Abstract:
The present invention generally relates to an apparatus and a method for computing an estimate position of a micro-image produced by a micro-lens of an array of micro-lenses of an optical acquisition system in order to calibrate said optical acquisition system. Existing solutions for estimating the position of the micro-image are not designed to be fast and are resource consuming. It is proposed to estimate the position of the micro-image by extracting the parameters D,θ,x 0,0 ,y 0,0  defining this position from an image in the Fourier domain. The four parameters D,θ,x 0,0 ,y 0,0  characterizing the position of the micro-image relatively to a pixel grid of a sensor of a plenoptic camera are estimated by knowing the accurate position of peaks of a Dirac comb in the image in the Fourier domain.

Description:
TECHNICAL FIELD 
       [0001]    The present disclosure generally relates to an apparatus and a method for computing an estimate position of a micro-image produced by a micro-lens of an array of micro-lenses of an optical acquisition system in order to calibrate said optical acquisition system. 
       BACKGROUND 
       [0002]    A plenoptic camera  100 , as represented on  FIG. 1 , is able to measure the amount of light traveling along each bundle of rays that intersects a sensor  101 , by arranging a microlens array  102  between a main lens  103  and the sensor  101 . The micro-lens array  102  comprises a plurality of micro-lenses  104  arranged in a periodic pattern such as an hexagonal pattern or a square pattern. 
         [0003]    The data acquired by such a camera  100  are called light-field data. These light-field data can be post-processed to reconstruct images of a scene from different viewpoints. Compared to a conventional camera, the plenoptic camera can obtain additional optical information components for achieving the reconstruction of the images of a scene from the different viewpoints and re-focusing depth by post-processing. 
         [0004]    The rendering of light-field data relies on a calibration step. Such a calibration step consists in an estimation of a position of micro-images produces by the micro-lenses  104  of the micro-lens array  102  relatively to the sensor  101 . The calibration parameters are then stored as metadata in an output light-field image header file. 
         [0005]    Theoretically, the properties of the micro-lens array  102  are provided by the manufacturer of the plenoptic camera. However, slight rotation e and shift (xo,o, yo,o) of the micro-lens array relatively to the sensor  101  might occur during the manufacturing of the plenoptic camera as shown in  FIG. 2 . Furthermore, the size of the micro-images is slightly larger than the size of the micro-lenses  104 , thus knowing the size of the micro-lenses  104  from the manufacturer does not give a reliable information about the size of the micro-images. This slight gap in the position of the micro-lens array  102  might lead to the generation of blurred re-focussed images. Thus, a calibration step has been commonly added in order to solve this problem. 
         [0006]    In “Modeling the calibration pipeline of the Lytro camera for high quality light-field image reconstruction” in  Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern recognition,  2013, pp. 3280-3287, Cho, D. et al. propose to use white images, i.e. images of a uniform white scene in order to compute the position of micro-images produced by the micro-lenses  104  of the micro-lens array  102 . 
         [0007]    A discrete Fourier transform of the white image is obtained and the value of the rotation angle of the micro-lens array  102  is computed in the Fourier domain. The value of the rotation angle is then used to rotate an image of a scene captured by the plenoptic camera  100 . The centres of the micro-images produced by the micro-lenses  104  of the micro-lens array  102  are then detected, in the spatial domain, by parabolic fitting and Delaunay triangulation. 
         [0008]    Such a method as described by Cho et al. is not designed to be fast and is resource consuming. Therefore, this method does not fit for monitoring the position of micro-images dynamically like on a plenoptic video camera or on a plenoptic camera mounted with a zoom or interchangeable lenses. 
         [0009]    The present invention has been devised with the foregoing in mind. 
       SUMMARY OF INVENTION 
       [0010]    According to a first aspect of the invention, there is provided an apparatus for computing an estimate position of a micro-image produced by a micro-lens of an array of micro-lenses of an optical acquisition system, said apparatus comprising a processor configured to :
       identify, in a discrete Fourier transform of an image captured by the optical acquisition system, at least one peak of a value of a module of a pixel of the discrete       
 
         [0012]    Fourier transform of the captured image,
       determine the position of said peak by adding a shift to a position of the pixel corresponding to the peak, said shift being a function of values of the modules of at least two pixels adjacent to said pixel in the discrete Fourier transform of the captured image,   compute the estimate position of the micro-image based on the position of the peak in the discrete Fourier transform of the captured image.       
 
         [0015]    According to an embodiment of the invention, the processor is configured to identify the peak in the discrete Fourier transform of the captured image by comparing the module of a pixel of the discrete Fourier transform of the captured image with the modules of adjoining pixels in the discrete Fourier transform of the captured image. 
         [0016]    According to an embodiment of the invention, the processor is configured to compare the module of said pixel with a threshold prior to comparing the module of said pixel with the modules of adjoining pixels when the module of said pixel is greater than or equal to the threshold. 
         [0017]    According to an embodiment of the invention, the processor is configured to compute the shift to be added to the position of the pixel by comparing the modules of at least two pixels adjacent to the pixel corresponding to the peak and calculating a ratio of the module of the adjoining pixel having the smaller value to the sum of the value of the module of the pixel corresponding to the peak with the module of one of the adjoining pixel. 
         [0018]    According to an embodiment of the invention, the processor is configured to compute a pitch of the array of micro-lenses based on a polar distance of the peak to a centre of the discrete Fourier transform of the captured image determined based on the position of the peak. 
         [0019]    According to an embodiment of the invention, the processor is configured to compute a rotation of the array of micro-lenses in relation to an array of pixels of a sensor of the optical acquisition system based on the polar distance of the peak to the centre of the discrete Fourier transform of the captured image determined based on the position of the peak. 
         [0020]    According to an embodiment of the invention, the processor is configured to compute a position of the micro-image produced by a micro-lens of the array of micro-lenses in relation to the array of pixels of the sensor based on the polar distance of the peak to the centre of the discrete Fourier transform of the captured image determined based on the position of the peak and a phase of the peak. 
         [0021]    Another aspect of the invention concerns a method for computing an estimate position of a micro-image produced by a micro-lens of an array of micro-lenses of an optical acquisition system, said method comprising :
       identifying, in a discrete Fourier transform of an image captured by the optical acquisition system, at least one peak of a value of a module of a pixel of the discrete Fourier transform of the captured image,   determining the position of said peak by adding a shift to a position of the pixel corresponding to the peak, said shift being a function of values of the modules of at least two pixels adjacent to said pixel in the discrete Fourier transform of the captured image,   computing the estimate position of the micro-image based on the position of the peak in the discrete Fourier transform of the captured image.       
 
