Keywords: energy method, variation, signal processing, Mumford-Shah functional, finite element method, multi-grid algorithm, symbolic computational software.

In this paper, we focus on the denoising and signal recovery problems within the framework of signal processing. For signal and image processing, one important task is the segmentation problem, i.e., to split a given signal or image into several disjoint regions where the signal or image is homogeneous. Nonlinear partial differential equations and variational principle has been shown very successfully for this task. For the specific cases in our paper, a initial signal is described by a one-dimensional function which values indicate the signal strength, and in general contaminated with noise. An appropriate denoising equation will reconstruct the unperturbed signals from the noise-contaminated one.

We use the Energy method on the denoising equation by minimizing an Mumford-shah functional [5] leading to a variational problem. This way has been shown to yield good results in many cases recently. We could discretize the obtained weak formulation in a finite element space [1], in order to analyze the reconstructed solutions.

The associated discrete equation in the finite element space is a parameter-dependent nonlinear form, and the numbers of its solutions depend on the evaluation of all parameters. However, since the discrete scheme is always nonconvex, usual Newton type methods or other numerical approaches are not sufficient to obtain all reconstructed results, especially if there exist any tight-together discrete solutions. Hence we use a new variational transformation which can turn the finite element scheme into an algebraic system, so that the discrete form can be simplified by eliminating most of the variables with aid of symbolic computational software [4]. Based on a coarse discretization, we can finally solve all the exact solutions inexpensively with least error. The obtained solutions are implicit functions which only depend on parameters, so that we can investigate their changing value on each partition grid nodes by plotting their function curve on Maple software. That also provide the possibilities for determining appropriate parameters in the denoising equations so that the finite element formulation can be uniquely solved by Newton type methods. And for the large-scale denoising problems, the obtained finite element solution on the coarse grid partition can be a good initial guess for further Newton recorrective iterations, e.g. using 2-grid [2,6] or multi-grid algorithms [3].

Acknowledgement

This joint work has been partially supported by the Austrian "Fonds zur Förderung der wissenschaftlichen Forschung (FWF)" under project nr. SFB F013/F1304 and F013/F1317.

1

P. Ciarlet, "The Finite Element Method for Elliptic Problems", North-Holland, 1978.

2

H. Gu, "The Finite Element Approximation to the Minimal Surfaces Subject to the Plateau Problems", Proceedings of The Sixth International Conference on Computational Structure Technology, Prague, Czech Rep., Civil-comp Press, 2002. doi:10.4203/ccp.75.19

3

W. Hackbusch, "Multi-Grid Methods and Applications", Springer, Berlin, 1985.

4

R. Hemmecke, E. Hillgarter, F. Winkler, "CASA, in Handbook of Computer Algebra: Foundations, Applications, Systems", J.Grabmeier, E.Kaltofen, V.Weispfenning (eds.), pp. 356-359, Springer-Verlag, 2003.

5

D. Mumford, J. Shah, "Optimal approximations by piecewise smooth functions and associated variational problems", Comm. Pure App. Math. XLII(1989), 577-685. doi:10.1002/cpa.3160420503

6

J. Xu, A. Zhou, "Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems", Adv. Comput. Math. 14, 393-327, 2001. doi:10.1023/A:1012284322811

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