On level set regularization for highly ill-posed distributed parameter estimation problems
Journal of Computational Physics
A piecewise constant level set method for elliptic inverse problems
Applied Numerical Mathematics
A parallel method for large scale time domain electromagnetic inverse problems
Applied Numerical Mathematics
Computational Optimization and Applications
A Multilevel Algorithm for Simultaneously Denoising and Deblurring Images
SIAM Journal on Scientific Computing
Multigrid Algorithm for High Order Denoising
SIAM Journal on Imaging Sciences
Intensity gradient based registration and fusion of multi-modal images
MICCAI'06 Proceedings of the 9th international conference on Medical Image Computing and Computer-Assisted Intervention - Volume Part II
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This paper considers the problem of reconstructing a piecewise smooth model function from given, measured data. The data are compared to a field which is given as a possibly nonlinear function of the model. A regularization functional is added which incorporates the a priori knowledge that the model function is piecewise smooth and may contain jump discontinuities. Regularization operators related to total variation (TV) are therefore employed. Two popular methods are modified TV and Huber's function. Both contain a parameter which must be selected. The Huber variant provides a more natural approach for selecting its parameter, and we use this to propose a scheme for both methods. Our selected parameter depends both on the resolution and on the model average roughness; thus, it is determined adaptively. Its variation from one iteration to the next yields additional information about the progress of the regularization process. The modified TV operator has a smoother generating function; nonetheless we obtain a Huber variant with comparable, and occasionally better, performance. For large problems (e.g., high resolution) the resulting reconstruction algorithms can be tediously slow. We propose two mechanisms to improve efficiency. The first is a multilevel continuation approach aimed mainly at obtaining a cheap yet good estimate for the regularization parameter and the solution. The second is a special multigrid preconditioner for the conjugate gradient algorithm used to solve the linearized systems encountered in the procedures for recovering the model function.