         [0025]    According to an embodiment of the invention, the method further comprises identifying the peak in the discrete Fourier transform of the captured image by comparing the module of a pixel of the discrete Fourier transform of the captured image with the modules of adjoining pixels in the discrete Fourier transform of the captured image. 
         [0026]    According to an embodiment of the invention, the method further comprises comparing the module of said pixel with a threshold prior to comparing the module of said pixel with the modules of adjoining pixels when the module of said pixel is greater than or equal to the threshold. 
         [0027]    According to an embodiment of the invention, the method further comprises computing the shift to be add to the position of the pixel by comparing the modules of at least two pixels adjacent to the pixel corresponding to the peak and calculating a ratio of the module of the adjoining pixel having the smaller value to the sum of the value of the module of the pixel corresponding to the peak with the module of one of the adjoining pixel. 
         [0028]    According to an embodiment of the invention, the method further comprises computing a pitch of the array of micro-lenses based on a polar distance of the peak to a centre of the discrete Fourier transform of the captured image determined based on the position of the peak. 
         [0029]    According to an embodiment of the invention, the method further comprises computing a rotation of the array of micro-lenses in relation to an array of pixels of a sensor of the optical acquisition system based on the polar distance of the peak to the centre of the discrete Fourier transform of the captured image determined based on the position of the peak. 
         [0030]    According to an embodiment of the invention, the method further comprises computing a position of the micro-image produced by a micro-lens of the array of micro-lenses in relation to the array of pixels of the sensor based on the polar distance of the peak to the centre of the discrete Fourier transform of the captured image determined based on the position of the peak and a phase of the peak. 
         [0031]    Some processes implemented by elements of the invention may be computer implemented. Accordingly, such elements may take the form of an entirely hardware embodiment, an entirely software embodiment (including firmware, resident software, micro-code, etc.) or an embodiment combining software and hardware aspects that may all generally be referred to herein as a “circuit”, “module” or “system”. 
         [0032]    Furthermore, such elements may take the form of a computer program product embodied in any tangible medium of expression having computer usable program code embodied in the medium. 
         [0033]    Since elements of the present invention can be implemented in software, the present invention can be embodied as computer readable code for provision to a programmable apparatus on any suitable carrier medium. A tangible carrier medium may comprise a storage medium such as a floppy disk, a CD-ROM, a hard disk drive, a magnetic tape device or a solid state memory device and the like. A transient carrier medium may include a signal such as an electrical signal, an electronic signal, an optical signal, an acoustic signal, a magnetic signal or an electromagnetic signal, e.g. a microwave or RF signal. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0034]    Embodiments of the invention will now be described, by way of example only, and with reference to the following drawings in which: 
           [0035]      FIG. 1  represents a plenoptic camera as mentioned in prior art; 
           [0036]      FIG. 2  represents the slight rotation θ and shift (x 0,0 , y 0,0 ) of a micro-lens array relatively to a sensor that might occur during the manufacturing of a plenoptic camera, as mentioned in the prior art, 
           [0037]      FIG. 3  is a schematic block diagram illustrating an example of an apparatus for computing an estimate position of a micro-image produced by a micro-lens of a micro-lens array of a plenoptic camera according to an embodiment of the invention, 
           [0038]      FIG. 4  represents a flow chart explaining a process for computing an estimate position of a micro-image produced by a micro-lens of the micro-lens array of a plenoptic camera according to an embodiment of the invention, 
           [0039]      FIG. 5  represents the Fourier transform of the image captured by the plenoptic camera, 
           [0040]      FIG. 6  represents N points located on a unitary circle regularly spaced between the angles [φ,φ+2πε[, 
           [0041]      FIG. 7  represents the L/2 cosine functions defined by the L first peaks of the Dirac comb in the direct space. 
       
    
    
