U.S. patent application number 12/755345 was filed with the patent office on 2010-08-05 for customized z-lens design program.
Invention is credited to Andreas W. DREHER, Bill Foote, Dave Sandler, Laurence Warden.
Application Number | 20100195047 12/755345 |
Document ID | / |
Family ID | 40788195 |
Filed Date | 2010-08-05 |
United States Patent
Application |
20100195047 |
Kind Code |
A1 |
DREHER; Andreas W. ; et
al. |
August 5, 2010 |
CUSTOMIZED Z-LENS DESIGN PROGRAM
Abstract
Embodiments of the invention pertain to a method for producing a
spectacle lens with optimal correction across the entire lens
taking into account the patient's complete measured wavefront.
Specific embodiments can also take into account one or more
additional factors such as vertex distance, SEG height, pantoscopic
tilt, and use conditions. The lens wavefront can be achieved by
optimizing a corrected wavefront, where the corrected wavefront is
the combined effect of the patient's measured wavefront and the
lens wavefront. The optimization of the corrected wavefront can
involve representing the measured wavefront and the lens wavefront
on a grid. In an embodiment, the grid can lie in a plane. During
the optimization, a subset of the grid can be used for the
representation of the measured wavefront at a point on the grid so
as to take into account the portions of the measured wavefront that
contribute to the corrected wavefront at that point on the
grid.
Inventors: |
DREHER; Andreas W.;
(Escondido, CA) ; Foote; Bill; (Poway, CA)
; Sandler; Dave; (San Diego, CA) ; Warden;
Laurence; (Poway, CA) |
Correspondence
Address: |
MORRISON & FOERSTER LLP
12531 HIGH BLUFF DRIVE, SUITE 100
SAN DIEGO
CA
92130-2040
US
|
Family ID: |
40788195 |
Appl. No.: |
12/755345 |
Filed: |
April 6, 2010 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
11963609 |
Dec 21, 2007 |
|
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12755345 |
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Current U.S.
Class: |
351/204 ;
351/246 |
Current CPC
Class: |
B33Y 80/00 20141201;
A61B 3/1015 20130101; G02C 7/027 20130101; G02C 2202/22
20130101 |
Class at
Publication: |
351/204 ;
351/246; 351/246 |
International
Class: |
A61B 3/11 20060101
A61B003/11; A61B 3/10 20060101 A61B003/10 |
Claims
1. A method for wavefront optimization process comprises, measuring
raw wavefront data, considering frame fitting data selected from
group comprising pupilary distance, vertex distance, pantascopic
tilt, SEG height, considering patient specific data such as desired
usage conditions, considering spectacle geometry such as its length
and width including location of the pupil in the frame, and
creating lens manufacturing instructions that is based on
determining optimum wavefront prescription, wherein the optimum
wavefront prescription comprises wavefront fitting and
processing.
2. The method of claim 1, wherein the lens manufacturing
instructions may include a surface map for front and/or back side
of a lens, a points file for freeform lens generator for custom
front and back surfaces, a refractive index profile that is created
in a changeable refractive index layer, a refractive index profile
that is created by ink-jet deposition, a stereolithography profile
in conjunction with casting, or combination of any of the said
techniques can be combined to achieve the custom lens
manufacturing.
3. The method of claim 1, wherein the lens manufacturing
instructions is applied to a lens where the wavefront is emphasized
along the central optical axis and de-emphasized outside the
central optical axis to produce aberration corrected single vision
or progressive addition lens.
4. The method of claim 1, wherein the wavefront fitting and
processing comprises method of determining lens wavefront.
5. The method of claim 4, wherein the method of determining lens
wavefront comprises, measuring a patient's wavefront to create a
pupil aberration, generating a plurality of corrected wavefronts,
generating a function of the plurality of corrected wavefronts,
determining a lens wavefront by optimizing the function of the
plurality of corrected wavefronts, and producing a lens taking into
account the lens wavefront.
6. A method for determining a wavefront for a lens from a patient's
measured wavefront comprises, measuring patient's wavefront
aberrations, optimizing the combined patient and lens wavefront
aberrations, considering frame fitting data selected from group
comprising pupilary distance, vertex distance, pantascopic tilt,
SEG height, and use conditions, and producing a spectacle lens with
optimal correction across the entire lens.
