U.S. patent application number 10/656690 was filed with the patent office on 2005-03-10 for multifocal optical device design.
This patent application is currently assigned to Regents of the University of Minnesota. Invention is credited to Gulliver, Robert D., Santosa, Fadil, Wang, Jing.
Application Number | 20050052615 10/656690 |
Document ID | / |
Family ID | 34226402 |
Filed Date | 2005-03-10 |
United States Patent
Application |
20050052615 |
Kind Code |
A1 |
Wang, Jing ; et al. |
March 10, 2005 |
Multifocal optical device design
Abstract
Multifocal optical device designs receive input parameters to
specify a desired power distribution function, a power deviation
weight function, and an astigmatism weight function over the design
field. A fourth-order partial differential variational equation is
linearized by defining the optical surface in terms of
perturbations from a base surface such as a sphere or a toric. The
solution may be expressed as piecewise quadratic splines superposed
over a triangulation of the field. Evaluation of the surface may
use a set of tensor-product splines. An astigmatic base surface
permits both multiple magnifying powers and a prescribed
astigmatism correction in a single optical surface.
Inventors: |
Wang, Jing; (Minneapolis,
MN) ; Santosa, Fadil; (St. Louis Park, MN) ;
Gulliver, Robert D.; (Minneapolis, MN) |
Correspondence
Address: |
Schwegman, Lundberg, Woessner & Kluth, P.A.
P.O. Box 2938
Minneapolis
MN
55402
US
|
Assignee: |
Regents of the University of
Minnesota
|
Family ID: |
34226402 |
Appl. No.: |
10/656690 |
Filed: |
September 5, 2003 |
Current U.S.
Class: |
351/159.52 |
Current CPC
Class: |
G02C 7/06 20130101; G06F
30/20 20200101; G02C 7/024 20130101; G02C 7/061 20130101; G02C
7/028 20130101 |
Class at
Publication: |
351/168 |
International
Class: |
G02C 007/06 |
Claims
We claim as our invention:
1. A machine-implemented method for a progressive optical device,
comprising: defining a triangulation grid over a design field;
receiving a set of parameters, including a desired power
distribution function, a power deviation weight function, and an
astigmatism weight function, the functions being defined over the
design field; defining a base surface over the design field;
solving a linearized variational partial differential equation on
the triangulation grid in terms of the power distribution function,
the power deviation weight function, and the astigmatism weight
function, to produce perturbations from the base surface; combining
the perturbations with the base surface to produce output data
representing a surface of the optical element.
2. The method of claim 1 where the optical element is a progressive
ophthalmic lens.
3. The method of claim 1 further comprising outputting the output
data to a medium.
4. The method of claim 3 where the medium is a digital storage
medium.
5. The method of claim 3 further comprising fabricating the optical
device from the output data.
6. The method of claim 5 where fabricating the medium comprises
grinding a lens blank.
7. The method of claim 1 where the power deviation weight function
represents relative importances of the power distribution function
over different areas of the design field.
8. The method of claim 7 where the power deviation weight function
is approximately piecewise linear.
9. The method of claim 8 where the power deviation weight function
is smoothed.
10. The method of claim 1 where the astigmatism weight function
represents relative importances of astigmatism over different areas
of the design field.
11. The method of claim 10 where the astigmatism weight function is
approximately piecewise linear.
12. The method of claim 11 where the astigmatism weight function is
smoothed.
13. The method of claim 1 further comprising dividing the design
field into a plurality of regions.
14. The method of claim 13 where the power distribution function is
approximately constant over at least some of the regions.
15. The method of claim 13 where the power deviation function is
approximately constant over at least some of the regions
16. The method of claim 13 where the astigmatism function is
approximately constant over at least some of the regions
17. The method of claim 13 where the regions include at least a
far-view region, a near-view region, and a corridor region.
18. The method of claim 1 where the base surface is non-planar.
19. The method of claim 18 where the base surface is spherical.
20. The method of claim 18 where the base surface is toric.
21. The method of claim 1 where the equation includes a linear term
and a bilinear term.
22. The method of claim 21 where the equation has the form
B(.delta.,v)=L(.delta.), where L is a linear functional of a test
function .delta., and B is a bilinear form depending on .delta. and
on a perturbation function v.
22.1. The method of claim 21 where B and L have substantially the
forms 15 B ( , v ) = [ 2 ( + ) H u 0 ( ) H u 0 ( v ) - K 0 ( , v )
] x y L ( ) = [ K u 0 ( , u 0 ) + 2 ( P - + R ) H u 0 ( ) ] x y
23. The method of claim 1 where the equation is solved as
superpositions of a set of overlapping splines.
24. The method of claim 23 where the splines are quadratic.
25. The method of claim 23 where each spline has compact support
over a different subset of the grid elements.
26. The method of claim 23 where some of the splines have a support
partially outside the design field.
27. The method of claim 1 where the boundary conditions of the
equation are less than fully clamped.
28. The method of claim 27 where the boundary conditions are
free.
29. The method of claim 1 where the triangulation grid is a uniform
type II grid.
30. where a size of the grid is one of the received parameters.
31. The method of claim 1 further comprising evaluating the output
data.
32. The method of claim 31 where evaluating includes interpolating
the output data with a set of splines having at least second-order
derivatives at points corresponding to points of the grid.
33. A machine-readable medium bearing instructions for causing a
digital computer to execute the method of claim 1.
34. A machine-implemented method for a progressive optical device
having integrated astigmatism correction, comprising: defining a
triangulation grid over a design field; receiving a set of
parameters, including a desired astigmatism correction, a desired
power distribution function, a power deviation weight function, and
an undesired-astigmatism weight function, the functions being
defined over the design field; defining over the design field a
base surface that includes the desired astigmatism correction;
solving a variational equation on the triangulation grid in terms
of the power distribution function, the power deviation function,
and the astigmatism function, to produce perturbations from the
base surface; combining the perturbations with the base surface to
produce output data representing a surface of the optical
element.
35. The method of claim 34 where the optical element is a
progressive ophthalmic lens with power correction and astigmatism
correction in a single surface.
