Abstract:
A method of approximating the inner or outer boundary of an iris comprises generating an approximate boundary representation ( 20 ) comprising a least squares approximation by a cosine transform series of a function of the angle (θ) about a fixed point (A) of the distance of measured points ( 10 ) on the boundary from the fixed point (A). More broadly, the method may be used to approximate the shape of any two-dimensional curve or figure.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     None. 
     TECHNICAL FIELD 
     The present invention relates to shape representation using Cosine Transforms. The invention finds particular although not exclusive application in biometrics, for example, in the generation of approximate representations of the outer and/or inner boundary of the iris in the human eye. 
     BACKGROUND OF THE INVENTION 
     Ghosh and Jain (“An Algebra of Geometric Shapes”,  IEEE Computer Graphics and Applications,  1993, 50) describe the use of Fast Fourier Transforms (FFTs) to model the outline of a shape by tracking around its outer periphery. 
     It is of considerable importance in biometric systems that rely on iris recognition to be able to identify and map accurately both the outer edge of the iris and also the inner edge (the periphery of the pupil). Many iris recognition systems assume that the shape of the pupil is always circular, an assumption which may be inaccurate in many cases. Indeed, even when pupils are circular, they tend to become elongate or oblong when viewed from an angle. 
     Some research into non-circular pupil localisation has been carried out: See B. Bonney, R. Ives, D. Etter, and D. Yingzi, “Iris pattern extraction using bit planes and standard deviations,”  Conference Record of the Thirty - Eighth Asilomar Conference on Signals, Systems and Computers,  2004, Y. Du, B. L. Bonney, R. W. Ives, D. M. Etter, and R. Schultz, “Analysis of Partial Iris Recognition Using a 1-D Approach,”  Proceedings of the  2005 IEEE International Conference on Acoustics, Speech, and Signal Processing , Mar. 18-23, 2005. However, in spite of these earlier approaches, there still remains a need for a system which can, in a straightforward way, approximate a boundary given a number of points (which may not be equally spaced) on that boundary. 
     SUMMARY OF THE INVENTION 
     According to a first aspect of the present invention there is provided a method of approximating an iris boundary, comprising the steps of:
         acquiring an image of an eye, including an iris boundary;   noting a plurality of spaced boundary points on the boundary;   selecting a fixed reference point; and,   generating an approximate boundary representation comprising a least squares approximation by a cosine transform series as a function of angle about said fixed point of the distance of said boundary points from said fixed point.       

     Because the points on the boundary may not be equally spaced, standard methods of calculating coefficients such as the Discrete Cosine Transform (DCT) cannot be used. 
     Preferably, this method is used to map the inner boundary of the iris (or, equivalently, the outer boundary of the pupil) of a human eye. Alternatively, it may be used to map the outer iris boundary. 
     In the method described, the use of higher harmonics provides excellent pupil localisation, both on general and on non-ideal eye images. The method provides excellent results on the vast majority of pupils which are significantly non-circular. 
     According to a second aspect of the present invention, there is provided a method of approximating a two-dimensional shape, comprising the steps of:
         noting a plurality of spaced measured points on the shape;   selecting a fixed reference point; and,   generating an approximate shape representation comprising a least squares approximation by a cosine transform series as a function of angle about said fixed point of the distance of said measured points from said fixed point.       

    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The invention may be carried into practice in a number of ways and one specific embodiment will now be described, by way of example, with reference to the accompanying drawings, in which: 
         FIG. 1 , shows a non-circular pupil shape, as imaged; and, 
         FIG. 2 , shows an approximation to that shape. 
     
    
    
     DETAILED DESCRIPTION 
     The way in which an embodiment of the invention may be applied to a particular problem, that of characterising a non-circular pupil shape, will now be described. 
     First, an eye may be imaged, and the image analysed to identify a plurality of points  10 , which occur on the imaged pupil/iris boundary  12 . To do this, an approximate pupil location may be first determined by searching for a dark area of significant size close to the image centre. A histogram analysis may then be carried out to find a more exact centre as well as the average pupil radius. This approximate circular pupil boundary may then be examined in detail to obtain the required number of edge points  10 . In the preferred embodiment, 16 such points are identified. It will be understood by those skilled in the art that other methods may be employed to locate points on the pupil/iris boundary and the scope of the claimed subject matter is not limited in this respect. 
     It will also be understood that the points  10  may not necessarily be equally spaced around the edge of the pupil. Indeed, in some images, part of the boundary  14  may be obscured by an eyelid and/or eyelashes  16 . 
     Once the boundary points  10  have been identified, those points can be used to generate a mathematical approximation  20  of the actual curve  12 , as is shown in  FIG. 2 . In the present invention, the fitted curve  20  is a Cosine Transform least squares approximation, as a function of angle θ, of the distance of the points  10  from a notional fixed point A (see  FIG. 1 ). 
     We wish to describe the distance d(θ) of a curve in the plane from a known or assumed centre as a harmonic function of the angle θ the form of a variation of the  1 D Discrete Cosine Transform (DCT). For simplicity, it will be assumed that the function d(θ) is single valued in θ. The inverse transform from N coefficients {C n ;n=0. . . N−1} is 
     