7. The method of claim 6, wherein the spectacle lens is produced by
one of the methods that include a surface map for front and/or back
side of a lens, a points file for freeform lens generator for
custom front and back surfaces, a refractive index profile that is
created in a changeable refractive index layer, a refractive index
profile that is created by ink jet deposition, a stereolithography
profile in conjunction with casting, or combination of any of the
said techniques can be combined to achieve the custom lens
manufacturing.
8. The method of claim 6, wherein the spectacle lens produced is a
single vision or progressive addition lens and comprises wavefront
that is emphasized along the central optical axis and de-emphasized
outside the central optical axis.
9. A method for determining lens wavefront comprises, optimizing a
corrected wavefront, wherein the optimization involves representing
the measured wavefront and the lens wavefront on a grid, wherein
the grid lies in a plane, a subset of grid is used for the
representation of the measured wavefront at a point on the grid so
as to take into account the portions of the measured wavefront that
contribute to the corrected wavefront at that point on the
grid.
10. The method of claim 9, wherein the corrected wavefront is the
combined effect of patient's measured wavefront and the lens
wavefront.
11. The method of claim 9, wherein the optimization involves hill
climbing optimization technique such as Gaussian Least Squares Fit
and Point Spread Optimization software to fit an optimal wavefront
across a specified surface larger than that of the measured
wavefront.
12. The method of claim 11, wherein the optimal wavefront across a
larger specified surface involves projection from a number of
points emanating in multiple directions from a nominal axis of
rotation representing the center of the eye.
13. The method of claim 12, wherein the wavefront as projected from
the center of the eye can be convolved with a weighting function
across the lens to enhance or emphasize the wavefront in certain
area while allowing other areas to be de-emphasized.
14. The method of claim 13, wherein the wavefront is best fit along
a surface representing a paraxial lens representing the neutral
axis of a lens, wherein the paraxial lens is fixed in space at a
specified vertex distance and follows the basic lens design
curvature of the chosen lens blank.
15. The method of claim 13, wherein the emphasized wavefront is in
the central region of the lens where the distortion is reduced and
the de-emphasized area is when the patient is looking off center
outside the central region.
16. The method of claim 9, wherein the wavefront pattern is solely
based upon the low order aberrations, the high order aberrations or
combination of both low and high order aberrations.
Description
BACKGROUND OF INVENTION
[0001] Ocular lenses are worn by many people to correct vision
problems. Vision problems are caused by aberrations of the light
rays entering the eyes. These include low order aberrations, such
as myopia, hyperopia, and astigmatism, and higher order
aberrations, such as spherical, coma, trefoil, and chromatic
aberrations. Because the distortion introduced by aberrations into
an optical system significantly degrades the quality of the images
on the image plane of such system, there are advantages to the
reduction of those aberrations.
[0002] Ocular lenses are typically made by writing prescriptions to
lens blanks. This is accomplished by altering the topography of the
surface of a lens blank.
[0003] Recently, attention has been given to methods of writing a
low order lens using a patient's measured wavefront information.
Currently, several techniques can be utilized to determine the
optimum low order refraction from the high order, including: the
Gaussian Least Squares Fit, point spread optimization, and neural
network analysis. Some of these techniques may be employed to not
only derive the best low order prescription from the high order
values, but may also be used to "fit" an optimum wavefront across
an entire spectacle lens based on the patient's measured
wavefront.
[0004] Using one or more of these fitting techniques may yield a
better refraction than conventional subjective refractions in the
center zone, but consideration must be given to off-axis gaze
angles. In particular, one considerable disadvantage of traditional
lens manufacturing is that that many people experience distortion
when looking off-center outside the central region, commonly called
"swim".
[0005] Accordingly, there is a need for a method of determining a
wavefront for a patient's spectacle based on the patient's measured
wavefront, in such a way to reduce distortion when the patient
looks off center outside the central region.
