Surface Reconstruction from Scattered Point via RBF Interpolation on GPU

In this paper we describe a parallel implicit method based on radial basis functions (RBF) for surface reconstruction. The applicability of RBF methods is hindered by its computational demand, that requires the solution of linear systems of size equal to the number of data points. Our reconstruction implementation relies on parallel scientific libraries and is supported for massively multi-core architectures, namely Graphic Processor Units (GPUs). The performance of the proposed method in terms of accuracy of the reconstruction and computing time shows that the RBF interpolant can be very effective for such problem.Comment: arXiv admin note: text overlap with arXiv:0909.5413 by other author

Scattered Data Interpolation on Embedded Submanifolds with Restricted Positive Definite Kernels: Sobolev Error Estimates

In this paper we investigate the approximation properties of kernel interpolants on manifolds. The kernels we consider will be obtained by the restriction of positive definite kernels on $\R^d$, such as radial basis functions (RBFs), to a smooth, compact embedded submanifold \M\subset \R^d. For restricted kernels having finite smoothness, we provide a complete characterization of the native space on \M. After this and some preliminary setup, we present Sobolev-type error estimates for the interpolation problem. Numerical results verifying the theory are also presented for a one-dimensional curve embedded in $\R^3$ and a two-dimensional torus

TVL₁ Planarity Regularization for 3D Shape Approximation

The modern emergence of automation in many industries has given impetus to extensive research into mobile robotics. Novel perception technologies now enable cars to drive autonomously, tractors to till a field automatically and underwater robots to construct pipelines. An essential requirement to facilitate both perception and autonomous navigation is the analysis of the 3D environment using sensors like laser scanners or stereo cameras. 3D sensors generate a very large number of 3D data points when sampling object shapes within an environment, but crucially do not provide any intrinsic information about the environment which the robots operate within. This work focuses on the fundamental task of 3D shape reconstruction and modelling from 3D point clouds. The novelty lies in the representation of surfaces by algebraic functions having limited support, which enables the extraction of smooth consistent implicit shapes from noisy samples with a heterogeneous density. The minimization of total variation of second differential degree makes it possible to enforce planar surfaces which often occur in man-made environments. Applying the new technique means that less accurate, low-cost 3D sensors can be employed without sacrificing the 3D shape reconstruction accuracy

TVL₁shape approximation from scattered 3D data

With the emergence in 3D sensors such as laser scanners and 3D reconstruction from cameras, large 3D point clouds can now be sampled from physical objects within a scene. The raw 3D samples delivered by these sensors however, contain only a limited degree of information about the environment the objects exist in, which means that further geometrical high-level modelling is essential. In addition, issues like sparse data measurements, noise, missing samples due to occlusion, and the inherently huge datasets involved in such representations makes this task extremely challenging. This paper addresses these issues by presenting a new 3D shape modelling framework for samples acquired from 3D sensor. Motivated by the success of nonlinear kernel-based approximation techniques in the statistics domain, existing methods using radial basis functions are applied to 3D object shape approximation. The task is framed as an optimization problem and is extended using non-smooth L1 total variation regularization. Appropriate convex energy functionals are constructed and solved by applying the Alternating Direction Method of Multipliers approach, which is then extended using Gauss-Seidel iterations. This significantly lowers the computational complexity involved in generating 3D shape from 3D samples, while both numerical and qualitative analysis confirms the superior shape modelling performance of this new framework compared with existing 3D shape reconstruction techniques

A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces

In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in $\mathbb{R}^d$. For two-dimensional surfaces embedded in $\mathbb{R}^3$, these types of problems have received growing interest in biology, chemistry, and computer graphics to model such things as diffusion of chemicals on biological cells or membranes, pattern formations in biology, nonlinear chemical oscillators in excitable media, and texture mappings. Our kernel method is based on radial basis functions (RBFs) and uses a semi-discrete approach (or the method-of-lines) in which the surface derivative operators that appear in the PDEs are approximated using collocation. The method only requires nodes at "scattered" locations on the surface and the corresponding normal vectors to the surface. Additionally, it does not rely on any surface-based metrics and avoids any intrinsic coordinate systems, and thus does not suffer from any coordinate distortions or singularities. We provide error estimates for the kernel-based approximate surface derivative operators and numerically study the accuracy and stability of the method. Applications to different non-linear systems of PDEs that arise in biology and chemistry are also presented