Abstract:
A method for generating a visual acuity metric, based on wavefront aberrations (WFAs), associated with a test subject and representing classes of imperfections, such as defocus, astigmatism, coma and spherical aberrations, of the subject&#39;s visual system. The metric allows choices of different image template, can predict acuity for different target probabilities, can incorporate different and possibly subject-specific neural transfer functions, can predict acuity for different subject templates, and incorporates a model of the optotype identification task.

Description:
ORIGIN OF THE INVENTION 
     This invention was made by one or more employees of the U.S. government. The U.S. government has the right to make, use and/or sell the invention described herein without payment of compensation, including but not limited to payment of royalties. 
    
    
     FIELD OF THE INVENTION 
     This invention relates to visual acuity of a human or other animal, based on wavefront aberrations associated with the animal&#39;s visual imaging system. 
     BACKGROUND OF THE INVENTION 
     It is now possible to routinely measure the monochromatic aberrations of the human eye. However, one cannot yet measure the visual acuity that will result from a given set of wavefront aberrations. One reason to seek a prediction of acuity from aberrations is the possibility of automated objective measurement of visual acuity, and of automated prescription of sphero-cylindrical corrections. However, it has been shown that correcting the spherical and cylindrical components of the aberrations (equivalent to minimizing the RMS error of the wavefront) does not provide best acuity. Thus these automated procedures must await a more sophisticated metric that can predict acuity from an arbitrary set of aberrations. 
     In the last decade there has been a revolution in measurement and treatment of visual optical defects. This revolution has included the development of aberrometers simple enough to be used in the clinic, refinement of methods of laser surgery for optical correction, and development of various optical implants, notably intra-ocular lenses (IOL). In all of these, measurement and interpretation of wavefront aberrations (WFAs) has played an important role. They are a simple and comprehensive way of describing the state of the optical system. In spite of this, there is at present no accepted, reliable way of converting WFAs to visual acuity, which is a standard measure of quality of vision. The WFA Metric allows calculation of visual acuity from wavefront aberrations. 
     What is needed is an approach, including one or more metrics, that allows a prediction of visual acuity, for a human or other animal, based on estimated wavefront aberrations (WFAs) measured or otherwise determined for the test subject. Preferably, the approach should allow acuity predictions for different optotypes, such as Sloan letters, Snellen e&#39;s, Landolt C&#39;s, Lea symbols, Chinese or Japanese characters and others. Preferably, the approach should permit incorporation of different, possibly subject-specific, neural transfer functions. 
     SUMMARY OF THE INVENTION 
     These needs are met by the invention, which develops and applies an optical-based and neural-based metric that allows prediction of visual acuity of the subject. For a given choice of an optotype set (e.g., Sloan letters), an optical transfer function OTF(x,y) is generated, using Zernike polynomials and the associated Zernike coefficients and a specification of a pupil aperture image PA(x,y) for two dimensional coordinates (x,y) for the subject. A generalized pupil image and associated point spread function PSF(x,y) is computed, from which an OTF is computed. 
     A neural transfer function NTF(x,y) is specified, and a total transfer function TTF(x,y) is computed as a product of the OTF and the NTF. A proportion correct function P(k) is estimated from the neural images and a noise value, using one of three or more methods for such estimation. A probability criterion P(target) for measurement of visual acuity is specified, normally between 0.5 and 0.8. A numerical procedure returns a final index value j (final), which is converted to an estimate of acuity using a standard logMAR calculation. The output of the logMAR computation is a WFA metric that provides an estimate of visual acuity for the subject. 
     The metric(s) developed here is designed to predict symbol acuity from wavefront aberrations. One embodiment of the metric relies on Monte Carlo simulations of a decision process and relies on an ideal observer, limited by optics, neural filtering, and neural noise. A second metric is a deterministic calculation involving optics, symbols, and a hypothetical neural contrast sensitivity function CSF. 
     A WFA Metric is an algorithm for estimating the visual acuity of an individual with a particular set of visual wavefront aberrations (WFAs). The WFAs represent arbitrary imperfections in an optical system, and can include low order aberrations, such as defocus and astigmatism, as well as high order aberrations, such as coma and spherical aberration. WFAs can now be measured routinely with an instrument called an aberrometer. In modern practice, the WFAs are represented as a sum of Zernike polynomials Z(x,y), each multiplied by a Zernike coefficient. A typical measurement on the eye of a subject will consist of a list of about 16 numbers, which are the coefficients of the polynomials. The WFA Metric converts the list of numbers into an estimate of the visual acuity of the subject. If changes are planned to the WFA of the subject (through surgery or optical aids) the predicted change in visual acuity can be calculated. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates an embodiment of a structure for computation of a WFA metric according to the invention. 
         FIG. 2  illustrates the ten Sloan letters, expressed in a sans serif font, that provide one of the optotype sets that can be used with the invention. 
         FIGS. 3A-3E  illustrate steps in creation of the OTF. 
         FIG. 4  graphically illustrates a representative radial neural transfer function. 
         FIG. 5  is an embodiment of a procedure for evaluation of a probability correct index for one size optotype. 
     
