CN104320598A - Aberration-free image reconstruction method based on CCD array pixel response function frequency domain calibration - Google Patents

Abstract

The invention discloses an aberration-free image reconstruction method based on CCD array pixel response function frequency domain calibration, and belongs to the technical field of detectors. A frequency domain model of pixel response functions is established, same-frequency lasers with high stability are utilized for generating sinusoidal interference fringes in a far field, fringe light fields at different spatial frequencies are acquired by changing the relative positions of the same-frequency lasers, frequency domain calibration is conducted on the pixel response functions, calibration coefficients in all orders of all the pixel response functions are acquired respectively, on the basis and in combination with a Fourier optics method, CCD imaging reconstruction is achieved, and aberration-free imaging of a CCD detector can be achieved. According to the method, the pixel frequency domain response functions can be calibrated, meanwhile, tiny offset of the relative positions of all pixels can be calibrated, reconstructed images are free of sample blurs, modulation transfer functions (MTF) approach 1, the contrast ratio is not decreased, and the aberration-free image reconstruction method has the advantages of being high in calibration accuracy and wide in application prospect.

Description

A kind of aberration free image reconstructing method demarcated based on ccd array pixel response function frequency domain

Technical field

The invention belongs to optical metrology detector calibration technique field, relate to the reconstruction in frequency-domain method of the detector aberrationless imaging that a kind of frequency domain based on pixel response function is demarcated.

Background technology

The appearance of photonics image detection device array, no matter be convenient to digitlization with it and store, transmit and process, be almost instead of film completely on army uses or be civilian.Charge coupled device (Charge Coupled Device, CCD) is one of the most frequently used detector array, and it can be converted into discrete digital signal optical image, and its minimum working cell is pixel.Along with the development of very lagre scale integrated circuit (VLSIC) technique, the technical indicator such as resolving power, sensitivity, stability of CCD has increased significantly, and has been widely used in the numerous areas such as industrial detection, machine vision, aerospace remote sensing, lll night vision, Imaging Guidance, digital hologram, automatic monitoring.

The rdaiation response characteristic of CCD reflect input light distribution and output gray level distribute between relation, this characteristic is directly connected to the precision of respective measurement system, rdaiation response performance parameter mainly comprises Photo-Response Non-Uniformity, nonlinearity, dark noise etc. between rdaiation response degree, pixel, be set up the distribution of CCD output gray level and the relativeness of corresponding input light distribution to its object of demarcating, the Energy distribution that the image for CCD collection is transformed to incident light provides inverting reference.Through years development, CCD scaling method in succession occurs and updates, and mainly contains bulk absorption method, wedge optical spectroscopy, double slit Fraunhofer diffraction method, pinhole Fraunhofer diffraction method etc.The least unit that these methods are considered is a pixel, namely simply think that different parts in a pixel is consistent for the response of light, what demarcate is the average response of single pixel, does not consider the crosstalk between the response distribution of pixel inside and neighbor.Due to the restriction of Pixel Dimensions, even desirable ccd array also can cause the decline (MTF<1) of the modulation contrast of imaging, affect image quality.And the device of reality is due to reasons such as the inhomogeneities of design feature, technique and the crosstalks of neighbor, in pixel, diverse location is inconsistent to the response of light, and the distribution of Pixel Dimensions, grid also exists small difference.Ignore these factors and will affect correctness and the authenticity of observed result that imaging data analyzes.

Usual pixel response function (PRF) characterizes the photosensitive unit at diverse location place in pixel and exports the response of incident light, and its response being defined as point source illumination some pixels when at array diverse location place of unit strength exports.Research shows, pixel center response is higher than pixel edge, and has certain crosstalk between the pixel of adjoining.The more theoretical modeling calculating of the PRF for CCD and the work of experiment measuring has just been carried out abroad as far back as latter stage in 20th century.By the PRF of measurement pixel that the different pixels position scanning CCD close to the focal beam spot (submicron-scale) of diffraction limit can be discrete, due to the restriction by Focused Optical system diffraction limit, the size of focal beam spot can not be infinitely small, therefore its resolution has certain restriction, and the scanning area of the method can only reach several pixel, measuring range is little, speed is slow, is unsuitable for practical engineering application.