36. The method of claim 35 where the single surface is a back
surface.
37. The method of claim 34 further comprising outputting the output
data to a medium.
38. The method of claim 37 where the medium is a digital storage
medium.
39. The method of claim 37 further comprising fabricating the
optical device in a lens blank from the output data by altering
only the single surface.
40. The method of claim 37 where fabricating the medium comprises
grinding only the single surface of a lens blank.
41. The method of claim 34 where the power deviation function
represents relative importances of the power distribution function
over different areas of the design field.
42. The method of claim 34 where the astigmatism function
represents relative importances of undesired astigmatism over
different areas of the design field.
43. The method of claim 34 where the base surface is toric.
44. The method of claim 34 where the equation includes a linear
term and a bilinear term.
45. The method of claim 44 where the equation has the form
B'(.delta.,v)=L'(.delta.), where L' is a linear functional of a
test function .delta. and B' is a bilinear form depending on
.delta. and on a perturbation function v.
46. The method of claim 45 where B' and L' have substantially the
forms 16 B ' ( , v ) = [ 2 ( + ) H u 0 ( v ) H u 0 ( ) - K u 0 ( ,
v ) ] x y L ' ( ) = 2 ( P - H u 0 ( u 0 ) ) H u 0 ( ) x y
47. The method of claim 34 where the equation is solved in terms of
a set of splines.
48. The method of claim 47 where the splines are piecewise
quadratic.
49. The method of claim 47 where each spline has compact support
over a different subset of the grid elements.
50. The method of claim 34 where the boundary conditions of the
equation are less than fully clamped.
51. The method of claim 50 where the boundary conditions are
free.
52. The method of claim 34 where the triangulation grid is a
uniform type II grid.
53. A machine-readable medium bearing instructions for causing a
digital computer to execute the method of claim 34.
54. An optical device having multiple different magnifying powers
and a desired astigmatism correction in the same optical
surface.
55. The device of claim 54 where the device is a lens having a
front surface and a back surface.
56. The device of claim 54 where the different magnifying powers
lie in two different regions of the device, and the astigmatism in
both regions is substantially the desired astigmatism
correction.
57. The device of claim 56 where the device includes a third region
having a magnifying power between those of the two regions.
58. The device of claim 56 where the astigmatism outside the two
regions differs from the desired astigmatism correction.
59. A progressive ophthalmic lens having front and back surfaces,
the lens having at least two different desired magnifying powers
and a desired astigmatism correction all in one of the surfaces of
the lens.
60. The lens of claim 59 where the one surface is the back
surface.
61. The lens of claim 59 where the lens comprises: a far-view
region where the one surface has a first of the desired power
corrections and the desired astigmatism correction; a near-view
region where the one surface has a second of the desired power
corrections and the desired astigmatism correction.
62. The lens of claim 61 where the lens further comprises a
corridor region where the one surface has a variable power between
the first and second power corrections and the desired astigmatism
correction.
63. The lens of claim 62 where the corridor region lies between the
far-view region and the near-view region.
64. The lens of claim 59 where the other surface has substantially
no astigmatism.
65. The lens of claim 59 where the maximum deviation of the total
astigmatism in the lens does not exceed about 75% of the difference
between the magnifying powers.
66. Ophthalmic spectacles comprising a frame and a pair of lenses
having front and back surfaces, each lens having at least two
different desired magnifying powers and at least one of the lenses
further including a desired astigmatism correction in the same
surface of the lens as the magnifying powers.
67. The spectacles of claim 66 where both of the lenses include a
desired astigmatism correction in the same surface of the lens as
the magnifying powers.
68. The spectacles of claim 66 where the same surface is a back
surface of the lens.
Description
TECHNICAL FIELD
[0001] The present invention relates to computer-implemented
methods and systems for designing optical devices, and particularly
concerns the design of multifocal devices such as progressive
ophthalmic lenses.
BACKGROUND
[0002] Multifocal optical devices find utility in a number of
technologies. For example, bifocal and trifocal lenses treat
presbyopia caused by loss of accommodative power of the eye.
Progressive addition lenses for this purpose include a far-view
zone with a low power for distance vision and a near-view zone with
higher power for reading and other close work. The power increases
progressively and smoothly in an intermediate zone or corridor
between the far and near zones. Normally, the near zone lies near
the bottom center of the lens, below the far zone; however, other
arrangements satisfy specialized purposes.
[0003] Lens design generates a specification of surface points that
a grinder can produce from a lens blank. Modern lens grinders are
numerically controlled machines that abrade the front and/or back
surfaces of the blank according to design data specifying various
points on the surface. Some lenses may alternatively be produced by
plastic casting or other techniques. Both the front and back
surfaces of a lens may have magnifying power; the effective power
is approximately the difference between the powers of the two
surfaces. Power at any point is related to the sum of a surface's
curvatures in two perpendicular directions.
[0004] Astigmatism is normally an undesirable property of lenses
that causes distortion and color fringing. Astigmatism, sometimes
called "cylinder," is related to the difference between the lens
curvatures in different directions. While a single power lens can
theoretically have zero astigmatism everywhere, a lens having
different powers at different locations necessarily has at least
some astigmatism. A major goal in designing progressive lenses is
to reduce astigmatism, especially in the region of the far, near,
and intermediate zones. Secondary zones outside this region are
less critical, although astigmatism reduction there is still
important.
[0005] Computer-aided design of progressive lens and similar
devices have employed both direct and indirect methods. A direct
method first prescribes a selected distribution of curvatures on a
vertical meridian line that follows the eye's up-down motion.
Prescribed horizontal curves across the meridian are chosen to have
the desired curvatures where they cross the meridian. Direct
methods usually result in a high astigmatism level with little
control over the astigmatism in different areas of the lens
surface.
[0006] Indirect methods balance a desired power distribution with
the unavoidable astigmatism produced. A variational approach
iterates toward the solution of a full non-linear fourth-order
Euler-Lagrange partial differential equation. Another approach
attempts to minimize a defined cost function with an optimization
algorithm. The cost function attains a low value when the desired
lens properties are approached. One such method specifies a desired
power distribution over an entire lens surface, then constructs a
surface close to this distribution while optimizing to achieve low
weighted amounts of astigmatism. Free-energy analogies and
variational approaches have also been investigated. Indirect
methods involve the solution of fourth-order nonlinear elliptic
equations, which involve very large amounts of computation--up to
several hours for a single lens design.