    
    
     DESCRIPTION OF THE INVENTION 
     0. Notation and Terminology 
     In this presentation, an “image” refers to a finite discrete digital image represented by a two-dimensional array of integers or real numbers. It has a width and height measured in pixels. Where the size is specified it will be given as a list {rows,columns}. The image has a resolution measured in pixels/degree. The pixel indices of the image are x (columns) and y (rows). Images will usually be an even number of pixels wide and tall. If the image size is {2h,2h}, then the indices x and y each follow the sequence {−h, . . . , 0, . . . , −h−1}. This places the origin of the image at the center. An image may be written with explicit row and column arguments A(x,y), or without the coordinates as A. 
     In this presentation, a dft refers to a two-dimensional finite discrete digital array of complex numbers representing a Discrete Fourier Transform (DFT). It has a width and height measured in pixels. Where the size is specified it will be given as a list {rows,columns}. A dft has a resolution measured in pixels/cycle/degree. The pixel indices of the dft are a (columns) and v (rows). Dfts will usually be an even number of pixels wide and tall. If the dft size is {2h,2h}, then the indices u and v each follow the sequence {0, . . . , h−1, −h . . . , -−1}. This places the origin of the dft at the first pixel. This is the conventional ordering of indices in the output of the Fast Fourier Transform (FFT) operator. The FFT is a particular algorithm for implementation of the DFT. In the body of this document we refer to the DFT, but this will usually be implemented by the FFT. 
     In this presentation, vectors will be written with one subscript A k , and matrices will be written with two subscripts A j,k , where the first subscript indicates the matrix row. Frequently, we will deal with vectors or matrices whose elements are images, in which case the image coordinates x,y are omitted. 
     We make use of the notation A:B to indicate Frobenius inner product of two matrices 
               A   :   B     =       ∑   y     ⁢       ∑   x     ⁢       A   ⁡     (     x   ,   y     )       ⁢     B   ⁡     (     x   ,   y     )                   
This is useful to describe a sum over pixels of the product of two images. The modulus or norm of an image is given by
 
∥A∥=√{square root over (A:A)}
 
     1. Inputs and Output 
     The WFA metric has four inputs. A first input is a set of wavefront aberrations, represented as a weighted sum of Zernike coefficients z n (x,y). A second input is a set of optotypes, represented in a standard graphic format, such as a font description, a set of raster images, or graphic language descriptors. One example set of optotypes is the Sloan font for the letters {C, D, H, K, N, O, R, S, V, Z}, a set often used in the measurement of acuity). A third input is a set of templates, equal in number to the number of optotypes in the set. By default, the templates are derived from the optotype set and are not a distinct input. A fourth input is a set of parameters, some of which may have default values that are permanently stored within the program. Some parameters may be changed on every calculation of the metric, while others are unlikely to be changed often. The parameters are described throughout this description. 
     A single output, the visual acuity, is expressed as a decimal acuity or log of decimal acuity (logMAR). An overall system structure is shown in  FIG. 1 . 
     2. Overview of the Algorithm 
     
         
         
           
             a. Generate the Optical Transfer Function (OTF) 
             b. Generate the Neural Transfer Function (NTF) 
             c. Generate the Total Transfer function (TTF) 
             d. Define the Proportion Correct function P(size) 
             e. Find the size for which P(size)≈P target  
 
Each of these steps is described in detail in the following.
 