     DETAILED DESCRIPTION 
       [0042]    As will be appreciated by one skilled in the art, aspects of the present principles can be embodied as a system, method or computer readable medium. Accordingly, aspects of the present principles can take the form of an entirely hardware embodiment, an entirely software embodiment, (including firmware, resident software, micro-code, and so forth) or an embodiment combining software and hardware aspects that can all generally be referred to herein as a “circuit”, “module”, or “system”. Furthermore, aspects of the present principles can take the form of a computer readable storage medium. Any combination of one or more computer readable storage medium(a) may be utilized. 
         [0043]    As previously described in relation to  FIG. 1 , the micro-lenses  104  of the micro-lens array  102  of an optical acquisition system such as a plenoptic camera  100  are arranged in a periodic pattern such as an hexagonal pattern or a square pattern. 
         [0044]    The coordinates (x i,j , y i,j ) of the center (i,j) of a micro-image in a pixel grid of the sensor  101  are defined as follows in case of a square pattern : 
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         [0046]    where (x 0,0 , u 0,0 ) are the coordinates of the first micro-lens centre; (i,j) is the micro-image coordinate within the micro-lens array  102 ; D the distance between two contiguous micro-images; and θ the rotation angle between the pixel grid of the sensor  101  and the micro-lens array  102 , as shown on  FIG. 2 . The four parameters D, θ, x 0,0 , y 0,0  fully characterize the position of the micro-images (x i , y j ) relatively to pixel grid of the sensor  101 . 
         [0047]    One way for obtaining the values of these four parameters D, θ, x 0,0 ,y 0,0  is to extract them from an image in the Fourier domain. An image in the Fourier domain is a representation of a discrete Fourier transform of an image captured by the plenoptic camera  100 . 
         [0048]    An image in the Fourier domain is made of N u  ×N v  pixels, hereafter called Fourier pixels to distinguish them from the pixels of the sensor  101  of the plenoptic camera  100 . Each Fourier pixel of the image in the Fourier domain has a set of coordinates (u; v) in a u-v coordinate system which enables to locate the Fourier pixel in the image in the Fourier domain. 
         [0049]    Since the micro-lenses  104  of the micro-lens array  102  are arranged in a periodic pattern, a discrete Fourier transform of an image captured by the plenoptic camera  100  is a Dirac comb. 
         [0050]    The four parameters D, θ, x 0,0 , y 0,0  characterizing the position of the micro-image relatively to the pixel grid of the sensor  101  may be estimated knowing the position of the peaks of the Dirac comb in the image in the Fourier domain. 
         [0051]      FIG. 3  is a schematic block diagram illustrating an example of an apparatus for computing an estimate position of a micro-image produced by a micro-lens  104  of the micro-lens array  102  of the plenoptic camera  100  according to an embodiment of the present invention. 
         [0052]    The apparatus  300  comprises a processor  301 , a storage unit  302 , an input device  303 , a display device  304 , and an interface unit  305  which are connected by a bus  306 . Of course, constituent elements of the computer apparatus  300  may be connected by a connection other than a bus connection. 
         [0053]    The processor  301  controls operations of the apparatus  300 . The storage unit  302  stores at least one program to be executed by the processor  301 , and various data, including data of 4D light-field images captured or provided by the plenoptic camera  100 , parameters used by computations performed by the processor  301 , intermediate data of computations performed by the processor  301 , and so on. The processor  301  may be formed by any known and suitable hardware, or software, or a combination of hardware and software. For example, the processor  301  may be formed by dedicated hardware such as a processing circuit, or by a programmable processing unit such as a CPU (Central Processing Unit) that executes a program stored in a memory thereof. 
         [0054]    The storage unit  302  may be formed by any suitable storage or means capable of storing the program, data, or the like in a computer-readable manner. Examples of the storage unit  302  include non-transitory computer-readable storage media such as semiconductor memory devices, and magnetic, optical, or magneto-optical recording media loaded into a read and write unit. The program causes the processor  301  to perform a process for computing an estimate position of a micro-image produced by a micro-lens  104  of the array of micro-lenses  102  of the optical acquisition system  100  according to an embodiment of the present invention as described hereinafter with reference to  FIG. 4 . 
         [0055]    The input device  303  may be formed by a keyboard, a pointing device such as a mouse, or the like for use by the user to input commands. The output device  304  may be formed by a display device to display, for example, a Graphical User Interface (GUI). The input device  303  and the output device  304  may be formed integrally by a touchscreen panel, for example. 
         [0056]    The interface unit  305  provides an interface between the apparatus  300  and an external apparatus. The interface unit  305  may be communicable with the external apparatus via cable or wireless communication. In an embodiment, the external apparatus may be a light-field camera  100 . In this case, data of 4D light-field images captured by the light-field camera  100  can be input from the light-field camera  100  to the apparatus  300  through the interface unit  305 , then stored in the storage unit  302 . 
         [0057]    In this embodiment the apparatus  300  is exemplary discussed as it is separated from the light-field camera  100  and they are communicable each other via cable or wireless communication, however it should be noted that the apparatus  300  can be integrated with such a light-field camera  100 . In this later case, the apparatus  300  may be for example a portable device such as a tablet or a smartphone embedding a light-field camera. 
         [0058]      FIG. 4  is a flow chart for explaining a process for computing an estimate position of a micro-image produced by a micro-lens  104  of the micro-lens array  102  of the plenoptic camera  100  according to an embodiment of the present invention. 
         [0059]    In a step  400 , an image is acquired by the apparatus  300 . In an embodiment of the invention, the image is captured by an external apparatus such as the plenoptic camera  100 . In this embodiment, the image is input from the plenoptic camera  100  to the apparatus  300  through the interface unit  305  and then stored in the storage unit  302 . 
         [0060]    In another embodiment of the invention, the apparatus  300  embeds the plenoptic camera  100 . In this case, the image is captured by the plenoptic camera  100  of the apparatus  300  and then stored in the storage unit  302 . 
         [0061]    In a step  401 , the processor  301  compute a discrete Fourier transform, or DFT, of the image captured by the plenoptic camera  100 . The image captured by the plenoptic camera  100  is made of [N x , N y ] pixels, i.e. the sensor  101  comprises N x ×N y  pixels. The discrete Fourier transform of the captured image comprises N u ×N v  pixels called Fourier pixels. For convenience and computation speed N u =N v =2 p  such that N u ≦min(N x , N y ). 
         [0062]    In a step  402 , a module F m (u, v) and a phase F m (u, v) of the Fourier pixels (u, v) with (u, v) ∈ [0, N u [×[0, N v [ of the discrete Fourier transform of the captured image are computed. The Fourier transform convention is such that (u, v)=(N u /2,N v /2) corresponds to the null frequency. 
         [0063]    A peak, and its negative counterpart in a discrete Fourier transform of a captured image is the discrete Fourier transform of a cosine function. A first peak in the Fourier domain is always observed with a second peak being symmetric around the image centre and having strictly the same amplitude and phase. 
         [0064]    The peaks, displayed with a negative intensity, observed in the discrete module of the Fourier transform of the image captured by the plenoptic camera  100  are shown on  FIG. 5 . These peaks have three possible origins. 
         [0065]    A central peak  500  is located at a position (u, v) =(N u /2, N v /2) on the discrete Fourier transform of the captured image. This central peak  500  indicates the energy of the null frequency which corresponds to the average value of the image. 
         [0066]    A plurality of symmetrical peaks  501  are located around the centre of the discrete Fourier transform of the captured image These peaks  501  belong to the Dirac comb the properties of which depend on the parameters D, θ, x 0,0 , y 0,0  of the micro-images produced by the micro-lens array  102 . The peaks  501  are arranged by group of 4 when the pattern of the micro-lens array  102  is a square pattern or 6 when the pattern of the micro-lens array  102  is a hexagonal pattern within a circle centred on the middle of the discrete Fourier transform of the captured image. Several circles encompass all the peaks  501  each of which corresponding to a harmonic frequency of the smallest circle. The positions of the peaks  501  of the smallest circle in the discrete Fourier transform of the captured image fully characterize the four parameters D, θ, x 0,0 , y 0,0  of the micro-images produced by the micro-lens array  102 . 
         [0067]    To record colours, the pixels of the sensor  101  are often mounted with a Color Filter Array (CFA) like the Bayer pattern for example. The captured image has not been de-mosaiced prior to computing the discrete Fourier transform, and the Bayer pattern is visible on the discrete Fourier transform of the captured image at the high frequency. The peaks  502  located close to the borders of the discrete Fourier transform of the captured image are discarded because they have no impact on the peaks  501 . 
         [0068]    In a step  403 , the processor  301  identifies, in the discrete Fourier transform of the image captured by the plenoptic camera  100 , at least one peak  501 . The peak  501  corresponds to a local maximum of a value of the module F m (u, v) of a Fourier pixel (u, v) of the discrete Fourier transform of the captured image. The processor  301  identifies the peaks  501  in the discrete Fourier transform of the captured image by comparing the module F m (u, v) of a Fourier pixel (u, v) of the discrete Fourier transform of the captured image with the modules of the eight adjoining Fourier pixels in the discrete Fourier transform of the captured image. In an embodiment of the invention, prior to comparing the modules of the adjoining Fourier pixels with the module F m (u, v) of the Fourier pixel (u, v), the processor  301  compares the module F m (u, v) the Fourier pixel (u, v) with a threshold T. This enables the processor  301  to only process Fourier pixels having a module higher than a background noise. If the module F m (u, v) the 
         [0069]    Fourier pixel (u, v) is below the threshold T, the processor  301  does not compare the module F m (u, v) the Fourier pixel (u, v) with the modules of the adjoining Fourier pixels. 
         [0070]    When the module F m (u, v) the Fourier pixel (u, v) is greater or equal to the modules of the adjoining Fourier pixels, then the Fourier pixel (u, v) corresponds to a local maximum. An accurate position of the peak  501  corresponding to the local maximum is given by (u n , v n )=(u+∈ u , v+∈ r ). The parameters E u , ∈ represent the shift to be add to the position (u, v) of the Fourier pixel corresponding the peak  501  and are given by the following equations : 
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                                 f 
                                 m 
                               