[0007] A lens design may deliberately include an astigmatic or
cylinder component to correct for astigmatism in the wearer's eyes.
Integrating the magnifying-power correction and the astigmatism or
cylinder correction in the same surface of a lens or other optical
device is desirable. Single-power lens conventionally fabricate
both corrections in the same surface of a toric (or atoric) lens.
However, progressive design techniques have heretofore not
integrated astigmatism correction, and fabricating the two
corrections sequentially on the same surface would unacceptably
thin the lens or device. Conventional methods therefore fabricate
the progressive component on one surface (usually the front), and
the astigmatism correction on the other surface. Although this
makes the two corrections approximately independent of each other,
some offset errors still exist, and structural weakness may still
occur, especially in currently fashionable thin, light ophthalmic
lenses.
SUMMARY OF THE INVENTION
[0008] We offer an indirect way to design progressive optical
devices that reduces computation time for accurate powers and small
undesired astigmatism. Input parameters comprise a prescribed power
distribution, a weight function for power deviation, and one for
astigmatism. We solve a linearized variational equation as
perturbations over a base surface, and may express the solution in
terms of superposed low-degree splines.
[0009] Another aspect of the invention permits progressive-power
and astigmatism corrections in the same optical surface by
incorporating a prescribed cylinder into the base surface.
DRAWING
[0010] FIGS. 1A-E depict an example of a progressive lens design
according to an embodiment of the invention.
[0011] FIG. 2 is a flowchart for an example of a method for
designing lenses such as that of FIG. 1.
[0012] FIG. 3 shows the partition of a design field into zones.
[0013] FIG. 4 illustrates a grid for finite element analysis.
[0014] FIGS. 5A-B show a spline function.
[0015] FIGS. 6A-B depict an example of a progressive lens design
incorporating astigmatism correction.
[0016] FIG. 7 is a block diagram of a system for hosting the
invention.
DETAILED DESCRIPTION
Progressive Correction
[0017] FIG. 1 shows an example ophthalmic progressive addition lens
(PAL) 100 inside an edge 110. This lens typifies designs made with
our procedure, although other optical devices can be produced in a
similar manner. The coordinate axes denote millimeters from the
center of the lens. FIG. 1A includes contour lines 120 having
constant optical power. The mean power of an optical surface at a
point x,y is proportional to the sum of the curvatures in two
perpendicular directions at that point; the mean curvature is
defined as H(x,y)=(k.sub.1(x,y)+k.sub.2(x,y))/2. In the example
lens 100, the add power is 0.00 diopter in the far-view upper part
of the lens. The near-view add power is 2.0 diopters in the bottom
center of the lens. ("Add power" is the power that must be added to
the far-view power to obtain the desired near-view power; in this
example, the base or far-view power is 4.94.) A vertical corridor
between the near-view and far-view portions has powers intermediate
the two specified powers.
[0018] FIG. 1B includes contours 130 of constant astigmatism,
sometimes called cylinder. Astigmatism is proportional to the
curvature difference between the two directions at the point,
A(x,y)=.vertline.k.sub.1(x,y)-k.- sub.2(x,y).vertline.. Unless
astigmatism is deliberately designed in to correct an astigmatic
condition, it should be lowest in the far-view, near-view, and
corridor portions. Astigmatism in other portions of the lens is not
as critical, although a good design attempts to reduce it
throughout the lens surface.
[0019] FIG. 2 is a flowchart showing an embodiment of a method 200
for designing a multifocal optical device. In this embodiment, the
device is a progressive ophthalmic lens.
[0020] Block 210 accesses a predefined design field. The field may
be partitioned into zones for the purpose of assigning weights
indicating the importance of various locations in the field. FIG. 3
shows a representative field 300 over which method 200 produces
output data. Most lenses have a generally circular shape, and the
field can be defined in a circular or other overall shape. However,
we have found that a square field allows us to simplify solution of
the equations. Dashed line 310 indicates the edge of a typical lens
within the field. Output data outside this edge is simply
discarded. The axes of field 300 indicate coordinates in
millimeters away from the lens center. A primary region 320 defines
areas of higher importance, in which the astigmatism should be low.
Region 320 includes a zone 321 roughly corresponding generally to
the base-power far-view area of an ophthalmic lens, a zone 322 in
the area of the add-power near-view zone, and an intermediate zone
323 forming a corridor of progressively increasing power between
zones 321 and 322. A secondary region 330 defines field areas of
lesser importance, such as outlying zones 331-334. Region 330 at
least partially surrounds region 320. More or fewer zones and
differently shaped zones are also useful.
[0021] Block 220 receives a data from the designer to specify
parameters for a particular design. Some of these concern
properties of the lens. For example, the designer specifies the
overall size of the lens and the index of refraction n of the lens
material. Typical diameters are about 60 to 80 mm. For common
optical glasses used in PALs, 1.5.ltoreq.n.ltoreq.1.7 is a typical
approximate range. An add power for near vision is specified in
diopters, commonly 0.75 to 3.5 in increments of 0.25. Some
parameters concern characteristics of the design, such as a base
radius of the lens. For ophthalmic applications, this radius
usually lies in the approximate range of 65-350 mm, smaller radii
being usual for high prescriptions (e.g., base power +3), and
larger radii for lower prescriptions (e.g., -10). A designer
commonly selects appropriate sizes, front-surface powers and
back-surface powers from standard charts for desired prescription
ranges. The grid size of field 300 may be allowed to vary as an
input parameter as explained below. Although there is no inherent
limitation on grid spacing, smaller spacings have higher
resolution, but larger ones favor computational efficiency;
acceptable compromises typically make each grid square or patch
approximately 1-4 mm.
