基于无抽样方向滤波器组的图像处理算法研究
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摘要
多速率数字信号处理(Multirate Digital Signal Processing)作为数字信号处理重要分支之一,在二十世纪70年代提出,迄今被国内外学者广泛重视和深入研究,其应用已从最初的语音子带编码(Subband Coding,SBC)领域扩展到数字通信、图像压缩、视频压缩、计算机视觉、噪声消除及天线系统等许多领域。特别是在多速率系统(Multirate System)中,最为典型的M带均匀最大抽取(Maximally Decimated)滤波器组(Filter Banks,FBs)系统,其研究经历了从基本概念的提出到理论体系的丰富、完善和发展,在小波变换(WaveletTransform,WT)与多尺度分析(Multiresolution Analysis,MRA)领域内更加体现了这一点,其研究内容涵盖了FBs理论、实现结构、设计方法及应用等诸多方面。在过去的二十年里,众多研究者集中于一维(One-dimensional,1-D)FBs理论的研究并取得丰硕的成果,集中表现在对具有完全重构(PerfectlyReconstruction,PR)特性的M带正交镜像滤波器组(Quadrature Mirror FilterBanks,QMFBs)、离散傅立叶变换滤波器组(Discrete Fourier Transform FilterBanks,DFTFBs)和余弦调制滤波器组(Cosine Modulated Filter Banks,CMFBs)等几种典型滤波器组的研究上。对于一维的M带均匀滤波器组,已形成相当成熟的理论与设计方法。
在近十年来,人们致力于2-D滤波器组的研究,取得了一定的成就与突破,但在2-D FBs理论和设计上至今仍存在尚未完全解决的问题,比如具有线性相位(Linear Phase)与完全重构特性的2-D余弦调制滤波器组(CMFBs),这在很大程度上依赖于一维线性相位和完全重构的CMFBs理论的发展。同时,在图像与视频处理领域,不可分离2-D滤波器组的应用问题也没有得到应有的重视。此外,尽管以多尺度几何分析为代表的一类新型的2-D函数最优表示方法正逐步成为该领域的研究热点,但是,如何根据图像包含的几何特征来自适应地实现图像任意方向的分解有待进一步解决。本论文研究工作着重从无抽样2-D方向滤波器组设计及其在图像处理中的应用上,丰富与完善多速率数字信号处理理论。
方向滤波器作为图像方向特征描述的重要工具,广泛应用于图像压缩、图像增强、边缘检测和图像去噪等图像处理中。比如,在经典边缘检测算法中,Sobel算子、Prewitt算子、LoG算子等都归属于方向滤波器,但它们都仅局限于水平、垂直和对角方向。在目前的图像表示方法中小波变换已被广泛接受,它对处理一维分段光滑信号提供了一个非常稀疏或者说有效的表示方法,而小波在高维应用中却表现出了局限性。在图像进行二维小波变换后,代表图像细节的小波系数,即高频成分,表现在图像亮度值与周围区域有着强烈对比的边缘上。值得注意的是,这些有意义的小波系数位置反映出图像几何关联性,形成了简单的曲线。因此,二维小波变换能有效地捕获图像的边界点。但是因这些边界点仅由一些孤立的点组成,很难形成描述这些边界的光滑曲线。这是因为小波基仅对于零维点奇异值的目标函数是最优的,而对一维奇异性且光滑的图像边界不是最优的。
Bamberger和Smith于1992年首次提出了方向滤波器组(Directional FilterBanks,DFB)设计,每个子带输出都具有特定的方向性,能较好地捕获图像的边缘信息,由于对原始图像要进行多倍抽样,造成各子带图像分辨率与原图像不同,并随着分解方向数的增加,子带图像分辨率越来越小,因此,它适用于图像压缩等少量系数能量集中描述,但在图像增强、边缘检测和图像去噪等应用中,还需对各子带图像进行插值恢复,获得相同分辨率的图像,便于各子带图像之间进行统计特性分析,图像的抽样与插值在一定程度上会造成图像信息损失,这在医学图像处理中,将影响临床诊断结果。因此,如何避免抽样与插值,是方向滤波器设计与应用的一个主要问题。本文在具有抽样的方向滤波器组设计基础上,提出一种无抽样方向滤波器组设计方法,即:由一维半带低通滤波器先变换为二维低通滤波器,此二维低通滤波器为后面所有多方向滤波器设计的基础,均是通过坐标变换或各滤波器之间的运算实现,由该二维低通滤波器分别变换为象限滤波器和平行四边形滤波器,扇形滤波器由象限滤波器变换而来,象限滤波器和扇形滤波器合成四方向滤波器,再与平行四边形滤波器合成八方向滤波器,八方向滤波器再与平行四边形滤波器合成十六方向滤波器,更多方向的滤波器也都可由以上各种滤波器经过不同的坐标变换以及各滤波器之间的加减乘等运算得到。并将各种多方向频率域滤波器经过反傅立叶变换转化成空间模板,此模板使用方法与Sobel算子、Prewitt算子相同,直接与图像卷积实现图像滤波,其滤波过程不需要对图像抽样、旋转等操作,只需设计滤波器本身,所以,对图像进行方向分解时就避免了由于对图像抽样引起的混叠现象以及其他失真现象,利用设计的各方向子带滤波器的空间模板对图像滤波实现多方向分解,每个方向子带图像的分辨率均与原图像保持一致,免除插值恢复过程,有利于在相同空间位置上对各子带图像像素进行统计特性的分析,同时,可以减少由于对图像的插值引起的视觉失真现象,并且在方向子带图像各自独立处理之后,直接相加合成,无需类似小波分析的反变换,操作简便。
