Perona‐Malik equation ‐ error estimates for explicit finite volume scheme

Error estimates in the L2 norm for the explicit fully discrete numerical finite volume scheme are derived and proved for Perona‐Malik equation. Numerical example is also presented. First Published Online: 14 Oct 201

3D early embryogenesis image filtering by nonlinear partial differential equations

We present nonlinear diffusion equations, numerical schemes to solve them and their application for filtering 3D images obtained from laser scanning microscopy (LSM) of living zebrafish embryos, with a goal to identify the optimal filtering method and its parameters. In the large scale applications dealing with analysis of 3D+time embryogenesis images, an important objective is a correct detection of the number and position of cell nuclei yielding the spatio-temporal cell lineage tree of embryogenesis. The filtering is the first and necessary step of the image analysis chain and must lead to correct results, removing the noise, sharpening the nuclei edges and correcting the acquisition errors related to spuriously connected subregions. In this paper we study such properties for the regularized Perona-Malik model and for the generalized mean curvature flow equations in the level-set formulation. A comparison with other nonlinear diffusion filters, like tensor anisotropic diffusion and Beltrami flow, is also included. All numerical schemes are based on the same discretization principles, i.e. finite volume method in space and semi-implicit scheme in time, for solving nonlinear partial differential equations. These numerical schemes are unconditionally stable, fast and naturally parallelizable. The filtering results are evaluated and compared first using the Mean Hausdorff distance between a gold standard and different isosurfaces of original and filtered data. Then, the number of isosurface connected components in a region of interest (ROI) detected in original and after the filtering is compared with the corresponding correct number of nuclei in the gold standard. Such analysis proves the robustness and reliability of the edge preserving nonlinear diffusion filtering for this type of data and lead to finding the optimal filtering parameters for the studied models and numerical schemes. Further comparisons consist in ability of splitting the very close objects which are artificially connected due to acquisition error intrinsically linked to physics of LSM. In all studied aspects it turned out that the nonlinear diffusion filter which is called geodesic mean curvature flow (GMCF) has the best performance

Designing 3-D Nonlinear Diffusion Filters for High Performance Cluster Computing

This paper deals with parallelization and implementation aspects of PDE based image processing models for large cluster environments with distributed memory. As an example we focus on nonlinear isotropic di#usion filtering which we discretize by means of an additive operator splitting (AOS). We start by decomposing the algorithm into small modules that shall be parallelized separately. For this purpose image partitioning strategies are discussed and their impact on the communication pattern and volume is analyzed. Based on the results we develop an algorithmic implementation with excellent scaling properties on massively connected low latency networks. Test runs on a high--end Myrinet cluster yield almost linear speedup factors up to 209 for 256 processors. This results in typical denoising times of 0.5 seconds for five iterations on a 256 128 data cube