摘要
由于其良好的性能,小波变换在图像处理、信号处理以及图像压缩等领域中得到了广泛的应用,然而其庞大的计算量制约了小波变换应用的进一步推广。因此,将光学方法与小波变换结合起来,形成光学小波变换方法,可以极大地减少小波变换所花的时间,这是很有理论和实用价值的。目前,光学小波变换已经被应用于边缘提取、特征提取、模式识别、图像分割、机器视觉等领域,显示出很好的应用前景。但是,现有的光学小波变换方法无法通过数值计算进行高精度的重构,这限制了它在图像压缩等领域中的应用。为此,本论文全面系统地阐述了小波变换的发展动态和基本理论,详细分析了光学小波变换的特性,深入研究了多分辨率分析的原理和光学实现方法。
研究光学4f系统的图像空间频率特性。阐述空间频率的基本理论,分析光学4f系统中图像实现方式和图像采集方式对图像空间频率特性的影响,研究输入图像频域能量集中度与对应图像重建质量及空间滤波半径的关系,比较不同尺寸的输入图像空间滤波后的图像重建质量的差异,给出关于光学4f系统中图像空间频率特性的计算和分析方法。通过理论分析、仿真计算和光学实验,表明用于图像压缩等需要高精度重构的领域的光学小波变换在频谱面上的空间滤波范围受到输入图像尺寸、近轴条件、光学4f系统的器件采样特性、图像频域能量集中度以及系统噪声的影响,并给出了输入图像尺寸的约束条件。
研究多分辨率分析的光学实现方法。分析小波滤波器系数与尺度函数和小波函数的关系,介绍由小波滤波器系数求尺度函数和小波函数的三种常用算法,并对其中的迭代卷积算法进行改进,同时给出其相应的收敛判定方法。深入分析多分辨率分析和光学4f系统的基本原理,研究利用光学4f系统实现基于连续小波变换的多分辨率分析的方法。针对CCD(Charge Coupled Device)只能直接记录光的强度,给出一种应用于光学4f系统的光学小波滤波器的设计方法及其后处理方法,并给出利用计算机通过数值计算实现光学多分辨率分析的重构方法。通过仿真实验和理论分析,说明连续小波变换的光学实现方法的局限性。
提出Mallat算法的光学实现方法。从多分辨率分析理论出发,分析Mallat算法的核心思想和光学4f系统的基本原理,论证利用光学4f系统实现Mallat算法的可行性,提出Mallat算法的光学实现方法。针对振幅型空间光调制器只能实现非负的实函数,且CCD只能直接记录光的强度,提出一种应用于光学4f系统的光学小波滤波器的设计方法。首先,根据采样间距,基于张量积方法由一维小波滤波器系数构造二维小波滤波函数。然后通过拆分、傅里叶变换与归一化,得到非负的实函数形式的频域小波滤波器。最后,给出相应的光学小波变换后处理方法。使用该种光学小波滤波器及其相应的后处理方法,利用光学4f系统实现Mallat算法的小波分解部分,并通过数值计算实现Mallat算法的小波重构部分。仿真实验结果表明,通过该方法能够极好地重构输入图像;在引入光学器件量化误差的条件下,通过该方法仍然能够高精度地重构输入图像。利用实际的光学4f系统进行光学实验,也能以良好的质量重构输入图像。此外,针对相干噪声问题,还实现了一种混合光学小波变换。研究空域和频域形式的光学小波滤波器的尺度一致性及其相应的设计方法,利用散焦系统实现Mallat算法的小波低通分解部分,利用光学4f系统实现Mallat算法的小波高通分解部分,并利用计算机通过数值计算实现Mallat算法的小波重构部分。
研究光学小波滤波器的优化设计方法。通过仿真实验,研究不同小波基构造频域形式光学小波滤波器的量化误差,比较将分解和重构滤波器交换对频域形式光学小波滤波器的量化误差的影响,并分析频域上的不同采样频率对光学小波滤波器的量化误差的影响。利用提升算法,基于空域和频域量化误差最小的原则构造出最优小波。
Due to its excellent performance, wavelet transform has been extensively applied to many fields, such as image processing , signal processing and image compression. However, its application is restricted by its computational complexity. Therefore, the optical method and wavelet transform are combined to implement optical wavelet transform, which can greatly reduce the time spent on wavelet transform and has good theoretical and practical value. At present, optical wavelet transform has been applied to many fields, such as edge extraction, feature extraction, pattern recognition, image segmentation as well as computer vision, and shows good prospects. But high precision reconstruction can not be implemented by the existing optical wavelet transform, which restricts its application in some fileds such as image compression. So, the fundamental theory and developments of wavelet transform were systematically described in this dissertation, with the characteristics of optical wavelet transform being analyzed in detail, the principle and optical implementation methods of multiresolution analysis being deeply studied.