BRIEF SUMMARY
[0006] The subject invention provides methods for determining a
wavefront for a lens from a patient's measured wavefront. The
wavefront can be used for producing a spectacle lens with optimal
correction across the entire lens taking into account the patient's
complete measured wavefront. Specific embodiments can also take
into account one or more additional factors such as vertex
distance, SEG height, pantoscopic tilt, and use conditions.
[0007] The lens wavefront can be achieved by optimizing a corrected
wavefront, where the corrected wavefront is the combined effect of
the patient's measured wavefront and the lens wavefront. In one
embodiment of the subject invention, the optimization of the
corrected wavefront involves representing the measured wavefront
and the lens wavefront on a grid. In an embodiment, the grid can
lie in a plane. During the optimization, a subset of the grid can
be used for the representation of the measured wavefront at a point
on the grid so as to take into account the portions of the measured
wavefront that contribute to the corrected wavefront at that point
on the grid.
BRIEF DESCRIPTION OF FIGURES
[0008] FIG. 1 shows the steps for a method for producing a
spectacle lens in accordance with an embodiment of the subject
invention.
[0009] FIG. 2 shows a flow chart in accordance with an embodiment
of the subject invention.
[0010] FIG. 3 shows a top view of spectacle and pupil samples as
images at particular shift (gaze).
[0011] FIG. 4 shows a side view of spectacle and eye, on the left
with corresponding dotted lines, with gaze rotation shown by the
curved arrow and the rotated eye and corresponding dotted lines the
curved arrow is pointing to and gaze shift shown by the straight
arrow and the shifted eye and corresponding dotted lines the
straight arrow is pointing to on the right.
[0012] FIG. 5 shows a schematic representation of an approximation
of representing the i-th direction as the i-th shift.
[0013] FIG. 6 shows a comparison of off-axis versus transverse
correction for spectacle lens applications.
[0014] FIG. 7 shows a schematic representation of a transverse
correction for a contact lens application.
[0015] FIG. 8 shows a schematic representation of a lens blank with
a monofocal higher-order region and a transition zone.
[0016] FIG. 9 shows a lens image.
[0017] FIGS. 10A-10D show an example of trefoil.
[0018] FIGS. 11A-11D show an example of coma.
[0019] FIGS. 12A-12D show an example of spherical aberration.
DETAILED DESCRIPTION
[0020] The subject invention provides methods for determining a
wavefront for a lens from a patient's measured wavefront. The
wavefront can be used for producing a spectacle lens with optimal
correction across the entire lens taking into account the patient's
complete measured wavefront. Specific embodiments can also take
into account one or more additional factors such as vertex
distance, SEG height, pantoscopic tilt, and use conditions.
[0021] The lens wavefront can be achieved by optimizing a corrected
wavefront, where the corrected wavefront is the combined effect of
the patient's measured wavefront and the lens wavefront. The
optimization of the corrected wavefront can involve representing
the measured wavefront and the lens wavefront on a grid. In one
embodiment, the grid can lie in a plane. During the optimization, a
subset of the grid can be used for the representation of the
measured wavefront at a point on the grid so as to take into
account the portions of the measured wavefront that contribute to
the corrected wavefront at that point on the grid.
[0022] One embodiment of the invention utilizes the hill climbing
optimization technique used in the Gaussian Least Squares Fit and
point spread optimization software to fit an optimal wavefront
across a specified surface larger than that of the measured
wavefront. The desired wavefront is projected from a number of
points emanating in multiple directions from a nominal axis of
rotation representing the center of the eye. The wavefront pattern
used can be solely based upon the low order, or can also include
some or all the high order as well.
[0023] Each position of the wavefront as projected from the center
of the eye can be convolved with a weighting function across the
lens to enhance or emphasize the wavefront in certain areas while
allowing other areas to be de-emphasized. The wavefront is best fit
along a surface representing a paraxial lens representing the
neutral axis of a lens. This paraxial lens is fixed in space at a
specified central vertex distance and follows the basic lens design
curvature of the chosen blank lens. The basic lens design curvature
may be simply derived from the central lower-order prescription or
may be used in conjunction with the high order and other factors
such as vertex distance.
