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Perona-Malik equation error estimates for numerical finite volume scheme Angela Handlovičová Zuzana Krivá Department of Mathematics Slovak University of.

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Presentation on theme: "Perona-Malik equation error estimates for numerical finite volume scheme Angela Handlovičová Zuzana Krivá Department of Mathematics Slovak University of."— Presentation transcript:

1 Perona-Malik equation error estimates for numerical finite volume scheme Angela Handlovičová Zuzana Krivá Department of Mathematics Slovak University of Technology, Bratislava, Slovakia

2 Initial noisy image Smooth initial noisy image and preserve edges

3 Perona-Malik Equation

4 Perona-Malik equation - properties of data g(s) is Lipschitz continuous decreasing function, g(0)=1, 0< g(s)  0 for s   is a smoothing kernel with compact support u 0 (x) is initial condition with and for Dirac function at point x

5 Results obtained for finite volume numerical scheme Existence of weak solutions and regularity-Catté, Lions, Morel, Coll (1992) Convergence of semi-implicit finite volume numerical scheme – Mikula Ramarossy (2001) Convergence of explicit finite volume scheme Krivá (2003) Error estimates for semi-implicit FV scheme Handlovičová, Krivá (2005) Error estimates for explicit scheme H,K (2005)

6 Finite volume p…volume of measure m(p), with representative point x p N(p) …neighbors of p e pq …common edge between volumes p and q of measure m(e pq ) d pq :=|x p -x q |, T pq :=m(e pq )/d pq u p n …numerical solution

7 Explicit scheme

8 Semi - implicit scheme

9 Regularity of the solution

10 Error estimates For explicit scheme For semi -implicit scheme

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16 Thank you Angela Handlovičová Zuzana Krivá Department of Mathematics Slovak University of Technology, Bratislava, Slovakia


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