The image spatial frequency characteristic in optical 4f system was studied. The fundamental theory of spatial frequency was described. Then effects on the spatial frequency characteristic were analyzed which caused by way of image realization and image acquisition in optical 4f system, and researches on the relationship among the characteristic of energy concentration of image in frequency domain, radius of spatial filter and the qualities of the reconstructed image were carried out. The qualities of the reconstructed image with different input sizes were compared. The relevant calculational and analysis methods were given. The theoretical analysis, simulations and optical experiments have shown that the spatial filtering radius of optical wavelet transform applied to some fileds requiring high precision reconstruction such as image compression is restricted by the size of input images, the adaxial condition, the sampling characteristic of apparatus in optical 4f system, the characteristic of energy concentration of image in frequency domain and the system noise. The restrictive conditions on the sizes of input image were given.
The optical implementation method of multiresolution analysis was studied. Connections between the scaling and wavelet functions and coefficients of wavelet filter were analyzed. Three algorithms on the computation of the scaling and wavelet function using coefficients of wavelet filter were introduced, and iterative convolution algorithm was improved, meanwhile the method of judging the convergence of iterative convolution was given. Principles of multiresolution analysis and optical 4f system were deeply analyzed, then the optical implementation method of multiresolution analysis based on continuous wavelet transform utilizing optical 4f system was studied. As CCD can only record light intensity, a design method of optical wavelet filters applied to optical 4f system and the relevant post-processing method were given, meanwhile the reconstruction method of multiresolution analysis by numerical computation utilizing computer was presented. Through simulations and theoretical analysis, the limitations on the optical implementation method of continuous wavelet transform were indicated.
Optical implementation method of Mallat algorithm was proposed. Based on the theory of multiresolution analysis, the core principle of Mallat algorithm and optical 4f system were analyzed, and the feasibility on Mallat algorithm implemented by optical 4f system was demonstrated, meanwhile optical implementation method of Mallat algorithm was proposed. As amplitude-only SLM (Spatial Light Modulator) can only implement non-negative real function and CCD can only record light intensity, a design method of optical wavelet filters applied to optical 4f system was presented. Firstly, according to the sampling interval, the two-dimensional wavelet filters were constructed with one-dimensional coefficients of wavelet filter in terms of tensor product method. Then the frequency domain wavelet filters in the form of non-negative real function were constructed by splitting, Fourier transform and normalization. At last, the relevant post-processing method of optical wavelet transform was given. With this kind of optical wavelet filter and its relevant post-processing method, the wavelet decomposition in Mallat algorithm was implemented utilizing an optical 4f system, and the wavelet reconstruction in Mallat algorithm was implemented by numerical computation. Simulation results indicated that input images can almost be perfectly reconstructed by the presented method, and input images can be reconstructed with high precision under the conditions of quantization errors introduced by optical devices. Input images can also be reconstructed with good quality by optical experiment utilizing an actual optical 4f system. Furthermore, a method of hybrid optical wavelet transform was presented to eliminate coherent noise. The scale consistency of optical wavelet filter in both frequency and spacial domain as well as the relevant design method was studied. The low-pass filtering of wavelet decomposition in Mallat algorithm was implemented utilizing a defocus system, while the high-pass filtering of wavelet decomposition in Mallat algorithm was implemented utilizing an optical 4f system. The wavelet reconstruction in Mallat algorithm was implemented by numerical computation utilizing computer.