[0024] The final wavefront can be fitted with one or more of the
following inputs:
[0025] Wavefront
[0026] Pupilary Distance
[0027] Vertex Distance
[0028] Pantoscopic Tilt
[0029] SEG Height
[0030] Pupil Diameter
[0031] Conditions under which the lens will be used (day, office,
night, etc)
[0032] Age
[0033] OD Subjective refraction
[0034] ADD Value
[0035] Spectacle Geometry
[0036] FIG. 1 shows the steps for one embodiment of a method for
producing a spectacle lens in accordance with the subject invention
and FIG. 2 shows a flowchart indicating the flow of information in
accordance with an embodiment of a wavefront optimization method.
In one embodiment, vertex distance and its effect on the lens power
and astigmatism can be compensated for in the wavefront fitting
process. The output of the wavefront fitting software process
(steps 1 & 2 in FIG. 1) is a set of instructions that
facilitates production of a custom lens.
[0037] Various techniques may be utilized to generate the actual
lens. For example; the instructions may include a surface map for
front and/or back surfaces of a lens, or a points file that can be
fed into a freeform lens generator, to cut custom front and back
surfaces. Other approaches may utilize a changeable refractive
index layer within the lens blank that can be customized with the
information from the fitting software. Yet another approach can use
an inkjet deposition of different refractive indices across a lens
surface to generate a corrected wavefront based on the fitting
software output. In yet another approach, stereo lithography may he
used in conjunction with casting, or combination of any of the
above techniques can be combined to achieve the custom lens
manufacturing.
[0038] Step 3 in FIG. 1 represents a freeform grinding approach to
lens manufacturing. Casting, inkjet, and sandwiched changeable
refractive index approaches as known in the art, can also be
utilized.
[0039] If utilizing the freeform grinding approach the final step
in the wavefront fitting software can generate shape of the front
and back surface of the lens to achieve the given wavefront.
Development of the shape of the front and back surface can also
take into account the distortions from lens thickness variations to
minimize distortions. The output of the fitted wavefront software
can, in an embodiment, be a points file, which can subsequently be
transferred into a freeform lens generator for manufacturing the
lens. The resulting lens can be essentially optimized across the
entire lens and customized for each patient based on all the input
parameters. This freeform grinding technique can be utilized in
conjunction with the refractive index changing material to further
tune or enhance the refractive properties after lens grinding and
polishing.
[0040] In one embodiment, a grid of shifts (rather than rotations)
for measured and target pupil wavefronts is used, represented
mathematically by images. The target wavefront can be used as the
lens wavefront. From the measured wavefront, the target wavefront
can be determined via one or more embodiments of the invention. A
variety of configurations can be used to implement the target
wavefront via an eyeglass for the patient.
[0041] As an example, a single lens with two surfaces can be used
to create an eyeglass for a patient, where one or both of the lens
surfaces can be controlled to effect the wavefront for the lens.
Alternatively, two lens blanks each having two surfaces can be used
with a variable index polymeric material between the two lens
blanks, where one or more of the four lens blank surfaces and/or
the polymeric material can be controlled to effect the wavefront
for the lens. The lens surface(s) and/or variable cured index of
the polymeric material of the polymeric material are described in a
two-dimensional plane corresponding to the height of the surface(s)
or the projection of the index layer(s) onto a plane.
[0042] Aberrations are measured as components in an orthogonal
expansion of the pupil sampled on the same grid spacing. In a
specific embodiment, the grid spacing is about 0.5 mm and in
another embodiment the grid spacing is about 0.1 mm. In an
embodiment using Zernike polynomials, the components can be made
orthogonal for the chosen pupil size due to discrete sampling. As
an example, the components can be made orthogonal through a process
such as Gram-Schmidt orthogonalization. Orthogonal components of
aberrations for pupils centered at a specific point on the
spectacle may then be computed by sample-by-sample multiplication
(inner product) of the aberration component image with the lens
(height or projection) image centered at the point of interest, as
in FIG. 3. FIG. 3 shows a top view of spectacle and pupil samples
as images at particular shift (gaze), which can be used for
computation of aberrations for pupils having a diameter of 3
samples, centered at a chosen coordinate on a spectacle grid having
a diameter of 8 samples.