本文针对两种不同的情况将无抽样方向滤波器的空间模板应用到图像增强,一种是比较干净但纹理较模糊的图像,一种是带噪声的图像。对于前者,主要利用本文设计的无抽样方向滤波器空间模板(decimation-free directional filterbanks spatial operators,DFDFBOs),提取最能代表原图像方向信息的方向子带系数来实现增强;对于后者,将DFDFBOs与多尺度分析相结合,针对不同的高频信息(分为强边缘、弱边缘和噪声)分别进行处理。其增强结果较传统方法及DFB方法都有很大改善。采用Contourlet变换的思想实现图像去噪,与其不同的是:在多方向分解时,利用本文设计的无抽样方向滤波器空间模板。实验结果表明本文方法不仅有效地去除图像噪声、保留图像的边缘纹理信息,并很好的减少了Contourlet变换去噪中无法避免的伪吉布斯现象(pseudo-Gibbsphenomenon)所引起的视觉失真,与现有阈值去噪方法相比,去噪后图像信噪比明显提高,视觉效果明显改善。
Multirate Digital Signal Processing theory,as one of the important branches in digital signal processing theory,was proposed in 20~(th) century 70s,and has been extensively studied by many domestic and foreign scholars.Its application extends from voice subband coding(SBC) to the field of digital communication,image compression,video compression,computer vision,noise cancellation,antenna systems and many other fields.Particularly,for the most typical M-band uniform Maximally Decimated Filter Banks(FBs) in Multirate system,the research has gone through from the proposition of basic theory to the rich,perfect and development. The process is more obvious in the fields of Wavelet Transform(WT) and Multiresolution Analysis(MRA),which mainly studied FBs theory,implementation structure,design method and applications.In the past two decades,many scholars focused on 1-D FBs theoretical studies and had obtained fruitful results,which focused on the research of some typical filter banks:M-band Quadrature Mirror Filter Banks(QMFBs) with perfect reconstruction(PR),Discrete Fourier Transform Filter Banks(DFTFBs) and Cosine Modulated Filter Banks(CMFBs) etc.The theory of 1-D M-band uniform Maximally Decimated Filter Banks and design methods have reached a very advanced stage.
In the past ten years,we have made significant achievements and breakthroughs in the study of 2-D FBs,but the problems about the theory and design of 2-D FBs have so far not well be resolved,such as CMFBs with linear phase(LP) and perfect reconstruction(PR).This largely depends on the further development of 1-D LP PR CMFBs theory.At the same time,in image and video processing,we have not attached importance to the nonseparable 2-D FBs.Besides,while 2-D function represented by multiscale geometric analysis becomes the hot area of research,there is no method can adaptively perform arbitrary direction decomposition in terms of geometric characteristics of image.In this paper,the work focused on the design of decimation-free directional filter banks and their application in image processing to develop Multirate Digital Signal Processing.
