| Abstract: |
| We present some segmentation/denoising problems in image and motion analysis. The framework is the variational PDEs models such as Mumford-Shah and Rudin-Osher-Fatemi which aim to perform the segmentation and preserving relevant geometric features of the image/scene. The discrete setting for this models are usually obtained after a phase-field reformulation step leading to a computable version.
However, the advantage of this approach allowing to use standard and efficient numerical methods comes up against its limitations of producing diffuse interfaces and somewhat ironically smoothing the edges and corners that the segmentation aims to capture.
We introduce another approach which consists of building a family of linear energies with variable diffusion coefficients -in space- that leads to simple and efficient numerical methods while preserving the singular set (edges, corners). Besides, the discrete counterparts of these energies under well suited control of the diffusion coefficients converge (in the Gamma-convergence sense) to the Mumford-Shah functional.
The control of the diffusion is performed at the discrete level and is combined with an adaptive approach that works with two ingredients: a geometric one which consists of mesh adaptation leading to a tight location of the singularities and a functional one where the diffusion is encouraged or stopped. Finally, we show several numerical examples. |
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