Analysis Method for Information Transmission Characteristics of Optical System in Frequency Domain

REN Zhibin，HU Jiasheng，ZHI Xiyang，YUE Shuai，TANG Honglang，LI Mingliang



Analysis Method for Information Transmission Characteristics of Optical System in Frequency Domain

REN Zhibin，HU Jiasheng，ZHI Xiyang，YUE Shuai，TANG Honglang，LI Mingliang

( School of Astronautics,Harbin Institute of Technology, Harbin 150001, China)

A novel method is proposed to study optical imaging system using the information theory. The light intensity function is decomposed into linearity combination of a series of harmonic waves without negative value by improved Fourier transform. The normalized harmonic wave coefficient set of object is regarded as information source and that of image is regarded as information home. The optical information channel matrix is obtained by eigenvalue of imaging integral equations. Moreover, average mutual information and information channel capacity are calculated that can evaluate the information transmission ability of the optical system.

information theory; information entropy; average mutual information; information channel capacity; improved Fourier transform

0 Introduction

Information theory is developed by Claude E. Shannon and it is widely used in communication field. From then on, information theory is applied in many other areas of data analysis. Task-specific information of an image is quantified based on information theory[1]. Evaluating method for barcode reading systems is provided by using average mutual information[2]. Information projection matrix is constructed to recover adaptive compressive sensing signal[3]. Digital focusing method of OCT images is proposed based on information entropy[4]. Improved fuzzy C-means algorithm is presented for image segmentation with local information[5]. However, the information transmission ability of optical imaging system can seldom be found.

In this paper, we use information theory to study optical imaging system. The object to be imaged by an optical system is regarded as information source, the image is regarded as information home, and the optical system is regarded as information channel. The light intensity is decomposed into linearity combination of a series of harmonic waves without negative value by improved Fourier transform. Moreover, the optical information channel matrix is obtained and the information transmission relationship between object and image through optical system is established. And computer simulation proves the validity of the proposed method.

1 Non-negative Harmonic Waves Based on Improved Fourier Transform

In an optical imaging system, the photoelectric detector on its focal plane is able to record the light intensity and unable to record the light amplitude. According to Fourier transform, the one-dimensional (1-D) light intensity function() can be expressed by its 1-D space spectrum function() as follow：

In Eq. (1),() is decomposed into linearity combination of harmonic waves of exp(i2p) with different spatial frequency, and() is the coefficient of exp(i2p). The real part cos(2p) and the imaginary part sin(2p) of exp(i2p) are oscillating waves with negative values. However, the light intensity is non-negative. The oscillating wave should be turned into non-negative harmonic wave and Fig. 1 shows the improved process demonstration.

Fig.1 Transition from cosine oscillating wave to non-negative light intensity harmonic wave

Improved non-negative light intensity harmonic wave() can expressed as

(2)

Where() is the coefficient of non-negative harmonic wave.

Since the modified zero frequency component is a constant, it can be given as

From Eqs. (2) and (3), the new expression of d() is obtained as

(4)

After substituting d() for()exp(i2p) in Eq. (1), the new expression of() is obtained as

(5)

Where d00() is zero frequency component of non-negative harmonic wave. By solving the Eqs. (1) and (5), the harmonic wave coefficients are obtained as

2 Eigenvalue of Optical Imaging System

In Fourier optics, the 1-D image light intensity functionimage() is equal to the convolution of 1-D object light distribution functionobject() and line spread function() of optical system, which can be expressed as

whereis the integral operator of convolution expression.

In Shift-Invariant Imaging Systems, complex exponential function is an eigenfunction of optical system. Therefore, exp(i2p) is an eigenfunction ofand the image of exp(i2p) is obtained as

whereis the optical transfer function (OTF) value at the spatial frequency. When the integral operatoris applied on zero frequency component d00() in Eq.(3), the image light intensity function is obtained as

(9)

When the integral operatoris applied on a certain non-negative harmonic wave d() in Eq.(4), the image light intensity wave is obtained as

(10)

From Eq. (10), the image light intensity contains two components ofd() and (1-)d0() after d() is transmitted by optical imaging system.

3 The Information Parameters for Optical Imaging System

3.1 The Entropy of Information Source and Information Destination

OTF of optical imaging system has a cut-off frequency, and the maximal integer which is not more than the cut-off frequency of the optical system is given as. For the convenience of numerical calculation, the harmonic waves with integral frequencyon a scale from 0 tocan be adopted as the samples of object light intensity functionobject().

According to information theory, the sum of elements in the information set is 1. And the sampled coefficients of functionobject() need to be normalized as

Similarly, the normalized image light intensity harmonic wave coefficient sample set is given as: {¢0,¢1,¢2,...,¢}, andcan be regarded as information destination. Moreover, the information destination entropy() is obtained as[6]

(12)

Similarly, the normalized image light intensity harmonic wave coefficient sample set is given as: {¢0,¢1,¢2,...,¢}, andcan be regarded as information home. Moreover, the information home entropy() is obtained as[6]

3.2 Average Mutual Information and Optical Information Channel Capacity

Substitute a normalized object light intensity harmonic wave sampleg() for d() in Eq. (10), the image light intensity wave sample is obtained as

Whereis the modulation transfer function (MTF) value of optical imaging system at the spatial frequency. And Fig.2 shows the processing of the normalized light intensity harmonic wave sample imaged by optical system.