[0022] The designer selects a desired or prescribed power
distribution function P(x,y) over design field 300. The value of
this function approaches the far-vision power in zone 321 and the
near-vision power in zone 322. The designer may encode into the
power function such specifications as the length and angle of the
corridor between the far-vision and near-vision zones, and the
point at which the power begins to change. This function may be
specified in a relatively arbitrary manner over the lens surface.
It is often convenient to specify it as a piecewise constant
function having different values in each zone. Because of the sharp
discontinuities of such a function, it may be smoothed, for example
by convolution-with a Gaussian function or a spline, if desired.
FIG. 1C graphs a power-distribution function 140 for example lens
100.
[0023] The designer also assigns two weight functions over the area
of field 300. A designer selects their magnitude distributions in
the x and y directions of field 300 for a particular design. The
weight functions may be in any form. It is often convenient to
assign them as piecewise-constant functions, with or without
smoothing, conforming to the boundaries of the zones defined above.
An astigmatism weight function .alpha.(x,y) is large where the
designer wishes small astigmatism. A power-deviation weight
function .beta.(x,y) is large where the designer wishes low
deviation from the power prescribed by P(x,y). In this example,
both weight functions are positive-valued, in order to guarantee
existence and uniqueness of the design solution. A designer
typically wishes to overlap areas of low astigmatism with areas
having low power deviation. Providing two independent weight
functions allows freedom to assign these areas in any desired
manner with respect to each other for different applications and
preferences, and allows choices as to power accuracy and
astigmatism to be made separately. A lens configuration depends
strongly upon these two functions. The high computation speed of
method 200 allows a user to pursue many variations of a design.
FIGS. 1D and 1E graph sample .alpha. and .beta. weight functions
141 and 142 for lens 100.
[0024] Blocks 230 comprehend operations for generating output data
representing the surface of an optical device such as progressive
or multifocal lens 100. For a surface defined parametrically as
u(x,y), we have selected a measure of the surface quality defined
as: 1 I ( u ) = { ( x , y ) ( k 1 - k 2 2 ) 2 + ( x , y ) ( k 1 + k
2 2 - P ( x , y ) ) 2 } x y
[0025] where the integral is taken over the entire design field 300
and P is the prescribed power at various points in the design
field. (All integrals of unspecified range herein are understood to
be taken over the entire design field 300.) An overall lens design
depends significantly upon the designer's choices of the weight
functions .alpha. and .beta.. This gives the designer freedom to
select among many options for different applications and
preferences, and allows choices as to power accuracy and
astigmatism to be made independently of each other.
[0026] In this context, optimizing the design means minimizing the
value of the surface-quality measure, which may also be thought of
as a cost function. Its global minimum specifies the height of a
surface z=u(x,y) within the design field 310 that has nearly
correct powers and low astigmatism in the areas selected as
important by the weight functions. For the present purpose, we have
modified the formulation of this measure so as to express it in
terms of the mean curvature H(x,y), as defined above, and the
Gaussian curvature K(x,y)=k.sub.1(x,y).times.k.sub.2(x,y). The
quality measure then becomes:
.intg.{.alpha.(H(x,y).sup.2-K(x,y))+.beta.(H(x,y)-P(x,y)).sup.2}dxdy
[0027] The explicit forms of H and K, in terms of u(x,y), then
become: 2 H ( u ) = ( 1 + u x 2 ) u yy - 2 u x u y u xy + ( 1 + u y
2 ) u xx 2 ( 1 + u 0 x 2 + u 0 y 2 ) 3 / 2 K ( u ) = u xx u yy - u
xy 2 ( 1 + u x 2 + u y 2 ) 2
[0028] where the subscript notation signifies partial
differentiation with respect to the subscripted variable. From the
variational calculus, the necessary condition for a minimum in I(u)
is: 3 t t = 0 I ( u + t ) = 0
[0029] for all infinitely differentiable functions .delta.(x,y)
having compact support. This condition translates to the following
form: 4 { 2 ( ( + ) H - P ) ( H u x x + H u y y + H u xx xx + H u
xy xy + H u yy yy ) - ( K u x x + K u y y + K u xx xx + K u xy xy +
K u yy yy ) } x y = 0
[0030] If .alpha., .beta., and P are at least twice differentiable,
then integration by parts twice yields the fourth-order partial
differential equation: 5 ( K u x ) x + ( K u y ) y - ( K u xx ) xx
- ( K u xy ) xy - ( K u yy ) yy - 2 ( ( H + H - P ) H u x ) x - 2 (
( H + H - P ) H u y ) y + 2 ( ( H + H - P ) H u xx ) xx + 2 ( ( H +
H - P ) H u xy ) xy + 2 ( ( H + H - P ) H u yy ) yy = 0
[0031] This is a form of the Euler-Lagrange equation corresponding
to this variational problem. This Euler-Lagrange form is nonlinear,
requiring excessive amounts of computational power and time for
solution. A finite-element solution of acceptable spatial
resolution might require several hours on the types of computers
now economically feasible for use in this type of application.
However, we have constructed a simplification that greatly reduces
the computational load while retaining accuracy and resolution. The
surfaces for most ophthalmic prescriptions, and for many other
kinds of practical lenses, are almost spherical. We therefore treat
the surface u(x,y) as a perturbation v(x,y) from a specified
spherical surface u.sub.0(x,y)={square
root}(R.sup.2-x.sup.2-y.sup.2) of radius R, and we treat the
gradients v.sub.x and v.sub.y of the perturbation as small.
(Although the equation could alternatively be linearized about a
plane or other fixed surface, this embodiment employs spherical
linearization as preferable for many ophthalmic applications.)
These modifications have the effect of linearizing the
Euler-Lagrange equation, which allows us to express H and K as: 6 H
u ( u ) = 1 R + ( 1 + u 0 x 2 ) v yy - 2 u 0 x u 0 y v xy + ( 1 + u
0 y 2 ) v xx 2 ( 1 + u 0 x 2 + u 0 y 2 ) 3 / 2 + E 1 K ( u ) = ( u
0 xx + v xx ) ( u 0 yy + v yy ) - ( u 0 xy + v xy ) 2 ( 1 + u 0 x 2
+ u 0 y 2 ) 2 + E 2
[0032] where E.sub.1 and E.sub.2 are small error terms. We select a
domain where x.sup.2+y.sup.2<R.sup.2, where R is the radius of
the spherical surface u.sub.0. This radius may be taken as the base
radius specified above as a parameter. Then, if
.vertline.v.sub.x,v.sub.y.vertline.<<- ;1 and if
.vertline.v.sub.xx, v.sub.xy, v.sub.yy.vertline. is bounded from
above, E.sub.1 and E.sub.2 become insignificant.
