Image Restoration Based on the Convolution Kernel Matrix Reconstructed by Kronecker Product

摘要

Image restoration is actually a deconvolution problem. In the restoration equation, the convolution kernel matrix is a large-scale Toeplitz matrix. In order to reduce the computational complexity of the iterative restoration algorithm, a preconditioned conjugate gradient iterative algorithm based on convolution kernel matrix reconstruction is proposed by analyzing the ill-posed character of the Toeplitz system and common fast solving methods. Firstly, according to the property that Toeplitz matrix can be decomposed into the sum of Kronecker products, the singular value decomposition of Point Spread Function is carried out. The left and right vectors corresponding to each singular value are used to construct the sub Toeplitz matrix, and the sub matrices are added by Kronecker product to obtain the decomposition formula of convolution kernel matrix. Then, according to the properties of Kronecker product, the decomposition formula is used to construct preconditioned operators, and finally the pre-conditional conjugate gradient method in solving iteration equation was proceed. Computational complexity analysis and experiments show that the algorithm is helpful to accelerate the convergence of iteration and obtain stable results.