           
         
       
    
     3. Select a Set of Optotypes 
     The optotypes are a set of graphic symbols that the human observer is asked to identify in the course of an acuity test. Examples are Sloan letters, Snellen e&#39;s, Landolt Cs, Lea symbols, Chinese or Japanese characters, or other pictograms of various sorts. Each optotype set will have a fixed number K of elements, and a defined size specification. 
     By way of example, the optotype set used here is the Sloan letters {C, D, H, K, N, O, R, S, V, Z}, with K=10. These letters are shown in  FIG. 2 . Each Sloan letter has a stroke width MAR, expressed in minutes of arc of visual angle, and each letter is 5 MAR tall by 5 MAR wide. The size specification used here is Log 10  MAR, expressed as
 
log  MAR ( mar )=log 10 ( mar )
 
     4. Determine the Usable Range of Optotype Sizes 
     The usable range will be limited by the resolution and size of the PSF image. As discussed below, these are determined by the pupil size, the wavelength (λ), and the pupil magnification (m). If the PSF image has a width of r, expressed in pixels, and d in degrees, the smallest stroke-width possible is one pixel, or 
               log   ⁢           ⁢     MAR   min       =       log   10     ⁡     (       60   ⁢   d     r     )             
The largest stroke-width will be one fifth width of the largest character, which will be one half the width of the PSF image; a margin is required to accommodate blur and to avoid wrap-around so that
 
log  MAR   max =log 10 (6 d )
 
It is sometimes convenient to adopt a positive integer index that corresponds to size. One example is computing logMAR in steps of 1/20. In that scheme, the minimum and maximum indices would be
 
index min =Ceiling(20 log  MAR   min )
 
index max =Floor(20 log  MAR   max )
 
The index l then extends from 1 to l max =index max −index min +1, and log MAR is given by
 
                 log   ⁢           ⁢   MAR     =       l   +     index     mi   ⁢   n       -   1     20       ,           ⁢       where   ⁢           ⁢   l     =   1     ,   …   ⁢           ,     l   max           
Using the default parameters, the PSF image will have a width of 256 pixels, and a width of 0.815525 deg. With these values
 
index min =−14
 
index max =13
 
The size index l will have values between 1 and l max =28 for this example.
 
     5. Generate the Optical Transfer Function (OTF) 
     The mathematical operations required to generate an optical transfer function (OTF) from a set of Zernike polynomials are well known. Graphs of the results at several stages are shown in  FIGS. 3A-3E .
         a. Create the Pupil Aperture Image PA(x,y). The image is of size {2h,2h}, where h is the half width of the pupil aperture image, expressed in pixels,       

     
       
         
           
             
               
                 
                   
                     
                       PA 
                       ⁡ 
                       
                         ( 
                         
                           x 
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                           y 
                         
                         ) 
                       
                     
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                           ⁢ 
                           
                               
                           
                         
                         ⁢ 
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 y 
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                           / 
                           h 
                         
                       
                       ≤ 
                       1 
                     
                   
                   , 
                   
                       
                   
                   ⁢ 
                   
                     
                       x 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       y 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       integers 
                     
                     ⁢ 
                     
                         
                     
                     ∈ 
                     
                       { 
                       
                         
                           - 
                           h 
                         
                         , 
                         
                           … 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           0 
                         
                         , 
                         
                           
                             … 
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                             h 
                           
                           - 
                           1 
                         
                       
                       } 
                     
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     0 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     otherwise 
                   
                 
               
             
           
         
       
         
         
           
             b. From the set of Zernike coefficients C={c 0 , c 1 , c 2 , . . . , c N } (expressed in microns), create a discrete digital image of the Wavefront Aberration Image WA(x,y), with image size {2h,2h}. If the Zernike polynomials z n (x,y) are identified by single index (the mode) n=0, . . . N; and if the c k  are the coefficients of the individual polynomials, then 
           
         
       
    
               WA   ⁡     (     x   ,   y     )       =       ∑     n   =   1     N     ⁢       c   n     ⁢       z   n     ⁡     (     x   ,   y     )                 
We make use of the standard form of the Zernike polynomials as defined by Thibos, 2002, Jour. Of Optical Society of America.
         c. Compute the Generalized pupil image GP(x,y)       

               GP   ⁡     (     x   ,   y     )       =       PA   ⁡     (     x   ,   y     )       ⁢     epx   ⁡     [         ⅈ   ⁢           ⁢   2   ⁢           ⁢   π       λ10     -   3         ⁢     WA   ⁡     (     xv   ,     )         ]               
where λ is the wavelength of light in nm used to illuminate the optotype set.
         d. Pad the image on the left and top with zeros to create an image of size {2hm, 2hm}. The parameter m is the pupil magnification.   e. Compute the Point Spread Function PSF(x,y)
 
PSF( x,y )=|DFT[GP( x,y )]| 2  
 
where DFT is the Discrete Fourier Transform operator.
   f. Normalize the PSF.       

