Abstract
The problem of out-of-focus image restoration can be modeled as an ill-posed integral equation, which can be regularized as a second kind of equation using the Tikhonov method. The multiscale collocation method with the compression strategy has already been developed to discretize this well-posed equation. However, the integral computation and solution of the large multiscale collocation integral equation are two time-consuming processes. To overcome these difficulties, we propose a fully discrete multiscale collocation method using an integral approximation strategy to compute the integral, which efficiently converts the integral operation to the matrix operation and reduces costs. In addition, we also propose a multilevel iteration method (MIM) to solve the fully discrete integral equation obtained from the integral approximation strategy. Herein, the stopping criterion and the computation complexity that correspond to the MIM are shown. Furthermore, a posteriori parameter choice strategy is developed for this method, and the final convergence order is evaluated. We present three numerical experiments to display the performance and computation efficiency of our proposed methods.
Highlights
Continuous integral equations are often used to model certain practical problems in image processing
Discrete models are piecewise constant approximations of integral equation models, and they introduce a bottleneck model error that cannot be addressed by any image processing method
A posteriori choice of the regularization parameter, which is related to the multilevel iteration method, is presented in Section 5, and we further prove that the MIM, combined with this posteriori parameter choice, makes our solution optimal
Summary
Continuous integral equations are often used to model certain practical problems in image processing. Inspired by [18], we further propose a fully discrete multiscale collocation method using an integral approximation strategy to compute the integration The idea of this strategy is the use of the Gaussian quadrature formula to efficiently compute the sparse coefficient matrix. Gauss-Legendre quadrature, we turn the calculation of the integration into the matrix operation, which will tremendously reduce the computation time Another challenging issue is that directly solving the large, fully discrete system obtained from the matrix compression strategy is time-consuming. These tests reveal the high efficiency of our proposed methods
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