基于离散径向Tchebichef矩的图像分析
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摘要
图像的矩函数和其不变性的定义起始于上世纪的60年代。自1961年Hu首次提出矩的不变性理论以来,矩和矩函数已经被广泛应用于图像处理和模式识别的各个领域。几何矩是最早出现的矩,它构造简单,计算方便。不足之处是在处理一些复杂的图像时,变换不是很方便,且随着阶数的增加,出现计算不稳和不易集中分析的现象。正交矩的显著特点就是可以描述图像的独立特征,具有最小的信息冗余度;由于其取值范围有一定的要求,常在计算前对定义域归一化。图像处理的计算方法通常是数值计算,采用离散求和的方式来逼近真实积分运算。而上述给出的几种矩函数都采用了连续积分的形式。
Zernike矩由于具有旋转不变性,低噪声灵敏度,正交性和最小的信息冗余度,在图像工程领域得到广泛的应用。然而,由于Zernike矩的基函数为连续函数,在离散化的过程中无法避免造成很大的离散误差。为了解决这个问题,Mukundan等提出了离散正交Tchebichef矩。随后,又提出了定义在极坐标上的径向Tchebichef矩,解决了离散正交矩定义在笛卡尔坐标系下不易获得旋转不变量的问题。但是Mukundan方法通过整数点采样,在半径较小时采样点数过多,而半径较大时采样点数不足,同时在处理大图像时效率比较低。本文先对大小为N×N的图像作方圆变换,然后在单位圆平面上作离散向量基,径向以N/2点的离散径向Tchebichef多项式为正交基,周向以4+8i点的离散Fourier为正交基,这样构成一类新的离散径向Tchebichef矩。实验结果分析,在处理大图像时本文所提方法比Mukundan方法好,同时消除了Mukundan方法在重构图像时出现的雪花点等现象。为了进一步提高图像重构效果,本文提出采用分段迭代法来减小多项式迭代时的误差累积,经过实验验证可以得到更好的图像重构效果。
The definition of the moment function and invariance of the image began from the 60’s in last century. The moment and its function have been widely used in image processing and pattern recognition in various fields since Hu firstly put up with the moment’s invariant theory in 1961.Geometric moment is the earliest moment, simple and convenient. But the Inadequacy thing is when in dealing with some complex image, the transformation is not very convenient. And with the order increasing, the phenomenon of instability in calculation and not easy to concentrate on analysis will appear.A distinctive feature of orthogonal moments is to describe the independent characteristics of the image, with minimum redundancy of information. because of its value range have certain requirements , the domain often be normalized before calculation.Current computational method of Image processing is numerical calculation, in a way of the discrete summation to approach the real integral operation. While the several given moment functions are all using a continuous integration form.
Because Zernike moments have features like rotational invariance, low noise sensitivity, orthogonality and the minimum redundancy of information, they are widely applied in image engineering field.However, since the basis functions of the Zernike moments are continuous functions, large discretization error is unavoidably caused during the process of discretization.To solve this problem, discrete orthogonal Tchebichef moments are proposed by Mukundan, etc. then another radial Tchebichef moments which are defined in polar coordinates are proposed. Solve the discrete orthogonal moments defined in the Cartesian coordinate system is not easy to obtain rotation invariant problem. Mukundan’s method adopted integer-point sampling which has defects of too many sampling points when the radius is small while the number of points are insurfficent when the radius is large and the efficiency of processing image is low.A square-to-circular image transformation is introduced to the image and make discrete vetor base which discrete radial Tchebichef polynomial orthgonal to discrete Fourier in circumferential direction constructing a kind of new discrete radial Tchebichef moments. The experimental results show that the suggested method is better than Mukundan’s method when processing large image and eliminates snowflake in the process of image reconstruction. In order to enhance the effect of the image reconstruction,in this paper the section iterative method is proposed to reduce the error accumulation. Experimental results show that we can get better image reconstruction results.