As the important tool describing directional information,directional filter has gained its popularity in areas like image compression,image enhancement,edge detection and image denoising.For example,in the classical edge detection algorithm, 'Sobel','Prewitt' and 'LOG' are normally used for directional analysis.But these masks are limited to local orientation of horizontal,vertical and diagonal directions. In the current methods of image representation,Wavelet Transform has been widely accepted.WT provides a very sparse and effective representation for dealing with one-dimensional piecewise smooth signals,but it has shown limitation for high-dimensional signals.The performance of the wavelet coefficients in the intensity is the strong contrast points(that is image edges) with surrounding region by analyzing image details obtained by image 2-D wavelet transforming.It is worth noting that the location of these meaningful coefficients shows geometric relevance. These points form simple curves.Therefore,the 2-D WT is effective in seizing the border points,but we can't see the smoothness along these boundary curves,that is to say,these points are isolated points but not smooth curves.The root cause of the disadvantage is that wavelet is optimal objective function for the 0-D points,but edges are usually smooth curves with 1-D singularity.Therefore,wavelet basis is not optimal for smooth curves.
In 1992,Bamberger and Smith developed directional filter banks(DFB) first.It is believed that each output of DFBs corresponds to global features in some particular direction in spatial domain.DFBs can capture the directional information easily. Decimation to the image leads to different resolution between subband images and input image.It was also mentioned in some literaturs that subband images' resolution will decrease with the direction number increasing,which is not convenient to analyze the statistic character of subband images.It suits for image impression,but it needs interpolation to obtain the image with the same resolution as the original image in image enhancement,edge detection and image denoising.Meanwhile, decimation and interpolation will lead to the image information losing,especially in medical image,which would affect the diagnosis results.Therefore,how to avoid decimation and interpolation is an important problem in directional filter banks' application.Decimation-free directional filter banks(DFDFB) presented in this paper satisfy the request.The design method of DFDFB is based on the design of DFB.The design process is as follows:first,we transform 1-D half-band filter into 2-D lowpass filter.This 2-D lowpass filter is the basis of all directional filters which are obtained by coordination transform and calculation between filters.Transform the 2-D lowpass filter into quadrant filters and parallelogram filters.Fan filters are obtained by transforming quadrant filters;4-band directional filters are synthesized by quadrant filters and fan filters;4-band directional filters and parallelogram filters synthesize 8-band directional filters;8-band directional filters and parallelogram filters synthesize 16-band directional filters;multi-band directional filters are obtained by transforming all directional filters mentioned above.Then,all filters in frequency domain were transformed to spatial operators,which are called DFDFBOs.No decimation lead to no aliasing and folding.We can get each subband image with the same size with input image by convolving input image with each spatial operator in the same bank.Meanwhile,the artifacts produced due to presence of decimators and interpolation will be avoided by using DFDFBOs.Synthesis by using DFDFBOs at any stage can be achieved by just simply adding all the subband images.
We apply the DFDFBOs to image enhancement for two different cases,the first case is for clean images but with blur textures,the other is images with noise.In the former case,we extracted the directional subband coefficients which were most representative of the direction information of original image;for the latter one,we combined the decimation-free directional filter banks spatial operators(DFDFBOs) with multiscale analysis to perform different processing for different high-frequency information which divided into strong edges,weak edges and noise.The results of the two methods outperform other classical method and DFB in obtaining clear structure. This paper applies the idea of Contourlet transform to image denoising.The difference with Contourlet is that directional analysis is performed by DFDFBOs.The experimental results indicate that the method avoids visual distortion caused by pseudo-Gibbs phenomenon,is better than the existing denoising algorithms in smoothing noises,preserves image textures and details and improves the SNR of image.Visual effects have been improved significantly.
