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

Abstract

Transmit a method for analyzing performance based on the optical system frequency domain information improving Fourier transform, belong to Fourier optics and optical system imaging performance evaluation technical field.Described method is: the first step, the improvement expression formula of Fourier transform and solving of light intensity harmonic constant at different levels; The imaging integral equation of second step, optical system and solving of frequency domain channel matrix thereof; 3rd step, optical system information transmit the calculating of performance parameter, and the information evaluating optical system according to these informations parameter transmits performance.Light intensity is launched into zero-frequency and the linear combination of non-negative light intensity harmonic wave with energy by improvement Fourier's analysis method provided by the invention, analyze the optical system imaging rule under new light intensity method of deploying, and then the analytical approach of application message opinion analyzes the information transmission performance of optical system.The present invention improves Fourier optics analytical approach, makes the theory of Fourier optics more perfect, and achieves the combination of optical analysis method and Information theoretic analysis method.

Description

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

Technical field

The invention belongs to Fourier optics and optical system imaging performance evaluation technical field, relate to a kind of based on improvement Fourier transform and information-theoretical optical system imaging method for analyzing performance.

Background technology

Modulation transfer function (MTF) is the important indicator that Optical System Design, image quality analysis and field of optical detection are commonly used, and its theoretical foundation derives from the correlation theory about incoherent imaging system optical transfer function in the monograph such as " information optics ", " Fourier optics ".Be normalized after the point spread function of light intensity can be made Fourier transform by the MTF of imaging optical system and obtain, analyze from the angle of optical system merely, this method is right-on in theory.

And in imaging analysis process, as the mould of light intensity frequency spectrum can be obtained by the mould of the light intensity frequency spectrum of thing and MTF product.For this reason, need the spatial domain light distribution of thing to do Fourier analysis, the spatial domain light distribution of thing is launched into the linear combination of the harmonic wave of different space frequency.But the harmonic waves at different levels launched according to Fourier's analysis method are cosine and the sine function of different frequency, and harmonic function at different levels is all containing negative value.Because harmonic function value does not meet nonnegativity, spectrum harmonics at different levels independently cannot represent light intensity (because light intensity can not be negative value), can only be superimposed upon correction zero-frequency component carrying out details to zero-frequency component, jointly represent light intensity with zero-frequency component.The existence of this problem result in Fourier optics utilize MTF analyze thing, as light intensity Spectrum Relationship time, thing, independently can not represent light intensity as spectrum harmonics at different levels, can not information be represented, thus cannot with the frequency domain information transfer law of information-theoretical analytical approach Optical system.

In addition, in the measurement links of optical system MTF, usually use the cosine light intensity transmitance grating of characteristic frequency as testee, determine the mtf value of optical system at this frequency place by the change of the degree of modulation detecting the picture of this cosine grating.This grating through light intensity there is nonnegativity, the light distribution of grating image has nonnegativity equally.But the harmonic waves at different levels in Fourier transform do not meet nonnegativity, make in the test experience of optical system MTF, also to there is the problem be not inconsistent with Fourier optics theory.

In traditional Fourier optics analytical approach, the imaging process of optical system can be described as two links, and one is that the spatial domain Light distribation of thing obtains thing frequency spectrum after Fourier transform, and two is moulds that MTF that the mould of thing frequency spectrum is multiplied by optical system can obtain picture frequency spectrum.The spatial domain Light distribation of thing is launched into the sine of zero-frequency and noenergy, the linear combination of cosine oscillation ripple by Fourier optics analytical approach, be set up in theory, but meet difficulty in the information transfer law of application message opinion methods analyst optical system and the theoretical explanation of MTF test experiments.Because sinusoidal, cosine oscillation ripple do not possess energy, do not meet nonnegativity, so cannot light intensity be represented, thus cannot information be represented; The transmitance of the cosine grating used in MTF test experiments is more than or equal to zero, through light intensity be non-negative, the light intensity of the cosine grating picture obtained also is non-negative, and Fourier optics theory cannot explain this problem exactly.

Fourier transform is improved, solve expansion harmonic wave at different levels in Fourier transform and do not meet the problem of nonnegativity, spectrum harmonics at different levels can be made to represent light intensity independently, the experiments of measuring link of MTF can be made to conform to theory, and the frequency domain information of the analytical approach Optical system of all right application message opinion transmits performance.The solution of technique, to the further investigation of optical system frequency domain imaging performance, the theoretical explanation of optical system MTF test experience, and optics and the architectonic combination of information theory all have great importance.

Summary of the invention

The object of this invention is to provide a kind of optical system frequency domain information based on improving Fourier transform and transmit method for analyzing performance, the spatial distribution functional expansion of light intensity can be become the linear combination of the light intensity harmonic waves at different levels of a series of non-negative by the method, and then information-theoretical analytical approach and parameter can be used for optical system imaging performance evaluation.