[0043] Zernike polynomials are orthogonal and when samples are
taken, approximations of Zernike polynomials can be created. In one
embodiment, the approximations of Zernike polynomials can then be
modified to make orthogonal polynomials, so as to create new
polynomials.
[0044] In an embodiment, points on the pupil outside the pupil
diameter are assumed zero. Non-squared pupil shapes may be formed
by zeroing select points within the square of pupil diameter.
Mathematically, the process of computing the inner product centered
at all possible locations on the grid is a cross correlation, which
may be implemented with a fast convolution algorithm. An image can
be produced for each Zernike via the cross-correlation. The image
for each Zernike can be used to create a target and an error. The
error can be used to produce an error discrimination, or a weighted
sum of all pixels in the image square.
[0045] In a specific embodiment, a grid size and spacing is chosen
to represent the lens and pupil in a plane. An example of such a
grid is shown in FIG. 3. The aberrations of interest are
orthogonalized on the grid at the chosen pupil size. Then a given
aberration centered at every point may be estimated by
cross-correlation of the orthogonalized Zernike image and the
spectacle image, resulting in an image for each Zernike component.
An error image for each point on the lens may be estimated as a
difference between the computed and desired Zernike aberration
centered at each point in the image.
[0046] The desired correction is, to a first approximation, assumed
to be constant in this plane with a shift corresponding to a given
gaze angle. The rotation is otherwise neglected as shown in FIG. 2.
For large gaze angles, the effect of rotation can be compensated by
providing a spatially-varying correction target. The
spatially-varying target can be approximated by rotating the
paraxial target.
[0047] Simple convolution may be replaced with a more exact
geometric calculation of the ray-surface intersection corresponding
to a ray-tracing-style algorithm over a fixed grid. Other grid
geometries may be used (e.g., hexagonal instead of rectangular).
The result is essentially a spatially varying sample spacing and
convolution, increasing computation time.
[0048] Other metrics of surface error may be computed from the
Zernike component error images, as done with single pupil
representations. Images of sphere/cylinder/values (or errors from
desired) may be computed by applying the usual conversion on a
pixel-by-pixel basis for example.
[0049] Total root-mean-square (rms) may be represented by either
the sum of all component terms squared for a particular pupil
location, or the sum of all pixels squared (and properly
normalized) within the pupil. This may be achieved by
cross-correlations of a pupil-sized aperture of ones with an image
of the lens values squared. Total high order may be computed by
subtracting the low order aberration images from the total rms
image. High order error may be computed by also subtracting the
target high order images, squared pixel-by-pixel. For certain error
choices optimizing error this may be mathematically equivalent to
known regularization algorithms.
[0050] A total error discriminant may be generated by summing
desired error images over the entire lens. A pixel-by-pixel
weighting may be incorporated to selectively weight the error at
various regions in the lens, and this may be done independently for
each Zernike component. Standard optimization procedures (e.g.,
convex programming) may be used to produce a lens image that
minimizes the error discriminant. If the lens image is sufficiently
small, the cross-correlation may be represented as a matrix
multiplication further simplifying the application of optimization
algorithms known in the art. For larger image sizes this may be
impractical, but may still be used to adapt the algorithm to the
problem before implementing with fast convolution algorithms.
[0051] Constraints on the error may also be used in the
optimization that would be represented by constraint images of max
and/or min Zernike components or functions thereof. An example of a
constraint that can be utilized is that the error for a certain
Zernike cannot be above a certain threshold for a certain area.
[0052] Free-floating points, such as boundaries, may be handled by
setting weights to zero or very small for those points. This allows
the optimized region to be smaller than the actual grid, the
optimized region to have an arbitrary shape, and/or the optimized
region to only be optimized for points that will ultimately be
used. In a specific embodiment, the patient-selected frame outline
may be input as the region of optimization. As there can be an
infinite number of solutions, an attempt can be made to optimize a
certain shape inside of the lenses, such as the spectacle shape. An
example of a certain shape that can be optimized is to optimize
within the shape of the frame that the lens will be used by, for
example, using a zero weight outside of the frame.