Summary of the invention

The problem that technology of the present invention solves is: the limitation overcoming traditional calibration technology, a kind of aberration free image reconstructing method demarcated based on ccd array pixel response function frequency domain is provided, pixel response function is demarcated in domain space, between consideration pixel while Photo-Response Non-Uniformity, emphasis considers the crosstalk between the Photo-Response Non-Uniformity of single pixel inside and pixel, the little deviation of the geometric position of pixel can be calibrated simultaneously, the imaging reconstruction in frequency-domain algorithm proposed on this basis, the distribution of light intensity distribution on CCD photosurface is incided in the reproduction of energy almost Perfect, make the modulation transfer function of detector close to 1, realize aberrationless imaging.

Technical solution of the present invention is:

Based on the aberration free image reconstructing method that ccd array pixel response function frequency domain is demarcated, step is as follows:

(1) detector array pixel response function frequency domain calibration system is built.Use that coherence is good, laser that power stage is stable, and in free space, in waveguiding structure or in optical fiber laser beam in single is divided into some bundles; Phase-detecting, phase shifting equipment are set in these laser beams, for detecting and the phase place of mobile laser beam; In these laser beams, select any two-way laser successively, they are interfered with each other, form interference field.Place target CCD camera undetermined and Standard Ratio degree meter at interference field plane place, regulate the photosurface of CCD plane and Standard Ratio degree meter coplanar, and be connected with computer by data wire and control line;

(2) data acquisition under fixed space frequency.Concrete process is as follows:

(2.1) open laser, regulate laser output power to be arranged on setting, select two-way outgoing laser beam, now corresponding spatial frequency is designated as (k _x1, k _y1), complete after Output of laser pattern and power stability until laser preheating, Standard Ratio degree note is moved into optical path, individually blocks two bundle shoot lasers, under the control of the computer the reading I of acquisition and recording Standard Ratio degree note ₁₁, I ₂₁; Removal is blocked, CCD to be calibrated is moved into optical path, the light intensity distributions of then now incident interference field is such as formula shown in (1), (x, y) be the spatial coordinate location on CCD, the time for exposure and the yield value that arrange CCD are setting, gather one group of image under the control of the computer and average, preserve this mean value view data, be designated as I _mn1for the pixel output of now (m, n) pixel;

(2.2) change the jack passage of optical fiber, repeat step (2.1), difference (k can be obtained _xj, k _yj) under I _1j, I _2j, I _mnj, wherein j=1,2,3 ... J, J represent the total number of combinations of the jack passage of optical fiber;

(3) nonlinear least square fitting demarcates each rank coefficient of pixel response function.The frequency-domain expression of pixel response function such as formula shown in (2), x in formula _mn, y _mnfor the theoretical center point coordinates of pixel (m, n), Δ x _m, Δ y _mfor the side-play amount of pixel virtual center point and theoretical center point, q _mn0, q _mn1, q _mn2... be each rank coefficient of pixel response function, actual level of getting is secondary more, precision higher;

${\tilde{R}}_{\mathrm{mn}}(k_x,k_y)=q_{\mathrm{mn}0}{e}^{i[k_x(x_{\mathrm{mn}}+\mathrm{\&Delta;}{x}_{\mathrm{mn}})+k_y(y_{\mathrm{mn}}+\mathrm{\&Delta;}{y}_{\mathrm{mn}})]}(1+q_{\mathrm{mn}1}{k_x}^2+q_{\mathrm{mn}2}{k_y}^2+q_{\mathrm{mn}3}{k}_x{k}_y+...)---\left(2\right)$

Pixel response output valve of the present invention meets the expression formula of formula (3), the data (2) step obtained substitute into formula (3), obtain one group of equation group be made up of J nonlinear equation, for ensureing that equation group has solution, calibration coefficient Δ x _m, Δ y _m, q _mn0, q _mn1, q _mn2... q _mnLnumber must be less than or equal to equation number, namely require J>=L+3, can calibration coefficient be obtained by nonlinear least square fitting;

(4) pair array scale is the CCD of N × N, based on the structure of pixel grid, uses the discrete frequency form shown in formula (4), discrete steps wherein p is Pixel Dimensions;

$\begin{array}{c}(k_g,k_h)=\frac{2\mathrm{\&pi;}}{N\&CenterDot;p}(g,h),\\ g=-N/2,-N/2+1,...,N/2-1;h=-N/2,-N/2+1,...,N/2-1\end{array}---\left(4\right)$