Optical imaging system belongs to incoherent imaging system, its focal plane photodetectors register be light distribution, cannot recording light wave amplitude.In order to carry out frequency-domain analysis, need the harmonic linear array configuration two-dimensional illumination intensity distribution on observation object plane being launched into inverse Fourier transform.

The strong I of object light (x, y) regards a series of primitive harmonic function exp (2 π if of different space frequency as _xx+2 π if _yy) linear combination, wherein, i is imaginary unit.Because two-dimensional spatial location variable x, y oppose mutually, two dimensional spatial frequency variable f _x, f _yalso be mutually oppose, therefore, only can consider one-dimensional case.Real part A (f) cos (2 π fx) in the primitive harmonic function exp (2 π ifx) of the one dimension of the strong I (x) of object light and imaginary part A (f) sin (2 π fx) represents cosine, sine-wave oscillation ripple, wherein, x is one-dimensional space location variable, and f is one-dimensional space frequency variable.There is negative value in cosine, sine-wave oscillation wave function value, independently can not exist as light intensity harmonic wave, can only be superimposed upon on zero-frequency component, stack result be existed as light intensity form.The present invention is directed to one dimension thing, as carrying out improvement Fourier analysis with optical system and information transmits performance evaluation, for two dimension thing, as with optical system, One Dimension Analysis method of the present invention can be adopted to carry out One Dimension Analysis to the information transfer law on different directions.

Analytical approach of the present invention is divided into three steps.

The first step, the improvement expression formula of Fourier transform and solving of light intensity harmonic constant at different levels:

(1) one dimension light intensity I (x) distribution on observation object plane is launched into inverse Fourier transform expression formula:

$I\left(x\right)={\∫}_{-\∞}^{\∞}A\left(f\right)\exp \left(2\mathrm{\πifx}\right)\mathrm{df}={\∫}_{-\∞}^{\∞}A\left(f\right)[\cos \left(2\mathrm{\πfx}\right)+i\sin \left(2\mathrm{\πfx}\right)]\mathrm{df};$

(2) give zero-frequency component to each harmonic component in step (1) expression formula, and ensure that the functional value of harmonic wave is nonnegative value, the expression formula with the non-negative light intensity harmonic wave of zero-frequency component after order improves is:

(3) one dimension light intensity I (x) distribution on observation object plane is launched into improvement Fourier transform expression formula according to the harmonic component that step (2) is improved:

(4) by the inverse Fourier transform expression formula simultaneous of the improvement Fourier transform expression formula of step (3) and step (1), solve and improve the coefficient B (f) that Fourier transform launches harmonic wave.

The imaging integral equation of second step, optical system and solving of frequency domain channel matrix thereof:

(1) object plane one dimension light distribution I _ox image planes one dimension light distribution I that () is formed after optical system _ix () can by object plane one dimension light distribution I _ox the line of () and optical system spreads h (x) carries out convolution algorithm and tries to achieve, that is:

$I_i\left(x\right)=I_o\left(x\right)*h\left(x\right)={\∫}_{-\∞}^{+\∞}{I}_o\left(\mathrm{\ξ}\right)\·h(x-\mathrm{\ξ})\mathrm{d\ξ}=K\left[I_o\left(x\right)\right];$

Wherein, ξ is the intermediate variable of convolution algorithm, and Κ is the integral operator of convolution expression formula,

$K={\∫}_{-\∞}^{\∞}h(x-\mathrm{\ξ})\mathrm{d\ξ};$

The eigen[value of integral operator Κ has following form:

Kφ _f(x)＝β _fφ _f(x)；

Wherein, φ _fx f rank eigenfunction that () is integral operator Κ, β _ffor the eigenwert of the f rank eigenfunction of integral operator Κ;

(2) eigenfunction of integral operator and eigenwert are solved, obtain optical system frequency domain channel matrix:

Imaging integral operator Κ is acted on light intensity harmonic wave , optical system can be obtained to the result after light intensity harmonic imaging, that is:

When the cutoff frequency of optical system is N, the improvement non-negative light intensity harmonic wave expression formula that the first step is tried to achieve normalization coefficient set X:{B (0), B (1), B (2), B (N) } as information source, export the photodistributed normalization coefficient set Y:{B ' (0) of picture, B ' (1), B ' (2), B ' (N) } as the stay of two nights, then the information transfering relation of thing, picture frequency spectrum can be expressed as:

X·P＝Y，

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

3rd step, optical system information transmit the computing method of performance parameter:

According to information-theoretical definition, the channel capacity improving mutual information that united information entropy, thing frequency spectrum and picture frequency that the thing frequency spectrum information source entropy of Fourier transform gained, picture frequency spectrum stay of two nights entropy, thing frequency spectrum and picture frequency compose compose and optical system is calculated, the information evaluating optical system according to these informations parameter transmits performance, wherein:

Thing frequency spectrum information source entropy is:

$H\left(X\right)=-\underset{m=1}{\overset{N+1}{\mathrm{\Σ}}}B\left(m\right)\·{\log }_{2}B\left(m\right);$

Picture frequency spectrum stay of two nights entropy is:

$H\left(Y\right)=-\underset{n=1}{\overset{N+1}{\mathrm{\Σ}}}{B}^{\′}\left(n\right)\·{\log }_2{B}^{\′}\left(n\right);$

The united information entropy that thing frequency spectrum and picture frequency are composed is:

$H\left(\mathrm{XY}\right)=-\underset{m=1}{\overset{N+1}{\mathrm{\Σ}}}\underset{n=1}{\overset{N+1}{\mathrm{\Σ}}}[B\left(m\right)\·p_{\mathrm{mn}}]\·{\log }_2[B\left(m\right)\·p_{\mathrm{mn}}];$

The mutual information that thing frequency spectrum and picture frequency are composed is:

I(X；Y)＝H(X)+H(Y)-H(XY)；

The channel capacity of optical system is:

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

Mutual information is the size of the part quantity of information of the information source that can correctly transmit in the channel, is also the size of the information source information amount comprised in the stay of two nights after channel transmission.Therefore, the size of mutual information can reflect the information transfer capacity of optical system to thing frequency spectrum.And channel capacity is the size of the maximum mutual information that optical system can be transmitted, channel capacity can characterize the limit capacity of a channel transmission information.

Light intensity is launched into zero-frequency and the linear combination of non-negative light intensity harmonic wave with energy by improvement Fourier's analysis method provided by the invention, analyze the optical system imaging rule under new light intensity method of deploying, and then the analytical approach of application message opinion analyzes the information transmission performance of optical system.The present invention improves Fourier optics analytical approach, makes the theory of Fourier optics more perfect, and achieves the combination of optical analysis method and Information theoretic analysis method.

Accompanying drawing explanation

Fig. 1 is the schematic diagram in Fourier optics analytical approach, light intensity being expanded into zero-frequency and cosine oscillation ripple, (a) certain light distribution I (x), (b) zero-frequency component, and (c) is without the cosine oscillation ripple of zero-frequency component;

Cosine oscillation ripple is added that zero-frequency component changes the principle of non-negative cosine light intensity harmonic wave into by Fig. 2, a () has the cosine light intensity harmonic wave of zero-frequency component, b (), without the cosine oscillation ripple of zero-frequency component, (c) needs the zero-frequency component giving cosine oscillation;

Light distribution I (x) is launched into the schematic diagram of non-negative light intensity harmonic wave by Fig. 3 in Fourier's analysis method for improving, (a) certain light distribution I (x), b () zero-frequency component distributes to cosine oscillation ripple, (c) has the cosine light intensity harmonic wave of zero-frequency component;

Fig. 4 is without the cosine wave (CW) wave of oscillation of zero-frequency through the schematic diagram of optical system imaging process, (a) without the cosine oscillation ripple of zero-frequency component, the cosine oscillation ripple without zero-frequency component after (b) imaging;

Fig. 5 is without the frequency spectrum of the cosine wave (CW) wave of oscillation of zero-frequency through the schematic diagram of optical system imaging process, the frequency spectrum of (a) cosine oscillation ripple 3, the MTF of (b) optical system, (c) cosine oscillation ripple as 7 frequency spectrum;

Fig. 6 is containing the non-negative light intensity harmonic wave of zero-frequency component through the schematic diagram of optical system imaging process, (a) containing the cosine light intensity harmonic wave of zero-frequency component, the picture that (b) cosine light intensity harmonic wave becomes through optical system;

Fig. 7 is containing the non-negative light intensity harmonic spectrum of zero-frequency component through the schematic diagram of optical system imaging process, the frequency spectrum of (a) cosine light intensity harmonic wave 4, the MTF of (b) optical system, (c) cosine light intensity harmonic wave as 11 frequency spectrum;

Fig. 8 is the schematic diagram being transferred to the former zero-frequency component of picture containing the light intensity harmonic wave of zero-frequency through optical system imaging rear section zero-frequency component, a picture 11 that () cosine light intensity harmonic wave becomes through optical system, the cosine light intensity harmonic wave of (b) zero-frequency and the decay of cosine harmonics equal proportion, (c) cosine light intensity harmonic wave 11 deducts the zero-frequency component of cosine light intensity harmonic wave 14 gained difference;

Fig. 9 is the schematic diagram being transferred to the former zero-frequency component of picture containing the frequency spectrum of light intensity harmonic wave of zero-frequency through optical system imaging rear section zero-frequency component, the frequency spectrum of (a) cosine light wave 11 after optical system imaging, the frequency spectrum of b cosine light intensity harmonic wave 14 that () zero-frequency and cosine harmonics equal proportion decay, c () is transferred to the part of picture zero-frequency component, i.e. the difference of 13 and 16;

Figure 10 is the graph of a relation between information source entropy, stay of two nights entropy, combination entropy and mutual information.