The optimized design method on optical wavelet filter was studied. Through computer simulation, the quantization error of optical wavelet filter constructed by different wavelet bases was studied, the effect on quantization error due to the exchange of decomposition wavelets with reconstruction wavelets was compared, and the effect on quantization error due to different sampling frequencies in frequency domain was analyzed. According to the least quantization error criteria of optical wavelet filter both in spacial domain and frequency domain, an optimal wavelet base was constructed based on lifting algorithm.
引文
[1]石钟慈.第三种科学方法—计算机时代的科学计算[M].北京:清华大学出版社,2000.
[2]李国杰.非传统的高性能计算技术[J].世界科技研究与发展,1998, 20(3):14-16.
[3]王玉堂.光子学与光子技术发展战略研究[J].光电子?激光,1998, 9(6):515-518.
[4]秦前清,杨宗凯编著.实用小波分析[M].西安:西安电子科技大学出版社,2002.
[5] D.Gabor. Theory of communication[J]. Inst.Elect.Eng., 1946, 93(26):429-457.
[6] F.T.S. Yu, G. Lu. Short-time Fourier transform and wavelet transform with Fourier-domain processing[J]. Appl.Opt., 1994, 33(8):5262-5270.
[7]王学文,丁晓青,刘长松.基于Gabor变换的高鲁棒汉字识别新方法[J].电子学报,2002, 30(9):1317-1322.
[8]丁裕峰,马利庄,聂栋栋,刘军波. Gabor滤波器在指纹图像分割中的应用[J].中国图象图形学报,2004, 9(9):1037-1041.
[9] I.Daubechies. The wavelet transform: time-frequency localization and signal analysis[J]. IEEE Trans. Info. Theory, 1990, 36(5):961-1005.
[10] [美]Ingrid Daubechies著,李建平,杨万年译.小波十讲[M].北京:国防工业出版社,2004.
[11] S. Mallat. A Wavelet Tour of Signal Processing[M].北京:机械工业出版社,2003.
[12]杨福生著.小波变换的工程分析与应用[M].北京:科学出版社,2003.
[13]孙延奎编著.小波分析及其应用[M].北京:机械工业出版社,2005.
[14]程正兴,杨守志,冯晓霞著.小波分析的理论、算法、进展和应用[M].北京:国防工业出版社,2007.
[15]程正兴.小波分析算法与应用[M].西安:西安交通大学出版社,2003.
[16]成礼智,郭汉伟.小波与离散变换理论及工程实践[M].北京:清华大学出版社,2005.
[17]冯象初,甘小冰,宋国乡编著.数值泛函与小波理论[M].西安:西安电子科技大学出版社,2003.
[18]崔锦泰著,程正兴译.小波分析导论[M].西安:西安交通大学出版社,1997.
[19]胡广书.数字信号处理——理论、算法与实现(第二版)[M].北京:清华大学出版社,2003.
[20] S.Mallat. A Theory for Multiresolution Signal Decomposition: The wavelet Representation[J]. IEEE Tran. On Pattern Analysis and Machine Intelligence. 1989, 11(7):674-693.
[21]陈家壁,苏显渝主编.光学信息技术原理及应用[M].北京:高等教育出版社,2002.
[22]郑光昭编著.光信息科学与技术应用[M].北京:电子工业出版社,2002.
[23] Ingrid Daubechies. Where Do Wavelets Come From?—A Personal Point of View[J].Proceeding of the IEEE, 1996, 84(4):510-513.
[24] J.O. Str?mberg. A modified Franklin system and higher order spline systems on Rn as unconditional bases for Hardy spaces. In W.Beckner et al.. editors. Proc. Conf. in Honor of Antoni Zygmund, Vol.2:475-493[C]. New York: Wadsworth, 1981.
[25] J. Morlet, G. Arens, E. Fourgeau, D. Giard. Wave propagation and sampling theory-Part 1: Complex signal scattering in multilayered media[J]. Geophysics, 1982, 47 (2): 203-221.
[26] J. Morlet, G. Arens, E. Fourgeau, D. Giard. Wave propagation and sampling theory-Part 2: Sampling theory and complex waves[J]. Geophysics, 1982, 47 (2): 222-236.