To the light distribution S (x, y) of arbitrary incident field, the pixel output I' of any pixel (m, n) _mnrepresent with the discrete form of frequency domain formula (5) Suo Shi, wherein for the Fourier transform value of incident field, demarcated by frequency domain and obtained, solving the system of linear equations be made up of N × N number of linear equation can try to achieve

$I_{\mathrm{mn}}^{\&prime;}=\underset{g}{\mathrm{\&Sigma;}}\underset{h}{\mathrm{\&Sigma;}}\tilde{S}(k_g,k_h)\&CenterDot;{\tilde{R}}_{\mathrm{mn}}(k_g,k_h)\&CenterDot;\mathrm{\&Delta;}{k}_x\&CenterDot;\mathrm{\&Delta;}{k}_y---\left(5\right)$

(5) the light distribution S (x, y) of incident field can by right inverse Fourier transform obtain, by what try to achieve in step (4) substitute into the discrete expression shown in formula (6), the intensity distributions of the incident field of reconstruct can be tried to achieve;

$S(x,y)=\underset{g}{\mathrm{\&Sigma;}}\underset{h}{\mathrm{\&Sigma;}}\tilde{S}(k_g,k_h)e^{i(k_{g}x+k_{h}y)}\&CenterDot;\mathrm{\&Delta;}{k}_x\&CenterDot;\mathrm{\&Delta;}{k}_y---\left(6\right)$

The principle of the invention: the inventive method constructs the frequency-domain model of pixel response function, the same frequency laser of high stable is utilized to produce sinusoidal interference striped in far field, by the striped light field changed with the relative position acquisition different space frequency of frequency laser, frequency domain demarcation is carried out to pixel response function, ask for each rank calibration coefficient of each pixel response function respectively, on this basis in conjunction with the method for Fourier Optics, propose a kind of restructing algorithm of CCD imaging, the aberrationless imaging of ccd detector can be realized.In the imaging process of CCD, due to respond inside and outside detector pixel inhomogeneities, CCD specification the factor such as not accuracy, the digital map image contrast of acquisition is compared theoretical value and is significantly declined, and corresponding transfer function is significantly less than 1.At present, people extensively adopt the response of the method for flat field correction to detector to calibrate, and namely by measurement detector array, the difference that uniform light field responds are demarcated to the response error of each pixel.The least unit that this method is demarcated is a pixel, can only calibrate the average response of whole pixel.Compare flat field correction method, method involved in the present invention then can calibrate the response distribution function of pixel inside on frequency domain, the minor shifts of each pixel relative position can also be calibrated simultaneously, the image of reconstruct is fuzzy without sampling, modulation transfer function (MTF) levels off to 1, contrast, without decline, has calibration precision high, the feature that application prospect is wide.

The present invention compared with prior art beneficial effect is:

(1) the present invention introduces the concept of pixel response function, and to characterize the crosstalk between the Photo-Response Non-Uniformity of single pixel inside and neighbor, the least unit of correcting than Photo-Response Non-Uniformity between traditional pixel is a pixel, and this model is more accurate.

(2) the invention provides a kind of method utilizing double excitation interference fringe to carry out the demarcation of pixel response function frequency domain, the frequency domain approximate expression of pixel response function can be calibrated, the little deviation of the geometric position of pixel can be calibrated simultaneously, more traditional flat field antidote, calibrating parameters of the present invention more horn of plenty, calibration result is more accurate.

(3) the present invention proposes a kind of frequency domain image reconstruction algorithm utilizing the pixel response function of frequency domain and the response output image of ccd array, the distribution of light intensity distribution on CCD photosurface is incided by the reproduction of the method energy almost Perfect, make the modulation transfer function of detector close to 1, realize the aberrationless imaging of CCD, thus reduce the aberration and image degradation that cause due to CCD in optical instrument optical train, improve the authenticity of observed result.

Accompanying drawing explanation

Fig. 1 is ccd detector pixelated array response function frequency domain calibration system composition diagram;

Fig. 2 is the double excitation interference field schematic diagram demarcated for frequency domain;

Fig. 3 be demarcate normalized flat field compensation coefficient figure, X of obtaining to and the position deflection graph of Y-direction;

Fig. 4 illustrates the contrast of distribution of light intensity distributed image, CCD response output image and the reconstruction in frequency-domain image inciding CCD photosurface.

Embodiment

Below in conjunction with accompanying drawing and preferred embodiment, the present invention is described in further detail.