Fig.2 Processing of the normalized light intensity harmonic wave sample imaged by optical system

According to Eq.(14), the information channel matrixof optical system is obtained as

Therefore, the relationship of information source, information channel and information destination is expressed as

(16)

And the average mutual information(;) can be obtained as[6]:

The average mutual information(;) means the undistorted amount of information in the information source. For a practical optical system, only part of information source can be transmitted to the information destination, and noise is added to the information destination. Therefore,(;) is less than either() or(). The less(;) compared to(), the less amount of information source is transmitted by the optical system. The less(;) compared to(), the more noise is added to the information destination by the optical system. Hence the value of(;) is an important parameter to evaluate the imaging quality of the optical system.

For a certain,(;) varies with different input information source {0,1,2,...,g} in Eq.(17). From the characteristics of average mutual information, the function curve of(;) is convex[7]. Therefore, we can find a certain information source that can make the optical system acquire maximum average mutual information, which is the information channel capacity. Andcan be given as

Therefore, the information channel capacityimplies the maximum undistorted information transmission ability of an optical imaging system.

3.3 Calculation Example for Information Transmit Parameters

A diffraction limited optical system with rectangular aperture,number of 10 and operating wavelength of 500 nm is adopted in this analysis. Fig.3 shows the object to be imaged. The normalized harmonic wave coefficient samples of object in +direction can be calculated according to Eq.(11) as shown in Fig.4. According to Eq.(15), the information channel matrixof the optical system is obtained. Thus, the normalized harmonic wave coefficient samples of image in +direction can be calculated according to Eq. (16) as shown in Fig.5.

Fig.3 The object to be imaged

Fig.4 Normalized harmonic wave coefficient samples of object in +x direction

Fig.5 Normalized harmonic wave coefficient samples of imagein +x direction

By using Eqs.(12), (13), (17) and(18), the values of(),(),(;) andare obtained as 6.089 7 bit, 4.647 0 bit, 4.271 1 bit and 5.303 5 bit/sign, respectively. The value of(;) is the information entropy that can be transmit without distortion in the information channel. The difference of 1.818 6 bit by() minus(;) indicates the lost information entropy from the information destination. And the difference of 0.375 9 bit by() minus(;) indicates the noise information entropy added to the information home. The value ofis the maximum average mutual information that can be transmitted by the information channel andindicates the best information transmission performance of the information channel.

4 Conclusions

In this paper, we propose a novel analysis method for optical imaging systems by adopting information theory. The information source and information destination can be obtained by improved Fourier transform of object and image light intensity functions. The information channel matrix can be obtained by the eigenvalue of optical imaging system. And information parameters can be calculated by information theory. For an optical imaging system with known object to be imaged, the average mutual information indicates the information transmission performance. And the information channel capacity evaluates the best information transmission performance for an optical imaging system. Computer simulation is presented to prove the validity of this method.

References：

[1] Neifeld M A，Ashok A，Baheti P K. Task-specific information for imaging system analysis [J]. Journal of the Optical Society of America A(S1084-7529)，2007，24(12)：B25-B41.

[2] Houni K，Sawaya W，Delignon Y. One-dimensional barcode reading：an information theoretic approach [J]. Applied Optics(S2155-3165)，2008，47(8)：1025-1036.

[3] Carson W R，Chen M，Rodrigues M R D. Communications-inspired projection design with application to compressive sensing [J]. SIAM Journal on Imaging Sciences(S1936-4954)，2012，5(4)：1185-1212.

[4] LIU Guozhong，ZHI Zhongwei，WANG Ruikang，. Digital focusing of OCT images based on scalar diffraction theory and information entropy [J]. Biomedical Optics Express(S2774-2783)，2012，3(11)：2774-2783.

[5] GONG Maoguo, LIANG Yan, SHI Jiao，. Fuzzy C-means clustering with local information and kernel metric for image segmentation[J]. IEEE Transactions on Image Processing(S1057-7149)，2013，22(2)：573-584.

[6] Shannon C E. A mathematical theory of communication [J]. ACM SIGMOBILE Mobile Computing and Communications Review(S1559-1662)，2001，5(1)：3-55.

[7] Gueguen L，Velasc S，Soille P. Local mutual information for dissimilarity-based image segmentation [J]. Journal of Mathematical Imaging and Vision(S0924-9907)，2014，48(3)：625-644.

本期组稿：杨淇名

责任编辑：谢小平

英文编辑：庞 洪

光学系统频域信息传递性能分析方法

任智斌，胡佳盛，智喜洋，岳 帅，唐洪浪，李明亮

( 哈尔滨工业大学航天学院，哈尔滨150001 )

本文提出了一种利用信息论研究光学成像系统的新方法。通过改进傅里叶变换，光强函数被分解成一系列非负谐波的线性组合。物的归一化谐波系数集合可看作信源，像的归一化谐波系数集合可看作信宿。利用成像积分方程的特征值可得到光学信道矩阵。进而通过计算平均互信息和信道容量来评估光学系统的信息传递能力。

信息论; 信息熵; 平均互信息; 信道容量; 改进傅里叶变换

2015-08-19；

2015-12-04

高分辨率对地观测系统重大专项资助项目

任智斌(1976-)，男(汉族)，黑龙江哈尔滨人。副教授，博士，主要研究工作是空间光学遥感技术。E-mail: rzb@hit.edu.cn。

A

10.3969/j.issn.1003-501X.2016.10.015

O438.2

智喜洋(1982-)，男(汉族)，黑龙江哈尔滨人。副研究员，博士，主要研究遥感图像处理技术。E-mail: zhixiyang@hit.edu.cn。

1003-501X(2016)10-0090-05