example is the wheelbase length. (The product
H*x is called `z` in the mathematical detail later.)
[0079] Different `H` matrices or functions multiplied by the
features `x` compute different lengths, positions, or other
quantities, on 1 or more features, by varying the numbers in H.
Examples of H that are considered important to constrain the model
may be defined in advance, and may be stored in a menu or list for
selection (see FIG. 5, Box 418).
[0080] The particular value of `z` for any particular function H
(e.g. wheelbase length) depends on the current value of the
features in the model `x`.
[0081] If the designer now wishes to change the design so that the
wheelbase length is different than the current value of `z`, he
provides a new value, for example `z0` through a slider, or other
means, indicating that the constraint must be applied with the new
value (see FIG. 5, Box 420), and the whole model updated in some
consistent fashion as a result.
[0082] This system uses a method to update the current design (see
FIG. 5, Box 422) based on satisfying the given constraints on a few
features, and finding a solution (among many possible solutions)
for values of all the rest of the features that additionally is
"best" or "optimal" in some sense. To do that, it uses the
statistics of the examples used to form the model space. The
current design is represented both by the values of the features in
`x`, and also the covariance matrix of the features--which
describes which features vary together, and by how much, in the
example set. We can define one kind of "optimal" solution as one
that minimizes the covariance of the features, after satisfying the
constraints. This is the "optimal estimation" procedure described
below. It is mentioned that if a deterministic (non-statistical)
interpretation is desired, the optimal estimation described is then
equivalent to minimizing the sum of weighted, squared deviation
errors between the solution feature set, and the examples. The
solution (from among many) that has the minimum total error in that
case is the chosen solution.
[0083] The constraint functions do not need to be linear functions.
They may also be non-linear--referenced as `h( )` in the
mathematical detail below. In that case the "optimal" solution, or
the solution that has minimum total error is not guaranteed to be
optimal or minimum over the entire design space, but only in the
neighborhood of the current design values.
[0084] Once the feature for manipulation is selected the next step
in the flowchart of FIG. 5 takes place. At step 420, the design
tool allows the feature constraint to be applied. Accordingly, if
the selected feature constraint is the rear window point, it may be
moved or dragged to a new position on the screen.
[0085] Selectable feature constraints may be provided in a drop
down menu, having a pick list for, e.g., H-point or wheelbase or
any other suitable user interface. The step at 420 in FIG. 5 may
also encompass, for example, selection from a library, inputting a
value, or adding a new definition. The selectable feature may also
be adjustable in value by the user using, for example, a slider or
drop down menu or any other suitable user interface.
[0086] The update step is shown in the flowchart of FIG. 5 at 422.
Typically other points of the design, besides the selected point,
are moved as a result of the update step, due to correlations among
geometric features in the design statistics. As part of the update
step, the resultant design may be rendered to the display. At this
point, a user may opt to save the design, as a completed design, as
shown at 424. The new design has a new associated feature
vector.
[0087] If the selected point has been moved outside the extent of
the usable design space, the user may see intersection of one or
more curves or triangles at an interior point of one of the curves
or triangles, signaling nonconformance to the design topology, and
breakdown of the morphable model. This caricaturing phenomenon
provides feedback to the user on the usability of the current
design.
[0088] A user may choose to pick a new point or feature constraint,
returning again to 418. Note that the "new" point or feature
constraint chosen in this step may, if the user wishes, be the same
point or feature constraint as one chosen earlier during the design
session. Once again, in the case of a selected point, for example,
the user may execute the step of dragging the selected point to a
new position as rendered on the display, with morphing of the
design to a new design taking place.
[0089] Another option, among many other possible options such as
those described and contemplated herein, is for the user after the
update step to add an additional design to the mix, or remove a
selected design from the mix, as shown at 416. As previously
discussed, either of these actions by the user at 416 generally
causes a return to the design space definition step at 406.
[0090] The current design as shown on a display screen 108 or
stored in a different step includes the chosen design that may be
the optimal design. An optimal design includes the design that
better or best meets given constraints and/or criteria. FIG. 5
shows a high level system diagram further illustrating the
cooperation of object shape design modules including an optimal
estimate update module 422.