     
       
         
           
             
               
                 PSF 
                 _ 
               
               ⁡ 
               
                 ( 
                 
                   x 
                   , 
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                 ) 
               
             
             = 
             
               
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                   ) 
                 
               
               
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                   PSF 
                   ⁡ 
                   
                     ( 
                     
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                       , 
                       y 
                     
                     ) 
                   
                 
                  
               
             
           
         
       
         
         
           
             g. Compute the Optical Transfer Function OTF(u,v)
 
OTF( u,v )=2 hm DFT[  PSF ( x,y )]
 
This result is a complex image of size {2hm, 2hm}.
 
The height and width of the PSF image in degrees of visual angle is given by
 
           
         
       
    
             d   =       h   ⁢           ⁢   360   ⁢     λ10     -   6           p   ⁢           ⁢   π             
where p is the pupil diameter in mm. The height and width of the PSF image in pixels is given by
 
r=2hm
 
where h is a half-width. The resolution of the PSF image in pixels/degree is
 
     
       
         
           
             v 
             = 
             
               
                 r 
                 d 
               
               = 
               
                 
                   2 
                   ⁢ 
                   
                       
                   
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                   π 
                   ⁢ 
                   
                       
                   
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                   m 
                   ⁢ 
                   
                       
                   
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                   p 
                 
                 
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                   ⁢ 
                   
                       
                   
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                   λ 
                   ⁢ 
                   
                       
                   
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                     10 
                     
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                       6 
                     
                   
                 
               
             
           
         
       
     
     6. Generate the Neural Transfer Function (NTF) 
     
         
         
           
             a. The Radial Neural Transfer Function RNTF(u,v) is a two-dimensional real dft given by 
           
         
       
    
               RNTF   ⁡     (     u   ,   v     )       =     gain   ⁡     (       exp   ⁡     [     -       (     f     f   0       )     b       ]       -     loss   ⁢           ⁢     exp   ⁡     [     -       (     f     f   1       )     2       ]           )             
where gain, f 0 , f 1 , b, and loss are parameters. An example of this function is shown graphically in  FIG. 4 .
         b, The Oblique Effect Filter OEF(u,v) is a two-dimensional real dft given by       

                     OEF   ⁡     (     u   ,   v     )       =       ⁢       OEF   ⁡     (     f   ,   θ     )       =       1   -       (     1   -     exp   ⁡     (     -       f   -   corner     slope       )         )     ⁢     sin   ⁡     (     2   ⁢           ⁢   θ     )       ⁢             ⁢             ⁢   if   ⁢           ⁢   f       ≥   corner                   =       ⁢     1   ⁢           ⁢   otherwise                   f =√{square root over ( u   2   +v   2 )}
 
θ=arctan( u,v )
 
     where corner and slope are parameters.
         c. Compute the Neural Transfer Function NTF(u,v), a two-dimensional real dft given by
 
NTF( u,v )=RNTF( u,v )OEF( u,v )
       

     7. Generate the Total Transfer Function (TTF) 
     The Total Transfer Function is given by
 
TTF( u,v )=OTF( u,v )NTF( u,v )
 
     8. Define the Proportion Correct Function P(k) 
     The steps in evaluation of the P(k) function are as follows, and are diagrammed in  FIG. 5 .
         a. Given a size index 1, create K optotype images O k (x,y). This may be done by rendering images from a graphic description, or the images may be pre-computed. Each image is of size {r,r}. See above for a definition of the optotype size index 1.   b. Create the K Neural Images S k (x,y) by computing the DFT of the each optotype image O k , multiplying by the TTF, and taking the inverse DFT,
 
S k =IDFT[DFT[O k ]TTF]
       

     where DFT is the DFT operation and IDFT is the Inverse DFT operation.
         c. Create the K template images T k . By default, these are identical to the Neural Images S k .   d. Compute the normalized templates. Each template is divided by its norm, equal to the square root of the sum of the squares of all its pixels.       