Embodiment

Below in conjunction with accompanying drawing, technical scheme of the present invention is further described; but be not limited thereto; everyly technical solution of the present invention modified or equivalent to replace, and not departing from the spirit and scope of technical solution of the present invention, all should be encompassed in protection scope of the present invention.

The invention provides a kind of optical system frequency domain information based on improving Fourier transform and transmit method for analyzing performance, be divided into following three steps:

The first step: improve the expression formula of Fourier transform and solving of each light intensity harmonic constant.

According to the expression way of one-dimensional Fourier transform, complex amplitude u (x) and its frequency spectrum function a (f) Fourier pair each other of the monochromatic light field that object plane sends, meet following relation:

$a\left(f\right)={\∫}_{-\∞}^{\∞}u\left(x\right)\exp (-2\mathrm{\πifx})\mathrm{dx}---\left(1\right);$

$u\left(x\right)={\∫}_{-\∞}^{\∞}a\left(f\right)\exp (-2\mathrm{\πifx})\mathrm{df}---\left(2\right).$

Formula (2) can be regarded as u (x) and is made up of the linear combination of the series of harmonic function exp of different space frequency f (2 π ifx), and the coefficient of each harmonic function is a (f).

Optical imaging system belongs to incoherent imaging system, its focal plane photodetectors register be light distribution, cannot recording light wave amplitude.Therefore, need one dimension light distribution I (x) on observation field scenery face to be launched into the form similar to the inverse Fourier transform of formula (2), shown in (3):

$I\left(x\right)={\∫}_{-\∞}^{\∞}A\left(f\right)\exp \left(2\mathrm{\πifx}\right)\mathrm{df}={\∫}_{-\∞}^{\∞}A\left(f\right)[\cos \left(2\mathrm{\πfx}\right)+i\sin \left(2\mathrm{\πfx}\right)]\mathrm{df}---\left(3\right).$

I (x) can regard the linear combination of the series of harmonic function exp (2 π ifx) of different space frequency as, the coefficient of each harmonic function is A (f), and A (f) is the Fourier transform frequency spectrum function of I (x).Real part A (f) cos (2 π fx) in harmonic function and imaginary part A (f) sin (2 π fx) is cosine, sine-wave oscillation ripple, there is negative value in oscillatory wave function value, can not independently exist as light intensity, can only be superimposed upon on zero-frequency component, stack result light intensity form is as a whole existed.Figure 1 shows that the form that the cosine oscillation ripple 3 expression formula I (x) of a certain light distribution 1 being in the x-direction launched into zero-frequency component 2 and certain frequency according to traditional Fourier optics analytical approach synthesizes.Cosine oscillation ripple 3 has negative value composition to exist, and is zero along the integrated value of x-axis, does not possess energy, cannot exist with independently light intensity harmonic wave form, can only to be superimposed upon on zero-frequency component 2 and jointly to represent light distribution 1 with zero-frequency component 2.The harmonic function of Fourier transform is for containing negative value composition cosine function and sine function, and sine function is identical with the analysis and processing method of cosine function.

If I (x) light intensity to be regarded as the linear combination result of each independently light intensity harmonic wave light component, then require that the functional value of each harmonic component can not occur negative value.In order to make each harmonic component have energy, and ensure that the functional value of harmonic component is non-negative, need to harmonic wave just, cosine Sasser adds certain zero-frequency component.The expression formula of the non-negative light intensity harmonic wave containing zero-frequency composition after order improves is:

Wherein, B (f) is the coefficient of each light intensity harmonic wave.

Because the zero-frequency component after improving is still constant, it can be made to be:

(5) formula is substituted into (4) formula obtain:

As shown in Figure 2, in order to make cosine oscillation wave function 3 can represent light intensity with the form of independently sub-light wave, needing to give certain zero-frequency component 5 to the string function wave of oscillation 3, thus the string function wave of oscillation 3 is transformed into the form of the independent light intensity harmonic wave 4 containing energy.The minimum value of light intensity harmonic wave 4 is 0, but there will not be negative value.