Zernike矩由于具有旋转不变性,低噪声灵敏度,正交性和最小的信息冗余度,在图像工程领域得到广泛的应用。然而,由于Zernike矩的基函数为连续函数,在离散化的过程中无法避免造成很大的离散误差。为了解决这个问题,Mukundan等提出了离散正交Tchebichef矩。随后,又提出了定义在极坐标上的径向Tchebichef矩,解决了离散正交矩定义在笛卡尔坐标系下不易获得旋转不变量的问题。但是Mukundan方法通过整数点采样,在半径较小时采样点数过多,而半径较大时采样点数不足,同时在处理大图像时效率比较低。本文先对大小为N×N的图像作方圆变换,然后在单位圆平面上作离散向量基,径向以N/2点的离散径向Tchebichef多项式为正交基,周向以4+8i点的离散Fourier为正交基,这样构成一类新的离散径向Tchebichef矩。实验结果分析,在处理大图像时本文所提方法比Mukundan方法好,同时消除了Mukundan方法在重构图像时出现的雪花点等现象。为了进一步提高图像重构效果,本文提出采用分段迭代法来减小多项式迭代时的误差累积,经过实验验证可以得到更好的图像重构效果。
The definition of the moment function and invariance of the image began from the 60’s in last century. The moment and its function have been widely used in image processing and pattern recognition in various fields since Hu firstly put up with the moment’s invariant theory in 1961.Geometric moment is the earliest moment, simple and convenient. But the Inadequacy thing is when in dealing with some complex image, the transformation is not very convenient. And with the order increasing, the phenomenon of instability in calculation and not easy to concentrate on analysis will appear.A distinctive feature of orthogonal moments is to describe the independent characteristics of the image, with minimum redundancy of information. because of its value range have certain requirements , the domain often be normalized before calculation.Current computational method of Image processing is numerical calculation, in a way of the discrete summation to approach the real integral operation. While the several given moment functions are all using a continuous integration form.
Because Zernike moments have features like rotational invariance, low noise sensitivity, orthogonality and the minimum redundancy of information, they are widely applied in image engineering field.However, since the basis functions of the Zernike moments are continuous functions, large discretization error is unavoidably caused during the process of discretization.To solve this problem, discrete orthogonal Tchebichef moments are proposed by Mukundan, etc. then another radial Tchebichef moments which are defined in polar coordinates are proposed. Solve the discrete orthogonal moments defined in the Cartesian coordinate system is not easy to obtain rotation invariant problem. Mukundan’s method adopted integer-point sampling which has defects of too many sampling points when the radius is small while the number of points are insurfficent when the radius is large and the efficiency of processing image is low.A square-to-circular image transformation is introduced to the image and make discrete vetor base which discrete radial Tchebichef polynomial orthgonal to discrete Fourier in circumferential direction constructing a kind of new discrete radial Tchebichef moments. The experimental results show that the suggested method is better than Mukundan’s method when processing large image and eliminates snowflake in the process of image reconstruction. In order to enhance the effect of the image reconstruction,in this paper the section iterative method is proposed to reduce the error accumulation. Experimental results show that we can get better image reconstruction results.
引文
[1] M. K. Hu.“Visual Pattern Recognition by Moment Invariants”. IRE TRANSACTIONS ON INFORMATION THEORY, 1962, pp. 179-187.
[2] M.R.Teague.“Image anaIysis via the general theory of moments”. Optical Society of America, 1980,pp. 920-930.
[3] R. Mukundan, S.H. Ong, P.A. Lee.“Image analysis by Tchebichef moments[J].”IEEE Transactions on Image Processing, 2001. 10(9): 1357-1364.
[4] R. Mukundan.“Some Computational Aspects of Discrete Orthonormal Moments”. IEEE Trans. Image Process, 2004,pp. 1055-1059.
[5] P.T. Yap, P. Raveendran, S.H. Ong.“Image analysis by Krawtchouk moments[J].”IEEE. Transactions on Image Processing, 2003. 12(11): 1367-1377.
[6] Z.L. Ping, R.G. Wu, Y.L. Sheng.“Image description with Chebyshev-Fourier moments[J].”J.Opt. Soc. Am. (A), 2002. 19(8): 1748-1754.