The present invention is implemented as follows:

Step 1: build detector array pixel response function frequency domain calibration system

The composition diagram of this scaling system as shown in Figure 1, is made up of the laser 1 of high stable, coupling fiber and divided beam system 2, optical fiber 3, multichannel optical fiber orifice plate 4, Standard Ratio degree meter 6, one-dimensional movement platform 7 and computer 8.Laser wavelength adopts 632.8nm laser, laser coupled is entered optical fiber and is divided into two bundles by coupling fiber and beam splitter, multichannel optical fiber orifice plate is used for fixed fiber outgoing head, can be approximately plane wave from two spherical waves of fiber exit within the scope of the small size of far field, produces and interferes.For ensureing stability and the repeatability of interference field, require that the bright dipping of laser has high stability, the optical fiber position of mounting hole simultaneously on orifice plate is stablized, and requires that the thermal coefficient of expansion of orifice plate material is low.The number of jack passage is relevant to the number of the coefficient needing the frequency domain pixels response function demarcated, and such as, frequency domain pixels response function gets its quadratic term coefficient, then need the coefficient demarcated always to have 6 (Δ x _m, Δ y _m, q _mn0, q _mn1, q _mn2, q _mn3), the interference fringe field being no less than 6 groups need be projected, at least need 4 passages at far-field position place, one dimension translation stage is installed, Standard Ratio degree meter and CCD to be calibrated are fixed on translation stage, regulate the position of three, make the photosurface of CCD and Standard Ratio degree meter coplanar, be connected with computer by data wire and control line, computer is for the data acquisition of the movement and CCD and Standard Ratio degree meter that control one dimension translation stage.

In the embodiment of the present invention, laser wavelength adopts 632.8nm laser, and demarcating other wavelength only need with changing corresponding laser, and the present invention stands good.

In the embodiment of the present invention, the CCD scope of demarcation gets the pel array of central area 30 × 30, and change the scale of demarcating CCD pixel, the present invention stands good.

In the embodiment of the present invention, laser coupled is entered optical fiber and is divided into two bundles by coupling fiber and beam splitter, is beamed into the multiple laser consistent with jack port number, is equipped with corresponding optical switch for easy to operate.

Step 2: the data acquisition under fixed space frequency

(2.1) open laser, regulate laser output power to be arranged on setting, select two passages on orifice plate to insert optical fiber, now corresponding spatial frequency is designated as (k _x1, k _y1), complete after Output of laser pattern and power stability until laser preheating, computer control translation stage, Standard Ratio degree note is moved into optical path, individually block two fiber exit ports, two uniform light fields will be obtained, the reading I of acquisition and recording Standard Ratio degree note under computer control in Standard Ratio degree meter photosurface position ₁₁, I ₂₁; Remove blocking optical fiber, CCD to be calibrated is moved into optical path, then the light intensity distributions of the interference field of now CCD photosurface place incidence is such as formula shown in (1):

Here requirement amplitude within the dynamic range of CCD, cross strong or cross weak, needing the power output readjusting laser.

The time of integration and the yield value that arrange CCD are setting, gather one group of image and average, preserve this mean value view data, be designated as I under computer control _mn1(m=1,2 ..., 30; N=1,2 ..., 30), i.e. the pixel output of (m, n) pixel.

(2.2) change the jack passage of optical fiber, repeat step (2.1), difference (k can be obtained _xj, k _yj) under I _1j, I _2j, I _mnj, wherein j=1,2,3 ... J, J represent the total number of combinations of the jack passage of optical fiber, and the port number used in this present embodiment is 5, then most multipotency demarcates cubic term coefficient, and Fig. 2 is this 10 groups of interference field schematic diagrames.

Step 3: nonlinear least square fitting demarcates each rank coefficient of pixel response function.The frequency-domain expression of pixel response function is such as formula shown in (2):

${\tilde{R}}_{\mathrm{mn}}(k_x,k_y)=q_{\mathrm{mn}0}{e}^{i[k_x(x_{\mathrm{mn}}+\mathrm{\&Delta;}{x}_{\mathrm{mn}})+k_y(y_{\mathrm{mn}}+\mathrm{\&Delta;}{y}_{\mathrm{mn}})]}(1+q_{\mathrm{mn}1}{k_x}^2+q_{\mathrm{mn}2}{k_y}^2+q_{\mathrm{mn}3}{k}_x{k}_y+...)---\left(2\right)$

Pixel response output valve of the present invention meets the expression formula of formula (3):

Data step 3 obtained substitute into formula (3), obtain one group of equation group be made up of J nonlinear equation, J=10 in the present embodiment, demarcate cubic term coefficient, by carrying out a nonlinear least square fitting to each pixel, the calibration coefficient of each pixel can be obtained, totally 30 × 30=900 pixel, 9000 coefficients, Fig. 3 is that the present embodiment demarcates first three the coefficient results figure obtained.

