This work deals with the transcription of continuous partial derivative equations to arbitrary discrete domains by exploiting the formalism of partial difference equations defined on weighted graphs. In the first part, we propose a transcription of the normalized p-Laplacian operator to the graph domains as a linear combination between the non-local infinity Laplacian and the normalized Laplacian (both in their discrete version). This adaptation can be considered as a new class of p-Laplacian operators on graphs that interpolate between non-local infinity Laplacian and normalized Laplacian. In the second part, we present an adaptation of fronts propagation equations on weighted graphs. These equations are obtained by the transcription of the continuous level sets method to a discrete formulation on the graphs domain. Beyond the transcription in itself, we propose a very general formulation and efficient algorithms for the simultaneous propagation of several fronts on a single graph. Both transcription of the p-Laplacian operator and level sets method enable many applications in image segmentation and data clustering that are illustrated in this manuscript. Finally, in the third part, we present a concrete application of the different tools proposed in the two previous parts for computer aided diagnosis. We also present the Antarctic software that was developed during this PhD.