[0091] The input 102 is receptive of a catalog module 118 of plural
object designs selected from datastore 116. Pattern extraction
module 460 extracts patterns and relationships from the plural
vehicle designs to develop a general statistical model 408 of a
vehicle based on previous designs in the datastore 116. Put
differently, the extraction module is adapted to extract patterns
and relationships from the predefined vehicle models to form a
statistical vehicle model providing plural, selectable vehicle
shapes. Moreover, the extraction module is adapted to extract
patterns from vehicles whose parameters, for example, might be
points, lines and curves. The vehicle model input 102 is receptive
of a set of predefined vehicle models 118 that are selected based
on a predetermination of a market segment for a new design.
[0092] As described above, the general statistical model provides a
variety of selectable shapes defined in terms of combinatorial
relationships between parameters. These relationships are
well-defined within the bounds of the extremes given by the input
designs as shown in FIG. 2. A modeling module is adapted to allow a
designer to manipulate the statistical model by applying designer
selections received from designer input such as by a mouse,
keyboard or touchscreen. The general statistical model provides an
infinite variety of derivable shapes that can be defined in terms
of continuously variable parameters.
[0093] The selected shape is stored in the datastore 116 and is
visually rendered to the designer via an active display. The
designer evaluates the selected shape and may save the shape in
datastore 116 as a new vehicle concept. Multiple vehicle concepts
may therefore be developed and employed in a vehicle design
process.
[0094] It will be understood that extraction module 460 (or a
collection of modules adapted to perform various functions) is
adapted to calculate means from correspondences between parameters
of the predefined vehicle models and to calculate a covariance
based on the means. The extraction model is further adapted to
perform dimensionality reduction of the design space. Moreover, the
extraction module is adapted to employ a principal component
analysis, or method providing similar results, to perform the
dimensionality reduction. Furthermore, the extraction module is
adapted to extract patterns from a plurality of vehicle parameters
represented in vector space, thereby generating the statistical
model in a design vector space.
[0095] Constraint module 467 may constrain the design by the
limitations of the original set of example designs. On either side
of the constraint module 467 are mapping modules 465 and 469.
Mapping module 465 maps a high dimensional space into a low
dimensional space of principal components. Mapping module 469 maps
a low dimensional space of principal component to a high
dimensional space. A user views on a display screen an image or
representation of a geometric design of a high dimensional space.
However, the large amount of data is prohibitive in providing
on-screen editing of such an image for an average CPU. By editing,
here it is meant constrained model editing and the calculations
needed to perform it. That is, an average desktop computer does not
have the computing power to quickly respond to high dimensional
constrained editing input by the user. Here, the dimensions of the
problem are reduced. Briefly referring to FIGS. 3 and 4, depicted
is a point that has been dragged by a user so that from FIG. 3 to
FIG. 4, the whole image is modified as the point is moved. That is,
in FIGS. 3 and 4, it is shown how a designer might drag a "hard
point" on a vehicle rear window while the vehicle continuously
changes. In order for the point to move and the rest of the image
to change accordingly, the high dimensional space of the image is
mapped to a low dimensional space where the morphing process
occurs. The high dimensional space may be displayed on a display
screen for a viewer to see while the low dimensional space is
transparent to the user. Once the morphing process has occurred in
the low dimensional space, the low dimensional space is mapped to
the high dimensional space. The speed at which this mapping,
morphing and mapping process occurs provides the appearance of a
seamless editing or morphing process on the display screen.
[0096] Constrained model editing provides the power to change the
design globally, and consistently within the model space, with only
a few interactions by the designer. As shown in FIGS. 3 and 4, the
effects of editing are not local to the feature being edited.
Constraint module 467 includes constraint feature module 420 and
optimization (which also may referred to as optimal estimation and
update) module 422. The constraint module is described in more
detail below.
[0097] Optimal estimation module 422 provides optimal estimation
techniques to generate a design within the statistical model that
meets preferred designer specifications. The optimal estimate
computation includes computing the hard points of the new design
shape given the constraints defined in the constraint feature
module 420. If the wheel centers are features of the design, a
matrix H can calculate the difference of those features along the
vehicle length axis, and a vector z, as detailed below, can be
designated by the designer as a desired wheelbase dimension,
perhaps by entering it numerically.
[0098] As explained in detail below, z may be assumed to have a
probability distribution characterized by additive noise v with
covariance R. The magnitude of the covariance (R) for v compared to
the magnitude of the system covariance used by H determines how
precisely the new value is adopted. In the case of moving a point,
the designer may want to specify the location exactly (R.fwdarw.0).
