Research On Beyond Wavelet Analysis And Variation Methods For Image Processing

In the field of mathematical imaging and computer visual, beyond wavelet analysisand variation method are the most representative methods at current．Beyond waveletanalysis is a new development based on wavelet theory, which come from the conceptand method in the application of modern harmonic analysis and theory of grouprepresentation. Beyond wavelet analysis aims to test, represent, and process the data insome high dimension space, but some of the important characteristics of these datafocus on low-dimensional subspace. For two or higher dimensional functions includingline or surface singular, beyond wavelet analysis shows the more sparseness of therepresentation ability than the wavelet. Difference from the calculation harmonicanalysis method, variation method is another effective tool in image processing. Therehave found a lot of successful applications in the image denoising and imageenhancement, edge detection and so on. For the background of image processingapplication, this dissertation has made some useful exploration and research around thebeyond wavelet analysis and variation methods and got some preliminary researchachevements.The main research results are as follows:1. Curvelet transform and reaction diffusion equation are discussed in detail.Twoimage denoising algorithms based on curvelet transform and reaction diffusion equationare presented.Nordstr m energy functional minimization problem is discussed in detail. From itsEuler equation, a new reaction diffusion filter model and related algorithm are given forimage denoising by new definition of appropriate control function. Our digitizedformulation leads to nonlinear algebraic equations instead of PDE’s and the analysisand application of the digital method needs no knowledge on numerical approximationof PDE’s. The advantages and disadvantages of the model are discussed. A smoothoperator and a new diffusion function are introduced to improve the model andeffectively avoid ill-posed diffusion coefficients and some spots in the process of usingthe reaction diffusion digital filter. The iterative process and the properties of the filtermodel are analyzed. The convergence of the iterative algorithm is proposed theoretically.The experiments demonstrate that the proposed model have preferable application forimage denoising with different types and degree noise．In order to avoid the "false" effect in the curvelet transform and reduce the spots inthe process of using the reaction diffusion digital filter, the denoising algorithm based on the proposed reaction diffusion filter model and the curvelet transform is presented.Experimental results show that the proposed algorithms reduce noise effectively andkeep edges well. At the same time, more important is that the algorithm can overcomethe Pseudo-Gibbs effect and “class curvelet” Pseudo curve phenomenon in the curvelettransform with the ideal visual effect.2. Basic characteristics, construction and digital realization of wave atoms arediscussed in detail, and its applications in image processing are introduced. Twoalgorithms for image denoising are proposed based on wave atoms transform.One algorithm is based on total variation minimization of wave atoms coefficients.Due to discontinuity of hard threshold operators and FFT cycle process of wave atoms,Pseudo Gibbs and new orientation texture in the neighborhood of discontinuous pointwas found in the denoised images. These distortions can be seen as a total oscillatoryand TV regularization can better suppress the oscillatory. Combined the minimizationof total variation, the algorithm based on wave atoms is presented for image denoising.Firstly, the nonlinear threshold strategy associated with wave atoms is applied to thetransform coefficients of noisy image．And then，the feasible domain of the proposedmodel is determined by the coefficients remained. Finally, the projected gradientalgorithm is used to solve the proposed model. Experiments results show that theproposed algorithm can reduce noise efficiently and preserve edges well while thePseudo-Gibbs aritifacts was suppressed with better visual effects.Another algorithm for image denoising is presented， which combine theadvantages of wave atoms transform and Cycle Spinning．Due to lack of translationinvariance of the wave atoms transform, image denoising method by coefficientthreshold would lead to Pseudo-Gibbs phenomena．Cycle Spinning is employed toavoid these artifacts．Experimental results show that the proposed algorithm can removenoise and remain edges，while Pseudo-Gibbs phenomena are suppressed efficiently withbetter visual effect and PSNR gains, compared with the methods like simple waveatoms or wavelet denoising using Cycle Spinning．4. Based on the dual tree complex wavelet transform, the structural similarity index(CW-SSIM) is proposed. Using multi-resolution analysis, the image can be decomposedinto sub-band images in different wavelet scales and image edges and structure isexpressed into wavelet coefficients. The dual tree complex wavelet transform has theadvantages of translation invariance and good direction selectivity, and it can respondmore detail information and edges of images. The index SSIM is very sensitive totranslation, scale change, rotating, so the structural similarity index based on dual tree complex wavelet transform is presented. The index is robust for small range oftranslation and rotation. When evaluating the image similarity, the image preprocessingis not needed. The effectiveness of the proposed index is amply illustrated on a varietyof examples.5. Some detailed discussions are made for image harmonic inpainting and totalvariation inpainting by considering some different image space. Three models based onwavelet transform and algorithms are discussed in image inpainting, which are totalvariation wavelet inpainting and image inpainting model based on curvature driven, aswell as the image domain realization for wavelet domain images inpainting. The TVwavelet inpainting model promotes the process of image repairing by total variationminimization and the minimization can also inhibit the noise of the image. Usingcurvature regularization standard, image inpainting model based on curvature driven haspunished the length of the edge line and the integral of curvature along the edge line,thus ensured the continuity of the curvature edge. The image domain realization forwavelet domain image inpainting is to use all inpainting technology in image domain inorder to finish the repairing of wavelet domain. Their effectiveness is illustrated on avariety of examples.