CN104102006B

CN104102006B - Method for analyzing performance is transmitted based on the optical system frequency domain information improving Fourier transform

Info

Publication number: CN104102006B
Application number: CN201410351397.9A
Authority: CN
Inventors: 任智斌; 马驰; 郑烁; 曲荣召
Original assignee: Harbin Institute of Technology Shenzhen
Current assignee: Harbin Institute of Technology Shenzhen
Priority date: 2014-07-23
Filing date: 2014-07-23
Publication date: 2016-03-30
Anticipated expiration: 2034-07-23
Also published as: CN104102006A

Abstract

An improved Fourier transform-based method for analyzing the performance of optical system frequency domain information transmission belongs to the technical field of Fourier optics and imaging performance analysis of optical systems. The method is as follows: the first step, the expression of the improved Fourier transform and the solution of the harmonic coefficients of light intensity at all levels; the second step, the solution of the imaging integral equation of the optical system and its channel matrix in the frequency domain; the third step 1. Calculation of the information transmission performance parameters of the optical system, and evaluate the information transmission performance of the optical system according to these information parameters. The improved Fourier analysis method provided by the present invention expands the light intensity into a linear combination of zero frequency and non-negative light intensity harmonics with energy, analyzes the imaging law of the optical system under the new light intensity expansion method, and then applies the information theory Analytical methods to analyze the information transfer performance of optical systems. The invention improves the Fourier optics analysis method, makes the theory of Fourier optics more perfect, and realizes the combination of the optical analysis method and the information theory analysis method.

Description

Performance Analysis Method of Optical System Frequency Domain Information Transmission Based on Improved Fourier Transform

技术领域technical field

本发明属于傅里叶光学与光学系统成像性能分析技术领域，涉及一种基于改进傅里叶变换与信息论的光学系统成像性能分析方法。The invention belongs to the technical field of Fourier optics and optical system imaging performance analysis, and relates to an optical system imaging performance analysis method based on improved Fourier transform and information theory.

背景技术Background technique

调制传递函数(MTF)是光学系统设计、成像质量分析及光学检测领域常用的重要指标，其理论基础来源于《信息光学》、《傅里叶光学》等专著中关于非相干成像系统光学传递函数的相关理论。成像光学系统的MTF可将光强的点扩散函数作傅里叶变换后进行归一化得到，单纯从光学系统的角度来分析，这种方法在理论上是完全正确的。Modulation transfer function (MTF) is an important indicator commonly used in the field of optical system design, imaging quality analysis and optical detection. related theories. The MTF of the imaging optical system can be obtained by normalizing the point spread function of the light intensity after Fourier transform. It is completely correct to analyze it purely from the perspective of the optical system.

而在成像分析过程中，像的光强频谱的模可由物的光强频谱的模与MTF乘积获得。为此，需要将物的空域光强分布作傅里叶分析，将物的空域光强分布展开成不同空间频率的谐波的线性组合形式。但是，按照傅里叶分析方法展开的各级谐波均为不同频率的余弦与正弦函数，且各级谐波函数均含有负值。由于谐波函数值不满足非负性，各级频谱谐波无法独立表示光强(因为光强不能为负值)，只能叠加在零频分量上对零频分量进行细节的修正，与零频分量共同表示光强。这个问题的存在导致了傅里叶光学中利用MTF分析物、像光强频谱关系时，物、像各级频谱谐波不能独立表示光强，也不能表示信息，从而无法用信息论的分析方法研究光学系统的频域信息传递规律。In the imaging analysis process, the mode of the light intensity spectrum of the image can be obtained by multiplying the mode of the light intensity spectrum of the object and the MTF. For this reason, it is necessary to perform Fourier analysis on the spatial light intensity distribution of the object, and expand the spatial light intensity distribution of the object into a linear combination form of harmonics of different spatial frequencies. However, the harmonics of all levels expanded according to the Fourier analysis method are cosine and sine functions of different frequencies, and the harmonic functions of all levels contain negative values. Since the harmonic function value does not satisfy the non-negativity, the spectrum harmonics at all levels cannot independently represent the light intensity (because the light intensity cannot be negative), and can only be superimposed on the zero-frequency component to correct the zero-frequency component in detail. The frequency components collectively represent light intensity. The existence of this problem leads to the use of MTF in Fourier optics to analyze the light intensity spectrum relationship between objects and images. The spectral harmonics of all levels of objects and images cannot independently represent light intensity, nor can they represent information, so they cannot be studied by information theory analysis methods. Frequency domain information transmission law of optical system.

另外，在光学系统MTF的测量环节中，通常使用特定频率的余弦光强透过率光栅作为被测物体，通过检测该余弦光栅的像的调制度的变化来确定光学系统在该频率处的MTF值。该光栅透过的光强具有非负性，光栅像的光强分布同样具有非负性。但傅里叶变换中的各级谐波不满足非负性，使得光学系统MTF的检测实验中也存在与傅里叶光学理论不符的问题。In addition, in the measurement link of the MTF of the optical system, the cosine light intensity transmittance grating of a specific frequency is usually used as the measured object, and the MTF of the optical system at this frequency is determined by detecting the change of the modulation degree of the image of the cosine grating value. The light intensity transmitted by the grating is non-negative, and the light intensity distribution of the grating image is also non-negative. However, the harmonics of all levels in the Fourier transform do not satisfy the non-negativity, which makes the MTF detection experiment of the optical system inconsistent with the Fourier optical theory.

在传统的傅里叶光学分析方法中，光学系统的成像过程可描述为两个环节，一是物的空域光分布经傅里叶变换后得到物频谱，二是物频谱的模乘以光学系统的MTF可求出像频谱的模。傅里叶光学分析方法将物的空域光分布展开成零频与无能量的正弦、余弦振荡波的线性组合，在理论上是成立的，但在应用信息论方法分析光学系统的信息传递规律及MTF测试实验的理论解释中遇到困难。由于正弦、余弦振荡波不具备能量，不满足非负性，所以无法表示光强，从而无法表示信息；在MTF测试实验中所使用的余弦光栅的透过率大于等于零，透过的光强是非负的，所获得的余弦光栅像的光强也是非负的，傅里叶光学理论无法准确地解释这个问题。In the traditional Fourier optical analysis method, the imaging process of the optical system can be described as two links, one is to obtain the object spectrum after the Fourier transform of the spatial light distribution of the object, and the other is to multiply the modulus of the object spectrum by the optical system The MTF can be calculated as the modulus of the image spectrum. The Fourier optical analysis method expands the spatial light distribution of the object into a linear combination of zero-frequency and energy-free sine and cosine oscillatory waves, which is theoretically valid, but when applying information theory methods to analyze the information transfer laws and MTF of optical systems Difficulties were encountered in the theoretical interpretation of test experiments. Since the sine and cosine oscillating waves do not have energy and do not satisfy non-negativity, they cannot express light intensity and thus cannot express information; the transmittance of the cosine grating used in the MTF test experiment is greater than or equal to zero, and the transmitted light intensity is non-negative. Negative, the light intensity of the obtained cosine grating image is also non-negative, and the theory of Fourier optics cannot explain this problem accurately.