[27] A. Grossmann, J. Morlet. Decomposition of Hardy functions into square integrable wavelets of constant shape[J]. SIAM J. of Math. Anal., 1984, 15(4): 723-736.
[28] A. Grossmann, J. Morlet, T. Paul. Transforms associated to square integrable group representations (I )-General results[J]. J. Math. Phys., 1985, 26 (10): 2473-2479.
[29] Y.Meyer. Principe d’incertitude, bases hillbertiennes et algebras d’operateurs. In Seminaire Bourbaki, vol.662[C]. Paris:1986.
[30] S.Mallat. An efficient image representation for multiscale analysis. Proc. of Mschine Vision Conference[C]. Lake Taho: 1987.
[31] PH.Tchamichian. Biothogonalire et theorie des operateurs[J]. Rev.Math, Iberoamer., 1987, 3(2):163-189.
[32] G.Battle. A block spin construction of ondelettes, Part I:Lamarie functions[J]. comm.Math..Phys., 1987, 110(4):601-615.
[33] P.G.Lemarie. Ondelettes a localisation exponentielle[J]. Math. Pures et Appl., 1988, 67:227-236.
[34] P.J.Burt, E.H.Adelson. The Laplacian Pyramid as a Compact Image Code[J]. IEEE Tran. On Comm, 1983, 31(4):532-540.
[35] S.Mallat. A Compact Multiresolution Representation: The Wavelet Model. Proc. IEEE Computer Society Workshop on Computer Vision[C]. Washington. D.C.:IEEE Computer Society Press, 1987.
[36] S.Mallat. Multiresolution Approximations and Wavelet Orthonormal Bases of L2(R)[J]. Trans. of the Amercian Math. Society, 1989, 315(1):69-87.
[37] S.Mallat. Multifrequency Channel Decompositions of Images and Wavelet Models[J]. IEEE trans on acoustics speech and signal processing, 1989, 37(12):2091-2110.
[38] I. Daubechies. Othonormal bases of compactly supported wavelets[J]. Comm, on Pure and Appl. Math, 1988, 41 (7): 909-996.
[39] A. Cohen, I.Daubechies and J.C.Feauveau. Biorthogonal bases of compactly supportedwavelets[J]. Commun. Pure Appl. Math., 1992, 45(5):485-560.
[40] R.R.Coifman and M.V.Wickerhauser. Entropy based algorithms for best basis selection[J]. IEEE Trans. Info. Theory, 1992, 38(2):713-718.
[41] T.N.T. Goodman and S.L.Lee. Wavelets of multiplicity r[J]. Trans. Amer. Math. Soc., 1994, 342(1):307-324.
[42] G. Donovan, J.S.Geronimo, D.Hardin and P.Massopust. Constrution of orthogonal wavelets using fractal interpolation function[J]. SIAM J. Math. Anal., 1996, 27(4):1158-1192.
[43] W.Sweldens. The lifting scheme: A custom-design construction of biorthogonal wavelets[J]. Appl. Comut. Harman. Appl., 1996, 3(2):186-200.
[44] W.Sweldens and Schr?der P. Building your own wavelets at home[C]. Wavelets in Computer Graphics, ACM SIGGRAPH Course Notes, 1996.
[45] W.Sweldens. The lifting Scheme: A construction of second generation wavelets[J]. SIAM J. Math. Anal., 1997, 29(2):511-546.
[46] I. Daubechies and W.Sweldens. Fractoring wavelet transform into lifting step[J]. J. Fourier Anal. Appl., 1998,4(3):245-267.
[47] R. Calderbank, I. Daubechies et al.. Wavelet transform that map integers to integers[J]. Appl. Comut. Harmon. Anal., 1998,5(3):332-369.
[48] G.M.Davis, V.Strela and R.Turcajova. Multiwavelet construction via the lifting scheme. Wavelet Analysis and Multiresolution Methods, He T.X., ED., Lecture Notes in Pure and Appl. Math., Marcel Dekker[C]. New York, 1999.
[49] F. Keinert. Raising multiwavelet approximation order through lifting[J]. SIAM J. Math. Anal., 2001, 32(5):1032-1049.
[50]倪伟,郭宝龙,杨缪.图像多尺度几何分析新进展:Contourlet[J].计算机科学,2006, 33(2):234-237.
