Computational & Technology Resources
an online resource for computational,
engineering & technology publications 

CivilComp Proceedings
ISSN 17593433 CCP: 79
PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and C.A. Mota Soares
Paper 242
Solution Analysis of Denoising Equations Related to the MumfordShah Functional H. Gu+ and S. Kindermann*
+RISCLinz Institute
H. Gu, S. Kindermann, "Solution Analysis of Denoising Equations Related to the MumfordShah Functional", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Seventh International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 242, 2004. doi:10.4203/ccp.79.242
Keywords: energy method, variation, signal processing, MumfordShah functional, finite element method, multigrid algorithm, symbolic computational software.
Summary
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
onedimensional function which values indicate the signal
strength, and in general contaminated
with noise. An appropriate denoising equation will reconstruct the
unperturbed signals from the noisecontaminated one.
We use the Energy method on the denoising equation by minimizing an Mumfordshah 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 parameterdependent 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 tighttogether 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 largescale 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 2grid [2,6] or multigrid algorithms [3].
AcknowledgementThis 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.References
purchase the fulltext of this paper (price £20)
go to the previous paper 