在近十年来,人们致力于2-D滤波器组的研究,取得了一定的成就与突破,但在2-D FBs理论和设计上至今仍存在尚未完全解决的问题,比如具有线性相位(Linear Phase)与完全重构特性的2-D余弦调制滤波器组(CMFBs),这在很大程度上依赖于一维线性相位和完全重构的CMFBs理论的发展。同时,在图像与视频处理领域,不可分离2-D滤波器组的应用问题也没有得到应有的重视。此外,尽管以多尺度几何分析为代表的一类新型的2-D函数最优表示方法正逐步成为该领域的研究热点,但是,如何根据图像包含的几何特征来自适应地实现图像任意方向的分解有待进一步解决。本论文研究工作着重从无抽样2-D方向滤波器组设计及其在图像处理中的应用上,丰富与完善多速率数字信号处理理论。
方向滤波器作为图像方向特征描述的重要工具,广泛应用于图像压缩、图像增强、边缘检测和图像去噪等图像处理中。比如,在经典边缘检测算法中,Sobel算子、Prewitt算子、LoG算子等都归属于方向滤波器,但它们都仅局限于水平、垂直和对角方向。在目前的图像表示方法中小波变换已被广泛接受,它对处理一维分段光滑信号提供了一个非常稀疏或者说有效的表示方法,而小波在高维应用中却表现出了局限性。在图像进行二维小波变换后,代表图像细节的小波系数,即高频成分,表现在图像亮度值与周围区域有着强烈对比的边缘上。值得注意的是,这些有意义的小波系数位置反映出图像几何关联性,形成了简单的曲线。因此,二维小波变换能有效地捕获图像的边界点。但是因这些边界点仅由一些孤立的点组成,很难形成描述这些边界的光滑曲线。这是因为小波基仅对于零维点奇异值的目标函数是最优的,而对一维奇异性且光滑的图像边界不是最优的。
Bamberger和Smith于1992年首次提出了方向滤波器组(Directional FilterBanks,DFB)设计,每个子带输出都具有特定的方向性,能较好地捕获图像的边缘信息,由于对原始图像要进行多倍抽样,造成各子带图像分辨率与原图像不同,并随着分解方向数的增加,子带图像分辨率越来越小,因此,它适用于图像压缩等少量系数能量集中描述,但在图像增强、边缘检测和图像去噪等应用中,还需对各子带图像进行插值恢复,获得相同分辨率的图像,便于各子带图像之间进行统计特性分析,图像的抽样与插值在一定程度上会造成图像信息损失,这在医学图像处理中,将影响临床诊断结果。因此,如何避免抽样与插值,是方向滤波器设计与应用的一个主要问题。本文在具有抽样的方向滤波器组设计基础上,提出一种无抽样方向滤波器组设计方法,即:由一维半带低通滤波器先变换为二维低通滤波器,此二维低通滤波器为后面所有多方向滤波器设计的基础,均是通过坐标变换或各滤波器之间的运算实现,由该二维低通滤波器分别变换为象限滤波器和平行四边形滤波器,扇形滤波器由象限滤波器变换而来,象限滤波器和扇形滤波器合成四方向滤波器,再与平行四边形滤波器合成八方向滤波器,八方向滤波器再与平行四边形滤波器合成十六方向滤波器,更多方向的滤波器也都可由以上各种滤波器经过不同的坐标变换以及各滤波器之间的加减乘等运算得到。并将各种多方向频率域滤波器经过反傅立叶变换转化成空间模板,此模板使用方法与Sobel算子、Prewitt算子相同,直接与图像卷积实现图像滤波,其滤波过程不需要对图像抽样、旋转等操作,只需设计滤波器本身,所以,对图像进行方向分解时就避免了由于对图像抽样引起的混叠现象以及其他失真现象,利用设计的各方向子带滤波器的空间模板对图像滤波实现多方向分解,每个方向子带图像的分辨率均与原图像保持一致,免除插值恢复过程,有利于在相同空间位置上对各子带图像像素进行统计特性的分析,同时,可以减少由于对图像的插值引起的视觉失真现象,并且在方向子带图像各自独立处理之后,直接相加合成,无需类似小波分析的反变换,操作简便。
本文针对两种不同的情况将无抽样方向滤波器的空间模板应用到图像增强,一种是比较干净但纹理较模糊的图像,一种是带噪声的图像。对于前者,主要利用本文设计的无抽样方向滤波器空间模板(decimation-free directional filterbanks spatial operators,DFDFBOs),提取最能代表原图像方向信息的方向子带系数来实现增强;对于后者,将DFDFBOs与多尺度分析相结合,针对不同的高频信息(分为强边缘、弱边缘和噪声)分别进行处理。其增强结果较传统方法及DFB方法都有很大改善。采用Contourlet变换的思想实现图像去噪,与其不同的是:在多方向分解时,利用本文设计的无抽样方向滤波器空间模板。实验结果表明本文方法不仅有效地去除图像噪声、保留图像的边缘纹理信息,并很好的减少了Contourlet变换去噪中无法避免的伪吉布斯现象(pseudo-Gibbsphenomenon)所引起的视觉失真,与现有阈值去噪方法相比,去噪后图像信噪比明显提高,视觉效果明显改善。
Multirate Digital Signal Processing theory,as one of the important branches in digital signal processing theory,was proposed in 20~(th) century 70s,and has been extensively studied by many domestic and foreign scholars.Its application extends from voice subband coding(SBC) to the field of digital communication,image compression,video compression,computer vision,noise cancellation,antenna systems and many other fields.Particularly,for the most typical M-band uniform Maximally Decimated Filter Banks(FBs) in Multirate system,the research has gone through from the proposition of basic theory to the rich,perfect and development. The process is more obvious in the fields of Wavelet Transform(WT) and Multiresolution Analysis(MRA),which mainly studied FBs theory,implementation structure,design method and applications.In the past two decades,many scholars focused on 1-D FBs theoretical studies and had obtained fruitful results,which focused on the research of some typical filter banks:M-band Quadrature Mirror Filter Banks(QMFBs) with perfect reconstruction(PR),Discrete Fourier Transform Filter Banks(DFTFBs) and Cosine Modulated Filter Banks(CMFBs) etc.The theory of 1-D M-band uniform Maximally Decimated Filter Banks and design methods have reached a very advanced stage.