                                
                               
                                 ( 
                                 
                                   u 
                                   , 
                                   v 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       else 
                     
                     
                       
                         
                           ε 
                           v 
                         
                         = 
                         
                           - 
                           
                             
                               
                                 f 
                                 m 
                               
                                
                               
                                 ( 
                                 
                                   u 
                                   , 
                                   
                                     v 
                                     - 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                             
                               
                                 
                                   f 
                                   m 
                                 
                                  
                                 
                                   ( 
                                   
                                     u 
                                     , 
                                     
                                       v 
                                       + 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   f 
                                   m 
                                 
                                  
                                 
                                   ( 
                                   
                                     u 
                                     , 
                                     v 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0071]    In a step  404 , the processor  301  computes the shift to be add to the position (u, v) of the Fourier pixel (u, v) by calculating the ratio defined by equations (3) and (4). Using equations (3) and (4), it is possible to estimate the position of a peak  501  with an accuracy of 0.01 pixel. 
         [0072]    Indeed, in the Fourier domain, a peak  500 ,  501 ,  502  and its negative counterpart is the discrete Fourier transform of a cosine function. 
         [0073]    In one dimension, the discrete Fourier transform v(k), where N is the size of the discrete Fourier transform, of a function f(x) is defined by : 
         [0000]    
       
         
           
             
               
                 
                   
                     v 
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         x 
                         = 
                         0 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         f 
                          
                         
                           ( 
                           x 
                           ) 
                         
                       
                        
                       
                          
                         
                           - 
                           
                             
                               2 
                                
                                
                                
                               
                                   
                               
                                
                               π 
                                
                               
                                   
                               
                                
                               xk 
                             
                             N 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0074]    The cosine function of period A can evaluated using the Euler formula : 
         [0000]    
       
         
           
             
               
