Symmetric box-splines on root lattices

AbstractRoot lattices are efficient sampling lattices for reconstructing isotropic signals in arbitrary dimensions, due to their highly symmetric structure. One root lattice, the Cartesian grid, is almost exclusively used since it matches the coordinate grid; but it is less efficient than other root lattices. Box-splines, on the other hand, generalize tensor-product B-splines by allowing non-Cartesian directions. They provide, in any number of dimensions, higher-order reconstructions of fields, often of higher efficiency than tensored B-splines. But on non-Cartesian lattices, such as the BCC (Body-Centered Cubic) or the FCC (Face-Centered Cubic) lattice, only some box-splines and then only up to dimension three have been investigated.This paper derives and completely characterizes efficient symmetric box-spline reconstruction filters on all irreducible root lattices that exist in any number of dimensions n≥2 (n≥3 for Dn and Dn∗ lattices). In all cases, box-splines are constructed by convolution using the lattice directions, generalizing the known constructions in two and three variables. For each box-spline, we document the basic properties for computational use: the polynomial degree, the continuity, the linear independence of shifts on the lattice and optimal quasi-interpolants for fast approximation of fields

Extensions of the Zwart-Powell Box Spline for Volumetric Data Reconstruction on the Cartesian Lattice

Abstract—In this article we propose a box spline and its variants for reconstructing volumetric data sampled on the Cartesian lattice. In particular we present a tri-variate box spline reconstruction kernel that is superior to tensor product reconstruction schemes in terms of recovering the proper Cartesian spectrum of the underlying function. This box spline produces a C 2 reconstruction that can be considered as a three dimensional extension of the well known Zwart-Powell element in 2D. While its smoothness and approximation power are equivalent to those of the tri-cubic B-spline, we illustrate the superiority of this reconstruction on functions sampled on the Cartesian lattice and contrast it to tensor product B-splines. Our construction is validated through a Fourier domain analysis of the reconstruction behavior of this box spline. Moreover, we present a stable method for evaluation of this box spline by means of a decomposition. Through a convolution, this decomposition reduces the problem to evaluation of a four directional box spline that we previously published in its explicit closed form [8]. Index Terms—Volumetric data interpolation, reconstruction, box splines.

Electron Beam X-Ray Computed Tomography for Multiphase Flows and An Experimental Study of Inter-channel Mixing

This thesis consists of two parts. In the first, a high speed X-ray Computed Tomography (CT) system for multiphase flows is developed. X-ray Computed Tomography (CT) has been employed in the study of multiphase flows. The systems developed to date often have excellent spatial resolution at the expense of poor temporal resolution. Hence, X-ray CT has mostly been employed to examining time averaged phase distributions. In the present work, we report on the development of a Scanning Electron Beam X-ray Tomography (SEBXT) CT system that will allow for much higher time resolution with acceptable spatial resolution. The designed system, however, can have issues such as beam-hardening and limited angle artifacts. In the present study, we developed a high speed, limited angle SEBXT system along with a new CT reconstruction algorithm designed to enhance the CT reconstruction results of such system. To test the performance of the CT system, we produced example CT reconstruction results for two test phantoms based on the actual measured sinograms and the simulated sinograms. The second part examines, the process by which fluid mixes between two parallel flow channels through a narrow gap. This flow is a canonical representation of the mixing and mass transfer processes that often occur in thermo-hydraulic systems. The mixing can be strongly related to the presence of large-scale periodic flow structures that form within the gap. In the present work, we have developed an experimental setup to examine the single-phase mixing through the narrow rectangular gaps connecting two rectangular channels. Our goal is to elucidate the underlying flow processes responsible for inter-channel mixing, and to produce high-fidelity data for validation of computational models. Dye concentration measurements were used to determine the time average rate of mixing. Particle Imaging Velocimetry (PIV) was used to measure the flow fields within the gap. A Proper Orthogonal Decomposition (POD) of the PIV flow fields revealed the presence of coherent flow structure. The decomposed flow fields were then used to predict the time averaged mixing, which closely matched the experimentally measured values.PHDNaval Architecture & Marine EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138666/1/seongjin_2.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/138666/2/seongjin_1.pd