[51] Candes E.J., Donoho D.L.. Curvelets-a surprisingly effective nonadaptive representation for objects with edges. In: Cohen A., Rabut C., Schumaker L.L. editors. Curve and surface fitting[C]. Saint-Malo: Vanderbilt University Press, 1999:105-120.
[52]焦李成,谭山.图像的多尺度几何分析:回顾与展望[J].电子学报, 2003, 31(12A):1975-1981.
[53] Starck J L, Candes E J, Donoho D L. The curvelet transform for image denoising[J]. IEEE Trans Image Proc., 2002, 11(6):670-684.
[54] Manjunath B S, Ma W Y. Texture features for browsing and retrieval of image data[J]. IEEE Trans Pattern Anal Machine Intell, 1996, 18(8):1148-1161.
[55] Kingsbury N. Complex wavelets for shift invariant analysis and filtering of signals[J]. Journalof Appl and Comput Harmonic Analysis 2001,10(3):234-253.
[56] Freeman W T, Simoncelli E P, Adelson E H. The design and use of steerable filters[J]. IEEE Trans Pattern Anal Machine Intell, 1991,13(9):891-906.
[57] Candes E J. Harmonic analysis of neural networks[J]. Applied and Computational Harmonic Analysis, 1999, 6(2): 197-218.
[58]宋菲君,S.Jutamulia.近代光学信息处理[M].北京:北京大学出版社,1998.
[59] E. Abbe. Beitrage zur Theorie des Mikroskops und der mikroskopichen Wahrnehmung[J]. Archiv. Mikroskop. Anat., 1873, 9:413- 468.
[60] F.Zernike. Das Phasenkontrastverfahren bei der mikroskoposchen Beobachtung[J]. Z.Tech.Phys., 1935, 16:454-456.
[61] D.Gabor. A new microscope principle[J]. Nature. 1948, 161:777-778.
[62] L.J.Cutrona, E.N.Leith, C.J.Palermo, L.J.Porcello. Optical data processing and filtering systems[J]. IRE Trans.Inf. Theory, 1960, IT-6:386- 400.
[63] A.Vander Lugt. Signal detection by complex spatial filtering[J]. IEEE Trans. Inf. Theory, 1964, IT-10:139-145.
[64] E. Freysz, B. Pouligny, F. Argoul, et al.. Optical wavelet transform of fractal aggregates[J]. Phys. Rev. Lett., 1990, 64 (7): 745-748.
[65] Yan.Zhang, Yao Li. Optical determination of Gabor coefficients of transient signals[J]. Opt. Lett., 1991, 16 (13): 1031-1033.
[66] Y.Sheng, D.Roberge, H.H.Szu. Optical wavelet transform[J].Opt.Eng.,1992,31(9): 1840-1845.
[67] H.Szu, Y.Sheng, J.Chen. Wavelet transform as a bank of matched filters[J]. Appl. Opt., 1992, 31(17):3267-3277.
[68] J.Caulfield, H.Szu. Parallel discrete and continuous wavelet transforms[J]. Opt. Eng., 1992, 31(9):1835-1839.
[69] X.Yang, H.H.Szu, Y.Sheng, H.J.Caulfield. Optical wavelet transform of binary images[J]. Opt. Eng., 1992, 31 (9):1846-1851.
[70] M.O.Freeman, A.Fedor, B.Bock, K.Duell. Optical Wavelet Processor for producing spatially localized ring-wedged-type information[J]. SPIE, 1992, 1772:241-250.
[71] Y.Sheng, D.Roberge, H.Szu, T. Lu. Optical wavelet matched filters for shift-invariant pattern recognition[J]. Opt. Lett.. 1993, 18 (4): 299-301.
[72] Jin Guofan, Yan Yingbai. Optical Harr wavelet transform for image feature extraction[C]. Proc SPIE Int Soc Opt Eng., 1993, 2034:371-380.
[73] D.Mendlovic. N. Konforti. Optical realization of the wavelet transform for two-dimensional objects[J]. Applied Optics, 1993, 32(32): 6542-6546.