对傅里叶变换进行改进，解决傅里叶变换中各级展开谐波不满足非负性的问题，可以使各级频谱谐波能独立地表示光强，可以使MTF的测量实验环节与理论相符，还可以应用信息论的分析方法研究光学系统的频域信息传递性能。此项技术的解决，对光学系统频域成像性能的深入研究，光学系统MTF检测实验的理论解释，及光学与信息论知识体系的结合都具有重要的意义。Improve the Fourier transform to solve the problem that the expansion harmonics at all levels in the Fourier transform do not meet the non-negativity, so that the spectrum harmonics at all levels can independently represent the light intensity, and the MTF measurement experiment link can be compared with the theory Correspondingly, the analysis method of information theory can also be used to study the frequency domain information transfer performance of the optical system. The resolution of this technology, the in-depth study of the frequency-domain imaging performance of the optical system, the theoretical explanation of the MTF detection experiment of the optical system, and the combination of optics and information theory knowledge systems are all of great significance.

发明内容Contents of the invention

本发明的目的是提供一种基于改进傅里叶变换的光学系统频域信息传递性能分析方法，该方法能将光强的空域分布函数展开成一系列非负的各级光强谐波的线性组合形式，进而可将信息论的分析方法和参数用于光学系统成像性能分析。The purpose of the present invention is to provide a method for analyzing the performance of optical system frequency domain information transmission based on the improved Fourier transform, which can expand the spatial domain distribution function of light intensity into a series of non-negative linear combinations of light intensity harmonics at all levels Form, and then the analysis methods and parameters of information theory can be used for the analysis of the imaging performance of the optical system.

光学成像系统属于非相干成像系统，其焦平面光电探测器记录的是光强分布，无法记录光波的振幅。为了进行频域分析，需要将观测物面上的二维光强分布展开成傅里叶逆变换的谐波线性组合形式。The optical imaging system is an incoherent imaging system, and its focal plane photodetector records the light intensity distribution, but cannot record the amplitude of the light wave. In order to perform frequency domain analysis, it is necessary to expand the two-dimensional light intensity distribution on the observed object surface into a harmonic linear combination form of the inverse Fourier transform.

物光强I(x，y)看作不同空间频率的一系列基元谐波函数exp(2πif_xx+2πif_yy)的线性组合，其中，i为虚数单位。由于二维空间位置变量x，y是相互对立的，二维空间频率变量f_x，f_y也是相互对立的，因此，可仅考虑一维情况。物光强I(x)的一维的基元谐波函数exp(2πifx)中的实部A(f)cos(2πfx)和虚部A(f)sin(2πfx)表示余弦、正弦振荡波，其中，x为一维空间位置变量，f为一维空间频率变量。余弦、正弦振荡波函数值存在负值，不能独立作为光强谐波存在，只能叠加在零频分量上，将叠加结果作为光强形式存在。本发明针对一维物、像与光学系统进行改进傅里叶分析与信息传递性能分析，对于二维的物、像与光学系统，可采用本发明的一维分析方法对不同方向上的信息传递规律进行一维分析。The object light intensity I(x, y) is regarded as a linear combination of a series of elementary harmonic functions exp(2πif _x x+2πif _y y) of different spatial frequencies, where i is the imaginary unit. Since the two-dimensional space position variables x, y are mutually opposite, and the two-dimensional spatial frequency variables f _x , f _y are also mutually opposite, therefore, only one-dimensional case can be considered. The real part A(f)cos(2πfx) and the imaginary part A(f)sin(2πfx) in the one-dimensional elementary harmonic function exp(2πifx) of the object light intensity I(x) represent cosine and sine oscillation waves, Among them, x is a one-dimensional spatial position variable, and f is a one-dimensional spatial frequency variable. The cosine and sine oscillatory wave function values have negative values, which cannot exist independently as light intensity harmonics, but can only be superimposed on the zero-frequency component, and the superposition result exists as light intensity. The present invention improves Fourier analysis and information transmission performance analysis for one-dimensional objects, images and optical systems. For two-dimensional objects, images and optical systems, the one-dimensional analysis method of the present invention can be used to transmit information in different directions Regular one-dimensional analysis.

本发明的分析方法分为三个步骤。The analysis method of the present invention is divided into three steps.

第一步、改进傅里叶变换的表达式及各级光强谐波系数的求解：The first step is to improve the expression of Fourier transform and solve the harmonic coefficients of light intensity at all levels:

(1)将观测物面上的一维光强I(x)分布展开成傅里叶逆变换表达式：(1) Expand the one-dimensional light intensity I(x) distribution on the observed object surface into an inverse Fourier transform expression:

$I I ((x x)) = = {&Integral; &Integral;}_{- - \infty \infty}^{\infty \infty} A A ((f f)) exp exp ((22 πifx πifx)) df df = = {&Integral; &Integral;}_{- - \infty \infty}^{\infty \infty} A A ((f f)) [[cos cos ((22 πfx πfx)) + + i i sin sin ((22 πfx πfx))]] df df;;$

(2)对步骤(1)表达式中的每个谐波分量赋予零频分量，且保证谐波的函数值均为非负值，令改进后的具有零频分量的非负光强谐波的表达式为：(2) Assign zero-frequency components to each harmonic component in the expression of step (1), and ensure that the function values of the harmonics are all non-negative values, so that the improved non-negative light intensity harmonics with zero-frequency components The expression is:

(3)将观测物面上的一维光强I(x)分布按照步骤(2)改进的谐波分量展开成改进傅里叶变换表达式：(3) Expand the one-dimensional light intensity I(x) distribution on the observed object surface into an improved Fourier transform expression according to the improved harmonic component of step (2):

(4)将步骤(3)的改进傅里叶变换表达式与步骤(1)的傅里叶逆变换表达式联立，求解改进傅里叶变换展开谐波的系数B(f)。(4) Combine the improved Fourier transform expression of step (3) with the inverse Fourier transform expression of step (1), and solve the coefficient B(f) of the expanded harmonic of the improved Fourier transform.

第二步、光学系统的成像积分方程及其频域信道矩阵的求解：The second step is the solution of the imaging integral equation of the optical system and its channel matrix in the frequency domain:

(1)物面一维光强分布I_o(x)经光学系统后形成的像面一维光强分布I_i(x)可由物面一维光强分布I_o(x)与光学系统的线扩散h(x)进行卷积运算求得，即：(1) The one-dimensional light intensity distribution I _o (x) on the object plane is formed by the optical system, and the one-dimensional light intensity distribution I _i (x) on the image plane can be obtained from the one-dimensional light intensity distribution I _o (x) on the object plane and the optical system Line diffusion h(x) is obtained by convolution operation, namely:

${I I}_{i i} ((x x)) = = {I I}_{o o} ((x x)) * * h h ((x x)) = = {&Integral; &Integral;}_{- - \infty \infty}^{+ + \infty \infty} {I I}_{o o} ((ξ ξ)) \cdot &Center Dot; h h ((x x - - ξ ξ)) dξ dξ = = K K [[{I I}_{o o} ((x x))]];;$

其中，ξ为卷积运算的中间变量，Κ为卷积表达式的积分算子，Among them, ξ is the intermediate variable of the convolution operation, and Κ is the integral operator of the convolution expression,

$K K = = {&Integral; &Integral;}_{- - \infty \infty}^{\infty \infty} h h ((x x - - ξ ξ)) dξ dξ;;$

积分算子Κ的本征方程具有如下形式：The eigenequation of the integral operator K has the following form:

Kφ_f(x)＝β_fφ_f(x)；Kφ _f (x) = β _f φ _f (x);

其中，φ_f(x)为积分算子Κ的第f阶本征函数，β_f为积分算子Κ的第f阶本征函数的特征值；Wherein, φ _f (x) is the fth order eigenfunction of integral operator K, and β _f is the eigenvalue of the fth order eigenfunction of integral operator K;