However, it is possible to let the design "pull back" and settle
into a state that is influenced less strongly by the manipulation
(in that case R is larger, allowing the dragged point position, for
example, some variability in subsequent steps). When a range of
values is permissible, e.g. a range of wheelbases, increasing the
value of R allows the design to find an optimum balance of this and
other constraints. This relative weighting is also a
designer-specified value, perhaps through a slider.
[0099] Turning now to a more detailed discussion of the three steps
418, 420, and the update step at 422 included in the constraint
module 467 of FIGS. 5, and 418a, 420a and 422a of FIG. 8, the
mathematics described below allows the very large amount of data to
be reduced for computations and then be expanded for display and
further manipulation. Accordingly, data in a high dimensional space
is mapped to a low dimensional space and then mapped back to the
high dimensional space once the user's manipulation of the design
on a display has been processed. In this way, a user is able to
watch morphing occur as the selectable features are processed.
[0100] The abstraction of the design process--in terms of the
mathematical model--may be given by the following steps: (1)
specify the model space for the new creation, which was discussed
above; (2) review the current shape, or read out geometric values
(e.g. dimensions); (3) apply constraints to parts of the geometric
shape, reducing freedom in the design space; and (4) add
innovation, adding freedom back into the design space.
[0101] In the first step, exemplars are chosen to populate the
object space with the right character. This is like the
pre-determination of a market segment for a new design, resulting
in a population of past, present, and concept vehicles as context
for the designer. The choice of exemplars for the object space is
part of the creative process. The discussion below shows how the
exemplars are processed to produce a usable low dimensional space
(with a compressed representation characterized by u and .LAMBDA.,
to be defined and discussed below).
[0102] The remaining steps are used iteratively, although at any
moment one will apply. Step 2, review the current shape, or read
out geometric values (e.g. dimensions), was discussed with
reference to the step for providing the average output 410 in FIG.
5. In these cases, it is preferable to maintain the low dimensional
space--so computations can be performed quickly--yet provide the
mapping to the original feature space (x) that is meaningful to the
designer.
[0103] To draw the current shape, or to query the model, mapping to
the original feature space is performed. For example, the points at
the center of the wheels may be defined in the original feature
vector (x), so wheelbase can be defined as the distance between the
points. However, the same points may not appear in the compressed
representation (u), although they can be reproduced from it. The
wheelbase calculation is automatically redefined to use u.
[0104] An example of a constraint is the specification that
wheelbase be changed to a particular value. The constraint uses the
same mapping as the query function. Another example is the
specification of the position of a particular model point (during
shape editing). In either case, the rest of the shape preferably
will assume some plausible values based on the specification.
[0105] Finally, it may be the case that the designer wants to
isolate part of the design for change, without globally affecting
the rest of the design. In that case, the features being
manipulated would preferably have their statistical correlations to
the rest of the model weakened so the impact on the rest of the
design is removed or lessened to a chosen degree.
[0106] A model space (x, .OMEGA.) for a design will be defined as a
topology .OMEGA., and an associated Gaussian-distributed
vector-valued random variable x with given means and covariance
matrix
[0107] P(x).about.N(.upsilon.,).
[0108] Estimates of the parameters .upsilon. and will be written
{circumflex over (x)} and C.sub.x, respectively. In the following,
the estimates {circumflex over (x)} and C.sub.x will be substituted
for .upsilon. and , in general and where appropriate.
[0109] In general, the covariance matrix C.sub.x may be
ill-conditioned. A singular value decomposition (SVD) is applied to
obtain the principal components characterizing the mix of designs.
The reduction to principal components allows the design space to be
explored in a generally stable, computationally economical way.
[0110] Within .OMEGA. a particular model--x.sub.1, for example--may
be defined by a vector of d model features in a canonical order, so
that the same features are present in corresponding order in the
models, as discussed previously. Geometric shape differences among
models in .OMEGA. are encoded in the feature vector values. For
example, the topology might be a mesh, and the features would be 3D
points at the mesh vertices with different values for different
shapes. The mathematics herein discussed is indifferent to the
contents of the feature vector, and other embodiments may include
other properties such as appearance information, engineering
properties, consumer scores, and other criteria. The common factor
is that features may be functionally or statistically related to
the geometric shape.
[0111] If a probabilistic interpretation is not warranted or
desired, the same mathematics can be developed from recursive,
weighted least-squares solutions to the problem. The "weights" may
be chosen to be the inverse of the "covariances" calculated in the
following.
