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 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 fig1 . 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 fig2 . 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 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 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 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 the size index l will have values between 1 and l max = 28 for this example . 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 fig3 a - 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 , 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 . 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 ) 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 where h is a half - width . the resolution of the psf image in pixels / degree is 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 fig4 . 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 )} c . compute the neural transfer function ntf ( u , v ), a two - dimensional real dft given by the steps in evaluation of the p ( k ) function are as follows , and are diagrammed in fig5 . 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 , 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 . 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 . g . at this point two or more methods are available , which we identify as methods 1 and 2 . 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 . p 1 , j = ∫ - ∞ ∞ ⁢ f ⁡ ( t ) ⁢ ∏ k ≠ j ⁢ ⁢ f ⁡ ( t - d 1 , j , k ) ⁢ ⁢ ⅆ t 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 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 . 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 ) 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 . fig5 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 . 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 .
a method for generating a visual acuity metric , based on wavefront aberrations , 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 .