With be integrated A (f) exp (2 π ifx) in alternate form (3), each light intensity harmonic component can be ensured do not occur negative value, then the expression formula of Fourier transform intensity distribution function I (x) that can be improved is such as formula shown in (7):

By solving the system of equations that formula (3) and formula (7) are formed, obtain and improve Fourier transform and launch harmonic constant B (f), the non-negative light intensity harmonic constant expression formula after solving is:

$\left\{\begin{array}{cc}B\left(f\right)=A\left(f\right),& f\≠0\\ B\left(0\right)=\frac{A\left(0\right)-(1+i){\∫}_{-\∞}^{0-}\left|B\left(f\right)\right|\mathrm{df}+(1+i){\∫}_{0+}^{\∞}\left|B\left(f\right)\right|\mathrm{df}}{(1+i)},& f=0\end{array}\right.---\left(8\right)$

If $A\left(0\right)<(1+i){\&Integral;}_{-\&infin;}^{0+}\left|B\left(f\right)\right|\mathrm{df}+(1+i){\&Integral;}_{0+}^{\&infin;}\left|B\left(f\right)\right|\mathrm{df},$ A (0) can be increased certain zero-frequency component to ensure B (0)=0.And A (0) is when increasing certain zero-frequency component, only changes the background luminance of thing light distribution I (x), do not change non-zero-frequency harmonic constant A (f) at different levels.B (f) is substituted into (7) formula, just can obtain by the light intensity harmonic waves at different levels of non-negative the expression formula of the light intensity I (x) of synthesized form.

As shown in Figure 3, light distribution 1 is launched into again the form that zero-frequency component 6 synthesizes with the cosine light intensity harmonic wave 4 with energy of corresponding frequencies.Wherein, zero-frequency component 6 equals former zero-frequency component 2 and deducts the zero-frequency component 5 that cosine harmonics light intensity 4 gets.

Second step: the imaging integral equation of optical system and solving of frequency domain channel matrix thereof.

The improvement light intensity harmonic wave expression formula that the first step is obtained be observed thing as optical system, utilize the intrinsic theory of optical system imaging integral equation to solve picture, thus write out the frequency domain channel matrix P of optical system.

Under one-dimensional case, object plane light distribution I _ox () forms image planes light distribution I after optical system _ix () can by object plane light distribution I _ox convolution that () spreads h (x) with the line of optical system is tried to achieve,

That is:

$I_i\left(x\right)=I_o\left(x\right)*h\left(x\right)={\&Integral;}_{-\&infin;}^{+\&infin;}{I}_o\left(\mathrm{\&xi;}\right)\&CenterDot;h(x-\mathrm{\&xi;})\mathrm{d\&xi;}=K\left[I_o\left(x\right)\right]---\left(9\right).$

Wherein, Κ is the integral operator of convolution expression formula,

$K={\&Integral;}_{-\&infin;}^{\&infin;}h(x-\mathrm{\&xi;})\mathrm{d\&xi;}---\left(10\right).$

The eigen[value of integral operator Κ has following form:

Κφ _f(x)＝β _fφ _f(x)(11)。

Wherein, φ _fx f rank eigenfunction that () is integral operator Κ, β _ffor the eigenwert of the f rank eigenfunction of integral operator Κ.

According to the basic theories of Fourier optics, optical system can regard linear empty invariant system as in isoplanatic region, and complex-exponential function is the eigenfunction of linear invariant system, and therefore, A (f) exp (2 π ifx) is the eigenfunction of integral operator Κ.As the strong I of object light _owhen () is for eigenfunction A (f) exp (2 π ifx) x, now light intensity I _ox () only comprises a spectrum component f, according to the frequency domain characteristic of optical system in Fourier optics, as light intensity I _ix () is α _fa (f) exp (2 π ifx), wherein, α _ffor optical system is at the mtf value at spatial frequency f place.So, corresponding to the eigenwert β of eigenfunction A (f) exp (2 π ifx) _fequal the functional value α of MTF at spatial frequency f place _f.

As shown in Figure 4, according to Fourier optics theory, what the cosine oscillation ripple 3 of noenergy obtained through optical system similarly is the cosine oscillation ripple 7 that amplitude is attenuated, and its amplitude fading scale factor is the mtf value α corresponding with its frequency content _f.Cosine oscillation ripple 3 and 7 is linear, and cosine oscillation ripple 3 meets the eigen[value (11) of integral operator Κ, so cosine oscillation ripple 3 is eigenfunctions of integral operator Κ, α _ffor its eigenwert, so α _f=β _f.Cosine oscillation ripple 3,7 is containing negative value composition, light intensity cannot be represented with the form of independently sub-light wave, can only be superimposed upon in zero-frequency light intensity with the form of the cosine oscillation ripple of noenergy and represent light distribution, so the imaging process shown in Fig. 4 is only set up in theory.Actual optical detection experiment adopts the grating of cosine function transmitance as measured object, cosine grating through light intensity become cosine distribution, but there will not be the Light distribation 3 containing negative light intensity shown in Fig. 4 (a), the picture of cosine grating also there will not be the Light distribation 7 containing negative light intensity shown in Fig. 4 (b).