[0033] We then recast the variational problem with these simplified
forms, using a constant spherical surface of radius R that is
selected by the designer or chosen in some other way. The above
necessary condition for a minimum of I(u) then becomes: 7 [ 2 ( + )
H u 0 ( v ) H u 0 ( ) - K u 0 ( , v ) ] x y - [ 2 ( P - 1 R ) H u 0
( ) ] x y = 0
[0034] The terms H.sub.u0 and K.sub.u0, which no longer represent
true curvatures, are defined as: 8 H u 0 ( v ) = ( 1 + u 0 x 2 ) v
yy - 2 u 0 x u 0 y v xy + ( 1 + u 0 y 2 ) v xx 2 ( 1 + u 0 x 2 + u
0 y 2 ) 3 / 2 K u 0 ( , v ) = xx v yy + yy v xx - 2 xy v xy ( 1 + u
0 x 2 + u 0 y 2 ) 2
[0035] Here again, .delta.(x,y) is an arbitrary test function
expressing the variation. This formulation demonstrates that
H.sub.u0 is linear, and K.sub.u0 is bilinear and symmetric. We then
defined two functions B(.delta.,v) and L(.delta.) as: 9 B ( , v ) =
[ 2 ( + ) H u 0 ( ) H u 0 ( v ) - K u 0 ( , v ) ] x y L ( ) = [ K u
0 ( , u 0 ) + 2 ( P - + R ) H u 0 ( ) ] x y
[0036] where B is a bilinear form depending on .delta. and the
perturbation function v, and where L is a linear functional of
.delta. alone. We then pose the minimum cost as a variational form
B(.delta.,v)=L(.delta.), for all .delta. in the relevant subspace
of functions having the desired properties.
[0037] Block 231 of method 200 solves the linearized partial
differential equation B(.delta.,v)=L(.delta.) as a boundary-value
problem for the parameters input in block 220. The solution v(x,y)
represents the perturbation or difference in height from the
specified spherical surface u.sub.0(x,y) over the entire design
field 300.
[0038] A number of general finite elements are known for solving
the present type of fourth-order equations on a digital computer,
such as the Argyris element and the Hsieh-Clough-Tocher (HCT)
element. However, these are computationally intensive. The Argyris
triangle, for example, employs polynomials of degree five. Although
the HCT element uses only cubic polynomials, each triangle must be
divided up into three smaller triangles, each computed
individually. As noted above, design field 300 is preferably
square, even though the shape of most lenses is generally round.
Defining the field in this shape rather than the usual round shape
permits the use of a more efficient element. FIG. 4 shows a
representative portion 400 of design field 300 divided by a grid of
horizontal, vertical, and diagonal lines 401. (Dashed line 400 in
FIG. 3 indicates the approximate relative size and location of
portion 400 with respect to field 300; for clarity of exposition,
the vertical lines i and the horizontal lines j are here taken to
form squares 402 that are 1 mm on a side, although normally they
are somewhat larger. The entire field 300 is divided into elements
in a similar manner. Equations 410 define these lines on the field,
assuming a center origin and normalization to unit length. For a
typical 80 mm diameter ophthalmic lens, a typical grid element size
(i.e., spacing between adjacent horizontal or vertical lines 401)
may be in the range of about 1-4 mm. A field 300 triangulated in
this manner may be described as having a uniform triangulation of
type II.
[0039] Block 232 introduces a set of basis functions for solving
the linearized partial differential equation. A triangulation of
type II permits the basis functions to comprise a particular type
of spline function having compact support and including piecewise
quadratic polynomials which are continuously differentiable.
Employing quadratic polynomials, rather than those of higher
degree, considerably eases the computational burden over previous
approaches.
[0040] FIG. 5A graphs a representative B-spline function 500 which
is piecewise quadratic over the octagon 510 in FIG. 5B. The
numerals at the vertices in FIG. 5A show the normalized coordinates
relative to the center [0,0]. This example quadratic B-spline is
defined over the coordinate range in FIG. 5B as:
B(x,y)=B.sup.0(x,y)+B.sup.x(x,y)+B.sup.y(x,y)+B.sup.xy(x,y)+B.sup.yx(x,y)
[0041] The spline is defined as identically zero outside octagon
510, giving it a compact support. The constituent terms of the
spline are defined as: 10 B 0 ( x , y ) = 1 2 - 1 2 x 2 - 1 2 y 2 B
x ( x , y ) = { 1 2 ( x + 1 2 ) 2 , - 3 2 x - 1 2 0 , - 1 2 x + 1 2
1 2 ( x - 1 2 ) 2 , + 1 2 x + 3 2 B y ( x , y ) = { 1 2 ( y + 1 2 )
2 , - 3 2 y - 1 2 0 , - 1 2 y + 1 2 1 2 ( y - 1 2 ) 2 , + 1 2 y + 3
2 B xy ( x , y ) = { 1 4 ( x + y + 1 ) 2 , - 2 x + y - 1 0 , - 1 x
+ y + 1 1 4 ( x + y - 1 ) 2 , + 1 x + y + 2 B yx ( x , y ) = { 1 4
( y - x + 1 ) 2 , - 2 y - x - 1 0 , - 1 y - x + 1 1 4 ( y - x - 1 )
2 , + 1 y - x + 2
[0042] This example of a piecewise quadratic spline function is
described in P. B. Zwart, "Multivariate Splines with Nondegenerate
Partitions," SIAM J. Numer. Anal. 10 (1973), pp. 665-673. (B-spline
function B(x,y) has no relation to bilinear function B(.delta.,v).