(2)对积分算子的本征函数与特征值求解，得到光学系统频域信道矩阵：(2) Solve the eigenfunction and eigenvalue of the integral operator to obtain the channel matrix in the frequency domain of the optical system:

将成像积分算子Κ作用于光强谐波，可得到光学系统对光强谐波成像后的结果，即：Applying the imaging integral operator K to the light intensity harmonics , the result of the optical system imaging the light intensity harmonics can be obtained, namely:

当光学系统的截止频率为N时，将第一步求得的改进非负光强谐波表达式的归一化系数集合X：{B(0)，B(1)，B(2)，…B(N)}作为信源，输出像的光分布的归一化系数集合Y：{B’(0)，B’(1)，B’(2)，…B’(N)}作为信宿，则物、像频谱的信息传递关系可表示为：When the cut-off frequency of the optical system is N, the improved non-negative intensity harmonic expression obtained in the first step is The set of normalized coefficients X: {B(0), B(1), B(2), ... B(N)} as the source, the set of normalized coefficients Y of the light distribution of the output image: {B' (0), B'(1), B'(2),...B'(N)} as the sink, the information transfer relationship of object and image spectrum can be expressed as:

X·P＝Y，X·P=Y,

其中，光学系统的频域信道矩阵P为：Among them, the frequency domain channel matrix P of the optical system is:

第三步、光学系统信息传递性能参数的计算方法：The third step, the calculation method of the optical system information transmission performance parameters:

根据信息论的定义，对改进傅里叶变换所得的物频谱信源熵、像频谱信宿熵、物频谱与像频谱的联合信息熵、物频谱与像频谱的互信息量及光学系统的信道容量进行计算，根据这些信息参数来评价光学系统的信息传递性能，其中：According to the definition of information theory, the source entropy of the object spectrum, the sink entropy of the image spectrum, the joint information entropy of the object spectrum and the image spectrum, the mutual information of the object spectrum and the image spectrum and the channel capacity of the optical system obtained by the improved Fourier transform are carried out. Calculate and evaluate the information transfer performance of the optical system based on these information parameters, where:

物频谱信源熵为：The material spectrum source entropy is:

$H h ((X x)) = = - - {Σ Σ}_{m m = = 11}^{N N + + 11} B B ((m m)) \cdot &Center Dot; {log log}_{22} B B ((m m));;$

像频谱信宿熵为：Like spectrum sink entropy is:

$H h ((Y Y)) = = - - {Σ Σ}_{n no = = 11}^{N N + + 11} {B B}^{' '} ((n no)) \cdot \cdot {log log}_{22} {B B}^{' '} ((n no));;$

物频谱与像频谱的联合信息熵为：The joint information entropy of object spectrum and image spectrum is:

$H h ((XY X Y)) = = - - {Σ Σ}_{m m = = 11}^{N N + + 11} {Σ Σ}_{n no = = 11}^{N N + + 11} [[B B ((m m)) \cdot &Center Dot; {p p}_{mn mn}]] \cdot &Center Dot; {log log}_{22} [[B B ((m m)) \cdot &Center Dot; {p p}_{mn mn}]];;$

物频谱与像频谱的互信息量为：The mutual information between object spectrum and image spectrum is:

I(X；Y)＝H(X)+H(Y)-H(XY)；I(X;Y)=H(X)+H(Y)-H(XY);

光学系统的信道容量为：The channel capacity of the optical system is:

$C C = = \underset{B B ((i i))}{max max} I I ((X x;; Y Y)) . .$

互信息量是能够在信道中正确传输的信源的那部分信息量的大小，也是经信道传递后的信宿中包含的信源信息量的大小。因此，互信息量的大小能反映出光学系统对物频谱的信息传递能力。而信道容量是光学系统能够传递的最大互信息量的大小，信道容量能够表征一个信道传输信息的极限能力。The amount of mutual information is the size of the part of the source information that can be correctly transmitted in the channel, and it is also the size of the source information contained in the sink after being transmitted through the channel. Therefore, the amount of mutual information can reflect the information transmission ability of the optical system to the object spectrum. The channel capacity is the maximum amount of mutual information that the optical system can transmit, and the channel capacity can represent the limit ability of a channel to transmit information.

本发明提供的改进傅里叶分析方法将光强展开成零频与具有能量的非负光强谐波的线性组合，分析了新的光强展开方法下的光学系统成像规律，进而应用信息论的分析方法来分析光学系统的信息传递性能。本发明对傅里叶光学分析方法进行了改进，使傅里叶光学的理论更完善，并实现了光学分析方法与信息论分析方法的结合。The improved Fourier analysis method provided by the present invention expands the light intensity into a linear combination of zero frequency and non-negative light intensity harmonics with energy, analyzes the imaging law of the optical system under the new light intensity expansion method, and then applies the information theory Analytical methods to analyze the information transfer performance of optical systems. The invention improves the Fourier optics analysis method, makes the theory of Fourier optics more perfect, and realizes the combination of the optical analysis method and the information theory analysis method.

附图说明Description of drawings

图1为傅里叶光学分析方法中将光强展开为零频与余弦振荡波的原理图，(a)某光强分布I(x)，(b)零频分量，(c)无零频分量的余弦振荡波；Figure 1 is a schematic diagram of expanding the light intensity into zero-frequency and cosine oscillation waves in the Fourier optical analysis method, (a) a certain light intensity distribution I(x), (b) zero-frequency component, (c) no zero-frequency component cosine oscillatory wave;

图2为将余弦振荡波加上零频分量转变为非负余弦光强谐波的原理，(a)有零频分量的余弦光强谐波，(b)无零频分量的余弦振荡波，(c)需要赋予余弦振荡的零频分量；Fig. 2 is the principle of converting cosine oscillation wave with zero frequency component into non-negative cosine light intensity harmonic, (a) cosine light intensity harmonic with zero frequency component, (b) cosine oscillation wave without zero frequency component, (c) need to endow the zero-frequency component of the cosine oscillation;

图3为改进傅里叶分析方法中将光强分布I(x)展开成非负光强谐波的原理图，(a)某光强分布I(x)，(b)零频分量部分分配给余弦振荡波，(c)有零频分量的余弦光强谐波；Figure 3 is a schematic diagram of expanding the light intensity distribution I(x) into non-negative light intensity harmonics in the improved Fourier analysis method, (a) a certain light intensity distribution I(x), (b) partial distribution of zero-frequency components Give the cosine oscillatory wave, (c) have the cosine light intensity harmonic wave of zero frequency component;

图4为无零频的余弦波振荡波经光学系统成像过程的原理图，(a)无零频分量的余弦振荡波，(b)成像后的无零频分量的余弦振荡波；Fig. 4 is the schematic diagram of the cosine wave oscillating wave without zero frequency through the imaging process of the optical system, (a) the cosine oscillating wave without the zero frequency component, (b) the cosine oscillating wave without the zero frequency component after imaging;

图5为无零频的余弦波振荡波的频谱经光学系统成像过程的原理图，(a)余弦振荡波3的频谱，(b)光学系统的MTF，(c)余弦振荡波像7的频谱；Figure 5 is a schematic diagram of the imaging process of the cosine wave oscillation without zero frequency through the optical system, (a) the spectrum of the cosine oscillation wave 3, (b) the MTF of the optical system, (c) the spectrum of the cosine oscillation wave image 7 ;

图6为含零频分量的非负光强谐波经光学系统成像过程的原理图，(a)含零频分量的余弦光强谐波，(b)余弦光强谐波经光学系统成的像；Figure 6 is a schematic diagram of the imaging process of non-negative light intensity harmonics containing zero frequency components through the optical system, (a) cosine light intensity harmonics containing zero frequency components, (b) cosine light intensity harmonics formed by the optical system picture;

