Research On Structured Sparse Optimization Algorithms In Image Processing

The image data is a very important type of high-dimensional data,which is the most important information carrier in our daily life.This dissertation focus on the im-age data,particularly the research of image restoration applications.The goal of image restoration aims to recover the underlying clean image from the degraded observation.Image restoration includes image denoising,deblurring,inpainting and superresolution etc.Mathematically,it is an inverse problem,which restores the original true data from the observed data.Over the past decades,the regularization-based methods are widely adopted in the field of image restoration.They recover the high quality images by regularizing the un-derlying solution spaces.The sparsity-based methods based on the image sparsity prior have achieved great success.They are built upon the assumption that image can be repre-sented sparsely under appropriate transforms.Specifically,the regularization process are often chosen by minimizing a sparsity-promoting functional,such as the ?₁ norm.For the regularization methods,exploring and exploiting more appropriate prior information is one of the most important research topics.This dissertation aims at refining the sparsity regularization models.Besides the sparsity prior,we would like to use other image priors to improve the final recovery quality.The main contents and novelties of this dissertation are summarized as follows:Firstly,we propose a novel wavelet frame based weighted ?₁ norm regularization model for image inpainting.By alternately performing the support detection and solving the weighted ?₁ norm minimization problem,we develop a multi-stage convex relaxation optimization method and gradually improve the recovery quality.More importantly,dur-ing the process of support detection,we further exploit the structural information of the frame coefficients.Therefore,we can acquire more accurate support set and attain higher quality recovery image.Secondly,we propose an efficient wavelet frame based truncated ?₀ regularization model for image deblurring.Most wavelet frame based image restoration models?such as the ?₁ and ?0 regularization model?only use the sparsity prior.In contrast,this disser-tation attains the partial support information by a self-learning process.We incorporate the support prior into the sparsity regularization model and propose a truncated ?₀ regu-larization model.The proposed algorithmic framework is able to seamlessly incorporate the existing state-of-the-art image restoration methods by taking their results as the initial reference image on which support detection of frame coefficients is performed.Thirdly,we propose a wavelet frame based truncated ?₀- ?₂ regularization model for image restoration.Compared with the truncated ?₀ regularization model,the added ?₂ regularizer term further exploits the self-recurrence prior of local image structures in spatial domain,i.e.,the nonlocal prior of frame coefficients in the transform domain.Therefore,the textures and tiny details can be well preserved in the restored image.Dif-ferent with the ?₀- ?₂ regularization model,where both sparsity prior and nonlocal prior of frame coefficients are applied,the support prior of frame coefficients is also respected in our proposed truncated ?₀- ?₂ regularization model,which leads to significant improve-ments in terms of sharp edges preservation.In contrast to classical wavelet frame based ?₁ or ?0 regularization model,where only single sparsity prior is utilized,both support and nonlocal priors of frame coefficients are added in our model,in the terms of truncated ?0 and ?2 regularizer,respectively.The proposed algorithm can produce a much better edge enhanced and texture preserved recovery image.Fourthly,we propose an efficient truncated ?₀ gradient regularization model for im-age smoothing.We will solve a standard ?₀ gradient regularization model or run other smoothing algorithms to obtain an initial smoothing image in the first stage.In the second stage,we first perform the support estimation on the smoothing result of the first stage,and then solve the resulting truncated ?₀ gradient regularization model and return a higher quality smoothing output.