Bandelet变换在图像去噪与增强中的应用
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摘要
小波在表示具有点奇异性的目标函数时是最优的基,但由于二维可分离小波由一维小波直接用张量积扩展得到,小波不能“最优”表示含线或者面奇异的二维图像,因此小波变换对图像去噪、增强的效果受到影响。
为了克服小波变换在处理高维信号时的不足,法国学者Pennec和Mallat、Peyre等人提出了Bandelet变换,Bandelet变换提供了一种新的基于边缘的图像表示方法。对于几何正则图像,采用Bandelet基函数可以实现最稀疏表示。但由于研究人员是从压缩编码的角度来构造的Bandelet基,因此将其用于去噪处理时存在明显不足:在带噪图像中,构造Bandelet基时易受噪声系数的影响,不能对基函数方向准确定位,这大大降低Bandelet基对图像的逼近性能。
论文从抑制噪声的目的出发,结合Bandelet变换思想,构造了具有抗噪性能的自适应Bandelet基(NA-Bandelet bases)。首先在带噪信号的二进剖分块内,利用软阈值函数和在SURE准则下获得的阈值T,计算出对无噪信号的逼近结果,再按照最小化逼近误差原则,寻找出二进剖分块的几何方向。然后在最小化均分误差(MMSE)的原则下合并二进剖分块,最终确定了NA-Bandelet基的方向。
在此基础上,论文提出了基于NA-Bandelet基的去噪算法。首先利用NA-Bandelet基对图像进行分解,再以NA-Bandelet块为单元,计算自适应阈值并进行阈值去噪处理。对光学图像和SAR图像进行去噪和降斑处理时,其去噪、降斑效果优与基于正交小波基、Bandelet基的去噪、降斑算法,并且具有更强地边缘保持能力。
针对带噪图像增强问题,论文提出了基于NA-Bandelet基的增强算法。该算法根据图像的几何特征将NA-Bandelet块分为两类:有几何方向的NA-Bandelet块和无几何方向的NA-Bandelet块,并分析了这两类NA-Bandelet块系数的不同特征,从而区分出噪声和信号、清晰边缘和脆弱边缘。在此基础上提出了一种新的增强函数,在抑制噪声的同时,增强较弱细节并保护图像中的清晰边缘不失真。实验结果同样表明,与传统的图像增强算法相比,该算法在抑制噪声和放大细节特征两方面均有明显改进。
论文以构建NA-Bandelet为主线,将其应用于对图像去噪与增强的处理中,并取得了良好的效果。
Wavelets is the optima bases in expressing the point singularity objective function. Because two dimensional separable wavelets is gained directly by tensor product of two independent one-dimensional wavelets,the wavelets fails to provide the optima representation of two-dimensional image with the "linear" or "face" singularity,And the effect of image denoising and enforcement with the wavelet transform will fall down.
In order to overcome the insufficiency of wavelet transform in processing high dimensional signal, the Bandelet transform which provides a sort of new image representation means based on the image edge is introduced by French scholar pennec,mallat and peyre. The Bandelet bases could realize best sparsity expression for the geometry regular image. Because the researcher constructs the Bandelet bases from the angle of compressed encoding, it is insufficient when the Bandelet bases perform image denoising. Because of the effect of noise coefficient, the direction of the Bandelet bases is incapable to be oriented exactly and the image approximation capability of the Bandelet bases is reduced greatly.
The noiseproof adaptive Bandelet bases Based on the Bandelet tansform idea are constructed in order to denoise in this paper. A new class of noiseproof adaptive Bandelet bases (NA-Bandelet bases) is proposed. First, employing soft-threshold function and the threshold obtained following SURE principle, the approximation of the original image in each dyadic square is computed, then the geometrical direction in each dyadic is found according to the principle of the minimal approximation error. secondly,the small dyadic square are combined based on the principle of Minimum Mean Square Error (MMSE), so the directions of NA-Bandelet bases are determined at last.
Moreover, a novel denoising method based on the NA -Bandelet bases is presented in this paper. First,the image is decomposed by the NA -Bandelet bases ,then the adaptive thesholding is computed and the thesholding denoising is performed in each NA-Bandelet block. Compared with the denoising methods based on Orthonormal wavelet bases and Bandelet bases, the result of experiment shows that the denoising methods based on the NA -Bandelet bases can reduce noise more effectively with improved edge-preserving ability for the desnoising and despeckling of optical image and SAR image.