图7为含零频分量的非负光强谐波频谱经光学系统成像过程的原理图，(a)余弦光强谐波4的频谱，(b)光学系统的MTF，(c)余弦光强谐波像11的频谱；Figure 7 is a schematic diagram of the imaging process of the non-negative light intensity harmonic spectrum containing zero frequency components through the optical system, (a) the spectrum of the cosine light intensity harmonic 4, (b) the MTF of the optical system, (c) the cosine light intensity Harmonics like the spectrum of 11;

图8为含零频的光强谐波经光学系统成像后部分零频分量转移给像的原零频分量的原理图，(a)余弦光强谐波经光学系统成的像11，(b)零频和余弦谐波等比例衰减的余弦光强谐波，(c)余弦光强谐波11减去余弦光强谐波14所得差值的零频分量；Fig. 8 is the schematic diagram of the original zero-frequency component transferred to the original zero-frequency component of the image after the light intensity harmonic containing zero frequency is imaged by the optical system, (a) the image 11 formed by the cosine light intensity harmonic through the optical system, (b ) cosine light intensity harmonics attenuated in equal proportion at zero frequency and cosine harmonics, (c) zero frequency component of the difference obtained by subtracting cosine light intensity harmonics 11 from cosine light intensity harmonics 14;

图9为含零频的光强谐波的频谱经光学系统成像后部分零频分量转移给像的原零频分量的原理图，(a)经光学系统成像后的余弦光波11的频谱，(b)零频与余弦谐波等比例衰减的余弦光强谐波14的频谱，(c)转移给像零频分量的部分，即13与16之差；Fig. 9 is the schematic diagram of the original zero-frequency component transferred to the original zero-frequency component of the image by the part of the zero-frequency component after the spectrum of the light intensity harmonic containing zero frequency is imaged by the optical system, (a) the spectrum of the cosine light wave 11 after the imaging by the optical system, ( b) the spectrum of the cosine light intensity harmonic 14 attenuated proportionally between zero frequency and cosine harmonic, (c) transferred to the part like zero frequency component, i.e. the difference between 13 and 16;

图10为信源熵、信宿熵、联合熵及互信息量之间的关系图。Fig. 10 is a diagram of the relationship among source entropy, sink entropy, joint entropy and mutual information.

具体实施方式detailed description

下面结合附图对本发明的技术方案作进一步的说明，但并不局限于此，凡是对本发明技术方案进行修改或者等同替换，而不脱离本发明技术方案的精神和范围，均应涵盖在本发明的保护范围中。The technical solution of the present invention will be further described below in conjunction with the accompanying drawings, but it is not limited thereto. Any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention should be covered by the present invention. within the scope of protection.

本发明提供了一种基于改进傅里叶变换的光学系统频域信息传递性能分析方法，分为以下三个步骤：The present invention provides an optical system frequency domain information transmission performance analysis method based on the improved Fourier transform, which is divided into the following three steps:

第一步：改进傅里叶变换的表达式及各光强谐波系数的求解。The first step: improve the expression of Fourier transform and the solution of the harmonic coefficients of each light intensity.

根据一维傅里叶变换的表达方式，物面发出的单色光场的复振幅u(x)与其频谱函数a(f)互为傅里叶变换对，满足如下关系：According to the expression of one-dimensional Fourier transform, the complex amplitude u(x) of the monochromatic light field emitted by the object surface and its spectral function a(f) are a Fourier transform pair, satisfying the following relationship:

$a a ((f f)) = = {&Integral; &Integral;}_{- - \infty \infty}^{\infty \infty} u u ((x x)) exp exp ((- - 22 πifx πifx)) dx dx - - - - - - ((11));;$

$u u ((x x)) = = {&Integral; &Integral;}_{- - \infty \infty}^{\infty \infty} a a ((f f)) exp exp ((- - 22 πifx πifx)) df df - - - - - - ((22)) . .$

式(2)可理解为u(x)是由不同空间频率f的一系列谐波函数exp(2πifx)的线性组合构成，各谐波函数的系数是a(f)。Formula (2) can be understood as u(x) is composed of a series of linear combinations of harmonic functions exp(2πifx) of different spatial frequencies f, and the coefficient of each harmonic function is a(f).

光学成像系统属于非相干成像系统，其焦平面光电探测器记录的是光强分布，无法记录光波的振幅。因此，需要将观测场景物面上的一维光强分布I(x)展开成与式(2)的傅里叶逆变换相似的形式，如式(3)所示：The optical imaging system is an incoherent imaging system, and its focal plane photodetector records the light intensity distribution, but cannot record the amplitude of the light wave. Therefore, it is necessary to expand the one-dimensional light intensity distribution I(x) on the object surface of the observed scene into a form similar to the inverse Fourier transform of formula (2), as shown in formula (3):

$I I ((x x)) = = {&Integral; &Integral;}_{- - \infty \infty}^{\infty \infty} A A ((f f)) exp exp ((22 πifx πifx)) df df = = {&Integral; &Integral;}_{- - \infty \infty}^{\infty \infty} A A ((f f)) [[cos cos ((22 πfx πfx)) + + i i sin sin ((22 πfx πfx))]] df df - - - - - - ((33)) . .$

I(x)可看作不同空间频率的一系列谐波函数exp(2πifx)的线性组合，各谐波函数的系数是A(f)，而A(f)是I(x)的傅里叶变换频谱函数。谐波函数中的实部A(f)cos(2πfx)和虚部A(f)sin(2πfx)为余弦、正弦振荡波，振荡波函数值存在负值，不能独立作为光强存在，只能叠加在零频分量上，将叠加结果作为整体的光强形式存在。图1所示为根据传统的傅里叶光学分析方法将某一沿x方向的光强分布1的表达式I(x)展开成零频分量2与某频率的余弦振荡波3合成的形式。余弦振荡波3有负值成分存在，且沿x轴的积分值为零，不具备能量，无法以独立的光强谐波形式存在，只能叠加在零频分量2上并与零频分量2共同表示光强分布1。傅里叶变换的谐波函数为含有负值成分余弦函数与正弦函数，而正弦函数与余弦函数的分析处理方法是相同的。I(x) can be regarded as a linear combination of a series of harmonic functions exp(2πifx) of different spatial frequencies, the coefficient of each harmonic function is A(f), and A(f) is the Fourier of I(x) Transform spectral function. The real part A(f)cos(2πfx) and the imaginary part A(f)sin(2πfx) in the harmonic function are cosine and sine oscillatory waves, and the value of the oscillatory wave function has negative values, which cannot exist independently as light intensity. Superimposed on the zero-frequency component, the superimposed result exists in the form of overall light intensity. Figure 1 shows that according to the traditional Fourier optical analysis method, the expression I(x) of a certain light intensity distribution 1 along the x direction is expanded into a form in which a zero-frequency component 2 and a cosine oscillation wave 3 of a certain frequency are synthesized. The cosine oscillatory wave 3 has a negative value component, and the integral value along the x-axis is zero, so it has no energy and cannot exist in the form of an independent light intensity harmonic, it can only be superimposed on the zero-frequency component 2 and combined with the zero-frequency component 2 collectively represent the light intensity distribution 1 . The harmonic functions of Fourier transform are cosine functions and sine functions with negative components, and the analysis and processing methods of sine functions and cosine functions are the same.

