GPU-based Parallel Computation of Integral Properties of Volumetrically Digitized Objects

Abstract

Integral properties of objects that have been created digitally or that are volumetrically digitized have an important role in many different applications. There are a few different approaches for the computation of integral properties. In this dissertation, direct integration methods, using a convergence proof that exists for twice differentiable functions, and basing the computations on numerical derivative operations that have been obtained by discretization of integral formulas have been proposed. We present the theoretical formulations for direct methods of computing integral properties, including surface, volume, and line integrals, based on grid data which represents an object. We show that while direct integration methods offer advantages by removing the need for random number generation or parametrization of data, they are still computationally intensive like the traditional methods such as Monte-Carlo computations and other integration methods that rely on parametrization of the given grid representation. While the computations are intensive they are parallelizable, and we explore overcoming the computational challenge by implementing parallel algorithms using general purpose graphics processing unit (GPGPU) computing techniques with the help of the parallel programming platform CUDA created by Nvidia. We present detailed accuracy analysis between the traditional and direct integration methods and detailed performance comparisons between traditional methods, serial implementation of direct methods, and parallel implementations of direct methods. In addition to the introduction and analysis of direct integration methods, we explore an application of direct integration. Motivated by research of alternative methods to Finite Elements and Boundary Elements methods, we apply our direct integration approach to the integral equation that is being used to solve a boundary value problem. The areas of application for direct integration method extend from 3D printing to medical applications and CAD software. Considering the application areas of these methods, one of the most crucial aspects is performance. Our tests suggest that using GPGPU techniques on direct integration methods reduces the computation times to levels which allow live visualization for the aforementioned application areas.