                 
                   
                     cos 
                      
                     
                       ( 
                       
                         
                           2 
                            
                           
                               
                           
                            
                           π 
                            
                           
                             x 
                             λ 
                           
                         
                         + 
                         ϕ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
                        
                       
                          
                         
                           
                             2 
                              
                             
                                 
                             
                              
                              
                              
                             
                                 
                             
                              
                             π 
                              
                             
                               x 
                               λ 
                             
                           
                           + 
                           
                             ϕ 
                              
                             
                                 
                             
                              
                              
                           
                         
                       
                     
                     + 
                     
                       
                         1 
                         2 
                       
                        
                       
                          
                         
                           
                             
                               - 
                               2 
                             
                              
                             
                                 
                             
                              
                              
                              
                             
                                 
                             
                              
                             π 
                              
                             
                               x 
                               λ 
                             
                           
                           - 
                           
                             ϕ 
                              
                             
                                 
                             
                              
                              
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0075]    The discrete Fourier transform of the cosine is equal to : 
         [0000]    
       
         
           
             
               
                 
                   
                     v 
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           x 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           1 
                           2 
                         
                          
                         
                            
                           
                             
                               - 
                               
                                 
                                   2 
                                    
                                    
                                    
                                   
                                       
                                   
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   x 
                                 
                                 N 
                               
                             
                              
                             
                               ( 
                               
                                 k 
                                 + 
                                 
                                   N 
                                   λ 
                                 
                               
                               ) 
                             
                           
                         
                          
                         
                            
                           
                             
                               - 
                               ϕ 
                             
                              
                             
                                 
                             
                              
                              
                           
                         
                       
                     
                     + 
                     
                       
                         1 
                         2 
                       
                        
                       
                          
                         
                           
                             - 
                             
                               
                                 2 
                                  
                                  
                                  
                                 
                                     
                                 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 x 
                               
                               N 
                             
                           
                            
                           
                             ( 
                             
                               k 
                               - 
                               
                                 N 
                                 λ 
                               
                             
                             ) 
                           
                         
                       
                        
                       
                          
                         
                           ϕ 
                            
                           
                               
                           
                            
                            
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0076]    If N /λ=k′ where k′ is an integer, equation (7) becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                     v 
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           x 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           1 
                           2 
                         
                          
                         
                            
                           
                             
                               - 
                               
                                 
                                   2 
                                    
                                    
                                    
                                   
                                       
                                   
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   x 
                                 
                                 N 
                               
                             
                              
                             
                               ( 
                               
                                 k 
                                 + 
                                 
                                   k 
                                   ′ 
                                 
                               
                               ) 
                             
                           
                         
                          
                         
                            
                           
                             
                               - 
                               ϕ 
                             
                              
                             
                                 
                             
                              
                              
                           
                         
                       
                     
                     + 
                     
                       
                         1 
                         2 
                       
                        
                       
                          
                         
                           
                             - 
                             
                               
                                 2 
                                  
                                  
                                  
                                 
                                     
                                 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 x 
                               
                               N 
                             
                           
                            
                           
                             ( 
                             
                               k 
                               - 
                               
                                 k 
                                 ′ 
                               
                             
                             ) 
                           
                         
                       
                        
                       
                          
                         
                           ϕ 
                            
                           
                               
                           
                            
                            
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0077]    The term 
         [0000]    
       
         
           
              
             
               
                 - 
                 
                   
                     2 
                      
                      
                      
                     
                         
                     
                      
                     π 
                      
                     
                         
                     
                      
                     x 
                   
                   N 
                 
               
                
               
                 ( 
                 
                   k 
                   + 
                   
                     k 
                     ′ 
                   
                 
                 ) 
               
             
           
         
       
     
         [0000]    is equal to 1 for k=−k′, respectively the term 
         [0000]    
       
         
           
              
             
               
                 
                   2 
                    
                    
                    
                   
                       
                   
                    
                   π 
                    
                   
                       
                   
                    
                   x 
                 
                 N 
               
                
               
                 ( 
                 
                   k 
                   - 
                   
                     k 
                     ′ 
                   
                 
                 ) 
               
             
           
         
       
     
         [0000]    is equal to 1 for 
         [0000]    
       
         
           
             k 
             = 
             
               
                 
                   k 
                   ′ 
                 
                 . 
                 
                   Σ 
                   
                     x 
                     = 
                     0 
                   
                   
                     N 
                     - 
                     1 
                   
                 
               
                
               
                  
                 
                   
                     - 
                     
                       
                         2 
                          
                          
                          
                         
                             
                         
                          
                         π 
                          
                         
                             
                         
                          
                         x 
                       
                       N 
                     
                   
                    
                   
                     ( 
                     
                       k 
                       + 
                       
                         k 
                         ′ 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    is equal to 0 if k#−k′, which corresponds to the well-known continuous Fourier transform of the cosine function made of two Dirac functions : 
         [0000]    
       
         
           
             
               
                 
                   
                     v 
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         N 
                         2 
                       
                        
                       
                         δ 
                          
                         
                           ( 
                           
                             - 
                             
                               k 
                               ′ 
                             
                           
                           ) 
                         
                       
                        
                       
                          
                         
                           - 
                           ϕ 
                         
                       
                     
                     + 
                     
                       
                         N 
                         2 
                       
                        
                       
                         δ 
                          
                         
                           ( 
                           
                             k 
                             ′ 
                           
                           ) 
                         
                       
                        
                       
                          
