Inverse Mathematical Problem And Its Applications In Image Restoration

Inverse problems have developed rapidly both in theory and practice as a hot topic in academic field.Among inverse problems,image restoration receives extensive attentions as a typical problem.In order to improve the image quality,image restoration is one of the most important and basic research topics.It has always received extensive attentions from a large number of researchers.Its theoretical and practical values are very prominent.In order to acquire images of high quality,based on the regularization and deep learning techniques,this paper focuses mainly on two inverse problems,one is the blind restoration of images and the other is single image super-resolution.Solutions for the related problems are put forward:1.We try to apply target information selected locally to the whole image's blind restoration.Though existing edge methods overcome the drawback that the edge methods can only be applied to straight edges,but will result in large estimation value error because of the stretching coordinate.An curve edge method of arbitrary shapes based on the edge method of projection is analysed and proposed using the idea of moving windows.And the feasibility is explained theoretically.The proposed method first fits the edge points linearly,then uses the projection method to sample the gray values of the rows or columns in a selected window and aligns the sampling centers of different windows,finally estimates the point spread function using interpolation and resampling after eliminating the improper sampling points.For the edges with curvatures from 0.001 to 0.01,the peak signal-to-noise ratio can still stay above 35 dB even though under strong blur.The error of peak of the point spread function can be controlled within20%.2.We try to restore the image blindly from the the whole image instead of using local information.An iterative algorithm of blind restoration based on alternating direction method of multipliers algorithm is analysed and proposed in order to overcome the low operating efficiency and poor reconstruction quality in the total variation blind deconvolution model of the regularization theory.It estimates restored image and point spread function alternatively by alternating iteration to improve the running speed and reconstruction quality through the way of controlling the iteration termination condition.The normalization and threshold conditions of the point spread function,the positive definite condition of the image were added while calculating.In the numerical experimentation,blind restoration of different types of blurred images were carried out.Further more we compared it with other existing blind image restoration methods.Through objective comparison,the peak signal to noise ratio of the proposed algorithm can raise1.2dB at most.The average structural similarity is increased by 1% and the computation time is saved about half of it.3.The traditional restoration algorithm can not effectively use the prior information,and the restoration algorithm based on the convolution neural network,which is based on the deep learning has poor edge restoration quality.In order to solve these problems,the paper improves the trainable reaction diffusion model.The interpolating step in the trainable reaction diffusion model is converted into a convolution neural network,and periodic operators are introduced to rearrange the feature maps of the last layer.The problem of introducing false information in the interpolating step of the trainable reaction diffusion model is solved to some extent.Compared with deep learning algorithm of main stream,the peak signal to noise ratio of the proposed algorithm can increase 0.1dB on average.