[74] Block P G., Steven K. Rogers, Dennis W. Ruck.. Optical wavelet transform from computer-generated holography[J]. Applied Optics. 1994, 33(23):5275-5278.
[75] Dong-Xue Wang, Ju-Wei Tai, Yan-Xin Zhang. Two-dimensional optical wavelet transform in space domain and its performance analysis[J]. Applied Optics, 1994, 33(23):5271-5274.
[76]王文陆,严瑛白,金国藩,邬敏贤.光学子波变换(1):基本理论[J].光电子.激光, 1994, 5(4):193-200.
[77]王文陆,严瑛白,金国藩,邬敏贤.光学子波变换(2):实验结构及技术[J].光电子.激光, 1994, 5(5):268-272.
[78]王文陆,严瑛白,金国藩,邬敏贤.光学子波变换(3):应用[J].光电子.激光, 1994, 5(6):331-339.
[79]华家宁,陈家壁.光学小波变换初探[J].南京师大学报(自然科学版), 1994, 17(4):31-37.
[80]涅守平,杨罕,李爱民.子波变换在光信息处理方面的应用[J].应用光学, 1994, 15(4):41-44.
[81]涅守平,杨罕,李爱民.一种新颖的光信息处理方法——子波变换[J].南京理工大学学报, 1995, 19(1):13-16.
[82]李骏,张阅春,胡家升.基于联合子波变换结构的实时目标识别研究[J].中国激光, 1995, 22(10):783-787.
[83]王文陆.光学子波变换及其在图像处理中的应用[D].清华大学博士学位论文,1995.
[84] D.Mendlovic, I.Ouzieli, I.Kiryuschev, E.Marom.Two-dimensional wavelet transform achieved by computer-generated multireference matched filter and Dammann grating[J]. Applied Optics, 1995,34(35): 8213-8219.
[85] I.Ouzieli, D.Mendlovic. Two-dimensional wavelet processor[J]. Applied Optics, 1996, 35(29):5839-5846.
[86] P.S.Erbach. Optical wavelet transform by the phase-only joint transform correlator[J]. Applied Optics, 1996,35(17): 3117-3126.
[87] J.Garcia, Z.Zalevsky, D.Mendlovic. Two-dimensional wavelet transform by wavelength multiplexing[J]. Appl. Opt., 1996, 34(35):7019-7024.
[88] Z.Zalevsky. Experimental implementation of a continuous two-dimensional on-axis optical wavelet transformer with white light illumination[J]. Opt.Eng., 1998, 37(4):1372-1375.
[89] Jose J.Esteve-Taboada, J.Garcia, C.Ferreira. Two-dimensional optical wavelet decomposition with white-light illumination by wavelength multiplexing[J]. Opt.Soc.Am., 2001, 18(1):157-163.
[90] D.Mendlovic, Z.Zalevsky, D.Mas, et al.. Fractional wavelet transform[J]. Appl.Opt., 1997, 36(20):4801-4806.
[91] D.Mendlovic. Continuous two-dimensional on-axis optical wavelet transformer and wavelet processor with white-light illumination[J]. Appl Opt.. 1998, 37(8):1279-1282.
[92] G.Shabtay, D.Mendlovic, Z.Zalevsky. Optical implementation of the continuous wavelet transform[J]. Appl.Opt., 1998, 37(14):2964-2966.
[93] Michinori Honma, Toshiaki Nose, Susumu Sato. Liquid crystal polarization-converting devices for edge and corner extractions of images using optical wavelet transforms[J]. Applied Optics, 2006, 45(13):3083-3090.
[94] Himadri S. Pal , Dinesh Ganotra, Mark A.Neifeld. Face recognition by using feature-specific imaging[J]. Applied Optics, 2005, 44(18):3784-3794.
[95] Yang-Hoi Doh, Jong-Soo Yoon, Kyung-Hyun Choi, Mohammad S.Alam. Optical security system for the protection of personal identification information[J]. Applied Optics, 2005, 44(5): 742-750 .
[96]冯文毅.光学子波并行处理技术及应用研究[D].清华大学博士学位论文,1999.
[97]岳宏,戴士杰,崔庆华.光学小波变换在视觉系统的应用研究[J].光子学报, 2005, 34 (3): 455-459
[98]才德,严瑛白,金国藩.光学可分离小波变换的研究[J].光学技术, 2006, 32(2):296-298.