[0112] The definition of the topology has been discussed above. The
current state of the design is defined by the topology, and the
current estimates of the shape vector and shape covariance matrix,
{circumflex over (x)} and C.sub.x. These estimates are made using
the statistics of the feature vectors of the selected exemplars.
The size d of the shape vector x can be very great, and the size of
matrix C.sub.x is thus the square of that dimension--often large
for computational purposes, especially when intended for use in an
interactive design tool. The matrix C.sub.x represents the
variability of geometric shape features within the set of selected
exemplars, and the inter-variability among features. Fortunately,
if the exemplars in the design space are cars, the shape features
have much more consistently defined relationships than if the space
also contained geometric objects from completely different classes
(e.g., spoons, or kitchen sinks). Due to this internal consistency,
or coherence, there is generally a far smaller, but nearly
equivalent set of features and associated covariance matrix (called
u and A in the following) that can be used instead to greatly
improve computational efficiency.
[0113] Referring to step (3)as described in a paragraph above, that
is, reducing the freedom in the design space, Principal Component
Analysis (PCA) may be used as a data dimensionality reduction
technique that seeks to maximize the retained variance of the data
in a (lower dimensional) projected space. The PCA space includes a
design space with a reduced number of dimensions via PCA. The
projection model can be derived in closed form from the data in a
set of examples. The following discussion illustrates the
technique.
[0114] The projection to principal components is implemented with
the eigenvectors and eigenvalues of the sample covariance of the
data, S.sub.x, with sample mean x. For n models, x.sub.1, . . . ,
x.sub.n let the d.times.n data matrix be defined as
[0115] {circumflex over (D)}=[x.sub.1-{overscore
(x)},x.sub.2-{overscore (x)}, . . . ,x.sub.n-{overscore (x)}].
[0116] The x.sub.i is a d-component non-random data vector, and
typically d >>n. The sample covariance of the data set is
then the d.times.d matrix 1 S x = 1 n - 1 [ D ~ D ~ T ] .
[0117] The eigenvectors of S.sub.x and {tilde over (D)}{tilde over
(D)}.sup.t are the same, and the eigenvalues differ by a scale
factor. Since it is sometimes useful to keep the original data
matrix, it is convenient to work with {tilde over (D)}{tilde over
(D)}.sup.t rather than S.sub.x.
[0118] The eigenvectors and eigenvalues of {tilde over (D)}{tilde
over (D)}.sup.t can be derived from the Singular Value
Decomposition (SVD) of D, avoiding altogether the storage and
manipulation of the elements of the matrix {tilde over (D)}{tilde
over (D)}.sup.t (or S.sub.x), which can be quite numerous. First,
the following factorization may be made.
[0119] Using SVD, factor the d.times.n matrix {tilde over (D)}:
[0120] {tilde over (D)}=U.SIGMA.V.sup.t.
[0121] Here, U is a d.times.n matrix, and .SIGMA. and V are
n.times.n matrices.
[0122] By the definition of SVD, .SIGMA. is a diagonal matrix.
Further, the matrix U is column orthonormal, and the matrix V.sup.t
is row and column orthonormal (since it is square); i.e.,
[0123] U.sup.tU=I
[0124] and
[0125] V.sup.tV=I.
[0126] From .SIGMA. and its transpose, define a new matrix
[0127] .LAMBDA.=.SIGMA..SIGMA.'.
[0128] The columns of U and the diagonal elements of .LAMBDA. are
the eigenvectors and corresponding eigenvalues of {tilde over
(D)}{tilde over (D)}.sup.t.
[0129] Next, in PCA the eigenvalues are sorted by size (largest
first). The columns of U and rows of V.sup.t (or columns of V) are
correspondingly sorted. If U', .SIGMA.', and V' are the reordered
matrices, the product of the new factors reproduces the original
matrix, i.e.
[0130] {tilde over (D)}=U'L'V'.sup.t=U.SIGMA.V.sup.t.
[0131] After reordering of the matrices, any zero eigenvalues will
be in the last diagonal elements of A (and thus last in .SIGMA.').
If there are m.ltoreq.n-1 non-zero eigenvalues from the n
exemplars, then rewriting the last equation in tableau 2 D ~ d , n
= U d , n ' n , n ' V n , n ' t = [ U d , m " A ] [ m , m " 0 0 0 ]
[ V m , n " t J ]
[0132] exposes the relevant sub-matrices and their dimensions. The
result after multiplying the sub-matrices is 3 D ~ d , n = U d , m
" m , m " V m , n " t .