In the past ten years,we have made significant achievements and breakthroughs in the study of 2-D FBs,but the problems about the theory and design of 2-D FBs have so far not well be resolved,such as CMFBs with linear phase(LP) and perfect reconstruction(PR).This largely depends on the further development of 1-D LP PR CMFBs theory.At the same time,in image and video processing,we have not attached importance to the nonseparable 2-D FBs.Besides,while 2-D function represented by multiscale geometric analysis becomes the hot area of research,there is no method can adaptively perform arbitrary direction decomposition in terms of geometric characteristics of image.In this paper,the work focused on the design of decimation-free directional filter banks and their application in image processing to develop Multirate Digital Signal Processing.
As the important tool describing directional information,directional filter has gained its popularity in areas like image compression,image enhancement,edge detection and image denoising.For example,in the classical edge detection algorithm, 'Sobel','Prewitt' and 'LOG' are normally used for directional analysis.But these masks are limited to local orientation of horizontal,vertical and diagonal directions. In the current methods of image representation,Wavelet Transform has been widely accepted.WT provides a very sparse and effective representation for dealing with one-dimensional piecewise smooth signals,but it has shown limitation for high-dimensional signals.The performance of the wavelet coefficients in the intensity is the strong contrast points(that is image edges) with surrounding region by analyzing image details obtained by image 2-D wavelet transforming.It is worth noting that the location of these meaningful coefficients shows geometric relevance. These points form simple curves.Therefore,the 2-D WT is effective in seizing the border points,but we can't see the smoothness along these boundary curves,that is to say,these points are isolated points but not smooth curves.The root cause of the disadvantage is that wavelet is optimal objective function for the 0-D points,but edges are usually smooth curves with 1-D singularity.Therefore,wavelet basis is not optimal for smooth curves.
In 1992,Bamberger and Smith developed directional filter banks(DFB) first.It is believed that each output of DFBs corresponds to global features in some particular direction in spatial domain.DFBs can capture the directional information easily. Decimation to the image leads to different resolution between subband images and input image.It was also mentioned in some literaturs that subband images' resolution will decrease with the direction number increasing,which is not convenient to analyze the statistic character of subband images.It suits for image impression,but it needs interpolation to obtain the image with the same resolution as the original image in image enhancement,edge detection and image denoising.Meanwhile, decimation and interpolation will lead to the image information losing,especially in medical image,which would affect the diagnosis results.Therefore,how to avoid decimation and interpolation is an important problem in directional filter banks' application.Decimation-free directional filter banks(DFDFB) presented in this paper satisfy the request.The design method of DFDFB is based on the design of DFB.The design process is as follows:first,we transform 1-D half-band filter into 2-D lowpass filter.This 2-D lowpass filter is the basis of all directional filters which are obtained by coordination transform and calculation between filters.Transform the 2-D lowpass filter into quadrant filters and parallelogram filters.Fan filters are obtained by transforming quadrant filters;4-band directional filters are synthesized by quadrant filters and fan filters;4-band directional filters and parallelogram filters synthesize 8-band directional filters;8-band directional filters and parallelogram filters synthesize 16-band directional filters;multi-band directional filters are obtained by transforming all directional filters mentioned above.Then,all filters in frequency domain were transformed to spatial operators,which are called DFDFBOs.No decimation lead to no aliasing and folding.We can get each subband image with the same size with input image by convolving input image with each spatial operator in the same bank.Meanwhile,the artifacts produced due to presence of decimators and interpolation will be avoided by using DFDFBOs.Synthesis by using DFDFBOs at any stage can be achieved by just simply adding all the subband images.