[99]才德,严瑛白,金国藩.光学小波包变换及其滤波器的研究[J].光子学报, 2006, 35 (7): 1076-1079.
[100]才德,严瑛白,金国藩.基于体全息的光电混合虹膜识别系统的研究[J].光电子.激光, 2005, 16(10): 1243-1247.
[101] A.V. Oppenheim, A.S.Willsky, S.H.Nawab. Signals and Systems(Second Edition)[M]. Beijing: Publishing House of Electronics Industry. 2002.
[102]郑君里,应启珩,杨为理.信号与系统[M].北京:高等教育出版社. 2000.
[103]吴大正主编.信号与线性系统分析[M].北京:高等教育出版社. 1994.
[104] [美]莫里斯.克莱因著,张理京,张锦炎,江泽涵译.古今数学思想[M].上海:上海科学技术出版社. 2002.
[105] Y.Meyer. Wavelets and operators[M]. Cambridge :Cambridge University Press. 1992.
[106] A.Croisier, D.Esteban, C.Galand. Perfect channel splitting by use of interpolation /decimation /tree decomposition techniques[C]. In int.conf. on info. Sciences and Systems. Patras,Greece, 1976.
[107] M.J.Smith, T.P.BarnwellⅢ. A procedure for designing exact reconstruction filter banks for tree structured subband coders[C]. In proc. IEEE int. Conf. Acoust., Speech, and Signal Proc. San Diego, CA, Match 1984.
[108] F.Mintzer. Filters for distortion-free two-band multirate filter banks[J]. IEEE Trans. Acoust.,Speech, and Signal Proc., 1985, 33(3):626-630.
[109] M.Vetterli. Splitting a signal into subsampled channels allowing perfect reconstruction[C]. In Proc., IASTED conf. on Appl. Sig. Proc. and Dig. Filt., Prais, 1985.
[110] M.Vetterli. Filter banks allowing perfect reconstruction[J]. Signal Proc. 1986, 10(3):219-244.
[111] P.P. Vaidyanathan, P.Q.Hoang. Lattice structures for optimal design and robust implementation of two-channel perfect reconstruction filter banks[J]. IEEE Trans. Acoust., Speech, and Signal Proc., 1988, 36(1):81-94.
[112] W.Sweldens. The lifting scheme : a new philosophy in biorthogonal wavelet constructions[C]. In : Laine A F , Unser M, eds. Wavelet Applications in Signal and Image Processing III Proc SPIE 2569 ,1995.
[113] J.W.Goodman[著].詹达三,董经武,顾本源[译].傅里叶光学导论[M].北京:科学出版社. 1976.
[114] Y.Li, H.H.Szu, Y.Sheng, H.J.Caufield. Wavelet processing and optics[J]. IEEE Proc., 1996, 84(5):720-732.
[115] X. J. Lu, A. Katz, E. G. Kanterakis, N. P. Caviris. Joint transform correlator that uses wavelet transform[J]. Opt. Lett., 1992, 17: 1700-1702.
[116] C.Hirliman, J.F.Morhange. Wavelet analysis of short light pulses[J]. Applied Optics, 1992, 31(17): 3263-3266.
[117]何俊发,赵选科,王红霞.基于光学小波微分预处理的联合变换相关目标识别[J].光子学报, 2002, 31 (12): 1538-1541.
[118] R.Tripathi, K.Singh. Pattern discrimination using a bank of wavelet filters in a joint transform correlator[J]. Opt. Eng., 1998, 37 (2): 532-538.
[119] A.Sinha, K.Singh. The design of a composite wavelet matched filter for face recognition using breeder genetic algorithm[J]. Optics and Lasers in Engineering, 2005, 43:1277-1291.
[120] Wenlu Wang, Guofan Jin, Yingbai Yan, et al.. Joint wavelet transform correlator for image features extraction[J]. Applied Optics, 1995, 34(2):370-376.
[121] C.S.Weaver, J.W.Goodman. A technique for optically convolving two functions[J]. Appl. Opt., 1966, 5:1248-1249.