若将I(x)光强看作各独立的光强谐波光分量的线性组合结果，则要求每个谐波分量的函数值不能出现负值。为了使每个谐波分量具有能量，且保证谐波分量的函数值均为非负，需对谐波的正、余弦震荡波加上一定的零频分量。令改进后的含零频成分的非负光强谐波的表达式为：If the light intensity of I(x) is regarded as the linear combination result of each independent light intensity harmonic light component, it is required that the function value of each harmonic component cannot have a negative value. In order to make each harmonic component have energy and ensure that the function values of the harmonic components are non-negative, it is necessary to add a certain zero-frequency component to the positive and cosine oscillation waves of the harmonic. Let the expression of the improved non-negative light intensity harmonic with zero frequency component be:

其中，B(f)为各光强谐波的系数。Among them, B(f) is the coefficient of each light intensity harmonic.

由于改进后的零频分量仍为常数，可令其为：Since the improved zero-frequency component is still constant, it can be made as:

将(5)式代入(4)式得：Substitute (5) into (4) to get:

如图2所示，为了使余弦振荡波函数3能以独立的子光波的形式表示光强，需要对弦函数振荡波3赋予一定的零频分量5，从而将弦函数振荡波3转变成为含有能量的独立光强谐波4的形式。光强谐波4的最小值为0，但不会出现负值。As shown in Fig. 2, in order to make the cosine oscillatory wave function 3 represent the light intensity in the form of an independent sub-light wave, it is necessary to assign a certain zero-frequency component 5 to the oscillating wave 3 of the sine function, so that the oscillating wave 3 of the sine function can be transformed into a wave containing Energy independent light intensity harmonic 4 form. Light Intensity Harmonic 4 has a minimum value of 0, but negative values cannot occur.

以替换式(3)中的被积分项A(f)exp(2πifx)，可保证每个光强谐波分量不出现负值，则可得到改进傅里叶变换光强分布函数I(x)的表达式如式(7)所示：by Replacing the integral term A(f)exp(2πifx) in formula (3) can ensure that each light intensity harmonic component If there is no negative value, the expression of the improved Fourier transform light intensity distribution function I(x) can be obtained as shown in formula (7):

通过求解式(3)和式(7)构成的方程组，求出改进傅里叶变换展开谐波系数B(f)，求解后的非负光强谐波系数表达式为：By solving the equations composed of formula (3) and formula (7), the improved Fourier transform expansion harmonic coefficient B(f) is obtained, and the non-negative light intensity harmonic coefficient expression after solving is:

$\{\begin{matrix} B B ((f f)) = = A A ((f f)),, & f f &NotEqual; &NotEqual; 00 \\ B B ((00)) = = \frac{A A ((00)) - - ((11 + + i i)) {&Integral; &Integral;}_{- - \infty \infty}^{00 - -} | | B B ((f f)) | | df df + + ((11 + + i i)) {&Integral; &Integral;}_{00 + +}^{\infty \infty} | | B B ((f f)) | | df df}{((11 + + i i))},, & f f = = 00 \end{matrix} - - - - - - ((88))$

若 $A (0) < (1 + i) {&Integral;}_{- \infty}^{0 +} | B (f) | df + (1 + i) {&Integral;}_{0 +}^{\infty} | B (f) | df,$ 可将A(0)增加一定的零频分量来保证B(0)＝0。而A(0)增加一定的零频分量时，仅仅改变了物光强分布I(x)的背景亮度，并不改变各级非零频谐波系数A(f)。将B(f)代入(7)式，便可得到由非负的各级光强谐波合成形式的光强I(x)的表达式。like $A (0) < (1 + i) {&Integral;}_{- \infty}^{0 +} | B (f) | df + (1 + i) {&Integral;}_{0 +}^{\infty} | B (f) | df,$ A certain zero-frequency component can be added to A(0) to ensure B(0)=0. When A(0) increases a certain zero-frequency component, it only changes the background brightness of the object light intensity distribution I(x), and does not change the non-zero-frequency harmonic coefficients A(f) of each level. Substituting B(f) into Equation (7), we can get non-negative harmonics of light intensity at all levels Expression for the light intensity I(x) in composite form.

如图3所示，将光强分布1重新展开成零频分量6与相应频率的具有能量的余弦光强谐波4合成的形式。其中，零频分量6等于原零频分量2减去余弦谐波光强4所分得的零频分量5。As shown in FIG. 3 , the light intensity distribution 1 is re-expanded into a form in which a zero-frequency component 6 is combined with a cosine light intensity harmonic 4 having energy of a corresponding frequency. Wherein, the zero-frequency component 6 is equal to the zero-frequency component 5 obtained by subtracting the cosine harmonic light intensity 4 from the original zero-frequency component 2 .

第二步：光学系统的成像积分方程及其频域信道矩阵的求解。The second step: the solution of the imaging integral equation of the optical system and its channel matrix in the frequency domain.

将第一步得到的改进光强谐波表达式作为光学系统的被观测物，利用光学系统成像积分方程的本征理论求解的像，从而写出光学系统的频域信道矩阵P。The improved light intensity harmonic expression obtained in the first step As the observed object of the optical system, use the eigen theory of the imaging integral equation of the optical system to solve , so as to write out the frequency domain channel matrix P of the optical system.

在一维情况下，物面光强分布I_o(x)经光学系统后形成像面光强分布I_i(x)可由物面光强分布I_o(x)与光学系统的线扩散h(x)的卷积求得，In the one-dimensional case, the light intensity distribution I _o (x) on the object plane passes through the optical system to form the light intensity distribution I _i (x) on the image plane, which can be obtained from the light intensity distribution I _o (x) on the object plane and the line diffusion h( The convolution of x) is obtained,

即：which is:

${I I}_{i i} ((x x)) = = {I I}_{o o} ((x x)) * * h h ((x x)) = = {&Integral; &Integral;}_{- - \infty \infty}^{+ + \infty \infty} {I I}_{o o} ((ξ ξ)) \cdot \cdot h h ((x x - - ξ ξ)) dξ dξ = = K K [[{I I}_{o o} ((x x))]] - - - - - - ((99)) . .$

其中，Κ为卷积表达式的积分算子，Among them, Κ is the integral operator of the convolution expression,

$K K = = {&Integral; &Integral;}_{- - \infty \infty}^{\infty \infty} h h ((x x - - ξ ξ)) dξ dξ - - - - - - ((1010)) . .$

Κφ_f(x)＝β_fφ_f(x)(11)。Κφ _f (x) = β _f φ _f (x) (11).

其中，φ_f(x)为积分算子Κ的第f阶本征函数，β_f为积分算子Κ的第f阶本征函数的特征值。Wherein, φ _f (x) is the fth order eigenfunction of the integral operator K, and β _f is the eigenvalue of the fth order eigenfunction of the integral operator K.

根据傅里叶光学的基本理论，光学系统在等晕区可看作线性空不变系统，复指数函数是线性不变系统的本征函数，因此，A(f)exp(2πifx)是积分算子Κ的本征函数。当物光强I_o(x)为本征函数A(f)exp(2πifx)时，此时光强I_o(x)只包含一个频谱成分f，根据傅里叶光学中光学系统的频域特性可知，像光强I_i(x)为α_fA(f)exp(2πifx)，其中，α_f为光学系统在空间频率f处的MTF值。所以，对应于本征函数A(f)exp(2πifx)的特征值β_f等于MTF在空间频率f处的函数值α_f。According to the basic theory of Fourier optics, the optical system can be regarded as a linear space-invariant system in the halo area, and the complex exponential function is the eigenfunction of the linear invariant system. Therefore, A(f)exp(2πifx) is an integral calculation The eigenfunction of sub-K. When the object light intensity I _o (x) is the eigenfunction A(f)exp(2πifx), the light intensity I _o (x) contains only one spectral component f, according to the frequency domain characteristics of the optical system in Fourier optics It can be seen that the image light intensity I _i (x) is α _f A(f)exp(2πifx), where α _f is the MTF value of the optical system at the spatial frequency f. Therefore, the eigenvalue β _f corresponding to the eigenfunction A(f)exp(2πifx) is equal to the function value α _{f of the MTF at the spatial frequency f} .