                         
                           + 
                           ϕ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0078]    In norm, ∥v(k′)∥=∥v(k)∥=N/2. The positions of the Dirac function in the Fourier spectra allows deducing the period λ with no bias. 
         [0079]    If N/λ=k′+ε, where k′ is an integer number and ε ∈ [0,1[ is a real number, the discrete Fourier transform becomes much more complex: 
         [0000]    
       
         
           
             
               
                 
                   
                     v 
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           x 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                           1 
                           2 
                         
                          
                         
                            
                           
                             
                               - 
                               
                                 
                                   2 
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   x 
                                 
                                 N 
                               
                             
                              
                             
                               ( 
                               
                                 k 
                                 + 
                                 
                                   k 
                                   ′ 
                                 
                                 + 
                                 ɛ 
                               
                               ) 
                             
                           
                         
                          
                         
                            
                           
                             - 
                             ϕ 
                           
                         
                       
                     
                     + 
                     
                       
                         1 
                         2 
                       
                        
                       
                          
                         
                           
                             - 
                             
                               
                                 2 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 x 
                               
                               N 
                             
                           
                            
                           
                             ( 
                             
                               k 
                               - 
                               
                                 k 
                                 ′ 
                               
                               - 
                               ɛ 
                             
                             ) 
                           
                         
                       
                        
                       
                          
                         
                           - 
                           ϕ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0080]    In this case, the discrete Fourier transform of the cosine function is not equal to two Dirac functions located at ±k′ any more. Instead all the v(k) are non-null. Two maxima&#39;s are located at ±k′. Thus, v(±k′±1) is the second maximal value just after v(±k′) : 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       v 
                        
                       
                         ( 
                         
                           k 
                           ′ 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             x 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                           
                             1 
                             2 
                           
                            
                           
                              
                             
                               
                                 - 
                                 
                                   
                                     2 
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     x 
                                   
                                   N 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     
                                       k 
                                       ′ 
                                     
                                   
                                   + 
                                   ɛ 
                                 
                                 ) 
                               
                             
                           
                            
                           
                              
                             
                               - 
                               ϕ 
                             
                           
                         
                       
                       + 
                       
                         
                           1 
                           2 
                         
                          
                         
                            
                           
                             
                               - 
                               
                                 
                                   2 
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   x 
                                 
                                 N 
                               
                             
                              
                             
                               ( 
                               
                                 - 
                                 ɛ 
                               
                               ) 
                             
                           
                         
                          
                         
                            
                           
                             - 
                             ϕ 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       v 
                        
                       
                         ( 
                         
                           
                             k 
                             ′ 
                           
                           + 
                           1 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             x 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                           
                             1 
                             2 
                           
                            
                           
                              
                             
                               
                                 - 
                                 
                                   
                                     2 
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     x 
                                   
                                   N 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     
                                       k 
                                       ′ 
                                     
                                   
                                   + 
                                   1 
                                   + 
                                   ɛ 
                                 
                                 ) 
                               
                             
                           
                            
                           
                              
                             
                               - 
                               ϕ 
                             
                           
                         
                       
                       + 
                       
                         
                           1 
                           2 
                         
                          
                         
                            
                           
                             
                               - 
                               
                                 
                                   2 
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   x 
                                 
                                 N 
                               
                             
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 ɛ 
                               
                               ) 
                             
                           
                         
                          
                         
                            
                           
                             - 
                             ϕ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0081]    It is interesting to evaluate ∈ and φ only using the values of v(k′) and v(k′+1) using the following formula: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ɛ 
                       ≈ 
                       
                         ɛ 
                         ^ 
                       
                     
                     = 
                     
                       
                          
                         
                           v 
                            
                           
                             ( 
                             
                               
                                 k 
                                 ′ 
                               
                               + 
                               1 
                             
                             ) 
                           
                         
                          
                       
                       
                         
                            
                           
                             v 
                              
                             
                               ( 
                               
                                 
                                   k 
                                   ′ 
                                 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                            
                         
                         + 
                         
                            
                           
                             v 
                              
                             
                               ( 
                               
                                 k 
                                 ′ 
                               
                               ) 
                             
                           
                            
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       ϕ 
                       ≈ 
                       
                         ϕ 
                         ^ 
                       
                     
                     = 
                     
                       
                         arg 
                          
                         
                           ( 
                           
                             v 
                              
                             
                               ( 
                               
                                 k 
                                 ′ 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                       - 
                       
                         π 
                          
                         
                           ɛ 
                           ^ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0082]    In order to demonstrate equation (12) one considers a geometrical representation of 
         [0000]    
       
         
           
             A 
             = 
             
               
                 Σ 
                 
                   x 
                   = 
                   0 
                 
                 
                   N 
                   - 
                   1 
                 
               
                
               
                 1 
                 2 
               
                
               
                  
                 
                   
                     - 
                     
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         x 
                       
                       N 
                     
                   
                    
                   
                     ( 
                     
                       - 
                       ɛ 
                     
                     ) 
                   
                 
               
                
               
                 
                    
                   ϕ 
                 
                 . 
               