We apply the DFDFBOs to image enhancement for two different cases,the first case is for clean images but with blur textures,the other is images with noise.In the former case,we extracted the directional subband coefficients which were most representative of the direction information of original image;for the latter one,we combined the decimation-free directional filter banks spatial operators(DFDFBOs) with multiscale analysis to perform different processing for different high-frequency information which divided into strong edges,weak edges and noise.The results of the two methods outperform other classical method and DFB in obtaining clear structure. This paper applies the idea of Contourlet transform to image denoising.The difference with Contourlet is that directional analysis is performed by DFDFBOs.The experimental results indicate that the method avoids visual distortion caused by pseudo-Gibbs phenomenon,is better than the existing denoising algorithms in smoothing noises,preserves image textures and details and improves the SNR of image.Visual effects have been improved significantly.
引文
[1]R.H.Bamberger.New subband decompositions and coders for image and video compression[C].IEEE.International Conference on Acoustics,Speech and Signal Processing,March 1992,3:217-220.
[2]S.Park,M.J.T.Smith,and J.J.Lee.Fingerprint enhancement based on the directional filter bank[C].IEEE.International Conference on Image Processing,Sept 2000:793-796.
[3]Ramin Eslami,and Hayder Radha.A New Family of Nonredundant Transforms Using Hybrid Wavelets and Directional Filter Banks[J].IEEE Transactions on image processing,April 2007,16(4):1152-1167.
[4]万洪林.基于方向的图像变换和滤波方法及其应用的研究[D].博士学位论文.济南:山东大学,2008.
[5]Roberto H.Bamberger and Mark J.T.Smith.A Filter Bank for the Directional Decomposition of Images:Theory and Design.IEEE transactions on signal processing[J].April 1992,40(4):882-892.
[6]RASHID ANSARI.Efficient IIR and FIR Fan Filters[J].IEEE transactions on circuits and systems,August 1987,CAS-34(8):941-945.
[7]D.L.Donoho,M.Vetterli,R.A.DeVore,et al.Data compression and harmonic analysis[J].IEEE Trans.Inform.Th.,October 1998,44(6):2435-2476.
[8]S.Mallat.A Wavelet Tour of Signal Processing[M].2nd ed.Academic Press,1999.
[9]M.N.Do and M.Vetterli.Contourlets:a directional multiresolution image representation[C].IEEE Proc.9th ICIR 2002:357-360.
[10]Mohammad A.U.Khan,M.Khalid Khan,M.Aurangzeb Khan.Comparative analysis of decimation-free directional filter bank with directional filter bank:In context of Image enhancement[C].IEEE 9th International Multitopic Conference,DEC.2005:1-8.
[11]C.H.Park,J.J.Lee,M.J.T.Smith,S.Park,and K.J.Park.Directional filter bank-based fingerprint feature extraction and matching[J].IEEE.Trans.On Circuits and Systems for Video Technology,Jan.2004,14(1):74-85.
[12]阮秋琦,阮宇智等译(R.C.Gonzalez,R.E.Woods著).数字图像处理(第二版)[M].北京:电子工业出版社,2004.
[13]梁晓云,曾卫明和罗立民.基于偏微分方程的医学磁共振图像去噪[J].信号处理.2004,20(3):318-321.
[14]Donoho D.L.,Vetterli M.,DeVore R.A.et al.Data compression and harmonic analysis[J].IEEE Trans.Inf.Theory,1998,44(6):2435-2476.
[15]Mallat S.A Wavelet Tour of Signal Processing[M].2nded.New York,Academic,1999.
[16]郭旭静.方向性数字图像表示方法及去噪算法研究[D].博士学位论文.天津:天津大学电子信息工程学院,2005.
[17]P.J.Burt and E.H.Adelson.The laplacian pyramid as a compact image code [J].IEEE Trans.Commun.,April 1983,31(4):532-540.
[18]Goyal V.K.,Kovacevic.Quantized Frame Expansions with Erasures[J].2001,10(3):203-233.
[19]Mallat S.Multiresolution approximation and wavelet orthonormal bases of L~2(R)[J].Trans.Amer.Math.Soc.1989,315:69-87.
[20]Horn R.A.,Johnson C.R.Matrix Analysis[M].Cambridge University Press,1985.
[21]Do M.N.,Vetterli M.Framing Pyramids[J].IEEE Trans.on Signal processing,2003,51(9):2329-2342.
