Abstract
Partial Differential Equations (PDEs)-based morphology offers a wide range of continuous operators to address various image processing problems. Most of these operators are formulated as Hamilton–Jacobi equations or curve evolution level set and morphological flows. In our previous works, we have proposed a simple method to solve PDEs on point clouds using the framework of PdEs (Partial difference Equations) on graphs. In this paper, we propose to apply a large class of morphological-based operators on graphs for processing raw 3D point clouds and extend their applications for the processing of colored point clouds of geo-informatics 3D data. Through illustrations, we show that this simple framework can be used in the resolution of many applications for geo-informatics purposes.
Highlights
Partial Differential Equations (PDEs) are widely used for modeling and solving many inverse problems in image processing and computer vision
We adopt the Partial difference Equation (PdE) framework, and we focus on some PDEs-based continuous morphological operators in the Euclidean domain: dilation/erosion, mean curvature flows and the Eikonal equation
Based on the discrete gradient on weighted graphs, we present a class of discrete equations that mimic PDEs-based definitions of erosion, dilation, ∞-Laplacian and mean curvature flows, the time level-set and the Eikonal equation
Summary
Partial Differential Equations (PDEs) are widely used for modeling and solving many inverse problems in image processing and computer vision. Since 3D point clouds and triangular meshes can have very different topologies, we proposed to rely on graph-based methods that directly work in any discrete domain. We adopt the PdE framework, and we focus on some PDEs-based continuous morphological operators in the Euclidean domain: dilation/erosion, mean curvature flows and the Eikonal equation. Other morphological PDEs, based on curve evolution, the level set or the Eikonal equation, are used for image segmentation, generalized distance computation or shape analysis. We consider the following general level set equation on surfaces or point clouds: To solve this PDE on a surface S, we replace the differential operators by their discrete analogous provided by the PdE framework on an adapted graph G(V, E, w).
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