             
           
         
       
     
         [0000]    This sum corresponds to N/2 times the barycenter of N points located on a unitary circle regularly spaced between the angles [φ, φ+2πε[ represented on  FIG. 6 . This sum is almost equal to the centroid of the arc of circle [1]. Thus : 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                            
                           A 
                            
                         
                         = 
                         
                            
                           
                             
                               ∑ 
                               
                                 x 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 1 
                                 2 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         x 
                                       
                                       N 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       - 
                                       ɛ 
                                     
                                     ) 
                                   
                                 
                               
                                
                               
                                  
                                 ϕ 
                               
                             
                           
                            
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                            
                           
                             
                               N 
                               2 
                             
                              
                             
                               
                                 sin 
                                  
                                 
                                   ( 
                                   πɛ 
                                   ) 
                                 
                               
                               πɛ 
                             
                              
                             
                                
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     πɛ 
                                     + 
                                     ϕ 
                                   
                                   ) 
                                 
                               
                             
                           
                            
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             N 
                             2 
                           
                            
                           
                             
                               sin 
                                
                               
                                 ( 
                                 πɛ 
                                 ) 
                               
                             
                             πɛ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0083]    The arc of circle of length 2πε and the arc of circle of length 2π(1−ε) are represented on  FIG. 6 . The two points A and B illustrate the centroid of the two arcs of circle respectively the arc of circle of length 2πε and the arc of circle of length 2π(1−ε) which lie on the same line of orientation φ+πε. 
         [0084]    It is now possible to demonstrate the 
         [0000]    
       
         
           
             
                
               B 
                
             
             
               
                  
                 A 
                  
               
               + 
               
                  
                 B 
                  
               
             
           
         
       
     
         [0000]    is equal to ε considering that sin(π(1−ε))=sin(πε) : 
         [0000]    
       
         
           
             
               
                 
                   
                     
                        
                       B 
                        
                     
                     
                       
                          
                         A 
                          
                       
                       + 
                       
                          
                         B 
                          
                       
                     
                   
                   = 
                   
                     
                       
                         
                           N 
                           2 
                         
                          
                         
                           
                             sin 
                              
                             
                               ( 
                               
                                 π 
                                  
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     ɛ 
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                           
                             π 
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 ɛ 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             N 
                             2 
                           
                            
                           
                             
                               sin 
                                
                               
                                 ( 
                                 πɛ 
                                 ) 
                               
                             
                             πɛ 
                           
                         
                         + 
                         
                           
                             N 
                             2 
                           
                            
                           
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   π 
                                    
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       ɛ 
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                             
                               π 
                                
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   ɛ 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     = 
                     ɛ 
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0085]    Also the phase of A allows computing φ: 
         [0000]      φ=arg(A)−πε  (15)
 
         [0086]    To finalize the demonstration, let us demonstrate that V(k′)=r(θ)+A and V(k′+1)=r(1)+B where 
         [0000]    
       
         
           
             
               r 
                
               
                 ( 
                 a 
                 ) 
               
             
             = 
             
               
                 
                   Σ 
                   
                     x 
                     = 
                     0 
                   
                   
                     N 
                     - 
                     1 
                   
                 
                  
                 
                   1 
                   2 
                 
                  
                 
                    
                   
                     
                       - 
                       
                         
                           2 
                            
                           π 
                            
                           
                               
                           
                            
                           x 
                         
                         N 
                       
                     
                      
                     
                       ( 
                       
                         
                           2 
                            
                           
                             k 
                             ′ 
                           
                         
                         + 
                         a 
                         + 
                         ɛ 
                       
                       ) 
                     
                   
                 
                  
                 
                    
                   
                     - 
                     ϕ 
                   
                 
               
               ≈ 
               0 
             
           
         
       
     
         [0000]    is negligible with α=0 or α=1. r (α) is equal to 
         [0000]    
       
         
           
             N 
             2 
           
         
       
     
         [0000]    times the barycenter of N points located on the unitary circle between the angles [−φ, −φ−4πk′−α−ε[. The corresponding arc of circle is equal to 2k′ complete circles plus α+E arc of circle. The complete centroid is a barycentre between the circle centre weighted by 2k′ and the centroid of an arc of circle of length TLE weighted by ε. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       r 
                        
                       
                         ( 
                         a 
                         ) 
                       
                     
                     ≈ 
                     
                       
                         ɛ 
                         
                           2 
                            
                           
                             k 
                             ′ 
                           
                         
                       
                        
                       
                         
                           sin 
                            
                           
                             ( 
                             πɛ 
                             ) 
                           
                         
                         πɛ 
                       
                     
                     ≈ 
                     
                       
                         sin 
                          
                         
                           ( 
                           πε 
                           ) 
                         
                       
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         
                           k 
                           ′ 
                         
                       
                     
                     &lt; 
                     
                       1 
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         
                           k 
                           ′ 
                         
                       
                     
                   
                   = 
                   
                     
                       1 
                       
                         2 
                          
                         π 
                       
                     
                      
                     
                       λ 
                       N 
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0087]    Thus when computing E from V(k′) and V(k′+1) the maximum error is in the order r. For a complete study, the term r should be considered with the angle φ versus the values A and B. experimentations shows that the errors between E and its approximation E depends on φ. In practice for cosine function with λ≈15 and a discrete Fourier of N=1024 values, the approximation of λ is in the order of 1/500 of a pixel. 
         [0088]    Back to  FIG. 4 , in a step  405 , knowing the position (u n , v n )=(u+ε u , v+ε v ) of the peak  501 , the processor  301  computes the polar distance (ρ n , θ n ) of the peak  501  to the centre of the discrete Fourier transform of the captured image: 
         [0000]    
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         ρ 
                         n 
                       
                       , 
                       
                         θ 
                         n 
                       
                     
                     ) 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   u 
                                   n 
                                 
                                 - 
                                 
                                   
                                     N 
                                     u 
                                   
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   u 
                                   n 
                                 
                                 - 
                                 
                                   
                                     N 
                                     v 
                                   
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                       , 
                       
                         atan 
                          
                         
                           ( 
                           
                             
                               ( 
                               
                                 
                                   v 
                                   n 
                                 
                                 - 
                                 
                                   
                                     N 
                                     v 
                                   
                                   2 
                                 
                               
                               ) 
                             