The difference will be apparent from the context.)
[0043] The generic spline B(x,y) is copied once for every square
402, FIG. 4, over the entire field 300 of FIG. 3, and for one layer
outside field 300 on all four sides, to form an extended field. If
the grid on field 300 is normalized to a unit square
[0,1].times.[0,1] and has m rows and n columns, the spline copies
are denoted as: 11 B ij ( x , y ) = B ( mx - i + 1 2 , ny - j + 1 2
)
[0044] for i=0, . . . , m+1 and j=0, . . . , n+1. The center of the
ij-th spline lies at the point [(i-1/2)/m, j-1/2)/n]. The spline
coordinates may then be scaled to fit the grid selected for a
particular design.
[0045] The square elements 511 of support 510 are the same size as
square elements 402 of field 300. The central square 512 of spline
500 is placed over each square 402 of the extended field 300. That
is, the outer rows and columns of the support extend one grid
spacing beyond the edge of design field 300 in every direction.
Using the splines as basis functions realizes the solution to the
variational equation as a linear superposition of a number of these
overlapping splines. That is, a computer solves the equation
B(.delta.,v)=L(.delta.) as a set of simultaneous equations. First,
we set v(x,y)=.SIGMA.c.sub.ijB.sub.ij(xy) over all ij. Substituting
this into B(.delta.,v)=L(.delta.) yields a system of linear
simultaneous equations with the c.sub.ij as unknowns. Solving this
linear system gives the values of the c.sub.ij, which are then
inserted back into .SIGMA.c.sub.ijB.sub.ij(x,y) to find the values
of v(x,y) for all values of x,y.
[0046] Block 233 imposes the boundary conditions on the linearized
equation B(.delta.,v)=L(.delta.). We consider three types of
boundaries: clamped, partially clamped, and natural (or free)
boundary conditions; free and partially clamped conditions may also
be grouped together as less than fully clamped conditions. A
clamped boundary specifies that u=0 and (u.sub.x,u.sub.y)=0 over
the entire boundary. A partially clamped boundary specifies u=0 and
(u.sub.x,u.sub.y)=0 for a designated part of the boundary. A free
boundary allows u, u.sub.x, and u.sub.y to have any values on the
boundary. Our methods permit a unique solution for all three types
of boundaries, as long as .alpha..ltoreq..alpha..sub.0>0 and
.beta..gtoreq.0. The desired boundary conditions are imposed by
selecting only certain of the simultaneous equations and/or by
setting certain variables to desired values. For many ophthalmic
applications, we have found that natural (free) boundary conditions
produce very good results. In this case, using all of the splines
overdetermines the solution, so a solution might not exist.
Dropping the splines in the four corners of design field 300 offers
a convenient way to achieve a unique solution for natural boundary
conditions. This is equivalent to specifying the value of v(x,y) at
each corner of field 300.
[0047] Should partially clamped conditions be desired for other
applications, a designer may, for example, set v(x,1)=0 to clamp
the upper edge of design field 300 to the far-vision spherical
surface. (y=1 here indicates the upper edge of a design field
normalized to a unit square.) In that case, the top row of splines
is discarded; that is, those splines whose support exist only
within the top surface participate in the solution. Fully clamped
boundaries may be imposed in a similar manner. We believe, however,
that a fully clamped boundary produces less desirable designs for
many ophthalmic purposes.
[0048] Finally, block 234 adds the variations v(x,y) to the
base-radius spherical surface u.sub.0(x,y) to produce a set of
heights u=u.sub.0+v above a plane at every point within design
field 300. (These heights may be easily converted to any other form
desired for device fabrication or other purposes.) Because the
splines exist throughout the design field, and not only at the grid
points of the triangulation, a computer may calculate and output
them for any set of points desired by a grinder, plastic casting
apparatus, or other tool that actually fabricates a lens. The data
may be truncated or discarded outside the edge 100, FIGS. 1 and 3,
of the desired lens diameter.
[0049] Block 240 evaluates the design produced in block 230. An
example of such an evaluation is to display or print images such as
100 and 110, FIGS. 1A and 1B for an operator to inspect. The spline
functions employed in the described embodiment of block 230 are
quadratic, in order to reduce the amount of computation. However,
such splines and their superpositions do not have well-defined
second derivatives at the grid points. Therefore, it is not in
general possible to calculate the lens power or astigmatism at some
points, because the curvature cannot be determined there. Block 240
may then interpolate between the grid points with a set of
higher-degree tensor-product splines. Fifth-degree tensor-product
splines, for example, have well-defined second derivatives at all
points, and can match the second derivatives of adjacent splines at
their ends, so that curvature and astigmatism is well-defined at
every point in design field 300. The evaluation data are thus
calculated from the higher-degree splines. Using splines of degree
higher than the quadratic splines of block 232 for evaluation does
not significantly increase overall computation time; the
computational effort for evaluation is generally much less than for
the design. Other evaluation methods, manual or automatic, are also
possible.
[0050] Blocks 250 output previously calculated data. Block 241 may
present the evaluation data from block 240 on a display or printer
for inspection by the designer. Block 242 outputs the design data
in a desired format. This format may comprise a data set for direct
fabrication of a lens according to the design. Outputting the
evaluation and design data may involve storing or communicating the
data, or manipulating it in other ways. For many applications,
method 200 is may operate on a one-pass basis, where the evaluation
data is primarily employed for comparison with other designs.
However, if desired, method 200 can easily be configured to repeat
blocks 220-252 for successive iterations of the same design, with
or without new parameters from the designer. If desired, the
computer itself may analyze the evaluation data, and specify new
design parameters for further iterations.
[0051] Block 260 represents manufacturing a physical lens from the
data output by block 252. This operation may employ any fabrication
technique, such as numerically controlled grinding and polishing of
glass blanks or molding of cast plastic.
