A Fast Algorithm for Deblurring Models with Neumann Boundary Conditions
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TLDR
It is shown that for symmetric blurring functions, these blurring matrices can always be diagonalized by discrete cosine transform matrices, and the cost of inversion is significantly lower than that of using the zero or periodic boundary conditions.Abstract:
Blur removal is an important problem in signal and image processing. The blurring matrices obtained by using the zero boundary condition (corresponding to assuming dark background outside the scene) are Toeplitz matrices for one-dimensional problems and block-Toeplitz--Toeplitz-block matrices for two-dimensional cases. They are computationally intensive to invert especially in the block case. If the periodic boundary condition is used, the matrices become (block) circulant and can be diagonalized by discrete Fourier transform matrices. In this paper, we consider the use of the Neumann boundary condition (corresponding to a reflection of the original scene at the boundary). The resulting matrices are (block) Toeplitz-plus-Hankel matrices. We show that for symmetric blurring functions, these blurring matrices can always be diagonalized by discrete cosine transform matrices. Thus the cost of inversion is significantly lower than that of using the zero or periodic boundary conditions. We also show that the use of the Neumann boundary condition provides an easy way of estimating the regularization parameter when the generalized cross-validation is used. When the blurring function is nonsymmetric, we show that the optimal cosine transform preconditioner of the blurring matrix is equal to the blurring matrix generated by the symmetric part of the blurring function. Numerical results are given to illustrate the efficiency of using the Neumann boundary condition.read more
Figures
Fig. 6.2. (a) Gaussian blur; (b) out-of-focus blur; noisy and blurred image (c) by Gaussian blur and (d) by out-of-focus blur. 
Fig. 6.6. (a) Restored image (µ∗ = 1× 10−4) and (b) the true image. 
Table 6.1 The numbers of iterations required for using the zero boundary condition. 
Fig. 6.5. (a) Observed image, (b) the guide star image, and (c) a cross section of the blurring function. 
Table 6.2 The number of iterations for convergence. 
Fig. 6.1. The Gatlinburg Conference image.
Citations
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References
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