       
         
           
             
               d 
               ⁡ 
               
                 ( 
                 θ 
                 ) 
               
             
             = 
             
               
                 
                   C 
                   0 
                 
                 
                   2 
                 
               
               + 
               
                 
                   ∑ 
                   
                     n 
                     = 
                     1 
                   
                   
                     N 
                     - 
                     1 
                   
                 
                 ⁢ 
                 
                   
                     C 
                     n 
                   
                   ⁢ 
                   cos 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     n 
                     ⁡ 
                     
                       ( 
                       
                         
                           θ 
                           2 
                         
                         + 
                         
                           π 
                           
                             2 
                             ⁢ 
                             N 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     A standard Discrete Cosine Transform is a fit of regularly spaced data related to the Chebyshev polynomial, which has equi-ripple properties between the sample points. 
     However, in the present application, the points to be fitted {r i ,θ i ;i=1. . . M} may be irregular in θ i . 
     If the error in the fit is
 
 E (θ i )= d ( 74   i )− r   i  
 
     Then we wish to find {C n ; n=0. . . N} which minimizes the sum of squares of the error, 
     
       
         
           
             
               
                 
                   
                     E 
                     2 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         M 
                       
                       ⁢ 
                       
                         
                           E 
                           ⁡ 
                           
                             ( 
                             
                               θ 
                               i 
                             
                             ) 
                           
                         
                         2 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         M 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               d 
                               ⁡ 
                               
                                 ( 
                                 
                                   θ 
                                   i 
                                 
                                 ) 
                               
                             
                             - 
                             
                               r 
                               i 
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       M 
                     
                     ⁢ 
                     
                       
                         [ 
                         
                           
                             { 
                             
                               
                                 
                                   C 
                                   0 
                                 
                                 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     1 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     C 
                                     n 
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     n 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           θ 
                                           2 
                                         
                                         + 
                                         
                                           π 
                                           
                                             2 
                                             ⁢ 
                                             N 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             } 
                           
                           - 
                           
                             r 
                             i 
                           
                         
                         ] 
                       
                       2 
                     
                   
                 
               
             
           
         
       
     
     To do this we differentiate with respect to C k  and equate to zero: 
     
       
         
           
             
               
                 ∂ 
                 
                   E 
                   2 
                 
               
               
                 ∂ 
                 
                   C 
                   0 
                 
               
             
             = 
               
             ⁢ 
             
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   M 
                 
                 ⁢ 
                 
                   
                     
                       2 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             C 
                             0 
                           
                           
                             2 
                           
                         
                         + 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               C 
                               n 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               n 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       θ 
                                       i 
                                     
                                     2 
                                   
                                   + 
                                   
                                     π 
                                     
                                       2 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         - 
                         
                           r 
                           i 
                         
                       
                       ] 
                     
                   
                   ⁢ 
                   cos 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       k 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               θ 
                               i 
                             
                             2 
                           
                           + 
                           
                             π 
                             
                               2 
                               ⁢ 
                               N 
                             
                           
                         
                         ) 
                       
                     
                     i 
                   
                 
               
               = 
               0 
             
           
         
       
       
         
           
             
               
                 
                   ∂ 
                   
                     E 
                     2 
                   
                 
                 
                   ∂ 
                   
                     C 
                     k 
                   
                 
               
               = 
                 
               ⁢ 
               
                 
                   
                     ∑ 
                     
                       i 
                       = 
                       1 
                     
                     M 
                   
                   ⁢ 
                   
                     
                       2 
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               C 
                               0 
                             
                             
                               2 
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 1 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 C 
                                 n 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 n 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       
                                         θ 
                                         i 
                                       
                                       2 
                                     
                                     + 
                                     
                                       π 
                                       
                                         2 
                                         ⁢ 
                                         N 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           - 
                           
                             r 
                             i 
                           
                         
                         ] 
                       
                     
                     ⁢ 
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         k 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 θ 
                                 i 
                               
                               2 
                             
                             + 
                             
                               π 
                               
                                 2 
                                 ⁢ 
                                 N 
                               
                             
                           
                           ) 
                         
                       
                       i 
                     
                   
                 
                 = 
                 0 
               
             
             , 
             
               
 
             
             ⁢ 
             
                 
             
             ⁢ 
             
               k 
               = 
               
                 
                   1 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   … 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   N 
                 
                 - 
                 1 
               
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               from 
               ⁢ 
               
                   
               
               ⁢ 
               which 
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       M 
                     
                     ⁢ 
                     
                       
                         C 
                         0 
                       
                       
                         2 
                       
                     
                   
                   + 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         1 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         C 
                         n 
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         n 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 θ 
                                 i 
                               
                               2 
                             
                             + 
                             
                               π 
                               
                                 2 
                                 ⁢ 
                                 N 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 = 
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       M 
                     