如图4所示，根据傅里叶光学理论，无能量的余弦振荡波3经光学系统得到的像是幅度被衰减的余弦振荡波7，其幅度衰减比例因子为与其频率成分相对应的MTF值α_f。余弦振荡波3与7成线性关系，余弦振荡波3满足积分算子Κ的本征方程(11)，所以，余弦振荡波3是积分算子Κ的本征函数，α_f为其特征值，所以α_f＝β_f。余弦振荡波3、7含有负值成分，无法以独立的子光波的形式表示光强，只能以无能量的余弦振荡波的形式叠加在零频光强上来表示光强分布，所以，图4所示的成像过程仅仅在理论上成立。实际的光学检测实验采用余弦函数透过率的光栅作为被测物，余弦光栅透过的光强成余弦分布，但不会出现图4(a)所示含负光强的光分布3，余弦光栅的像也不会出现图4(b)所示含负光强的光分布7。As shown in Figure 4, according to the theory of Fourier optics, the energy-free cosine oscillation wave 3 is obtained by the optical system as a cosine oscillation wave 7 whose amplitude is attenuated, and its amplitude attenuation scale factor is the MTF value corresponding to its frequency component α _f . Cosine oscillating wave 3 is linearly related to 7, and cosine oscillating wave 3 satisfies the eigenequation (11) of integral operator Κ, so cosine oscillating wave 3 is the eigenfunction of integral operator Κ, and α _f is its eigenvalue, So α _f =β _f . Cosine oscillatory waves 3 and 7 contain negative value components, so they cannot represent the light intensity in the form of independent sub-light waves, and can only be superimposed on the zero-frequency light intensity in the form of energy-free cosine oscillatory waves to represent the light intensity distribution. Therefore, Figure 4 The imaging process shown is only theoretical. The actual optical detection experiment uses a grating with a cosine function transmittance as the measured object, and the light intensity transmitted by the cosine grating has a cosine distribution, but the light distribution with negative light intensity shown in Figure 4(a) does not appear 3, cosine The image of the grating will not appear the light distribution 7 with negative light intensity as shown in Fig. 4(b).

如图5所示，根据傅里叶光学理论，将图4(a)所示的无能量的余弦振荡波3作傅里叶变换得到图5(a)所示的物频谱函数8，物频谱函数8乘以光学系统的调制传递函数(MTF)9可得到像7的频谱10。由于图4(a)中的余弦振荡波3与图4(b)中的余弦振荡波7均不满足非负性，不能以独立的光强谐波的形式存在，则图5中描述的光学系统对输入物频谱8与输出像频谱10的传递规律也仅仅是对无能量的余弦振荡波3、7的传递规律，不能描述具有能量的余弦光强谐波的传递规律，不能准确描述光学检测实验中的余弦光栅的成像规律。As shown in Figure 5, according to the theory of Fourier optics, the energy-free cosine oscillation wave 3 shown in Figure 4 (a) is Fourier transformed to obtain the object spectrum function 8 shown in Figure 5 (a), the object spectrum Multiplying the function 8 by the modulation transfer function (MTF) 9 of the optical system results in a spectrum 10 like 7 . Since neither the cosine oscillation wave 3 in Figure 4(a) nor the cosine oscillation wave 7 in Figure 4(b) satisfies non-negativity, and cannot exist in the form of independent light intensity harmonics, the optical The transmission law of the system to the input object spectrum 8 and the output image spectrum 10 is only the transmission law of the cosine oscillation wave 3 and 7 without energy, and cannot describe the transmission law of the cosine light intensity harmonic with energy, and cannot accurately describe the optical detection The imaging law of the cosine grating in the experiment.

将成像积分算子Κ作用于式(4)所示的光强谐波，可得到光学系统对光强谐波成像后的结果，即：Apply the imaging integral operator K to the light intensity harmonic shown in formula (4) , the result of the optical system imaging the light intensity harmonics can be obtained, namely:

式(12)表明，当输入物光强分布为时，输出的像光强分布有和两种频率成分，由于β_f≤1，所以可认为的能量在光学系统传递的过程中，的幅度被衰减，其被衰减的零频成分转移给像的零频分量 Equation (12) shows that when the light intensity distribution of the input object is When , the output image light intensity distribution has and Two frequency components, since β _f ≤ 1, it can be considered In the process of energy transfer in the optical system, The amplitude of is attenuated, and its attenuated zero-frequency component is transferred to the zero-frequency component of the image

如图6所示，光强谐波4可作为独立的子光波的形式表示光强谐波，经光学系统成像后，光强谐波4的零频分量5保持不变，而余弦谐波的幅度会受到衰减而变为余弦谐波11，衰减的比例因子为相应频率处的MTF值α_f。As shown in Figure 6, the optical intensity harmonic 4 can be used as an independent sub-light wave to represent the optical intensity harmonic. After imaging by the optical system, the zero-frequency component 5 of the optical intensity harmonic 4 remains unchanged, while the cosine harmonic The amplitude will be attenuated to the cosine harmonic 11 by a scaling factor of the MTF value α _f at the corresponding frequency.

如图7所示，将图6(a)所示的余弦波光强谐波4作傅里叶变换得到图7(a)所示的物频谱函数12，物频谱函数12乘以光学系统的调制传递函数(MTF)9可得到像11的频谱13。As shown in Figure 7, the cosine wave light intensity harmonic 4 shown in Figure 6 (a) is Fourier transformed to obtain the object spectrum function 12 shown in Figure 7 (a), and the object spectrum function 12 is multiplied by the modulation of the optical system A transfer function (MTF) 9 results in a spectrum 13 like 11 .

如图8所示，光强谐波的像11与图6(a)的输入光强谐波4不存在线性比例关系。将光强谐波的像11的零频分量去除掉一部分(零频分量15)，使光强谐波的像11转变为最小值为零的光强谐波14的形式，此时，输入的余弦光强谐波4与输出的余弦光强谐波14是符合线性比例关系的，谐波4与谐波14的比例因子为相应频率处的MTF值α_f。被去除掉的部分零频分量15可认为转移给了像光波原有的零频分量。As shown in FIG. 8 , there is no linear proportional relationship between the image 11 of the light intensity harmonic and the input light intensity harmonic 4 in FIG. 6( a ). Remove a part (zero frequency component 15) of the zero-frequency component (zero-frequency component 15) of the image 11 of the light intensity harmonic, so that the image 11 of the light intensity harmonic is converted into the form of light intensity harmonic 14 whose minimum value is zero. At this time, the input The cosine light intensity harmonic 4 and the output cosine light intensity harmonic 14 conform to a linear proportional relationship, and the proportional factor between the harmonic 4 and the harmonic 14 is the MTF value α _f at the corresponding frequency. The removed part of the zero-frequency component 15 can be considered as transferred to the original zero-frequency component of the image-like light wave.

