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Denoising with higher order derivatives of bounded variation and an application to parameter estimation. (English) Zbl 0891.65103
The paper is devoted to the generalization of bounded variation regularization to functions with higher-order derivatives of bounded variation. This concept is applied to locate discontinuities in derivatives, which has important applications in parameter estimation problems. The main objective is the analysis of an algorithm for locating discontinuities in derivatives of a function, which is an important question arising in inverse conductivity problems. Based on an overview on denoising algorithms the idea of the work is presented. In contrast to other papers on bounded variation regularization the main emphasis is to reconstruct discontinuities of the derivatives in smooth images (images which do not have edges). The numerical solution of a one-dimensional denoising problem is considered for both higher-order bounded variation regularization and standard Tikhonov regularization. A comparison of the numerical results is performed. Two-dimensional denoising problems are also studied. The way to extend all results theoretically for problems which involve determination of jumps in higher-order derivatives is proposed for future exercise.

65M30Improperly posed problems (IVP of PDE, numerical methods)
35R30Inverse problems for PDE
35K05Heat equation
Full Text: DOI
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