As shown in Figure 5, according to Fourier optics theory, the cosine oscillation ripple 3 of the noenergy shown in Fig. 4 (a) is obtained the thing frequency spectrum function 8 shown in Fig. 5 (a) as Fourier transform, and the modulation transfer function (MTF) 9 that thing frequency spectrum function 8 is multiplied by optical system can obtain the frequency spectrum 10 of picture 7.Because the cosine oscillation ripple 3 in Fig. 4 (a) does not all meet nonnegativity with the cosine oscillation ripple 7 in Fig. 4 (b), can not exist with the form of independently light intensity harmonic wave, the optical system then described in Fig. 5 to input frequency spectrum 8 with export the transfer law that transfer law that picture frequency composes 10 is also only the cosine oscillation ripple 3,7 to noenergy, the transfer law of the cosine light intensity harmonic wave with energy can not be described, can not the imaging law of cosine grating in accurate description optical detection experiment.

Imaging integral operator Κ is acted on the light intensity harmonic wave shown in formula (4) , optical system can be obtained to the result after light intensity harmonic imaging, that is:

Formula (12) shows, when input light distribution is time, having as light distribution of output with two kinds of frequency contents, due to β _f≤ 1, so can think the process transmitted in optical system of energy in, amplitude be attenuated, the zero-frequency composition that it is attenuated is transferred to the zero-frequency component of picture

As shown in Figure 6, the form that light intensity harmonic wave 4 can be used as independently sub-light wave represents light intensity harmonic wave, after optical system imaging, the zero-frequency component 5 of light intensity harmonic wave 4 remains unchanged, and the amplitude of cosine harmonics can be decayed and become cosine harmonics 11, the scale factor of decay is the mtf value α at corresponding frequencies place _f.

As shown in Figure 7, cosine wave (CW) light intensity harmonic wave 4 shown in Fig. 6 (a) is obtained the thing frequency spectrum function 12 shown in Fig. 7 (a) as Fourier transform, and the modulation transfer function (MTF) 9 that thing frequency spectrum function 12 is multiplied by optical system can obtain the frequency spectrum 13 of picture 11.

As shown in Figure 8, there is not linear ratio relation with the input light intensity harmonic wave 4 of Fig. 6 (a) in the picture 11 of light intensity harmonic wave.By light intensity harmonic wave as 11 zero-frequency component get rid of a part (zero-frequency component 15), what make light intensity harmonic wave changes as 11 the form that minimum value is the light intensity harmonic wave 14 of zero into, now, the cosine light intensity harmonic wave 4 of input meets linear ratio relation with the cosine light intensity harmonic wave 14 exported, and harmonic wave 4 and the scale factor of harmonic wave 14 are the mtf value α at corresponding frequencies place _f.The part zero-frequency component 15 be removed can think that transfer gives picture light wave original zero-frequency component.

As shown in Figure 9, cosine wave (CW) light intensity harmonic wave shown in Fig. 8 (a) is obtained the picture frequency spectral function 13 shown in Fig. 9 (a) as 11 as Fourier transform, part zero-frequency component 17 in picture frequency spectrum 13 is removed, just the frequency spectrum 16 of cosine harmonics 14 correspondence can be obtained, frequency spectrum 16 is linearly proportional with the input spectrum 12 in Fig. 7 (a), and scale factor is the mtf value α at corresponding frequencies place _f.

From information-theoretical angle analysis, when the cutoff frequency of optical system is N, get normalization light intensity harmonic constant X:{B (0) of thing, B (1), B (2) ... B (N) } as information source.After information source is input to optical system, known according to formula (12), the photodistributed information of normalization exporting picture 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 ₁b (1), β ₂b (2) ..., β _nb (N) }, wherein, β ₁, β ₂... β _nfor 1,2 of integral operator Κ ..., N rank eigenwert, equals optical system in spatial frequency 1,2 ..., the mtf value α at N place ₁, α ₂... α _n.According to information-theoretical model, can regard the normalization coefficient set (vector) of strong for object light harmonic wave as information source, the stay of two nights is regarded in the normalization coefficient set (vector) as light overtone as, and optical system can regard channel as.According to formula (12), the channel matrix P of optical system can be write as the form of formula (13):

Wherein, p _mnfor the element of channel matrix P, subscript m, n are the element numbers in matrix P.

Then the information transfering relation of thing, picture frequency spectrum can be expressed as

X·P＝Y(14)。

3rd step: optical system frequency domain information transmits the computing method of performance parameter

The improvement light intensity harmonic wave expression formula that the first step is tried to achieve normalization coefficient set (vector) X as information source, the normalization coefficient set (vector) of the optical system frequency domain channel matrix P obtained in conjunction with second step and picture, for Y is as the stay of two nights, can obtain the information evaluation index mutual information I (X of optical system; Y) with channel capacity C.