[122] J.E.Rau. Detection of differences in real distributions[J]. Opt.Soc.Am., 1966, 56:1490-1494.
[123]冯文毅,黄高贵,严瑛白,等.多通道体全息子波相关器及其在指纹识别中的应用[J].光学学报,2000, 20(3):370-375.
[124]曾孝平,许晓诣,田逢春.胶片制作光学滤波器研究[J].光电子技术,2007, 27(2):82-87.
[125] A.W.Lohmann, D.P.Paris. Binary Fraunhofer holograms generared by computer[J]. Appl.Opt., 1967, 6(10):1739-1748.
[126]易开红,田逢春,冯文江.计算全息再现像的误差源分析[J].计算机技术与发展,2007, 17(10):20-23.
[127]印勇,陈梨,田逢春.利用光栅闪耀特性优化低通滤波器[J].光电工程,2006, 33(9): 124-127,132.
[128] Z.He, M.Honma, S.Masuda, et al.. Optical Haar wavelet transforms with liquid crystal elements[J]. Japanese Journal of Applied physics, 1995, 34(12A):6433-6438.
[129] Y.Itoh, C.Hiraide, K.Kodate, et al.. Applications of high-resolution active-matrix liquid crystal spatial light modulators to parallel optical processing[J]. Proc. of SPIE, 1994, 2175:200-207.
[130] C.Decusatis, J.Koay, and P.Das. Hybrid optical implementation of discrete wavelet transforms: a tutorial[J]. Optics & Laser Technology, 1996, 28(2): 51-58.
[131] A.W.Snyde, J.D.Love著,周幼威等译.光波导理论[M].北京:人民邮电出版社,1991.
[132]印勇,谭勇,田逢春.基于单模光纤耦合器的Haar小波滤波器系数的实现[J].光学精密工程,2007, 15(3):417-421.
[133]容观澳.计算机图像处理[M].北京:清华大学出版社,2000.
[134] L.G.Shapiro, G.C.Stockman著,赵清杰,钱芳,蔡利栋译.计算机视觉[M].北京:机械工业出版社,2005.
[135]游明俊.傅里叶光学[M].北京:兵器工业出版社,2000.
[136] Fengchun Tian, Xin Xu, Feng Xue, Simon X. Yang. The Size Limitation on Input Image of 4f System in Applications of Optical Image Data Compression[C]. Proceedings of International Conference on Sensing, Computing and Automation (ICSCA 2006). Waterloo, Canada: Watam Press, 2006, (5):2208~2211.
[137]韩亮,田逢春,徐鑫,刘伟,王宇.光学4f系统的图像空间频率特性[M].重庆大学学报(自然科学版),2008年3月录用.
[138] G.Strang, T.Nguyen. Wavelets and Filter Banks[M]. Wellesley-Cambridge Press, 1996.
[139]张平文,刘法启,张宇.小波函数值的计算[J].计算数学,1995, (2):173-185.
[140]韩亮,田逢春,王宇.由小波滤波器系数求尺度函数和小波函数[J].重庆大学学报(自然科学版),2007, 30(4):79-82.
[141]宋丰华编著.现代光电器件技术及应用[M].北京:国防工业出版社,2004.
[142]宋丰华编著.现代空间光电系统及应用[M].北京:国防工业出版社,2004.
[143]田逢春,陈修建,孟书萍.用光学方法实现图像数据压缩时对滤波器的精度要求[J].中国图像图形学报,2005, 10(9):1190-1196.
[144] MENG Shu-ping, TIAN Feng-chun, XU Xin. Comparison of fast discrete wavelet transform algorithms[J].Chongqing Univ.-Eng. Ed.. 2005, 4(2):84-89.
[145]韩亮,田逢春,徐鑫,李立. Mallat算法的光学实现方法[M].光学精密工程,2007年12月录用.
[146]王仕璠编著.信息光学理论与应用[M].北京:北京邮电大学出版社,2003.
[147]苏显渝,李继陶编著.信息光学[M].北京:科学出版社,1999.
[148]田逢春,韩亮,王宇,徐鑫.空频域量化误差最小的光学小波滤波器设计[J].光电工程,2007年12月录用.
