Laplacian interpolation on triangulated tissue surface models

3D triangulated meshes are used ubiquitously in biomedicine to model tissue surfaces, owing to their geometric flexibility and non-parametric nature. One often needs to interpolate scalar functions over such meshes, either because the interpolant itself provides the solution to a problem, a visualization of the interpolant on top of the mesh is desired, or, quite commonly, both. A common example is the production of electroanatomical maps of the left atrium in the study, diagnosis, and treatment of atrial fibrillation. One appealing method that has been used is Laplacian interpolation because of its unique property of maximizing smoothness. However, the only documented method to date to adopt Laplacian interpolation to 3D triangulated meshes uses matrix formulations that may impose undesirable constraints. In this paper, we present a simple, novel numerical method for Laplacian interpolation on 3D triangulated surfaces. It converges quickly, requires relatively few computations compared to a matrix solution, and can be easily implemented. The method was successfully used in clinical studies of atrial fibrillation, and can be adopted to many problem domains involving 3D meshes.