     
       
         
           
             
               
                 T 
                 _ 
               
               k 
             
             = 
             
               
                 T 
                 k 
               
               
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                   T 
                   k 
                 
                  
               
             
           
         
       
         
         
           
             e. Compute the matrix of normalized template cross-correlations W j,k 
 
W j,k =  T   j :  T   k  
 
             f. Create an array of cross-correlations between each neural images and each template. Note that the row indexes the neural image and the column, the template.
 
R j,k =S j :  T   k  
 
             g. At this point two or more methods are available, which we identify as methods 1 and 2. 
           
         
       
    
     Method 1.
         i. Subtract each value from the main diagonal entry in the same row, and divide by a factor that includes the parameter σ (default value≈1). There are two possible versions of a matrix D, identified by subscripts 1 and 2.       

     
       
         
           
             
               D 
               
                 1 
                 , 
                 j 
                 , 
                 k 
               
             
             = 
             
               
                 
                   R 
                   
                     j 
                     , 
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                 - 
                 
                   R 
                   
                     j 
                     , 
                     k 
                   
                 
               
               
                 σ 
                 ⁢ 
                 
                   
                     1 
                     - 
                     
                       W 
                       
                         j 
                         , 
                         k 
                       
                       2 
                     
                   
                 
               
             
           
         
       
         
         
           
             ii. The probability correct for optotype j is given by 
           
         
       
    
               P     1   ,   j       =       ∫     -   ∞     ∞     ⁢       f   ⁡     (   t   )       ⁢       ∏     k   ≠   j       ⁢           ⁢       F   ⁡     (     t   -     D     1   ,   j   ,   k         )       ⁢           ⁢     ⅆ   t                   
where f(t) and F(t) are probability density function and cumulative probability
 
     Method 2 
               D     2   ,   j   ,   k       =         R     j   ,   j       -     R     j   ,   k           σ   ⁢     2     ⁢       1   -     W     j   ,   k                             P     2   ,   j       =       ∏     k   ≠   j       ⁢           ⁢     F   ⁡     (     D     2   ,   j   ,   k       )               
The final value of P is given by
 
     
       
         
           
             P 
             = 
             
               
                 1 
                 K 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     k 
                     = 
                     1 
                   
                   K 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   P 
                   k 
                 
               
             
           
         
       
     
     9. Find the Size for which P≈P target    
     The parameter P target  is the criterion probability for measurement of visual acuity. It is usually set to a value between 0.5 and 0.8. This value will depend upon the number K of optotypes and must be greater than 1/K (the probability of getting the right answer by guessing). For the Sloan letters, a default value P target =0.55 is used. Various efficient iterative procedures may be used to locate the value of size for which P≈P target . Here we describe the method of bisection, though other methods may be used.
 
l low =1
 
l high =l max  
 
 P   low   =P ( l   low )
 
 P   high   =P ( l   high )
 
begin loop
 
                   If   ⁢           ⁢     l   high       -     l   low       =   1     ,           ⁢       exit   ⁢           ⁢   and   ⁢           ⁢   return   ⁢           ⁢     l   final       =       l   low     +           l   high     -     l   low           p   high     -     p   low         ⁢     (       p   t     -     p   low       )                         l   mid     =     Round   ⁢           [         l   high     +     l   low       2     ]             P   mid   =P ( l   mid ) If P mid &lt;P target , l low =l mid      P   low =( l   low ) otherwise l high =l mid      P   high   =P ( l   high ) 
     Go to begin loop 
     The returned value of l final  can then be converted to an acuity in logMAR using the Equation above. This is the output of the WFA Metric. 
       FIG. 5  illustrates a sequence of steps of a procedure for practicing the invention. In step  51 , an OTF is generated. In step  52 , an NTF is generated and is multiplied by the OTF, to form a TTF (step  53 ). In step  54 , a set of optotypes is and is subjected to a DFT process, in step  55 . In step  56 , the processed optotypes are used to form images S j  of the optotypes. In step  57 , the images S j  are used to create a set of templates T k , and normalized templates T k * are created in step  58 . Cross-correlations R j,k  of the images S j  and the normalized templates T k * are formed, in step  59 . In step  60 , cross-correlations W k  of the normalized templates T k *. Normalized difference matrices D j,k  are formed from the cross-correlation matrix R j,k  are formed in step  61 , using information from the cross-correlations W k . and a statistical parameter σ, in step  62 . In step  63 , a probability P associated with measurement of visual acuity is computed. 
     10. Unique Features of the WFA Metric 
     The WFA metric is the only known metric to compute acuity from wavefronts that:
         (i) incorporates a model of the optotype identification task   (ii) can predict acuity for different target probabilities   (iii) can predict acuity for different optotypes   (iv) allows user specification of optotypes   (v) can incorporate different and possibly subject-specific neural transfer functions   (vi) can predict acuity for different subject templates
 
The template matching algorithm that is fundamental to this metric may have other uses in predicting performance in identification tasks.
       