Step 4: in the present embodiment, pair array scale is the CCD of 30 × 30, and based on the structure of pixel grid, use the discrete frequency form shown in formula (4), in formula, p is Pixel Dimensions:

$(k_g,k_h)=\frac{2\mathrm{\&pi;}}{30\&CenterDot;p}(g,h),g=-15,-14,...,14;h=-15,-14,...,14---\left(4\right)$

Step 5: for the incident field intensity distributions S of the unknown on CCD photosurface ₀(x, y), under the time of integration and gain condition of regulation, CCD camera will export piece image I, and the pixel output of (m, n) pixel is I' _mn, shown in available formula (5), the discrete form of frequency domain represents:

$I_{\mathrm{mn}}^{\&prime;}=\underset{g}{\mathrm{\&Sigma;}}\underset{h}{\mathrm{\&Sigma;}}\tilde{S}(k_g,k_h)\&CenterDot;{\tilde{R}}_{\mathrm{mn}}(k_g,k_h)\&CenterDot;\mathrm{\&Delta;}{k}_x\&CenterDot;\mathrm{\&Delta;}{k}_y---\left(5\right)$

In the embodiment of the present invention, solve the system of linear equations be made up of 30 × 30 linear equations can try to achieve $\tilde{S}(k_g,k_h).$

Step 6: by what try to achieve in step 5 substitute into the discrete expression shown in formula (6), the intensity distributions of the incident field of reconstruct can be tried to achieve:

$S(x,y)=\underset{g}{\mathrm{\&Sigma;}}\underset{h}{\mathrm{\&Sigma;}}\tilde{S}(k_g,k_h)e^{i(k_{g}x+k_{h}y)}\&CenterDot;\mathrm{\&Delta;}{k}_x\&CenterDot;\mathrm{\&Delta;}{k}_y---\left(6\right)$

Fig. 4 illustrates in the embodiment of the present invention, incides the distribution of light intensity distribution S of CCD photosurface ₀(x, y), CCD respond output image I, and distribution of light intensity distribution S (x, y) that reconstruction in frequency-domain obtains.Reconstructed image eliminates the image blur of CCD discrete sampling, and contrast is improved, and has reproduced the intensity distributions of incident field.

Frequency domain of the present invention is demarcated and image reconstructing method, multi-wavelength, the calibration of pixel response function of multiple ccd array scale and the reprocessing of digital picture can be expanded to, the aberration and image degradation that cause due to CCD in the optical train of optical instrument can be reduced largely, improve the authenticity of observed result, in high accuracy imaging measurement (measurement as position and luminosity) field, there is good application prospect.

There is provided above embodiment to be only used to describe object of the present invention, and do not really want to limit the scope of the invention.Scope of the present invention is defined by the following claims.Do not depart from spirit of the present invention and principle and the various equivalent substitutions and modifications made, all should contain within the scope of the present invention.

Claims (1)

1., based on the aberration free image reconstructing method that ccd array pixel response function frequency domain is demarcated, it is characterized in that comprising the following steps:

(1) detector array pixel response function frequency domain calibration system is built

Use monochrome and the laser of frequency stabilization, stable output power, in free space, some bundles are divided into by beam splitter in waveguide or in optical fiber, in every Shu Jiguang, use phase-detecting, control phase device carries out phase-locked and phase shift, to produce the interference fringe light field or not beam splitting with different space frequency in space, the method for spatial phase modulation is adopted to produce interference field; Place target CCD camera undetermined and Standard Ratio degree meter at interference field plane place, regulate the photosurface of CCD plane and Standard Ratio degree meter coplanar, and be connected with computer by data wire and control line;