[7] R. Mukundan.“A New Class Of Rotational Invariants Using Discrete Orthogonal Moments”. SIGNAL AND IMAGE PROCESSING, Honolulu, Hawai,USA, 2004
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[9] S.X. Liao, M. Pawlak.“On image analysis by moments[J].”IEEE Transactions on Pattern Analysis and Machine Intelligence. 1996. 18(3): 254-266.
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[11] B. Fu, J.Z Zhou, Y.H. Li, G.J. Zhang, C. Wang.“Image analysis by modified Legendre moments[J].”Pattern Recognition. 2007, 40(2): 691-704.
[12] G.L. Amu, S.R. Hasi, X.Y. Yang, Z.L. Ping.“Image Analysis by Pseudo-Jacobi (p=4, q=3)-Fourier Moments[J].”Applied Optics. 2004. 43(10):2093-2101.
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[18]阿木古楞,杨性愉,平子良.用变形雅可比(p=4, q=3)-傅立叶矩进行图像描述[J].光电子.激光, 2003,14(9):981-985.
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[22]王海波,平子良.“几种正交矩描述图像的性能比较[J]”.内蒙古师范大学学报. 2005,34(1).35-39.
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[32]金丰华,秦磊,汪蕙,罗立民.“几何矩不变量在基于内容医学图像检索中的应用[J]”.山东生物医学工程. 2002,4(2).7-10
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[37] Bo W u-rigen,Ping Zi-fang,Ren Hai-ping,Sheng Yun-lon .“Multi-distorted Invariant Pattern Recognition with Chebyshev-Fourier moments[J].”2 003.1 4 (8) :846-850
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[2] M.R.Teague.“Image anaIysis via the general theory of moments”. Optical Society of America, 1980,pp. 920-930.
[3] R. Mukundan, S.H. Ong, P.A. Lee.“Image analysis by Tchebichef moments[J].”IEEE Transactions on Image Processing, 2001. 10(9): 1357-1364.
[4] R. Mukundan.“Some Computational Aspects of Discrete Orthonormal Moments”. IEEE Trans. Image Process, 2004,pp. 1055-1059.
[5] P.T. Yap, P. Raveendran, S.H. Ong.“Image analysis by Krawtchouk moments[J].”IEEE. Transactions on Image Processing, 2003. 12(11): 1367-1377.
[6] Z.L. Ping, R.G. Wu, Y.L. Sheng.“Image description with Chebyshev-Fourier moments[J].”J.Opt. Soc. Am. (A), 2002. 19(8): 1748-1754.
[7] R. Mukundan.“A New Class Of Rotational Invariants Using Discrete Orthogonal Moments”. SIGNAL AND IMAGE PROCESSING, Honolulu, Hawai,USA, 2004
[8] R. Mukundan, K. R. Ramakrishnan,“Fast computation of Legendre and Zernike moments”, Pattern Recognition, 1995, pp. 1433-1442.
[9] S.X. Liao, M. Pawlak.“On image analysis by moments[J].”IEEE Transactions on Pattern Analysis and Machine Intelligence. 1996. 18(3): 254-266.
[10] The C H, Chin R T.“On image analysis by the methods of moments[J]”. IEEE Trans Pattern Anal Machine Intell, 1988, 10(4): 496-513.
[11] B. Fu, J.Z Zhou, Y.H. Li, G.J. Zhang, C. Wang.“Image analysis by modified Legendre moments[J].”Pattern Recognition. 2007, 40(2): 691-704.
[12] G.L. Amu, S.R. Hasi, X.Y. Yang, Z.L. Ping.“Image Analysis by Pseudo-Jacobi (p=4, q=3)-Fourier Moments[J].”Applied Optics. 2004. 43(10):2093-2101.
[13] H.P. Ren, Z.L. Ping, W. Bo, W. Wu, Y.L. Sheng.“Multidistortion-invariant image recognition with radial harmonic Fourier moments[J].”J. Opt. Soc. Am. (A), 2003. 20(4): 631-637.
[14] Nico M. Temme, Jose L. Lopez. The Askey Scheme for Hypergeometric Orthogonal Polynomials Viewed from Asymptotic Analysis[J]. Classical Analysis and ODEs.February 1,2008.