According to information-theoretical definition, quantity of information and the information source entropy of the improvement Fourier transform frequency spectrum (information source) of thing are:

$H\left(X\right)=-\underset{m=1}{\overset{N+1}{\mathrm{\&Sigma;}}}B\left(m\right)\&CenterDot;{\log }_{2}B\left(m\right)---\left(15\right),$

Wherein, m is the sequence number of element in X.

Correspondingly, as the quantity of information of improvement Fourier transform frequency spectrum (stay of two nights) and stay of two nights entropy be:

$H\left(Y\right)=-\underset{n=1}{\overset{N+1}{\mathrm{\&Sigma;}}}{B}^{\&prime;}\left(n\right)\&CenterDot;{\log }_2{B}^{\&prime;}\left(n\right)---\left(16\right),$

Wherein, n is the sequence number of element in Y.

The united information entropy of information source and the stay of two nights is:

$H\left(\mathrm{XY}\right)=-\underset{m=1}{\overset{N+1}{\mathrm{\&Sigma;}}}\underset{n=1}{\overset{N+1}{\mathrm{\&Sigma;}}}[B\left(m\right)\&CenterDot;p_{\mathrm{mn}}]\&CenterDot;{\log }_2[B\left(m\right)\&CenterDot;p_{\mathrm{mn}}]---\left(17\right).$

The mutual information of information source and the stay of two nights is:

I(X；Y)＝H(X)+H(Y)-H(XY)(18)。

Mutual information I (X; Y) be the parameter of size of the effective information describing channel.If information source entropy is H (X), wish that the quantity of information received at the output terminal of channel is also H (X).But because channel is undesirable, generally can only receive a part of H (X) at output terminal, i.e. mutual information I (X; Y).From the angle of information source, mutual information I (X; Y) be the size of part quantity of information of the information source H (X) that can correctly transmit in the channel; From the angle of the stay of two nights, mutual information I (X; Y) after referring to that channel transmits, the size of the quantity of information of the information source H (X) comprised in stay of two nights H (Y).

H (X), H (Y), H (XY), I (X; Y) relation between can be represented by Figure 10.For perfect optical system, information source passes to the stay of two nights undistortedly, then H (X)=H (Y)=H (XY)=I (X; Y).And for the optical system of reality, due to the existence of diffraction effect and aberration, information source and the stay of two nights can not be completely the same, then mutual information I (X; Y) be the important indicator evaluating optical system information transfer capacity.Mutual information I (X; Y) can describe thing after optical system imaging, thing information is undistorted passes to the size of quantity of information of picture information.

From the character of mutual information, I (X; Y)≤H (X).Mean that output terminal Y often can only obtain a part of information about input X.I (X; Y) be information source distribution { B (m) } and channel transfer coefficients distribution { p _mnbinary function, as optical channel characteristic { p _mnfixing after, I (X; Y) source distribution with the letter { B (m) } change and change.Adjustment { B (m) }, just can obtain different mutual informations in the stay of two nights.Known by the character of mutual information, I (X; Y) be the Convex Functions of { B (m) }, therefore, can find a kind of thing spectrum distribution { B (m) }, the mutual information that optical channel can be transmitted is maximum.Defining this maximum mutual information is channel capacity, that is:

$C=\underset{B\left(m\right)}{\max }I(X;Y)---\left(19\right).$

Therefore, channel capacity can characterize the limit capacity of optical system transmission information.

Claims (6)

1. transmit a method for analyzing performance based on the optical system frequency domain information improving Fourier transform, it is characterized in that described method step is as follows:

The first step, the improvement expression formula of Fourier transform and solving of light intensity harmonic constant at different levels:

(1) one dimension light intensity I (x) distribution on observation object plane is launched into inverse Fourier transform expression formula:

$I\left(x\right)={\&Integral;}_{-\mathrm{\&infin;}}^{\mathrm{\&infin;}}A\left(f\right)\exp \left(2\mathrm{\&pi;}ifx\right)df={\&Integral;}_{-\mathrm{\&infin;}}^{\mathrm{\&infin;}}A\left(f\right)\&lsqb;\cos \left(2\mathrm{\&pi;}fx\right)+i\sin \left(2\mathrm{\&pi;}fx\right)\&rsqb;df,$

In formula, the Fourier transform frequency spectrum function that A (f) is I (x), x is one-dimensional space location variable, and f is one-dimensional space frequency variable;

(2) give zero-frequency component to each harmonic component in step (1) expression formula, and ensure that the functional value of harmonic wave is nonnegative value, the expression formula with the non-negative light intensity harmonic wave of zero-frequency component after order improves is:

In formula, i is imaginary unit, and B (f) improves the coefficient that Fourier transform launches harmonic wave;