A image enhancement arithmetic based on the NA -Bandelet bases is presented for the noise image in this paper. Acording to the geometry characters of image, The Bandelet blocks are classified into two sorts: the Bandelet blocks with geometrical directions and the Bandelet blocks without geometrical directions, and the different characters of the two sorts of Bandelet blocks are analyzed in order to distinguish the noise and signal,the weak contrast features and the strong contrast features. Image enhancement is realized by a new strategy that can emphasize features with low contrast and protect the strong contrast features from distortions while restraining noise. Results of experiments show that this method offers significant improved performance in restraining noise and emphasizing features with low contrast over the conventional image enhancement algorithms.
The construction of the NA-Bandelet bases is the leading focus of the whole thesis. and it has gained better effect that the NA-Bandelet is applied in the image denoise and enhancement.
为了克服小波变换在处理高维信号时的不足,法国学者Pennec和Mallat、Peyre等人提出了Bandelet变换,Bandelet变换提供了一种新的基于边缘的图像表示方法。对于几何正则图像,采用Bandelet基函数可以实现最稀疏表示。但由于研究人员是从压缩编码的角度来构造的Bandelet基,因此将其用于去噪处理时存在明显不足:在带噪图像中,构造Bandelet基时易受噪声系数的影响,不能对基函数方向准确定位,这大大降低Bandelet基对图像的逼近性能。
论文从抑制噪声的目的出发,结合Bandelet变换思想,构造了具有抗噪性能的自适应Bandelet基(NA-Bandelet bases)。首先在带噪信号的二进剖分块内,利用软阈值函数和在SURE准则下获得的阈值T,计算出对无噪信号的逼近结果,再按照最小化逼近误差原则,寻找出二进剖分块的几何方向。然后在最小化均分误差(MMSE)的原则下合并二进剖分块,最终确定了NA-Bandelet基的方向。
在此基础上,论文提出了基于NA-Bandelet基的去噪算法。首先利用NA-Bandelet基对图像进行分解,再以NA-Bandelet块为单元,计算自适应阈值并进行阈值去噪处理。对光学图像和SAR图像进行去噪和降斑处理时,其去噪、降斑效果优与基于正交小波基、Bandelet基的去噪、降斑算法,并且具有更强地边缘保持能力。
针对带噪图像增强问题,论文提出了基于NA-Bandelet基的增强算法。该算法根据图像的几何特征将NA-Bandelet块分为两类:有几何方向的NA-Bandelet块和无几何方向的NA-Bandelet块,并分析了这两类NA-Bandelet块系数的不同特征,从而区分出噪声和信号、清晰边缘和脆弱边缘。在此基础上提出了一种新的增强函数,在抑制噪声的同时,增强较弱细节并保护图像中的清晰边缘不失真。实验结果同样表明,与传统的图像增强算法相比,该算法在抑制噪声和放大细节特征两方面均有明显改进。
论文以构建NA-Bandelet为主线,将其应用于对图像去噪与增强的处理中,并取得了良好的效果。
Wavelets is the optima bases in expressing the point singularity objective function. Because two dimensional separable wavelets is gained directly by tensor product of two independent one-dimensional wavelets,the wavelets fails to provide the optima representation of two-dimensional image with the "linear" or "face" singularity,And the effect of image denoising and enforcement with the wavelet transform will fall down.
In order to overcome the insufficiency of wavelet transform in processing high dimensional signal, the Bandelet transform which provides a sort of new image representation means based on the image edge is introduced by French scholar pennec,mallat and peyre. The Bandelet bases could realize best sparsity expression for the geometry regular image. Because the researcher constructs the Bandelet bases from the angle of compressed encoding, it is insufficient when the Bandelet bases perform image denoising. Because of the effect of noise coefficient, the direction of the Bandelet bases is incapable to be oriented exactly and the image approximation capability of the Bandelet bases is reduced greatly.
