Image Deblurring, Gaussian Markov Random Fields, and Neumann Boundary Conditions
نویسندگان
چکیده
In this paper we consider the inverse problem of image deblurring with Neumann boundary conditions. Regularization is incorporated by using Gaussian Markov random fields (GMRFs) to model an appropriate prior on the image pixel values. We provide a linear algebraic framework for GMRFs, and we establish an important connection between GMRFs studied in the statistical literature, and negative-Laplacian-based Tikhonov regularization used in the inverse problems and imaging communities. This connection allows us to show that the negative Laplacian Tikhonov regularization method corresponds to concrete statistical assumptions about the unknown pixel values conditioned on those of its neighbors. For image reconstruction, we implement a Markov Chain Monte Carlo (MCMC) method that yields samples of the unknown image x and of the regularization parameter α. From the samples, we compute the reconstructed image and quantify uncertainty in both x and α. Furthermore, we show that the approach can be implemented very efficiently by exploiting matrix structure, through the use of the discrete cosine transform.
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