                     ⁢ 
                     
                       
                         r 
                         i 
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           M 
                         
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   C 
                                   0 
                                 
                                 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     1 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     C 
                                     n 
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     n 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           
                                             θ 
                                             i 
                                           
                                           2 
                                         
                                         + 
                                         
                                           π 
                                           
                                             2 
                                             ⁢ 
                                             N 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             k 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     θ 
                                     i 
                                   
                                   2 
                                 
                                 + 
                                 
                                   π 
                                   
                                     2 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       M 
                     
                     ⁢ 
                     
                       
                         r 
                         i 
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         k 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 θ 
                                 i 
                               
                               2 
                             
                             + 
                             
                               π 
                               
                                 2 
                                 ⁢ 
                                 N 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               , 
               
                 
 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 k 
                 = 
                 
                   
                     1 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     … 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     N 
                   
                   - 
                   1 
                 
               
             
           
         
       
     
     This gives N linear equations in the unknowns {C n ; n=0. . . N −1} which can be expressed in matrix form as
 
 PC=S  
 
     Where the unknowns are given by 
     
       
         
           
             C 
             = 
             
               [ 
               
                 
                   
                     
                       C 
                       0 
                     
                   
                 
                 
                   
                     
                       C 
                       1 
                     
                   
                 
                 
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       C 
                       N 
                     
                   
                 
               
               ] 
             
           
         
       
     
     and the right hand side is given by 
     
       
         
           
             S 
             = 
             
               [ 
               
                 
                   
                     
                       S 
                       0 
                     
                   
                 
                 
                   
                     
                       S 
                       1 
                     
                   
                 
                 
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       S 
                       N 
                     
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               where 
               ⁢ 
               
                   
               
               ⁢ 
               
                 S 
                 k 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   0 
                 
                 M 
               
               ⁢ 
               
                 
                   r 
                   i 
                 
                 ⁢ 
                 cos 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   k 
                   ⁡ 
                   
                     ( 
                     
                       
                         
                           θ 
                           i 
                         
                         2 
                       
                       + 
                       
                         π 
                         
                           2 
                           ⁢ 
                           N 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     The N by N matrix P is then 
     
       
         
           
             
               P 
               
                 k 
                 , 
                 0 
               
             
             = 
             
               1 
               
                 2 
               
             
           
         
       
       
         
           
             
               P 
               
                 k 
                 , 
                 n 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 M 
               
               ⁢ 
               
                 
                   C 
                   n 
                 
                 ⁢ 
                 cos 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   n 
                   ⁡ 
                   
                     ( 
                     
                       
                         
                           θ 
                           i 
                         
                         2 
                       
                       + 
                       
                         π 
                         
                           2 
                           ⁢ 
                           N 
                         
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 cos 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   k 
                   ⁡ 
                   
                     ( 
                     
                       
                         
                           θ 
                           i 
                         
                         2 
                       
                       + 
                       
                         π 
                         
                           2 
                           ⁢ 
                           N 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               for 
               ⁢ 
               
                   
               
               ⁢ 
               k 
             
             = 
             
               
                 
                   0 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   … 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   N 
                 
                 - 
                 
                   1 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   n 
                 
               
               = 
               
                 
                   1 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   … 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   N 
                 
                 - 
                 1 
               
             
           
         
       
     
     The matrix P is symmetric. This can be solved using standard numerical methods for any M and N, giving an approximation by N coefficients to M given points:
 
 C=P   −1   S  
 
     Where M≧N, the equations above minimise the Root Mean Square (RMS) error between each measured point  100  on the actual boundary, and the closest corresponding point  101  of the fitted boundary  20 . When the number of coefficients is 1, the fitted curve is a circle, and as the number of coefficients increases the RMS error generally decreases. It has been found in practice that good results in iris approximation can be obtained by using 5 coefficients. 
     In addition to modelling the boundary of the pupil (or, equivalently, the inner boundary of the iris), the present embodiment may also be used to model the shape of the outer boundary of the iris. Once the inner and outer boundaries have been determined, biometric identification can proceed in the normal way based on the characteristics of the iris image between the inner and outer boundaries. 
     The position of the fixed point A ( FIG. 1 ) is not of great importance, and although the approximate centre of the pupil is a convenient point to take other points are not excluded—even points which lie outside the boundary being fitted. Of course, if the fixed point lies outside the boundary, the resulting function will no longer be single valued in θ, and a corresponding allowance for that will need to be made. 
     If there is a large variability in distances, an improved fit may sometimes be achieved using a multi pass approach: carry out a first fit, exclude any outliers which are greater than a cut-off value, and repeat the calculation. The cut-off value may be fixed, or may be data dependent, for example, a given number of standard deviations. 
     It will be understood that the method described above may find application in the fitting of a variety of other curves and/or boundaries, in addition to fitting of the inner and outer iris boundaries. With a suitable choice of fixed reference points, the method could even be used to approximate open shapes such as simple curve fragments.