The noiseproof adaptive Bandelet bases Based on the Bandelet tansform idea are constructed in order to denoise in this paper. A new class of noiseproof adaptive Bandelet bases (NA-Bandelet bases) is proposed. First, employing soft-threshold function and the threshold obtained following SURE principle, the approximation of the original image in each dyadic square is computed, then the geometrical direction in each dyadic is found according to the principle of the minimal approximation error. secondly,the small dyadic square are combined based on the principle of Minimum Mean Square Error (MMSE), so the directions of NA-Bandelet bases are determined at last.
Moreover, a novel denoising method based on the NA -Bandelet bases is presented in this paper. First,the image is decomposed by the NA -Bandelet bases ,then the adaptive thesholding is computed and the thesholding denoising is performed in each NA-Bandelet block. Compared with the denoising methods based on Orthonormal wavelet bases and Bandelet bases, the result of experiment shows that the denoising methods based on the NA -Bandelet bases can reduce noise more effectively with improved edge-preserving ability for the desnoising and despeckling of optical image and SAR image.
A image enhancement arithmetic based on the NA -Bandelet bases is presented for the noise image in this paper. Acording to the geometry characters of image, The Bandelet blocks are classified into two sorts: the Bandelet blocks with geometrical directions and the Bandelet blocks without geometrical directions, and the different characters of the two sorts of Bandelet blocks are analyzed in order to distinguish the noise and signal,the weak contrast features and the strong contrast features. Image enhancement is realized by a new strategy that can emphasize features with low contrast and protect the strong contrast features from distortions while restraining noise. Results of experiments show that this method offers significant improved performance in restraining noise and emphasizing features with low contrast over the conventional image enhancement algorithms.
The construction of the NA-Bandelet bases is the leading focus of the whole thesis. and it has gained better effect that the NA-Bandelet is applied in the image denoise and enhancement.
引文
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[8]G Peyre and Stephane Mallat,Discrete Bandelets with Geometric Orthogonal Filters,Proceedings of ICIP,September,2005
[9]G Peyre and Stephane Mallat,Surface Compression with Geometric Bandelets,ACM Transactions on Graphics(SIGGRAPH'05),vol.14,no.3,2005
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[2]Starck J,Candes E J,Donoho D L.The Curvelet Transform for Image Denoising[J].IEEE Transactions on Image Processing,2002,11(6):670-684
[3]EJ Candes.Ridgelets:Theory And Applications D.USA:Department of Statistics,Stanford University,1998
[4]Candes EJ,DL Donoho.Curvelets[R].USA:Department of Statistics,Stanford University,1999
[5]Donoho D L.De-noising by soft-thresholding[J].IEEE Transactions on Information Theory,1995,41:613-627
[6]Erwan Le Pennec and Stephane Mallat,Bandelet Image Approximation and Compression,SIAM Journal of Multiscale Modeling and Simulation,vol.4,no.3,pages 992-1039,2005
[7]Erwan Le Pennec and Stephane Mallat,Sparse Geometric Image Representation with Bandelets,IEEE Trans.on Image Processing,vol 14,no.4,pages 423-438,April 2005
[8]G Peyre and Stephane Mallat,Discrete Bandelets with Geometric Orthogonal Filters,Proceedings of ICIP,September,2005
[9]G Peyre and Stephane Mallat,Surface Compression with Geometric Bandelets,ACM Transactions on Graphics(SIGGRAPH'05),vol.14,no.3,2005
[10]Mallat S.(著),杨力华等译.信号处理的小波导引.机械工业出版社,北京,2003
[11]Donoho D L.Sparse component analysis and optimal atomic decomposition.Constructive Approximation,1998,17:353-382
[12]Olshausen B A.and Field D J.Emergence of simple-cell receptive field properties by learning a sparse code for natural images.Nature.1996.381:607-609