     
       
         
               
               
               
               
             
           
               
                   
               
               
                   
                 Default  
                   
                   
               
               
                 Parameter 
                 value 
                 Unit 
                 Definition 
               
               
                   
               
             
             
               
                 K 
                   
                   
                 number of optotypes 
               
               
                 k 
                   
                   
                 index of optotpye, 1, . . . , K 
               
               
                 └ 
                 556 
                 nm 
                 wavelength 
               
               
                 p 
                 5 
                 mm 
                 diameter of pupil 
               
               
                 h 
                 64 
                 pixels 
                 half width of pupil image 
               
               
                 m 
                 2 
                   
                 magnification 
               
               
                 d 
                 derived 
                 degrees 
                 size of the PSF image 
               
               
                 r 
                 2 m h 
                 pixels 
                 size of the PSF image 
               
               
                 v 
                 r/d 
                 pixels/deg 
                 resolution of PSF image 
               
               
                 corner 
                 13.5715 
                 cycles/deg 
                 oblique effect parameter 
               
               
                 slope 
                 3.481 
                   
                 oblique effect parameter 
               
               
                 gain 
                 3.149614 
                   
                 NTF parameter 
               
               
                 loss 
                 0.9260249 
                   
                 NTF parameter 
               
               
                 f 0   
                 35.869213 
                   
                 NTF parameter 
               
               
                 f 1   
                 5.412887 
                   
                 NTF parameter 
               
               
                 b 
                 1.064181 
                   
                 NTF parameter 
               
               
                 l 
                   
                   
                 optotype size index 
               
               
                 WA 
                   
                 image 
                 wavefront aberration 
               
               
                 PSF 
                   
                 image 
                 point spread function 
               
               
                 PA 
                   
                 image 
                 pupil aperture 
               
               
                 GP 
                   
                 complex image 
                 generalized pupil 
               
               
                 TTF 
                   
                 dft 
                 total transfer function 
               
               
                 NTF 
                   
                 dft 
                 neural transfer function 
               
               
                 OTF 
                   
                 dft 
                 optical transfer function 
               
               
                 OEF 
                   
                 dft 
                 oblique effect transfer function 
               
               
                 O k   
                   
                   
                 optotype with index k 
               
               
                 S k   
                   
                   
                 neural image of optotype 
               
               
                 T k   
                   
                   
                 template with index k 
               
               
                 
                   T 
                   k 
                 
                   
                   
                 normalized template with index k 
               
               
                 W j,k   
                   
                   
                 cross-correlation between  
               
               
                   
                   
                   
                 normalized templates 
               
               
                 R j,k   
                   
                   
                 cross-correlation between  
               
               
                   
                   
                   
                 normalized templates 
               
               
                   
                   
                   
                 and neural images 
               
               
                 D j,k   
                   
                   
                 template response distribution  
               
               
                   
                   
                   
                 means 
               
               
                 P j   
                   
                 probability 
                 probability correct for optotype  
               
               
                   
                   
                   
                 with index k 
               
               
                 P 
                   
                 probability 
                 probability correct for optotypes  
               
               
                   
                   
                   
                 of one size 
               
               
                 σ 
                   
                   
                 noise standard deviation 
               
               
                 P target   
                 0.55 
                   
                 criterion proportion correct 
               
               
                 mar 
                   
                 minutes 
                 optotype stroke size 
               
               
                 x,y 
                   
                   
                 image pixel coordinates 
               
               
                 u,v 
                   
                   
                 dft pixel coordinates 
               
               
                 j,k 
                   
                   
                 row, column indices of matrices 
               
               
                 C n   
                   
                   
                 coefficient of Zernike  
               
               
                   
                   
                   
                 polynomial n 
               
               
                 Z n   
                   
                   
                 Zernike polynomial n