[0053] Fixed points, which are given prior to optimization and
remain unchanged, may be provided by using the points to compute
correction but not applying it to them in the optimization
algorithm. This can be used for boundaries, so as to only optimize
for certain portions of lenses.
[0054] Grid(s) of constraints may be converted into a weighting
and/or target (for unconstrained optimization) via a separate
optimization procedure.
[0055] Multiple surfaces may be optimized simultaneously. As an
example, two grids can be optimized simultaneously or each grid
point can have two numbers associated with it to be optimized.
[0056] The patient's prescription (including high order) may be
used as target, including deterministic variations with gaze if
available.
Example 1
Simple Alignment Tolerant Lens
[0057] For purposes of this example, lower case bold letters denote
matrices (or equivalently, images), which describe either the
two-dimensional (single-surface) lens as a optical-path delay (OPD)
map, or a two-dimensional wavefront, also as an OPD map. [0058]
a.sub.i,j Corrected wavefront aberration in the (i,j).sup.th
position. [0059] i, j Indices i.sup.th horizontal and j.sup.th
vertical positions. [0060] n, m Indices of matrix elements. [0061]
L Total number of pixels on side of (square) pupil and aberration
matrices. [0062] N Total number of pixels on side of (square)
spectacle OPD matrix. [0063] p, Pupil aberration OPD, [0064]
p.sub.i,j pupil aberration in the (i,j).sup.th position if it
changes with gaze. [0065] s Spectacle OPD over entire range of
lens, e.g. 50 mm diameter [0066] g Matrix describing weighting of
error over pupil [0067] z.sub.k k-th (sampled) Zernike wavefront
matrix [0068] w(i,j), Weighting over angle, [0069] w.sub.k(i,j)
Weighting over angle for the k.sup.th Zernike term [0070] f
Objective function
[0071] An assumption can be made that an optimization of image
quality for a lens over a range of gaze angles can be well
approximated by an optimization of image quality over a range of
translations. The rotation of the eye relative to the lens is,
therefore, ignored in this example. The desired OPD calculated
through optimization can be converted to, for example, a surface or
pair of surfaces via a ray-tracing application.
[0072] The corrected wavefront aberration in the (i,j).sup.th
position, including both the pupil aberation p and the
corresponding apertured portion of the spectacle s, is described
as
a i , j ( n , m ) = p ( n , m ) + s ( n + i , m + j ) ; n , m = - L
2 , , L 2 ( 1 ) ##EQU00001##
a, p, and s are matrices. The total error to be optimized will be a
function of all the a.sub.i,j. The matrices can have zeroes as
entries for input data or output data that is, for example,
circular, rectangular, or has a non-square pattern. The simplest
function is the total squared error over all shifts with weightings
over both the shifts and the pupil, which can be represented as
shown in Equation (2).
f = i , j w ( i , j ) n , m g ( n , m ) [ a i , j ( n , m ) ] 2 ( 2
) ##EQU00002##
[0073] One approach to optimize the corrected wavefront is to
preferentially select certain Zernike terms to be corrected or
excluded. An example of preferential selection of certain Zernike
terms is to only correct astigmatism for a Progressive-Addition
Lens (PAL) design. To select the component of the chosen Zernike
term(s) from the full wavefront aberration we make use of the
orthonomality of Zernikes and simply take the inner product of each
Zernike with the wavefront in Equation (1).
a i , j ( n , m ) = z 0 n , m { z 0 ( n , m ) a i , j ( n , m ) } +
z 1 n , m { z 1 ( n , m ) a i , j ( n , m ) } + = k = 0 .infin. z k
c k ( i , j ) ( 3 ) ##EQU00003##
The coefficient matrix for the k-th Zernike, e.sub.k(i, j), can be
computed as
c k ( i , j ) = n , m { z k ( n , m ) a i , j ( n , m ) } = z k * a
i , j ( 4 ) ##EQU00004##
which uses the cross-correlation operation or, equivalently, the
convolution operation implemented appropriately.
[0074] Particular Zernike components of the aberration can be
selectively weighted by weighting the component in (3)
a i , j ' ( n , m ) = k = 0 .infin. w k ( i , j ) c k ( i , j ) z k
( 5 ) ##EQU00005##
The appropriate w.sub.k may be set to zero to ignore certain
components.