[15]杨性愉,平子良.“用变形雅可比(p=4, q=3)-傅立叶矩进行图像描述[J].”光电子.激光. 2003, 14(9): 981-985.
[16]梁俊,舒华忠,朱宏擎,罗立民.“一种离散正交Hahn矩的分析方法[J].”上海交通大学学报. 2006.40(5): 796-800
[17]周卫平,李松毅,舒华忠.“Chebyshev-Fourier矩的快速计算方法[J].”计算机学报. 2005.28(8): 1393-1397.
[18]阿木古楞,杨性愉,平子良.用变形雅可比(p=4, q=3)-傅立叶矩进行图像描述[J].光电子.激光, 2003,14(9):981-985.
[19]李金泉,王建伟,陈善本,吴林.“一种改进的Zernike正交矩亚像素边缘检测算法[J]”.光学技术,2003,29(4):500-503.
[20]王耀明.Tchebichef矩及其在图像重建中的应用[J].上海电机学院学报学报. 2007.10(3):42-44.
[21]朱娜,张辉,舒华忠.“基于径向Krawtchouk矩的医学图像检索[J]”.生物医学工程研究, 2008,27(1). 40-44.
[22]王海波,平子良.“几种正交矩描述图像的性能比较[J]”.内蒙古师范大学学报. 2005,34(1).35-39.
[23] C.W. Chong, P. Raveendran, R. Mukandan.“A comparative analysis of algorithm for fast conputation of Zernike moments[J] .”Pattern Recognition, 2003,36(3):731- 742.
[24] A. Khotanzad, Y.H. Hong.“Invariant image recognition by Zernike moments[J].”IEEE Trans Pattern Anal Machine Intell, 1990,12(5):489-498.
[25]邓晓红等著.“神经Chebyshev正交多项式均衡器及自适应算法[J]”.西南交通大学学报.2005年02期.
[26]梁俊著.“离散正交矩的图像分析方法研究(硕士学位论文)[D]”.东南大学. 2006年3月.
[27]沈澍等著.“基于矩的图像变化检测[J]”.计算机仿真. 2008年07期.
[28]王晓红著.“矩技术及其在图像处理与识别中的应用研究(博士学位论文)[D]”.西北工业大学. 2001年4月.
[29]阿木古楞,哈斯苏荣,任爱珍.“不变矩图像分析进展[D].”内蒙古农业大学学报(自然科学版). 2005, 26(4): 146-150.
[30]商立群.“基于Zernike矩的图像识别[J]”.西安科技学院学报. 2000,20(6).
[31]张里,韦岗,张基宏.“Tchebichef矩在图像数字水印技术中的应用[J]”.通信学报. 2003,24(9).
[32]金丰华,秦磊,汪蕙,罗立民.“几何矩不变量在基于内容医学图像检索中的应用[J]”.山东生物医学工程. 2002,4(2).7-10
[33]张航,罗大庸.“基于Tchebichef不变矩的图形识别[J]”.计算机工程2006,14.52-54
[34] L.Kotoulas , I.“Andreadis.Image analysis using moments[J]”.
[35]王耀明编著图像的矩函数—原理,算法及应用[M].华东理工大学出版社
[36]徐杰主编数字图像处理[M].华中科技大学出版社
[37] Bo W u-rigen,Ping Zi-fang,Ren Hai-ping,Sheng Yun-lon .“Multi-distorted Invariant Pattern Recognition with Chebyshev-Fourier moments[J].”2 003.1 4 (8) :846-850
[38] Li Li,Bo Fu,Wen Xu,Bo Li,Guojun Zhang.“Image Analysis by Discrete Radial Tchebichef Moments [C]”. 2010 Fuzzy Systems And Knowledge Discovery(FSKD2010), IEEE Computer Society, 2010,pp 569-572
[39] Guojun Zhang, Bo Li, Bo Fu,Li Li et al.“A comparative image analysis of discrete radial Fourier transforms [J]”. Optics and Lasers in Engineering 48(2010),pp 1186-1192
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