[13]Donoho D L.and Flesia A G Can recent innovations in harmonic analysis explain' key findings in natural image statistics? Network:Computation in Neural Systems,2001,12(3):371-393
[14]焦李成,谭山.图像的多尺度几何分析:回顾和展望.电子学报,2003.12,31(12A):43-50
[15]Carlsson S.Sketch based coding of grey level images.IEEE Trans.IP.,1988,15(1):57-83
[16]Mallat S G and Zhong S.Characterization of signals from multiscale edges.IEEE Trans Patt.Anal.And Mach.Intell.,1992.7,14(7):710-732
[17]Elder J.Are edges incomplete? International Journal of Computer Vision,1999,34(2):97-122
[18]Xue X.and Wu X.Image representation based on mufti-scale edge compensation.IEEE Internat.Conf.On Image Processing,1999
[19]Cohen A.and Matei B.Nonlinear subdivisions scheme:Application to imageprocessing.Tutorial on multiresolution in geometric modeling,Springer,2002
[20]Cohen A.and Matei B.Compact representation of images by edge adapted multiscale transforms.Proc.IEEE Int.Conf.on Image Proc.,Special Session on Image Processing and Non-Linear Approximation,Thessaloniki,Greece,2001.10
[21]Dragotti P L.and Vetterli M.Shift-Invariant gibbs free denoising algorithm based on wavelet transfoem foot-prints.In Proc.SPIE2000,Wavelet Application in Signal and Image Processing,2000.8
[22]Dragotti P L.and Vetterli M.Footprints and edgeprints for image denoising andcompression.Proc.IEEE Int.Conf.on Image Proc.,Thessaloniki,Greece,2001,10,23 7-240
[23]Donoho D L.Wedgelets:nearly-minimax estimation of edges.Ann.Statist.,1999,27:859-897
[24]Wakin M B.Romberg J K.Choi H and et al.Rate-distortion optimized image compression using wedgelets.Proc.IEEE Int.Conf.on Image Proc.,Rochester,New York,2001.10
[25]Shukla R.Dragotti P L.Do M N.and et al.Rate-distortion optimized tree structured compression algorithms for piece smooth images.Submittd to IEEE Trans.IP.,2003,http://www.ifp.uiuc.edu/mindo/publications
[26]Candes E J.and Donoho D L.Ridgelets:The key to high-dimensional intermittency? Phil.Trans.R.Soc.Lond.A.1999,2495-2509
[27]Donoho D L.Orthonormal ridgelets and linear singularities.SIAM,Math Anal.2000,31(5),1062-1099
[28]Candes E J.and Donoho D L.Curvelets-A surprisingly effective nonadaptive representation for objects with edges.1999,.Technical report,Stanford Univ.
[29]Do M N.and Vetterli M.Contourlets.Stoeckler J.Welland G V,Beyond Press,2002
[30]Naghdy G Wang J.and Ogunbona P.Texture analysis using Gabor wavelets.Proceeding of SPIE.,1996,2657:74-85
[31]Cohen LRaz Sand Malah D.Translation-invariant denoising using the minimum description length criterion.[J]Signal Processing,1999,75(3):201-223
[32]Krim H,Schick I C.Minimax.description length for signal denoising and optimized representation.[J]IEEE Trans.Inform.Theory,1999,45(3):898-908
[33]Hansen M.and Yu B.Wavelet thresholding via MDL for nature images.[J]IEEE Trans.on Inform.Theory,2000,46(5):1778-788
[34]Bruce A G,Hong-Ye G.Understanding waveshrink:variance and bias estimation.http://wvv.mathsoft.com/wavelets.html
[35]Donoho D L.De-noising by soft-threshoding.IEEE Trans.On Information Theory,1995 41(3):617627
[36]Donoho D L,Johnstone I M.Ideal spatial adaptation via wavelet shrinkage.Biometrika,1994,81:425455
[37]Chang S G,Yu B,Vetterli,M.Adatpive wavelet thresholding for image denoising and compression.IEEE Trans.Image Processing,2000,9(9):15321546
[38]Donoho D L,Johnstone I M.Adapting to unknown smoothness via wavelet shrinkage,J.Am stat.Ass.,1995,90:12001224
[39]Stein C.Estimation of the mean of a multivariate normal distribution.Annala of Statistics,1981,9:11351151
[40]Donoho D L.De-noising by soft-thresholding[J].IEEE Trans.Information1995,41(3):613-627
[41]D.L.Donoho.Wavelet Thresholding and W.V.D:A 10-minute Tour[C].Int.Conf.on Wavelets and Applications,Taulouse.France:1992
[42]Jansen M,Malfait M,Signal Processing,1997 Bultheel A.Generalized cross validation for wavelet thresholding.56(1):33-44
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