Shape Reconstruction from Point Clouds Using Closed Form Solution of a Fourth-Order Partial Differential Equation.

Abstract

Partial differential equation (PDE) based geometric modelling has a number of advantages such as fewer design variables, avoidance of stitching adjacent patches together to achieve required continuities, and physics-based nature. Although a lot of papers have investigated PDE-based shape creation, shape manipulation, surface blending and volume blending as well as surface reconstruction using implicit PDE surfaces, there is little work of investigating PDE-based shape reconstruction using explicit PDE surfaces, specially satisfying the constraints on four boundaries of a PDE surface patch. In this paper, we propose a new method of using an accurate closed form solution to a fourth-order partial differential equation to reconstruct 3D surfaces from point clouds. It includes selecting a fourth-order partial differential equation, obtaining the closed form solutions of the equation, investigating the errors of using one of the obtained closed form solutions to reconstruct PDE surfaces from differential number of 3D points.