We present a unified method for numerical evaluation of volume, surface, and path integrals of smooth, bounded functions on implicitly defined bounded domains. The method avoids both the stochastic nature (and slow convergence) of Monte Carlo methods and problem-specific domain decompositions required by most traditional numerical integration techniques. Our approach operates on a uniform grid over an axis-aligned box containing the region of interest, so we refer to it as a grid-based method. All grid-based integrals are computed as a sum of contributions from a stencil computation on the grid points. Each class of integrals (path, surface, or volume) involves a different stencil formulation, but grid-based integrals of a given class can be evaluated by applying the same stencil on the same set of grid points; only the data on the grid points changes. When functions are defined over the continuous domain so that grid refinement is possible, grid-based integration is supported by a convergence proof based on wavelet analysis. Given the foundation of function values on a uniform grid, grid-based integration methods apply directly to data produced by volumetric imaging (including computed tomography and magnetic resonance), direct numerical simulation of fluid flow, or any other method that produces data corresponding to values of a function sampled on a regular grid. Every step of a grid-based integral computation (including evaluating a function on a grid, application of stencils on a grid, and reduction of the contributions from the grid points to a single sum) is well suited for parallelization. We present results from a parallelized CUDA implementation of grid-based integrals that faithfully reproduces the output of a serial implementation but with significant reductions in computing time. We also present example grid-based integral results to quantify convergence rates associated with grid refinement and dependence of the convergence rate on the specific choice of difference stencil (corresponding to a particular genus of Daubechies wavelet).

Image reconstruction is the transformation process from a reduced-order representation to the original image pixel form. In materials characterization, it can be utilized as a method to retrieve material composition information. In our previous work, a surfacelet transform was developed to efficiently represent boundary information in material images with surfacelet coefficients. In this paper, new constrained-conjugate-gradient based image reconstruction methods are proposed as the inverse surfacelet transform. With geometric constraints on boundaries and internal distributions of materials, the proposed methods are able to reconstruct material images from surfacelet coefficients as either lossy or lossless compressions. The results between the proposed and other optimization methods for solving the least-square error inverse problems are compared.

This paper describes a new formulation of solid modeling for treating parts derived from volumetric scans (computed tomography, magnetic resonance, etc.) along with parts from traditional computer-aided design operations. Recent advances in segmentation via level set methods produce voxel grids of signed distance values, and we interpolate the signed distance values using wavelets to produce an implicit or function-based representation called wavelet signed distance function representation that provides inherent support for data compression, multiscale modeling, and skeletal-based operations.

The use of multiresolution control toward the editing of freeform curves and surfaces has already been recognized as a valuable modeling tool [1–3]. Similarly, in contemporary computer aided geometric design, the use of constraints to precisely prescribe freeform shape is considered an essential capability [4,5]. This paper presents a scheme that combines multiresolution control with linear constraints into one framework, allowing one to perform multiresolution manipulation of nonuniform B-spline curves, while specifying and satisfying various linear constraints on the curves. Positional, tangential, and orthogonality constraints are all linear and can be easily incorporated into a multiresolution freeform curve editing environment, as will be shown. Moreover, we also show that the symmetry as well as the area constraints can be reformulated as linear constraints and similarly incorporated. The presented framework is extendible and we also portray this same framework in the context of freeform surfaces.