如图9所示，将图8(a)所示的余弦波光强谐波像11作傅里叶变换得到图9(a)所示的像频谱函数13，将像频谱13中的部分零频分量17去除，便可得到余弦谐波14对应的频谱16，频谱16与图7(a)中的输入频谱12的成线性比例关系，比例因子为相应频率处的MTF值α_f。As shown in Figure 9, the cosine wave light intensity harmonic image 11 shown in Figure 8 (a) is made Fourier transform to obtain the image spectrum function 13 shown in Figure 9 (a), and part of the zero frequency in the image spectrum 13 By removing the component 17, the spectrum 16 corresponding to the cosine harmonic 14 can be obtained. The spectrum 16 is linearly proportional to the input spectrum 12 in FIG. 7(a), and the scaling factor is the MTF value α _f at the corresponding frequency.

从信息论的角度分析，当光学系统的截止频率为N时，取物的归一化光强谐波系数X：{B(0)，B(1)，B(2)，…B(N)}作为信源。信源输入到光学系统后，根据公式(12)可知，输出像的归一化光分布的信息为Y：{B’(0)，B’(1)，B’(2)，…B’(N)}＝{B(0)+(1-β₁)B(1)+(1-β₂)B(2)+…+(1-β_N)B(N)，β₁B(1)，β₂B(2)，…，β_NB(N)}，其中，β₁，β₂，…β_N为积分算子Κ的1，2，…，N阶特征值，等于光学系统在空间频率1，2，…，N处的MTF值α₁，α₂，…α_N。根据信息论的模型，可将物光强谐波的归一化系数集合(向量)看作信源，像光谐波的归一化系数集合(向量)看作信宿，光学系统可看作信道。根据式(12)，可将光学系统的信道矩阵P写成式(13)的形式：From the perspective of information theory, when the cut-off frequency of the optical system is N, the normalized light intensity harmonic coefficient X of the object is: {B(0), B(1), B(2),...B(N) } as the source. After the signal source is input into the optical system, according to the formula (12), the normalized light distribution information of the output image is Y: {B'(0), B'(1), B'(2),...B' (N)}={B(0)+(1-β ₁ )B(1)+(1-β ₂ )B(2)+…+(1-β _N )B(N), β ₁ B( 1), β ₂ B(2), ..., β _N B (N)}, wherein, β ₁ , β ₂ , ... β _N are 1, 2, ..., N-order eigenvalues of the integral operator Κ, equal to optical MTF values α ₁ , α ₂ , ... α _N of the system at spatial frequencies 1, 2, ..., N. According to the model of information theory, the normalized coefficient set (vector) of the object light intensity harmonic can be regarded as the source, the normalized coefficient set (vector) of the optical harmonic can be regarded as the sink, and the optical system can be regarded as the channel. According to formula (12), the channel matrix P of the optical system can be written in the form of formula (13):

其中，p_mn为信道矩阵P的元素，下角标m，n为矩阵P中的元素序号。Wherein, p _mn is an element of the channel matrix P, subscript m, and n is an element number in the matrix P.

则物、像频谱的信息传递关系可表示为Then the information transfer relationship between object and image spectrum can be expressed as

X·P＝Y(14)。X·P=Y (14).

第三步：光学系统频域信息传递性能参数的计算方法Step 3: Calculation method of optical system frequency domain information transmission performance parameters

将第一步求得的改进光强谐波表达式的归一化系数集合(向量)X作为信源，结合第二步得到的光学系统频域信道矩阵P与像的归一化系数集合(向量)为Y作为信宿，可求出光学系统的信息评价指标互信息量I(X；Y)与信道容量C。The improved light intensity harmonic expression obtained in the first step The normalization coefficient set (vector) X of the optical system is used as the source, and the optical system frequency domain channel matrix P obtained in the second step and the normalization coefficient set (vector) of the image are Y as the sink, and the information of the optical system can be obtained Evaluation index mutual information I(X; Y) and channel capacity C.

根据信息论的定义，物的改进傅里叶变换频谱(信源)的信息量即信源熵为：According to the definition of information theory, the information amount of the improved Fourier transform spectrum (information source) of the object, that is, the information source entropy, is:

$H h ((X x)) = = - - {Σ Σ}_{m m = = 11}^{N N + + 11} B B ((m m)) \cdot &Center Dot; {log log}_{22} B B ((m m)) - - - - - - ((1515)),,$

其中，m为X中元素的序号。Among them, m is the serial number of the element in X.

相应地，像的改进傅里叶变换频谱(信宿)的信息量即信宿熵为：Correspondingly, the information amount of the improved Fourier transform spectrum (sink) of the image, that is, the sink entropy is:

$H h ((Y Y)) = = - - {Σ Σ}_{n no = = 11}^{N N + + 11} {B B}^{' '} ((n no)) \cdot &Center Dot; {log log}_{22} {B B}^{' '} ((n no)) - - - - - - ((1616)),,$

其中，n为Y中元素的序号。Among them, n is the serial number of the element in Y.

信源与信宿的联合信息熵为：The joint information entropy of source and destination is:

$H h ((XY X Y)) = = - - {Σ Σ}_{m m = = 11}^{N N + + 11} {Σ Σ}_{n no = = 11}^{N N + + 11} [[B B ((m m)) \cdot &Center Dot; {p p}_{mn mn}]] \cdot \cdot {log log}_{22} [[B B ((m m)) \cdot \cdot {p p}_{mn mn}]] - - - - - - ((1717)) . .$

信源与信宿的互信息量为：The mutual information between the source and the sink is:

I(X；Y)＝H(X)+H(Y)-H(XY)(18)。I(X;Y)=H(X)+H(Y)-H(XY) (18).

互信息量I(X；Y)是描述信道上传输的有效信息量的大小的参数。如果信源熵为H(X)，希望在信道的输出端接收的信息量也是H(X)。但是由于信道不理想，一般情况下在输出端只能接收到H(X)的一部分，即互信息量I(X；Y)。从信源的角度来说，互信息量I(X；Y)是能够在信道中正确传输的信源H(X)的那部分信息量的大小；从信宿的角度来说，互信息量I(X；Y)是指经信道传递后，信宿H(Y)中包含的信源H(X)的信息量的大小。The mutual information I(X; Y) is a parameter describing the size of the effective information transmitted on the channel. If the source entropy is H(X), the amount of information expected to be received at the output of the channel is also H(X). But because the channel is not ideal, generally only a part of H(X) can be received at the output end, that is, the mutual information I(X; Y). From the point of view of the information source, the amount of mutual information I(X; Y) is the size of the part of the information amount of the source H(X) that can be correctly transmitted in the channel; from the point of view of the sink, the amount of mutual information I (X; Y) refers to the amount of information of the source H(X) contained in the sink H(Y) after being transmitted through the channel.

H(X)、H(Y)、H(XY)、I(X；Y)之间的关系可由图10表示。对于理想光学系统，信源无失真地传递给信宿，则H(X)＝H(Y)＝H(XY)＝I(X；Y)。而对于实际的光学系统，由于衍射效应和像差的存在，信源与信宿不能完全一致，则互信息量I(X；Y)是评价光学系统信息传递能力的重要指标。互信息量I(X；Y)能描述物经光学系统成像后，物信息无失真传递给像信息的信息量的大小。The relationship between H(X), H(Y), H(XY), and I(X;Y) can be represented by FIG. 10 . For an ideal optical system, the source is transmitted to the sink without distortion, then H(X)=H(Y)=H(XY)=I(X; Y). For the actual optical system, due to the existence of diffraction effects and aberrations, the source and the sink cannot be completely consistent, so the mutual information I(X; Y) is an important index to evaluate the information transmission ability of the optical system. The mutual information I(X; Y) can describe the amount of information that the object information transfers to the image information without distortion after the object is imaged by the optical system.