[22]S.Park.New directional filter banks and their applications in image processing [J].Ph.D.thesis,Georgia Institute of Technology,Nov.1999.
[23]S.Park,M.J.T.Smith,and R.M.Mersereau.Improved structures of maximally decimated directional filter banks for spatial image analysis[J].IEEE Trans.IP.,Nov.2004,13(11):1424-1431.
[24]宋亮.二维滤波器(组)及其在图像分析中的应用[D].硕士学位论文.西安:西安电子科技大学,2007.
[25]胡广书 编著.现代信号处理教程[M].北京:清华大学出版社,2004.11.
[26]胡广书 编著.数字信号处理--理论、算法与实现[M].第2版.北京:清华大学出版社,2003.
[27]Ansari R,Guillemot C,et al.Wavelet construction using lagrange halfband filters[J].IEEE Trans CAS,1991,38:116-118.
[28]Vaidyanathan P P.Multirate Systems and Filter Banks[M].Englewood Cliffs,NJ:Pretice-Hall,1993.
[29]Yuan-Pei Lin and P.P.Vaidyanathan.Theory and Design of Two-Parallelogram Filter Banks[J].IEEE transactions on signal processing,November 1996,44(11):2688-2706.
[30]Yuan-Pei Lin and P.P.Vaidyanathan.On the Study of Four-Parallelogram Filter Banks[J].IEEE transactions on signal processing,November 1996,44(11):2707-2717.
[31]高贇.图像灰度增强算法的研究[D].硕士学位论文.西安:西安电子科技大学,2007.
[32]Minh N.Do.And Martin Vetterli.The Contourlet Transform:An efficient Directional Multiresolution Image Representation[J].IEEE Transctions on Image Processing,Dec.2005,14:2091-2106.
[33]Dunncan D.-Y.Po and Mihn N.D.Directional Multiscale Modeling of Images using the Contourlet Transform[J].IEEE Transactions on Image Processing,Jun.2006,15(6):1610-1620.
[34]Donoho D L.De-noising by soft-thresholding[J].IEEE Transaction on Information Theory,1995,41(3):613-627.
[35]D.Donoho and I.Johnstone.Ideal spatial adaption via wavelet shrinkage[J].Biometrika,Dec.1994,31:425-455.
[36]戴维,于盛林等人.基于Contourlet变换自适应阈值的图像去噪算法.电子学报,Oct.2007,35(10):1939-1943.
[37]Mink N Do,Martin Vetterli.The finite Ridgelet transform for image representation[J].IEEE Transaction on Image Precessing,2002,1(12):16-28.
[38]J-L Starck,E J Candes,D L Donoho.The curvelet transform for image denoising[J].IEEE Transaction on Image Processing,2002,6(11):670-684.
[2]S.Park,M.J.T.Smith,and J.J.Lee.Fingerprint enhancement based on the directional filter bank[C].IEEE.International Conference on Image Processing,Sept 2000:793-796.
[3]Ramin Eslami,and Hayder Radha.A New Family of Nonredundant Transforms Using Hybrid Wavelets and Directional Filter Banks[J].IEEE Transactions on image processing,April 2007,16(4):1152-1167.
[4]万洪林.基于方向的图像变换和滤波方法及其应用的研究[D].博士学位论文.济南:山东大学,2008.
[5]Roberto H.Bamberger and Mark J.T.Smith.A Filter Bank for the Directional Decomposition of Images:Theory and Design.IEEE transactions on signal processing[J].April 1992,40(4):882-892.
[6]RASHID ANSARI.Efficient IIR and FIR Fan Filters[J].IEEE transactions on circuits and systems,August 1987,CAS-34(8):941-945.
[7]D.L.Donoho,M.Vetterli,R.A.DeVore,et al.Data compression and harmonic analysis[J].IEEE Trans.Inform.Th.,October 1998,44(6):2435-2476.
[8]S.Mallat.A Wavelet Tour of Signal Processing[M].2nd ed.Academic Press,1999.
[9]M.N.Do and M.Vetterli.Contourlets:a directional multiresolution image representation[C].IEEE Proc.9th ICIR 2002:357-360.
[10]Mohammad A.U.Khan,M.Khalid Khan,M.Aurangzeb Khan.Comparative analysis of decimation-free directional filter bank with directional filter bank:In context of Image enhancement[C].IEEE 9th International Multitopic Conference,DEC.2005:1-8.