[0133] The remaining columns of U" and rows of V".sup.t are still
orthonormal, though V".sup.t is no longer square. The matrix
.SIGMA." remains diagonal. Finally, the original value of {tilde
over (D)} is unchanged, though its constituent matrices are
potentially much smaller--a benefit for storage and computational
efficiency.
[0134] The columns of U are the eigenvectors of {tilde over
(D)}{tilde over (D)}.sup.t needed for PCA, and the diagonal
elements of .SIGMA. are the square roots of the eigenvalues.
[0135] There are at most m=n-1 non-zero eigenvalues in the solution
of {tilde over (D)}{tilde over (D)}.sup.t, but m<n-1 if "small"
eigenvalues are set to zero, as is typical in PCA. These small
values correspond to the "singular values" found by SVD, which may
be set to zero to improve numerical stability in subsequent
calculations. PCA also prescribes setting small eigenvalues to zero
as a form of data compression. The existence of zero eigenvalues
allows for compressed representation (in u and .LAMBDA., below) to
be used.
[0136] Assuming no small eigenvalues are discarded (i.e., if m=n-1)
the (unbiased) estimate of the sample covariance (S.sub.x) can be
written
[0137] S.sub.x=U.LAMBDA.U.sup.t
[0138] using the definition 4 = 1 n - 1 t
[0139] (this has been scaled from the earlier definition of
.LAMBDA., above).
[0140] This defines a transformation from the small (m.times.m)
diagonal matrix .LAMBDA. to S.sub.x. The inverse transformation
is
[0141] .LAMBDA.=U.sup.tS.sub.xU.
[0142] PCA defines the model covariance (estimated as C.sub.x) to
be equal to the sample covariance S.sub.x, so this last equation
can be rewritten. A is the covariance matrix of a new random
variable vector u: 5 = U t C x U t = U t E [ x ~ x ~ t ] U = E [ (
U t x ~ ) ( U t x ~ ) t ] = E [ uu t ]
[0143] where {tilde over (x)}=x-{overscore (x)}, and E is the
operator for taking the statistical expectation value.
[0144] Thus
[0145] u.ident.U.sup.t{tilde over (x)}=U.sup.t(x-{overscore
(x)})
[0146] and
[0147] E[u]==0.
[0148] Further, the expression for C.sub.x in terms of A and U can
be rewritten to show the relation of x to u: 6 C x = U U t = UE [
uu t ] U t = E [ Uu ) ( Uu ) t ] = E [ x ~ x ~ t ] Thus x ~ = x - x
_ = Uu and x = Uu + x _ .
[0149] The vector u is the projection of x in the m-dimensional
principal component space. This PCA projection maximizes the
retained variance of S.sub.x.
[0150] The design state may be defined in terms of the
low-dimensional space as
[0151] P(x).about.N({tilde over (x)},C.sub.x)=N(U+b,
U.LAMBDA.U.sup.t)
[0152] P(u).about.N(, .LAMBDA.),
[0153] with initial values
[0154] b={overscore (x)},
[0155] .sub.0=0
[0156] .LAMBDA..sub.0=U.sub.tS.sub.xU
[0157] This discussion of the reformulation of the statistics of
the feature vectors of the selected exemplars in the
low-dimensional space characterized by u and A provides further
detail of step 406 shown in the flowchart of FIG. 5.
[0158] Constraints may be incorporated through introduction of a
random vector z, with r components, taken to be a linear function
of the d-component random vector x, with constant r.times.d
coefficient matrix H, and r-component random noise vector v, with
mean 0 and covariance R.
[0159] z=Hx+v
[0160] with components of v uncorrelated with components of the
deviation of x from its mean .upsilon.:
[0161] E[(x-u)v.sup.t]=0.
[0162] The choice of a value for z, and the choice of H, is
completely equivalent, in this context, to choosing a feature
constraint. This is step 418 in FIG. 5, and also shown at 418a in
FIG. 6.
[0163] The mean and covariance of z, and the cross-covariances of z
with x, regardless of the distributions of x and v, are:
[0164] {circumflex over (z)}=H{circumflex over (x)}
[0165] C.sub.z=HC.sub.xH.sup.t+R
[0166] C.sub.z,x=HC.sub.x
[0167] C.sub.x,z=C.sub.xH.sup.t.
[0168] If x is Gaussian-distributed, so are the marginal (P(z)) and
conditional (P(z.vertline.x)) densities of z:
[0169] P(z).about.N({circumflex over (z)},C.sub.z)=N(H{circumflex
over (x)},HC.sub.xH.sup.t+R)
[0170] P(z.vertline.x).about.N(Hx,R).