[0075] The simplest approach, and one of very few that can be
solved analytically, is to optimize a weighted total squared error
over all gaze positions, or shifts, where the error is defined
according some carefully chosen and/or excluded combination of
Zernikes.
[0076] It is aparrent that the minimum least-squares optimum over
an angle-range larger than the pupil diameter with no weighting
will result in a purely low-order solution. However if a weighting
is used it is possible to trade off improvement in some areas for
others.
[0077] Combining equations (2) and (5), the squared error objective
is then
f = i , j n , m g ( n , m ) [ k = 0 .infin. w k ( i , j ) c k ( i ,
j ) z k ] 2 = i , j n , m g ( n , m ) [ k = 0 .infin. w k ( i , j )
( z k * ( p ( n , m ) + s ( n + i , m + j ) ) ) z k ] 2 ( 6 )
##EQU00006##
which may be solved via standard optimization algorithms for the
optimal s.
[0078] Making the weighting w (n,m) the same for every Zernike term
allows an analytical solution. Setting the gradient of equation (6)
with respect to s equal to zero yields an optimized result which
can be written as follows:
s=-{w*(g.times.p)}/{w*g} (7)
[0079] The ".times." and "/" operators indicate element-by-element
multiplication or division respectively, and the "*" operator
indicates two-dimensional convolution.
Using a uniform pupil-weighting further simplifies equation (7)
to
s=-{w*p}/{w*l} (8)
[0080] Where l is simply a matrix of ones, resulting in a low-pass
filtered w in the denominator term. So the result is a convolution
of the weighting with the aberration that is normalized by a
filtered version of the weighting.
[0081] Some simulated examples for certain zernike terms are
provided in FIGS. 10A-10D, FIGS. 11A-11D, and FIGS. 12A-12D, where
FIGS. 10A-10D show an example of trefoil, FIGS. 11A-11D show an
example of coma, and FIGS. 12A-12D show an example of spherical
aberration. Note that the amplitudes are normalized for a 1 um rms
aberration, and the y-axis is normalized to the zernike diameter.
One-dimensional cross-sections of the lens, error are provided.
Again, FIGS. 10A-10D show the results for trefoil; FIGS. 11A-11D
show the results for coma; and FIGS. 12A-12D show the results for
spherical aberration.
[0082] Referring to Example 1, several different methods can be
reflected in different choices of g, the matrix describing
weighting or error over pupil, with q.sub.ij being effective pupil
aberration for the i, j position and h being a matrix describing
weighting of wavefront over pupil. Examples of several methods are
provided in Table 1.
TABLE-US-00001 TABLE 1 Mathematical Step Description of Method g =
I, the identity matrix Standard Least Squares g.sub.w = diag{g},
diagonal matrix Least Squares weighted over pupil g = hg.sub.wh, h
= z.sub.3z.sub.3.sup.T + z.sub.5z.sub.5.sup.T Minimum Astigmatism -
PAL design g = hg.sub.wh, h = I - z.sub.0z.sub.0.sup.T Free Varying
Piston g = hg.sub.wh, Free Varying Tilt - distortion allowed h = I
- z.sub.0z.sub.0.sup.T - z.sub.1z.sub.1.sup.T -
z.sub.2z.sub.2.sup.T q.sub.ij = p.sub.ij - i c.sub.1z.sub.1 - i
c.sub.2(i)z.sub.2 Constantly Changing Tilt - Magnification
[0083] Another approach can require an "unknown prescribed
Zernike". This can be used, for example, to yield a tilt that is
constant (as possible) over the lens but not necessarily zero. For
certain aberrations this can result in an improved vision quality
by allowing some magnification, which in this treatment can be
referred to as linear distortion. This can be achieved by
iteratively varying the values for c.sub.1,desired and
c.sub.2,desired with some allowable range and choosing the pair
that results in the lowest error minimum.