[11]C.H.Park,J.J.Lee,M.J.T.Smith,S.Park,and K.J.Park.Directional filter bank-based fingerprint feature extraction and matching[J].IEEE.Trans.On Circuits and Systems for Video Technology,Jan.2004,14(1):74-85.
[12]阮秋琦,阮宇智等译(R.C.Gonzalez,R.E.Woods著).数字图像处理(第二版)[M].北京:电子工业出版社,2004.
[13]梁晓云,曾卫明和罗立民.基于偏微分方程的医学磁共振图像去噪[J].信号处理.2004,20(3):318-321.
[14]Donoho D.L.,Vetterli M.,DeVore R.A.et al.Data compression and harmonic analysis[J].IEEE Trans.Inf.Theory,1998,44(6):2435-2476.
[15]Mallat S.A Wavelet Tour of Signal Processing[M].2nded.New York,Academic,1999.
[16]郭旭静.方向性数字图像表示方法及去噪算法研究[D].博士学位论文.天津:天津大学电子信息工程学院,2005.
[17]P.J.Burt and E.H.Adelson.The laplacian pyramid as a compact image code [J].IEEE Trans.Commun.,April 1983,31(4):532-540.
[18]Goyal V.K.,Kovacevic.Quantized Frame Expansions with Erasures[J].2001,10(3):203-233.
[19]Mallat S.Multiresolution approximation and wavelet orthonormal bases of L~2(R)[J].Trans.Amer.Math.Soc.1989,315:69-87.
[20]Horn R.A.,Johnson C.R.Matrix Analysis[M].Cambridge University Press,1985.
[21]Do M.N.,Vetterli M.Framing Pyramids[J].IEEE Trans.on Signal processing,2003,51(9):2329-2342.
[22]S.Park.New directional filter banks and their applications in image processing [J].Ph.D.thesis,Georgia Institute of Technology,Nov.1999.
[23]S.Park,M.J.T.Smith,and R.M.Mersereau.Improved structures of maximally decimated directional filter banks for spatial image analysis[J].IEEE Trans.IP.,Nov.2004,13(11):1424-1431.
[24]宋亮.二维滤波器(组)及其在图像分析中的应用[D].硕士学位论文.西安:西安电子科技大学,2007.
[25]胡广书 编著.现代信号处理教程[M].北京:清华大学出版社,2004.11.
[26]胡广书 编著.数字信号处理--理论、算法与实现[M].第2版.北京:清华大学出版社,2003.
[27]Ansari R,Guillemot C,et al.Wavelet construction using lagrange halfband filters[J].IEEE Trans CAS,1991,38:116-118.
[28]Vaidyanathan P P.Multirate Systems and Filter Banks[M].Englewood Cliffs,NJ:Pretice-Hall,1993.
[29]Yuan-Pei Lin and P.P.Vaidyanathan.Theory and Design of Two-Parallelogram Filter Banks[J].IEEE transactions on signal processing,November 1996,44(11):2688-2706.
[30]Yuan-Pei Lin and P.P.Vaidyanathan.On the Study of Four-Parallelogram Filter Banks[J].IEEE transactions on signal processing,November 1996,44(11):2707-2717.
[31]高贇.图像灰度增强算法的研究[D].硕士学位论文.西安:西安电子科技大学,2007.
[32]Minh N.Do.And Martin Vetterli.The Contourlet Transform:An efficient Directional Multiresolution Image Representation[J].IEEE Transctions on Image Processing,Dec.2005,14:2091-2106.
[33]Dunncan D.-Y.Po and Mihn N.D.Directional Multiscale Modeling of Images using the Contourlet Transform[J].IEEE Transactions on Image Processing,Jun.2006,15(6):1610-1620.
[34]Donoho D L.De-noising by soft-thresholding[J].IEEE Transaction on Information Theory,1995,41(3):613-627.
[35]D.Donoho and I.Johnstone.Ideal spatial adaption via wavelet shrinkage[J].Biometrika,Dec.1994,31:425-455.
[36]戴维,于盛林等人.基于Contourlet变换自适应阈值的图像去噪算法.电子学报,Oct.2007,35(10):1939-1943.
[37]Mink N Do,Martin Vetterli.The finite Ridgelet transform for image representation[J].IEEE Transaction on Image Precessing,2002,1(12):16-28.
[38]J-L Starck,E J Candes,D L Donoho.The curvelet transform for image denoising[J].IEEE Transaction on Image Processing,2002,6(11):670-684.