An Efficient Implementation of a 3D CeVeFE DDFV Scheme on Cartesian Grids and an Application in Image Processing

Niklas Hartung (1) , Florence Hubert (1)

1 I2M - Institut de Mathématiques de Marseille

In this work we describe the implementation of a 3D Center-Vertex-Face/Edge Discrete Duality Finite Volume (CeVeFE DDFV) scheme using only the degrees of freedom (DOF) disposed on a Cartesian grid. These DOF are organised in a three-mesh structure proper to the CeVeFE DDFV setting. Reposing on a diamond structure, the approach presented here greatly simplifies the implementation, also in the case of grids topologically equivalent to the uniform Cartesian one. The numerical scheme is then applied to a problem in image processing, where uniform Cartesian structure of the DOF is naturally imposed by the pixel/voxel structure. A semi-implicit DDFV scheme is used for solving a nonlinear advection-diffusion equation, the subjective surfaces equation, in order to reconstruct the volume of a tumour from noisy 3D SPECT images with signal intensity on the tumour boundary. The matrix of the linear system has a band structure and the method is fast and able to successfully reconstruct the tumour volume.

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Niklas Hartung, Florence Hubert. An Efficient Implementation of a 3D CeVeFE DDFV Scheme on Cartesian Grids and an Application in Image Processing. Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, Jun 2014, Berlin, Germany. pp.637-645, ⟨10.1007/978-3-319-05591-6_63⟩. ⟨hal-01107682⟩