由互信息的性质可知，I(X；Y)≤H(X)。意味着输出端Y往往只能获得关于输入X的一部分信息。I(X；Y)是信源分布{B(m)}和信道转移系数分布{p_mn}的二元函数，当光学信道特性{p_mn}固定后，I(X；Y)随信源分布{B(m)}的变化而变化。调整{B(m)}，在信宿就能获得不同的互信息量。由互信息的性质已知，I(X；Y)是{B(m)}的上凸函数，因此，总能找到一种物频谱分布{B(m)}，使光学信道所能传送的互信息量最大。定义这个最大的互信息为信道容量，即：According to the nature of mutual information, I(X; Y)≤H(X). Means that the output Y can often only get a part of the information about the input X. I(X; Y) is a binary function of source distribution {B(m)} and channel transfer coefficient distribution {p _mn }. When the optical channel characteristic {p _mn } is fixed, I(X; Y) follows the source Changes in the distribution {B(m)}. By adjusting {B(m)}, different amounts of mutual information can be obtained at the sink. It is known from the nature of mutual information that I(X; Y) is an upward convex function of {B(m)}, therefore, an object spectral distribution {B(m)} can always be found so that the optical channel can transmit The amount of mutual information is the largest. Define this maximum mutual information as the channel capacity, namely:

$C C = = \underset{B B ((m m))}{max max} I I ((X x;; Y Y)) - - - - - - ((1919)) . .$

因此，信道容量可表征光学系统传输信息的极限能力。Therefore, the channel capacity can represent the limit ability of the optical system to transmit information.

Claims

1. an optical system frequency domain information transfer performance analysis method based on improved Fourier transform, it is characterized in that described method step is as follows:

The first step is to improve the expression of Fourier transform and solve the harmonic coefficients of light intensity at all levels:

(1) Expand the one-dimensional light intensity I(x) distribution on the observed object surface into an inverse Fourier transform expression:

I I ((x x)) = = {&Integral; &Integral;}_{- - ∞ ∞}^{∞ ∞} A A ((f f)) exp exp ((22 π π i i f f x x)) d d f f = = {&Integral; &Integral;}_{- - ∞ ∞}^{∞ ∞} A A ((f f)) [[cos cos ((22 π π f f x x)) + + i i sin sin ((22 π π f f x x))]] d d f f,,

In the formula, A(f) is the Fourier transform spectrum function of I(x), x is a one-dimensional spatial position variable, and f is a one-dimensional spatial frequency variable;

(2) Assign zero-frequency components to each harmonic component in the expression of step (1), and ensure that the function values of the harmonics are all non-negative values, so that the improved non-negative light intensity harmonics with zero-frequency components The expression is:

In the formula, i is the imaginary number unit, and B(f) is the coefficient of the expanded harmonic of the improved Fourier transform;

(3) Expand the one-dimensional light intensity I(x) distribution on the observed object surface into an improved Fourier transform expression according to the improved harmonic component of step (2):

(4) the improved Fourier transform expression of step (3) is connected with the Fourier inverse transform expression of step (1), solves the coefficient B (f) of improved Fourier transform expansion harmonic;

The second step is the solution of the imaging integral equation of the optical system and its channel matrix in the frequency domain:

(1) The one-dimensional light intensity distribution I _o (x) on the object plane is formed by the optical system, and the one-dimensional light intensity distribution I _i (x) on the image plane can be obtained from the one-dimensional light intensity distribution I _o (x) on the object plane and the optical system Line diffusion h(x) is obtained by convolution operation, namely:

{I I}_{i i} ((x x)) = = {I I}_{o o} ((x x)) * * h h ((x x)) = = {&Integral; &Integral;}_{- - ∞ ∞}^{+ + ∞ ∞} {I I}_{o o} ((ξ ξ)) \cdot &Center Dot; h h ((x x - - ξ ξ)) d d ξ ξ = = K K [[{I I}_{o o} ((x x))]],,

In the formula, ξ is the intermediate variable of the convolution operation, K is the integral operator of the convolution expression, and the eigenequation of the integral operator K has the following form:

Kφ _f (x) = β _f φ _f (x),

Among them, φ _f (x) is the fth order eigenfunction of the integral operator K, and β _f is the eigenvalue of the fth order eigenfunction of the integral operator K;

(2) Solve the eigenfunctions and eigenvalues of the integral operator to obtain the channel matrix in the frequency domain of the optical system:

Applying the imaging integral operator K to the light intensity harmonics The result of imaging the light intensity harmonics by the optical system can be obtained, namely:

When the cut-off frequency of the optical system is N, the improved non-negative intensity harmonic expression obtained in the first step is The set of normalized coefficients X: {B(0), B(1), B(2),...B(N)} is used as the source, and the set of normalized coefficients Y of the light distribution of the output image: {B' (0), B'(1), B'(2),...B'(N)} as the sink, the information transfer relationship of object and image spectrum can be expressed as:

X·P=Y,

Among them, the frequency domain channel matrix P of the optical system is:

Among them, β _f is the eigenvalue of the fth order eigenfunction of the integral operator K;

The third step, the calculation method of the optical system information transmission performance parameters:

According to the definition of information theory, the source entropy of the object spectrum, the sink entropy of the image spectrum, the joint information entropy of the object spectrum and the image spectrum, the mutual information of the object spectrum and the image spectrum and the channel capacity of the optical system obtained by the improved Fourier transform are carried out. Calculate and evaluate the information transfer performance of the optical system based on these information parameters.

2. The optical system frequency domain information transfer performance analysis method based on the improved Fourier transform according to claim 1, characterized in that in the third step, to improve the non-negative light intensity harmonic expression The normalization coefficient set X of X is used as the information source, then the material spectrum information source entropy is:

H h ((X x)) = = - - {Σ Σ}_{m m = = 11}^{N N + + 11} B B ((m m)) \cdot \cdot {log log}_{22} B B ((m m)) . .

3. The optical system frequency domain information transfer performance analysis method based on the improved Fourier transform according to claim 1, characterized in that in the third step, the normalization coefficient set Y of the light distribution of the image is used as the sink , then the spectrum sink entropy is:

H h ((Y Y)) = = - - {Σ Σ}_{n no = = 11}^{N N + + 11} {B B}^{' '} ((n no)) \cdot \cdot {log log}_{22} {B B}^{' '} ((n no)) . .

4. The optical system frequency-domain information transfer performance analysis method based on the improved Fourier transform according to claim 1, characterized in that in the third step, to improve the light intensity harmonic expression The set of normalized coefficients X of X is used as the source, and the channel matrix P is used as the sink, then the joint information entropy of the object spectrum and the image spectrum is:

H h ((X x Y Y)) = = - - {Σ Σ}_{m m = = 11}^{N N + + 11} {Σ Σ}_{n no = = 11}^{N N + + 11} [[B B ((m m)) \cdot \cdot {p p}_{m m n no}]] \cdot &Center Dot; {log log}_{22} [[B B ((m m)) \cdot \cdot {p p}_{m m n no}]] . .

5. the optical system frequency domain information transfer performance analysis method based on improved Fourier transform according to claim 1, is characterized in that in the described 3rd step, the mutual information amount of object spectrum and image spectrum is:

I(X;Y)=H(X)+H(Y)-H(XY).

6. the optical system frequency domain information transmission performance analysis method based on improved Fourier transform according to claim 1, is characterized in that in the described 3rd step, the channel capacity of optical system is:

C C = = \underset{B B ((i i))}{m m a a x x} I I ((X x;; Y Y)) . .