[0052] FIG. 6 illustrates an example of a computerized system 700
for hosting the invention. Bus 710 connects a number of components,
such as a processor 720, internal storage 730, and input/output
devices 740. Blocks 740 may include input devices 741 such as
keyboard and mouse, and output devices 742 such as a display and
printer. Storage 750 may accept an external medium such as 751 for
holding instructions and data for carrying out methods according to
the invention, for holding data for designs of the optical devices
produced by the invention, and/or for other data, such as input
parameters. One or more interfaces 760 communicate between
components 710-750 and external computing facilities, such as
server 770 and lens fabricating machinery 780. Server 770 may
constitute a remote facility for holding design programs and data,
and/or other items pertaining to system 700. Block 780 may comprise
a numerically controlled (NC) grinding engine of any type, molding
apparatus for cast plastic lenses, or any similar apparatus. It may
accept design data from system 100, by communication over a network
761 or from removable medium 751, to produce PAL lenses or other
optical devices 782 from blanks 781. In some cases, it may be
advantageous to combine machinery 780 physically with some or all
of the components 710-750, receiving data directly rather than
communicating over an external network 761. Machinery 780 may also
supply data back to other components for design verification or
other purposes.
Astigmatism Correction
[0053] The basic design methods described above attempt to reduce
all astigmatism in the lens to zero. However, many ophthalmic
lenses include a prescribed amount of astigmatism or cylinder in
order to correct a measured amount of astigmatism in a patient's
eyes. Single-power lenses commonly employ a spherical (or
aspherical) surface, and a toric or atoric surface to give both
power and astigmatism correction in the same surface of the
lens.
[0054] In conventional designs, the complex surfaces of progressive
lenses preclude astigmatism correction in the same surface as power
correction. Progressive lens designs, including those using the
basic methods above, grind or mold the progressive magnifying
powers into one surface of a lens (usually the front), and
fabricate the prescribed astigmatism or cylinder correction with
toric curves in the other surface, because the power of a thin lens
is always the difference in powers between the two surfaces.
Disadvantages of this technique include added time and expense in
fabricating both lens surfaces, and optical errors arising from the
distance between the two surfaces. Further, researchers in recent
years have found that placing progressive power correction on the
back surface of a lens offers advantages such as increasing the
area of clear vision; placing prescribed astigmatism on the back
surface denies this feature to patients who require astigmatism
correction.
[0055] An extension to the basic methods of the present invention,
however, permit a prescribed amount of astigmatism correction to be
designed and fabricated into the same surface as the progressive
power correction. The basic methods design progressive powers as
perturbations from a base surface such as a sphere, or even a
plane. However, the equation forms chosen for the basic methods do
not significantly limit the shape of the base surface. Therefore,
the extended methods place a prescribed astigmatism correction into
the base surface, and design the progressive power correction as a
perturbation to that surface. The base surface may constitute a
surface that also incorporates part or all of the far-view
correction for the progressive prescription. Although the following
examples employ toric base surfaces, atoric surfaces are also
suitable.
[0056] A torus is a volume of revolution, part of whose surface can
be expressed as: 12 z ( x , y ) = ( R + R 1 2 - x 2 ) 2 - y 2
[0057] In this example, the astigmatism correction is horizontal
(angle .pi./2) if R>R.sub.1. Rotating the surface produces
corrections at any other prescribed angle.
[0058] Following the same approach as in the basic methods, we
assume that the final surface is a base surface with a
perturbation, u(x,y)=u.sub.0(x,y)+v(x,y). In this case, however,
the base surface is a toric surface that includes at least the
astigmatism correction, and may include some power correction as
well. That is, u.sub.0=z on a rotation of z.
[0059] The expressions for H(u) and K(u) in the previous section
could have assumed several different forms; the particular forms
were selected in order to facilitate the present purpose as well.
Here again, the error terms E.sub.1 and E.sub.2 are insignificant
as long as x.sup.2+y.sup.2.ltoreq.cR.sup.2, where c<1. The
expressions then become:
H.apprxeq.H.sub.u.sub..sub.0(u.sub.0)+H.sub.u.sub..sub.0(v)
K(u).apprxeq.K.sub.u.sub..sub.0(u.sub.0)+K.sub.u.sub..sub.0(v)+K.sub.u.sub-
..sub.0(u.sub.0,v)
[0060] where: 13 K u 0 ( w ) = 1 2 K u 0 ( w , w ) ,
[0061] w being a dummy variable. We now represent the
cylinder--i.e., the measure of its astigmatism--as
A(u)=A(u.sub.0)+A.sub.u0(v), where the first term represents the
cylinder of the original surface u.sub.0 and the second term is the
change in cylinder resulting from a perturbation.
[0062] In contrast to the basic methods described in the previous
section, we desire to keep the prescribed amount of cylinder of the
original surface, at least in the critical regions 321-323, FIG. 3,
for far, near, and intermediate vision. Therefore, the goal is to
minimize A.sub.u0(v).sup.2 instead of A(u).sup.2. Clearly,
A(u.sub.0).sup.2=H.sub.- u0(u.sub.0).sup.2-K.sub.u0(u.sub.0), and
approximately
A.sub.u0(v).sup.2.apprxeq.H.sub.u0(v).sup.2-K.sub.u0(v). We then
define the following functional, integrating as usual over the
entire design domain 300, FIG. 3:
I(u)=.intg.[.alpha.(x,y)(H.sub.u.sub..sub.0(v).sup.2-K.sub.u.sub..sub.0(v)-
)+.beta.(x,y)(H(u)-P(x,y)).sup.2]dxdy
[0063] Assume that the surface u(x,y) minimizes the functional. As
before, let the test function .delta.(x,y) be infinitely
differentiable over the continuum. Calculating the first variation
of I(u) at u and using the notations H.sub.u0(v) and
K.sub.u0(.delta.,v) as in the previous section yields: 14 [ 2 ( + )
H u 0 ( v ) H u 0 ( ) - K u 0 ( , v ) ] x y = 2 ( P - H u 0 ( u 0 )
) H u 0 ( ) x y
[0064] This result yields bilinear and linear terms that are
similar to those in the equations for B(.delta.,v) and L(.delta.)
in the preceding section, the only difference residing in the
second line. A spherical surface is a special case of a toric
surface, and it can indeed be shown that these two equations match
when u.sub.0 is a spherical surface of radius R.
