Research On Image Restoration Using Wavelet-based Method

Wavelet transform is an effective method for signal processing,which is widely used in feature extraction and sparse reconstruction of signals.Based on the prior knowledge that the image has sparse representation under wavelet multiscale transforms,the wavelet-based method are used to study the image sparse restoration problem in this dissertation.From the perspective of sparse representation,firstly,a kind of geometric multiscale system with specific directional properties is constructed to describe the direc-tional singularities of images,which is used to fully explore the inherent sparse property of images.Then,the corresponding sparse regularizer is introduced to establish a convex optimization model for image reconstruction.In order to further improve the quality of the restoration,a directional total variation is also introduced as the joint regularizer to constrain the edge smoothness of the reconstructed result.Finally,the fast algorithm is utilized to solve the model.Aiming at three types of sparse restoration problems:image deblurring,image inpainting and image deraining,the main research results of this thesis are listed as follows:1.For the problem of image deblurring,a joint regularized convex model with weighted?₁ norm is proposed for image sparse reconstruction.Based on the sparse structure of shearlets coefficients in high frequency,the?₁ norm of shearlets coefficients is used as a regularizer to constrain the inherent sparseness of the reconstructed result,and the directional total variational regularizer is introduced to constrain the smoothness of the restoration.The model solving includes two parts:weights updating and solving the weighted model.The weights are updated by continuously learning the sparse structure of the wavelet coefficients through a multi-step optimization algorithm with the relaxation strategy.The weighted model is solved by the split Bregman algorithm.Due to the convexity of the model,the proposed algorithm converges quickly.The effectiveness of proposed method is tested by three different types of blurry images.We compared the reconstructed results of three state-of-the-art methods with our method in two aspects:the quality evaluation index and the visualization,to illustrate the effectiveness of the proposed method.2.For the problem of image inpaiting,firstly,a joint regularized convex model with?₁ norm is proposed for image sparse restoration.Based on the energy distribution of images in frequency domain,mainly centering around the axes?i.e.low frequency domain and part of high frequency domain?,a directional geometric multi-scale system related to energy distribution is constructed to enhance the sparseness of wavelet decomposition coefficients in high frequency.Then,the?₁ norm of the proposed wavelet coefficient and the directional total variational are used as joint regularizers to detect the directional singularities of restoration and constrain the smoothness of edges.Finally,the model is solved by the split Bregman algorithm.The algorithm converges quickly since the proposed model is convex and the global optimal solver can be obtained.Moreover,the proposed model can also be used for image deblurring.In the numerical experiments,the magnetic resonance image and blurry images are used to test the effectiveness of the proposed method for image inpaiting and image deblurring respectively.The reconstructed result shows that our method performs well both in image deblurring and image inpainting.3.For the problem of rain removal from single image,firstly,a convex optimization model with multiple regularizers is proposed for rain removal.Based on the directionality of rain streaks,a geometric wavelet multi-scale system with particular direction is constructed to describe the inherent sparse structure of rain streaks and the background layer.Then,the sparse deraining model is proposed by following three sparse priors:?1?The wavelet decomposition coefficient of the background layer in the rain direction is sparse;?2?The wavelet decomposition coefficient of rain streaks in its perpendicular direction is sparse;?3?the rain streak is sparse.The model is solved by the split Bregman algorithm.Since the model is a convex function,the proposed algorithm converges quickly.Finally,the simulation data and the real data are used to test the effectiveness of the proposed method for rain removal,and the restoration of our method is evaluated in two aspects:the visualization and the quality evaluation index.