S-WT Research And The Application In The Image De-noising

This article is mainly directed against the S-WT transformation,analysis and research the algorithm of the original , and put forward another method of implementation. At the same time, I have proposed quadtree-based decompositionmethod of adaptive S-WT transform.The first chapter reviews the history and basic content of the study of the multi-scale geometric analysis (MGA).First of all, introduce the Fourier analysisand wavelets analysis, two methods of non-linear function approximation of the effect of such basic Content,by contrast, showing that the wavelet functionapproximation of the total variation of the good results(Approximation order to achieve the O(N^-2)).At the same time, also pointed out that if unitarywavelets directly to the tensor form of wavelet applied to the binary functionon,We have found that the non-linear approximation is only O(N^-1). In order to be able to get better results, people start developing a new analyicalinstrument, and multi-scale geometric analysis is requested in this situation.Chapter I, section I introduce several major multi-scale geometric analysis tools Ridgelet,Curvelet,Bandelet,Contourlet, Simple description the their constructionideaand their own characteristics, which indicate the advantages of the multi-scale geometric analysis in the treatment of high-dimensional space of data.Section II, devoted to the S-WT, mainly introduce construction idea, the basic concepts, implementation methods, the effect on the function approxima- tion and so on.More important is the integer lattice and the concept of coset: from two linearly independent integer vectors:d₁ = (a₁,b₁),d₂ = (a₂,b₂) their linear combination constitute a integer lattice,to simplify notation,the two slopesare jointly denoted by the matrix M =(?).The integer lattice partitionsthe discrete space Z² to |det(M)|cosets , where each coset is determined by the shift vector s_k.The intersection between coset and digital line is co-lines , the most important of these on the S-WT is the 1-D wavelet transform on these co-lines o Non-linear approximation from S-WT is O(N^-1.55), although there is no approach to achieve optimal results 0(N^-2), but is better than the separable wavelet 0(N^-1.The second chapter is devoted to the image denoising and various denoisingmethods,focused on threshold denoising and how to determine the threshold.Image denoising is a very important issue in image processing,the beginning of this chapter,we introduce the mathematical model of image noise.section I,we give some functions which are used in threshold denoising frequently. In the application,mainly are the soft threshold function and the hard threshold function.Then we need to determine the threshold,ther are several classical algorithms,such as VisuShrink threshold, SUREShrink threshold, GCV threshold and so on.They are the optimal solution based on different rules.The third chapter is the content of improving the existing methods for S-WT.In accordance with the original image changed coordinates by rotating using the two transform ejections ,to be a new image, tnen,do the separable orthogonal wavelet transform on this new image.Still,the two transform directions are d₁ = (a₁,b₁),d₂ = (a₂,b₂),and the two slopes are jointly denoted by the matrix M as above,Well, the formula for the CORDIC isThrough the formula (1),we get |det(M)|new images.Do the separable orthogonalwavelet transform on these new images,it is equivalent to the original S-WT.In sectionⅡ,We propose a adaptive decomposition algorithm based on the edge,making the S-WT more applicable to complex images.The idea of adaptive algorithm is based on certain rules ,make image divide into four continuously.All sub-images have a strong local orientation information,so that the S-WT transformon these sub-images more conveniently.Chapter IV is the using of the above-mentioned algorithm to a denoising image, the experimental results with the standard separable wavelet denoising results were compared.The results of the experimental,the simpley S-WT (based on coordinate transformation's much better than the standard separable wavelet, and the adaptive algorithm is slightly better than the simple S-WT.The adaptive S-WT to meet the complex image,but it must pay the cost of computing time.From the point of view on the effect,the adaptive S-WT transform indeed is a effective algorithm.