(3) one dimension light intensity I (x) distribution on observation object plane is launched into improvement Fourier transform expression formula according to the harmonic component that step (2) is improved:

(4) by the inverse Fourier transform expression formula simultaneous of the improvement Fourier transform expression formula of step (3) and step (1), solve and improve the coefficient B (f) that Fourier transform launches harmonic wave;

The imaging integral equation of second step, optical system and solving of frequency domain channel matrix thereof:

(1) object plane one dimension light distribution I _ox image planes one dimension light distribution I that () is formed after optical system _ix () can by object plane one dimension light distribution I _ox the line of () and optical system spreads h (x) carries out convolution algorithm and tries to achieve, that is:

$I_i\left(x\right)=I_o\left(x\right)*h\left(x\right)={\&Integral;}_{-\mathrm{\&infin;}}^{+\mathrm{\&infin;}}{I}_o\left(\mathrm{\&xi;}\right)\&CenterDot;h(x-\mathrm{\&xi;})d\mathrm{\&xi;}=K\&lsqb;I_o\left(x\right)\&rsqb;,$

In formula, ξ is the intermediate variable of convolution algorithm, and K is the integral operator of convolution expression formula, and the eigen[value of integral operator K has following form:

Kφ _f(x)＝β _fφ _f(x)，

Wherein, φ _fx f rank eigenfunction that () is integral operator K, β _ffor the eigenwert of the f rank eigenfunction of integral operator K;

(2) eigenfunction of integral operator and eigenwert are solved, obtain optical system frequency domain channel matrix:

Imaging integral operator K is acted on light intensity harmonic wave optical system can be obtained to the result after light intensity harmonic imaging, that is:

When the cutoff frequency of optical system is N, the improvement non-negative light intensity harmonic wave expression formula that the first step is tried to achieve normalization coefficient set X:{B (0), B (1), B (2), B (N) } as information source, export the photodistributed normalization coefficient set Y:{B ' (0) of picture, B ' (1), B ' (2), B ' (N) } as the stay of two nights, then the information transfering relation of thing, picture frequency spectrum can be expressed as:

X·P＝Y，

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

Wherein, β _ffor the eigenwert of the f rank eigenfunction of integral operator K;

3rd step, optical system information transmit the computing method of performance parameter:

According to information-theoretical definition, the channel capacity improving mutual information that united information entropy, thing frequency spectrum and picture frequency that the thing frequency spectrum information source entropy of Fourier transform gained, picture frequency spectrum stay of two nights entropy, thing frequency spectrum and picture frequency compose compose and optical system is calculated, evaluates the information transmission performance of optical system according to these informations parameter.

2. the optical system frequency domain information based on improving Fourier transform according to claim 1 transmits method for analyzing performance, it is characterized in that in described 3rd step, to improve non-negative light intensity harmonic wave expression formula normalization coefficient set X as information source, then thing frequency spectrum information source entropy is:

$H\left(X\right)=-\underset{m=1}{\overset{N+1}{\&Sigma;}}B\left(m\right)\&CenterDot;{\log }_{2}B\left(m\right).$

3. the optical system frequency domain information based on improving Fourier transform according to claim 1 transmits method for analyzing performance, it is characterized in that in described 3rd step, and using the photodistributed normalization coefficient set Y of picture as the stay of two nights, then picture frequency spectrum stay of two nights entropy is:

$H\left(Y\right)=-\underset{n=1}{\overset{N+1}{\&Sigma;}}{B}^{\&prime;}\left(n\right)\&CenterDot;{\log }_2{B}^{\&prime;}\left(n\right).$

4. the optical system frequency domain information based on improving Fourier transform according to claim 1 transmits method for analyzing performance, it is characterized in that in described 3rd step, to improve light intensity harmonic wave expression formula normalization coefficient set X as information source, channel matrix P is as the stay of two nights, then the united information entropy that thing frequency spectrum and picture frequency are composed is:

$H\left(XY\right)=-\underset{m=1}{\overset{N+1}{\&Sigma;}}\underset{n=1}{\overset{N+1}{\&Sigma;}}\&lsqb;B\left(m\right)\&CenterDot;p_{mn}\&rsqb;\&CenterDot;{\log }_2\&lsqb;B\left(m\right)\&CenterDot;p_{mn}\&rsqb;.$

5. the optical system frequency domain information based on improving Fourier transform according to claim 1 transmits method for analyzing performance, it is characterized in that in described 3rd step, the mutual information that thing frequency spectrum and picture frequency are composed is:

I(X；Y)＝H(X)+H(Y)-H(XY)。

6. the optical system frequency domain information based on improving Fourier transform according to claim 1 transmits method for analyzing performance, it is characterized in that in described 3rd step, the channel capacity of optical system is:

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