Aberration Bandwidth
[0084] An approach at optimizing a metric based on the point-spread
function (PSF) can be performed via consideration of the spatial
"bandwidth" as a metric of the PSF at a specific gaze angle. If a
wavefront can be described as a two-dimensional single-component FM
signal, then its local spatial frequency content at a given point
can be estimated form the spatial derivatives of the phase. By
estimating the average mean squared value of the spatial derivative
of the wavefront, an estimate of the bandwidth of the PSF can be
produced.
[0085] An estimate for the averaged bandwidth of the PSF can be
described as Da,
where D is a matrix that computes the averaged local derivative via
finite differences. There is a variety of possible choices for D to
approximate a derivative.
[0086] An approach similar to the use of a matrix that computes the
averaged local derivative via finite differences approach of
Example 1, but using derivatives of the pupil and spectacle, can be
used.
In another embodiment, D can be selected to approximate a second
derivative, in order to approximate power matching.
Region-Based Optimization
[0087] In an embodiment, optimization can be performed using a
standard iterative approach such as Gradient Descent with some
chosen boundary conditions. These boundary conditions can describe
the required outcome at the edges of the lens as well as in zones
in its interior. A variety of different applications can be
addressed by selecting these regions.
Transition Zones
[0088] Transition zone based lens designs have the requirement of
perfection for the axial ray and for the correction at some outer
radius to be a constant (or low-order). Therefore, the single
zone's correction can be predetermined as some high-order
correction, then the optimization can vary only a narrow region
outside this zone to find the optimum lens. The remaining region
outside this transition zone can be required to be some flat or
low-order correction.
Progressive-Addition Lenses
[0089] Progressive-Addition Lenses (PAL's) have required low-order
correction in a pair of zones, with some varying power along a line
connecting them. Typically, the rest of the lens is then optimized
to reduce distortion. In an embodiment of the subject invention,
the lens can be optimized to similarly reduce distortion. The lens
can be optimized to reduced distortion via, for example, power
matching, matching second order wavefronts only, or full-wavefront
matching with a varying tilt.
Example 2
Simple Rotation and Alignment Tolerant Lens Design
[0090] This Example uses a method for optimizing the correction
"programmed" onto a higher-order contact lens. Contact lenses can
be designed to be in a certain orientation, but can still rotate
with respect to this orientation and can slide as well so as to
become decentralized. The unpredictable rotation and decentration
of the lens during normal wearing can be addressed. Given a range
of rotations and decentrations, the contact lens design is
optimized to improve vision throughout the entire range by
minimizing the total wavefront error summed over the ranges. The
optimum over decentrations may be computed as in Equation (7)
provided in Example 1. The optimum over rotations is computed as
follows.
The error discriminant is:
f = .theta. w ( .theta. ) n , m [ p ( n , m ) + s ( n cos .theta. +
m sin .theta. , - n sin .theta. + m cos .theta. ) ] 2 ( 9 )
##EQU00007##
The optimal result is similar conceptually to the decentration
case.
s = - 1 .theta. w ( .theta. ) { .theta. w ( .theta. ) p ( n cos
.theta. - m sin .theta. , n sin .theta. + m cos .theta. ) } ( 10 )
##EQU00008##
[0091] In a specific embodiment of the subject invention, the
desired final programmed size can be determined and the information
describing the eye aberration can be projected into a larger
"transition" radius. FIG. 9 shows an example of the pupil region
and the transition region. Preferably, the extrapolation is
continuous while decaying toward zero. Alternatively, these can be
achieved after the optimization.
[0092] Optimizing with respect to rotation and decentration can be
performed independently. They may be done in either order depending
on the expectation of the physical behavior, yielding results that
may not be exactly the same depending on the rotational symmetry of
the decentration range.
[0093] In an embodiment, the transition region may be deemphasized
by applying a decaying apodization. Further, the entire image may
be refit with Zernike polynomials.
[0094] T All patents, patent applications, provisional
applications, and publications referred to or cited herein are
incorporated by reference in their entirety, including all figures
and tables, to the extent they are not inconsistent with the
explicit teachings of this specification.
[0095] It should be understood that the examples and embodiments
described herein are for illustrative purposes only and that
various modifications or changes in light thereof will be suggested
to persons skilled in the art and are to be included within the
spirit and purview of this application.
* * * * *