[0171] Using Bayes' Rule 7 P ( x z ) = P ( z x ) P ( x ) P ( z )
,
[0172] a posteriori density of x given a particular z (both
Gaussian) is:
[0173] P(x.vertline.z)=N({circumflex over
(x)}+C.sub.x,zC.sub.z.sup.-1(z-{- circumflex over
(z)})C.sub.x-C.sub.x,zC.sub.z.sup.-1C.sub.z,x).
[0174] With z given as the particular function of x above, the
conditional mean and covariance are:
[0175] {overscore (x.vertline.z)}={circumflex over
(x)}+C.sub.xH.sup.t[HC.- sub.xH.sup.t+R].sup.-1(z-H{circumflex over
(x)})
[0176]
C.sub.x.vertline.z=C.sub.x-C.sub.xH.sup.t[HC.sub.xH.sup.t+R].sup.-1-
HC.sub.x.
[0177] These last two equations are the Kalman Filter Measurement
Update equations. As mentioned, deterministic arguments can be used
instead of probabilistic ones, and an equivalent recursive weighted
least squares solution can be obtained if desired.
[0178] To briefly summarize the above description, the reduction to
principal components by the Principal Component Analysis (PCA)
allows the design space to be explored by morphing and other
changes in a generally stable, computationally economical way.
[0179] Again referring to FIG. 5, in the notation below,
superscript .sup.(+) represents the quantity post update, and
.sup.(-) represents the quantity before the update step 422.
[0180] With the definition,
[0181] G=HU
[0182] the Kalman Filter Measurement Update formulae can be
re-written 8 x ^ ( + ) = x ^ ( - ) + U ( - ) G t [ G ( - ) G t + R
] - 1 ( z 0 - H x ^ ( - ) ) C x ( + ) = U [ ( - ) - ( - ) G t [ G (
- ) G t + R ] - 1 G ( - ) ] U t = U ( + ) U t ( + ) = ( - ) - ( - )
G t [ G ( - ) G t + R ] - 1 G ( - ) = U t C x ( + ) U Also , u ^ (
+ ) = ( - ) G t [ G ( - ) G t + R ] - 1 ( z 0 - H x ^ ( - ) )
[0183] can be substituted above, showing the relationship of
principal components (u) to the original space, and its
re-projection:
[0184] {circumflex over (x)}.sup.(+)={circumflex over
(x)}.sup.(-)+U.sup.(+).
[0185] The projection to principal components space is then
[0186] .sup.(+)=U.sup.t({circumflex over (x)}.sup.(+)-{circumflex
over (x)}.sup.(-).
[0187] In the above set of equations, the update is driven by the
user's choice of a new value z.sub.0 for z, shown in FIG. 8 at
420a. As explained further below, joint adjustment of both the
feature vector x and the "noise" or variability allowed in z.sub.0,
provide an optimal estimate for the new feature vector in the
presence of the constraint.
[0188] It should be noted that when C.sub.x does not have full rank
the constraint function H is projected into the subspace as G,
which can contain zeros. In such cases, no update of the
corresponding components occurs. Those combinations of components
have zero variance, however, and are not expected to change. For
instance, if an earlier exact constraint had been enforced (with
R=0), then the result remains true even in the presence of later
constraints. Alternatively, if there was no variation in some
combinations of variables in the original basis exemplars, then no
manipulation of those combinations is allowed by this
mechanism.
[0189] As presented, the update equations are the optimal, minimum
variance Bayesian estimate, which is equal to the a posteriori
conditional density of x given the prior statistics of x and the
statistics of the measurement z. A non-linear estimator may not
produce estimates with smaller mean-square errors. If the noise
does not have a Gaussian distribution, then the update is not
optimal, but produces the optimal linear estimate (no linear
estimator does better, but a non-linear estimator may).
[0190] If the measurement function is non-linear in x, then H is a
partial derivative matrix (not constant) and will have to be
evaluated. A one-step evaluation of H on {circumflex over
(x)}.sup.(-) yields the Extended Kalman Filter, a sub-optimal
non-linear estimator that is widely used because of its similarity
to the optimal linear filter, its simplicity of implementation, and
its ability to provide accurate estimates in practice. There is
also an Iterated Extended Kalman Filter that can be used to
significantly reduce errors due to non-linearity.