[0065] Method 200, FIG. 2, can now be adapted to design progressive
lenses with astigmatism correction on the same surface.
[0066] Block 220 adds another input parameter, a conventional
specification of a prescribed astigmatism power (cylinder) and axis
angle. From this, method 200 may easily calculate an appropriate
toric base surface u.sub.0(x,y).
[0067] Block 231 solves an equation B'(.delta.,v)=L'(.delta.) in
the form written directly above, rather than in the form described
in the previous section. This equation is already linearized,
therefore greatly reducing the computing time and power required
for its solution. The solution still minimizes the astigmatism
toward zero, using the weight function .alpha., because this
solution includes only the undesired astigmatism from the
progressive powers, and does not carry any part of the prescribed
correctional astigmatism. The splines 232 of FIG. 5 or others may
serve as basis functions as in the previous section. Boundary
conditions 233 are employed in the same way.
[0068] Operation 234 adds the perturbations from the toric base
surface u.sub.0(x,y) defined in modified block 220 to produce a set
of heights u=u.sub.0+v above a plane at every point within design
field 300. Again, these heights may be converted to other units and
formats if desired.
[0069] The output design data 252 now represents a single surface
incorporating both progressive power corrections and astigmatism
corrections. Block 260 thus need only fabricate a single lens
surface for both corrections.
[0070] FIG. 6 shows an example ophthalmic progressive addition lens
(PAL) 600 that incorporates both multiple magnifying powers and
prescribed astigmatism in a single surface according to the
invention. This lens, having an edge 610, typifies designs made
with our procedure, although other optical devices can be produced
in a similar manner. The coordinate axes denote millimeters from
the center of the lens. FIG. 6A includes contour lines 620 having
constant optical power. FIG. 6B includes contours 630 of constant
astigmatism or cylinder. Short bars 640 indicate the direction of
maximum principal curvature, that is, the direction of the total
astigmatism, both for the prescribed correction and for errors
induced by the progressive powers.
[0071] This example illustrates a lens design of radius 30 mm,
having an add power of 2.0 diopters, and a prescribed (i.e.,
desired) astigmatism correction of 3.0 diopters at an angle of
.pi./4. (northwest in FIGS. 6A and B). This design achieves the add
power, and the total astigmatism is close to the prescribed value
in the far-vision, near-vision, and corridor regions. The direction
of the total astigmatism remains close to the prescribed direction,
and both the power and the cylinder vary smoothly, without visibly
annoying sharp changes. There is a clear corridor connecting the
far and near regions in the cylinder contour in FIG. 6B. The
maximum deviation in total astigmatism from the prescribed value is
only 75% of the add power, that is, 75% of the difference between
the magnifying powers of the far- and near-vision regions. This is
a remarkable result in comparison to previous progressive designs
known to us. Typically, a design is considered good when its total
astigmatism deviates from the prescribed value by less than 100% of
the add power over the lens surface.
System and Products
[0072] FIG. 7 illustrates an example of a system 700 for hosting
embodiments of the invention. A bus 710 interconnects a digital
processor 720, internal storage such as RAM and/or ROM 730, and
input/output equipment 740, including display and/or printer
devices 741 and keyboard and/or pointing devices 742. I/O equipment
740 may receive input parameters for methods according to the
invention, and display prompts, evaluations, and other data for the
methods. Bus 710 also couples external storage devices 750 such as
magnetic and/or optical drives capable of reading and/or writing
removable media such as disk 751. One or more interfaces 760
mediate data transfer among bus 710 and other equipment such as
remote server 770 and/or fabrication machinery such as lens grinder
780. One or more links 661 may be realized as direct wire or
wireless connections, local or wide-area networks, or public
networks such as the Internet.
[0073] Instructions and data for practicing methods according to
the invention may be held and/or transferred by various media, such
as external storage 651, communication links 661, and storage 771
associated with a remote server.
[0074] Numeral 790 illustrates an optical product according to the
invention, in this example ophthalmic spectacles or eyewear.
Machinery 680 accepts lens blanks 791 or other raw materials,
fabricating therefrom finished progressive lenses 792, possibly
with prescribed astigmatism correction. Fabricating lenses 792 may
include cutting them to a shape different from that of blanks 791.
Lenses 792 each have a front surface 793 away from the wearer, and
a back surface 794 toward the wearer.
[0075] If astigmatism correction is included, the basic methods
described above may place the multiple magnifying powers on one
surface such as front surface 793, and the prescribed astigmatism
on a cylindrical, toric, or atoric back surface 794. Extended
methods such as those described in the Astigmatism Correction
section are capable of grinding, molding, or otherwise fabricating
both the progressive magnifying powers and the prescribed
astigmatism correction directly in a single surface of blanks or
materials 791, in a single operation.
[0076] Some researchers believe that placing the progressive powers
on back surface 794 has advantages, such as a larger clear-vision
area. Methods according the present invention permit this where
astigmatism correction is also included, by integrating both the
magnifying-power corrections and the astigmatism correction into
the same surface. Placing both on the same surface also eliminates
undesirable offsets between the two forms of correction when they
are placed on physically separated surfaces.
[0077] Where astigmatism correction is not included, or where it is
included on the same surface as the magnifying corrections, the
other surface of lenses 792 may have a constant power, such as a
spherical surface. The two lenses may of course differ from each
other in magnifying and/or astigmatism corrections, as a particular
wearer may require different corrections in each eye.
[0078] Spectacles 790 also include any type of frame 795, attached
to lenses 792 by conventional means such as fasteners or rims (not
shown).
Conclusion
[0079] The foregoing description and the drawings illustrate
specific embodiments of the invention sufficiently to enable those
skilled in the art to practice the invention in those embodiments
and in other embodiments. Such other embodiments may incorporate
structural, logical, electrical, process, and other changes.
Examples described herein merely typify possible variations.
Individual components and functions are optional unless explicitly
required, and operations may be performed in any order, including
in parallel. Portions and features of some embodiments may be
included in or substituted for those of others. The scope of the
invention encompasses the full ambit of the claims set forth below,
and all available equivalents.
* * * * *