                             / 
                             
                               ( 
                               
                                 
                                   u 
                                   n 
                                 
                                 - 
                                 
                                   
                                     N 
                                     u 
                                   
                                   2 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
         [0089]    The phase φ n =F p (u, v)−π*(ε u +ε v ) of the peak  501  is also computed by the processor  301 . 
         [0090]    Thus, N Fourier peaks  500 ,  501  have been extracted by the processor  301 , each of them being characterized by the following parameters ρ n , θ n , φ n  with n ∈ [0, N[. The N peaks  500 ,  501  are then sorted in a list, by the processor  301 , by increasing polar distance ρ n  in a step  406 . The first peak in the list has a polar distance Σ 0 =0 as it corresponds to the central peak  500 . By sorting the peaks by polar distances, it is easy to isolate the peaks  501  which correspond to the first harmonic frequency. 
         [0091]    Within the list of sorted Fourier peaks, the L=4 first peaks  501  when the pattern of micro-lens array  102  is a square pattern, or the L=6 first peaks when the pattern of micro-lens array  102  is an hexagonal pattern have the same polar distance p within typically 1/N u  Fourier pixel. 
         [0092]    In a step  407 , the processor  301  computes the values of the parameters D and θ by averaging the L first ρ i  and θ i  of the L first peaks  501 : 
         [0000]    
       
         
           
             
               
                 
                   D 
                   = 
                   
                     
                       L 
                        
                       
                           
                       
                        
                       
                         N 
                         u 
                       
                     
                     
                       
                         Σ 
                         
                           i 
                           = 
                           1 
                         
                         
                           i 
                           = 
                           L 
                         
                       
                        
                       
                         ρ 
                         i 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
             
               
                 
                   θ 
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       
                         i 
                         = 
                         L 
                       
                     
                      
                     
                       mod 
                        
                       
                         ( 
                         
                           
                             θ 
                             i 
                           
                           , 
                           
                             
                               2 
                                
                               π 
                             
                             L 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where D represents the pitch of the array of micro-lenses which corresponds to the distance between three consecutive micro-lens  104  and θ represents the rotation of the array of micro-lenses  102  in relation to the sensor  101 . 
         [0093]    In a step  408 , the processor  301  derives the L/2 cosine functions which are characterized by the parameters ρ i , θ i ,φ i , from the L first peaks  501 . 
         [0094]    The L first peaks  501  define L/2 cosine functions. In the direct space, each cosine function defines a corrugated sheet as illustrated in  FIG. 7 . The sum of this L/2 cosine functions defines approximatively the pattern of the micro-lens array  102 . Using the phase φ i  of the L peaks  501  it is possible for the processor  301  to compute the position of one micro-images (x 0,0 , y 0,0 ) relatively to the sensor  101  of the plenoptic camera  100  in a step  409 . The position of one micro-images (x 0,0 , y 0,0 ) relatively to the sensor  101  corresponds to the position in the captured image where the sum of the L/2 cosine functions is maximal. 
         [0095]    The parameter θ i , estimated by the processor  301  from the positions of the peaks  501 , indicate the orientations of the cosine functions in the captured image as represented by the arrows  700  on  FIG. 7 . The intersection of the perpendicular directions θ i +π/2 as represented by the arrows  701  gives the position of the centres of the micro-images produced by the micro-lenses  104 . 
         [0096]    The position (x 0,0 , y 0,0 ) of the micro-image is estimated as the intersection of 2 arrows  601 , and is computed by the processor  301  by using least square estimation. 
         [0000]    
       
         
           
             
               
                 Let 
                  
                 
                     
                 
                  
                 A 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               θ 
                               1 
                             
                             + 
                             
                               π 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         - 
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 θ 
                                 1 
                               
                               + 
                               
                                 π 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               θ 
                               2 
                             
                             + 
                             
                               π 
                               2 
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         - 
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 θ 
                                 2 
                               
                               + 
                               
                                 π 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               
                 and 
                  
                 
                     
                 
                  
                 B 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         D 
                          
                         
                             
                         
                          
                         
                           ϕ 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         D 
                          
                         
                             
                         
                          
                         
                           ϕ 
                           2 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
         [0000]    being 2 matrices defined by the two first cosine function parameters. The result of the computation of a least square estimation by the processor  301  gives : 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             x 
                             
                               0 
                               , 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             y 
                             
                               0 
                               , 
                               0 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             A 
                             T 
                           
                            
                           A 
                         
                         ) 
                       
                       
                         - 
                         1 
                       
                     
                      
                     
                       A 
                       T 
                     
                      
                     B 
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0097]    The four parameters D,θ,x 0,0 ,x 0,0 ,y 0,0  are thus fully computed from the position in the discrete Fourier transform of the captured image of the peaks  501 . 
         [0098]    Although the present invention has been described hereinabove with reference to specific embodiments, the present invention is not limited to the specific embodiments, and modifications will be apparent to a skilled person in the art which lie within the scope of the present invention. 
         [0099]    Such a method for computing an estimate position of a micro-image produced by a micro-lens  104  of the array of micro-lenses  102  of the plenoptic camera is robust and does not require the use of a white image in order to obtain the position of the micro-image. Thus, the method according to an embodiment of the invention can be used to monitor the position of the micro-image dynamically, e.g. on a plenoptic video or in the case of a plenoptic camera mounted with a zoom or interchangeable lenses. 
         [0100]    Many further modifications and variations will suggest themselves to those versed in the art upon making reference to the foregoing illustrative embodiments, which are given by way of example only and which are not intended to limit the scope of the invention, that being determined solely by the appended claims. In particular the different features from different embodiments may be interchanged, where appropriate.