[0191] Thus, assuming a linear function of the state with mean
zero, additive noise (v):
[0192] z=Hx+v
[0193] P(v).about.N(0,R)
[0194] The marginal density of z is then
[0195] P(z).about.N({circumflex over (z)},C.sub.z)=N(H{circumflex
over (x)},HC.sub.xH.sup.t+R).
[0196] Given the current estimate of the model ({circumflex over
(x)}) the value z is expected. Forcing the model to any other given
value changes the current model estimate. The minimum variance
estimate seeks to minimize the variance of (z-H{circumflex over
(x)}) jointly; i.e., both the estimate of x, and the independent
noise v of z are adjusted so that (z.sub.0-H{circumflex over
(x)}.sup.(+)-v.sub.0)=0. The solution is the mean of the
conditional density of x, given the information z=z.sub.0.
[0197] Using the linear composition z(u)=z(x(u)) the marginal
density for z(u) is
[0198] P(z).about.N({circumflex over (z)}.sup.(-),
C.sub.z.sup.(-))=N(H(U.- sup.(-)+b),
H(U.LAMBDA..sup.(-)U.sup.t)H.sup.t+R)
[0199] and the desired conditional distribution for u is
[0200] P(u.vertline.z=z.sub.0).about.N(.sup.(+),
.LAMBDA..sup.(+))=N(.sup.- (-)+K.sub.u(z.sub.0-{circumflex over
(z)}), .LAMBDA..sup.(-)-K.sub.uC.sub.- z,u.sup.(-))
[0201] K.sub.u=C.sub.u,zC.sub.z.sup.-1,
C.sub.u,z=.LAMBDA..sup.(-)U.sup.tH- .sup.t,
C.sub.z,u=C.sub.u,z.sup.t.
[0202] The model estimate is then updated as
[0203] P(x.vertline.z)=N({circumflex over (x)}.sup.(+),
C.sub.x.sup.(+))=N(U.sup.(+)+b, U.LAMBDA..sup.(+)U.sup.t)
[0204] shown in FIG. 8 at 422a. This update is equivalent to the
direct update on x, but without ever computing the large matrix
C.sub.x. Inspection reveals that the low-dimensional (i.e.,
r.times.d) constraint function H can be distributed across the
calculation of the vector u.sup.(+) and matrix .LAMBDA..sup.(+)
(which is no longer diagonal). The matrix C.sub.x need not be
created, since it appears in C.sub.z which can be re-written 9 C z
= H ( U U t ) H t + R = G G t + R G rxm HU .
[0205] Constrained model editing provides the power to change the
design globally, and consistently within the model space, with a
few interactions by the designer. The effects of editing are not
local to the feature being edited.
[0206] FIGS. 3 and 4 show how a designer might drag a "hard point"
on a vehicle roof, while the vehicle changes continuously from a
sedan to a truck. In this case, the matrix H is composed of zeros
and ones, and selects the single feature being dragged. The vector
z.sub.0 is the desired, continuously changing value of the feature
point while it is being dragged.
[0207] If the wheel centers are features in the design, then a
different matrix H can calculate the difference of those features
along the vehicle length axis, and z.sub.0 can be designated by the
designer as a desired wheelbase dimension--perhaps by entering it
numerically.
[0208] The magnitude of the covariance (R) for v compared to the
magnitude of the system covariance used by H determines how
precisely the new value is adopted. In the case of moving a point,
the designer may want to specify the location exactly (R.fwdarw.0).
However, it is possible to let the design "pull back" and settle
into a state that is influenced less strongly by the manipulation
(in that case R is larger, allowing the dragged point position, for
example, some variability in subsequent steps). When a range of
values is permissible--e.g. a range of wheelbases--increasing the
value of R allows the design to find an optimum balance of this and
other constraints. This relative weighting may also be a
designer-specified value, perhaps through a slider.
[0209] To recapitulate the update discussion in brief, in general,
the covariance matrix may be ill-conditioned. A singular value
decomposition (SVD) may be applied to obtain the principal
components characterizing the mix of designs. The reduction to
principal components may then allow the design space to be explored
in a generally stable, computationally economical way.
[0210] The average array and covariance matrix completely
characterize data, in the case where the data were drawn from an
underlying Gaussian distribution. Linear constraints imposed on the
data then result in different Gaussian distributions, as can be
determined by evaluating the conditional probabilities for the data
array elements in the presence of a linear constraint. Even in the
case of non-linear constraints, however, this approach may be used
to advantage in exploring the design space.
