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 PDEbased 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 different number of 3D points.

PDE-Based 3D Surface Reconstruction from Multi-View 2D Images

“…In our previous work [35], we proposed a fourth-order partial differential equation, obtained its closed form solutions, and used one of the obtained closed form solutions to test the applicability of the proposed method in surface reconstruction by reconstructing several simple surface patches from several sets of a few ordered points uniformly sampled from a known surface. In this paper, we extend the closed form solutions obtained in [35] from 16 vector-valued variables to 64 vector-valued variables to greatly raise the capacity in surface reconstruction and improve reconstruction quality, propose a pipeline of surface reconstruction from multi-view 2D images, and make a comparison between the method proposed in this paper and a polygon-based surface reconstruction method to demonstrate the effectiveness and advantages of the proposed method.…”

“…According to [35], the partial differential equation used for surface reconstruction can be a fourth-order partial differential equation given in Equation ( 1) below. The solution to the partial differential equation represents a parametric surface.…”

“…Inserting Equation (2) into Equation ( 1) and solving it analytically under different conditions, we obtain four closed-form solutions. More detailed solving procedures can be referenced in our previous work [35]. We choose one of the solutions as the PDE model to reconstruct the PDE-based surface from multi-view 2D images, shown in Equation (3).…”