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

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.