[0211] As discussed above, in the probability calculation from the
populated design space, one gets an average or mean array and a
covariance matrix. The user can choose to constrain the geometric
shape delivered by the system and method, thereby obtaining a new
geometric shape. The user may choose to continue to use the
original, unconstrained covariance matrix. That is, when the method
is implemented, the average array is updated, and not the
covariance matrix. In this way, some information about the original
mix of designs is still available to the designer for further use
in exploring the design space from a new starting design.
[0212] Another manipulation possible is shifting. As illustrated in
FIG. 15, from input 102, in this manner one space is defined 602
and then a second space is defined 604, while the manipulations
take place in the first space, but have the boundaries of the
second space. This process enables the user to operate in a more
extensive space than an initial space. For example, the user may
mix the CORVETTEs but then mix the CADILLACs. The resultant average
CORVETTE is then manipulated in the resultant CADILLAC space. More
designing freedom results.
[0213] Accordingly, a user can choose a covariance matrix derived
from one mix of designs from the catalog 606, and apply that
covariance matrix to a particular design 608, even one not in the
selected mix to generate an output 610. In this way stylistic, or
other salient features of a mix, could be imported or applied to, a
particular design, to impart to it a flavor or sense of the
selected mix.
[0214] FIG. 15 shows a system diagram for the method shown in FIG.
14. There is an input module 620 for receiving a plurality of at
least two predefined models defined as parameters in a common
coordinate system, a calculating module 622 for calculating means
from correspondences between parameters of the predefined models, a
covariance module 624 for calculating a covariance based on the
means and an application module 626 applying the covariance to a
predefined model that may or may not be part of the plurality of
predefined models. The covariance may be applied to some design
other than the average. That design might have been in the same
class of object used in computing the covariance, or not.
[0215] This is shifting--without rotation--of the Gaussian
distribution derived from the selected mix, away from the average
of the selected mix of designs, to instead be centered at the
particular design.
[0216] In summary, and with all of the qualifications and those
that may be inferred by the preceding disclosure incorporated
herein, a segmentation tool method and system may be incorporated
in or separate from the design tool apparatus and product.
Adjustments to the methods described herein are within the scope of
this disclosure so that the output results are substantially
achieved accordingly. The output generated by the use of the
general statistical and mathematical models described herein
provides a substantial variety of derivable shapes defined in terms
of continuously variable parameters. Referring to FIGS. 9 to 13, it
is further to be noted that the elements on the display are
different sections of the design, and they may be accessed
collaboratively. So one user may be running this on perhaps a wall
display at one site, as shown at 500 in FIG. 9, another designer
may be updating another part of the design on a tablet PC at a
second site, and third user may be on a personal computer working
remotely drawing the colors or doing different things; these
activities may be done around the world. Collaborative tools
sharing 3D models are contemplated by this disclosure. Multiple
parties in different parts of the world can be sharing parts of the
interface, in real time.
[0217] FIGS. 9-13 show various possible user interface
configuration. Briefly referring to FIG. 1, it is within the scope
of this disclosure that user 110 may be more than one user, users
may access the processor 104 through LAN or WAN so that users may
access the processor 104 remotely. FIG. 9 shows an embodiment of a
user interface providing for interactive sketching at 502, and a
mouse and monitor for input at 504. FIG. 10 shows a collaborative
full size display at 506, with an interactive plasma display at 508
and multi-collaborative workstations at 510. FIG. 11 shows an
embodiment of a user interface providing for a cordless stylus and
a 3D spaceball at 512, and another embodiment providing for a force
feedback stereoscopic display at 514. FIG. 12 shows an embodiment
including a workbench display at 516, and a 3D interactive
interface at 518. FIG. 13 shows an embodiment including a walk-up
kiosk at 520.
[0218] While the invention has been described with reference to
exemplary embodiments, it will be understood by those skilled in
the art that various changes may be made and equivalents may be
substituted for elements thereof without departing from the scope
of the invention. In addition, many modifications may be made to
adapt a particular situation or material to the teachings of the
invention without departing from the essential scope thereof.
Therefore, it is intended that the invention not be limited to the
particular embodiment disclosed as the best mode contemplated for
carrying out this invention, but that the invention wilt include
all embodiments falling within the scope of the appended claims.
Moreover, the use of the terms first, second, etc. do not denote
any order or importance, but rather the terms first, second, etc.
are used to distinguish one element from another.
* * * * *