Research In Image Processing Based On Decimation-free Directional Filter Bank

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.