(2) data acquisition under fixed space frequency

(2.1) open laser, regulate laser output power to be arranged on setting, select two-way outgoing laser beam and interfere, now corresponding spatial frequency is designated as (k _x1, k _y1), complete after Output of laser pattern and power stability until laser preheating, Standard Ratio degree note is moved into optical path, individually blocks two fiber exit ports, under the control of the computer the reading I of acquisition and recording Standard Ratio degree note ₁₁, I ₂₁; Remove blocking optical fiber, CCD to be calibrated is moved into optical path, then the light intensity distributions of now incident interference field shown in (1), (x, y) is the spatial coordinate location on CCD, and the time for exposure and the yield value that arrange CCD are setting, gather one group of image under the control of the computer and average, preserving this mean value view data, be designated as I _mn1, i.e. the pixel output of (m, n) pixel;

In formula it is the phase difference of two bundle laser beams;

(2.2) select any two-way laser beam to interfere successively, repeat step (2.1), difference (k can be obtained _xj, k _yj) under I _1j, I _2j, I _mnj, wherein j=1,2,3 ... J, J represent total number of combinations of any two-way laser beam;

(3) nonlinear least square fitting demarcates each rank coefficient of pixel response function

The frequency-domain expression of pixel response function such as formula shown in (2), x in formula _mn, y _mnfor the theoretical center point coordinates of pixel (m, n), Δ x _m, Δ y _mfor the side-play amount of virtual center point and theoretical center point, q _mn0, q _mn1, q _mn2... be each rank coefficient of pixel response function, actual level of getting is secondary more, precision higher,

${\tilde{R}}_{\mathrm{mn}}(k_x,k_y)=q_{\mathrm{mn}0}{e}^{i[k_x(x_{\mathrm{mn}}+\mathrm{\&Delta;}{x}_{\mathrm{mn}})+k_y(y_{\mathrm{mn}}+\mathrm{\&Delta;}{y}_{\mathrm{mn}})]}(1+q_{\mathrm{mn}1}{k_x}^2+q_{\mathrm{mn}2}{k_y}^2+q_{\mathrm{mn}3}{k}_x{k}_y+...)---\left(2\right)$

Pixel response output valve I _mnmeet the expression formula of formula (3), the data (2) step obtained substitute into formula (3), obtain one group of equation group be made up of J nonlinear equation, for ensureing that equation group has solution, and calibration coefficient Δ x _m, Δ y _m, q _mn0, q _mn1, q _mn2... q _mnLnumber must be less than or equal to equation number, namely require J>=L+2, namely obtain L+2 calibration coefficient Δ x by nonlinear least square fitting _m, Δ y _m, q _mn0, q _mn1, q _mn2... q _mnL, thus the frequency-domain expression of pixel response function can be determined concrete form, provide known parameter, to be embodied as the reconstruction in frequency-domain of picture for calculating each rank frequency domain components of incident field in domain space;

(4) pair array scale is the CCD of N × N, based on the structure of pixel grid, uses the discrete frequency form shown in formula (4), (k _g, k _h) be got spatial frequency, discrete steps wherein p is Pixel Dimensions;

$(k_g,k_h)=\frac{2\mathrm{\&pi;}}{N\&CenterDot;p}(g,h),g=-N/2,-N/2+1,...,N/2-1;h=-N/2,-N/2+1,...,N/2-1---\left(4\right)$

To the light distribution S (x, y) of arbitrary incident field, the discrete picture that output one secondary N × N pixel forms by ccd array, the pixel output I' of any pixel (m, n) _mnfor known quantity, and represent with the discrete form of frequency domain formula (5) Suo Shi, solving the system of linear equations be made up of N × N number of linear equation can try to achieve

$I_{\mathrm{mn}}^{\&prime;}=\underset{g}{\mathrm{\&Sigma;}}\underset{h}{\mathrm{\&Sigma;}}\tilde{S}(k_g,k_h)\&CenterDot;{\tilde{R}}_{\mathrm{mn}}(k_g,k_h)\&CenterDot;\mathrm{\&Delta;}{k}_x\&CenterDot;\mathrm{\&Delta;}{k}_y---\left(5\right)$

(5) the light distribution S (x, y) of incident field is by right inverse Fourier transform obtain, by what try to achieve in step (4) substitute into the inverse discrete Fourier transform expression formula shown in formula (6), namely try to achieve the intensity distributions of the incident field of reconstruct:

$S(x,y)=\underset{g}{\mathrm{\&Sigma;}}\underset{h}{\mathrm{\&Sigma;}}\tilde{S}(k_g,k_h)e^{i(k_{g}x+k_{h}y)}\&CenterDot;\mathrm{\&Delta;}{k}_x\&CenterDot;\mathrm{\&Delta;}{k}_y---\left(6\right).$