Colliding Interfaces in Old and New Diffuse-interface Approximations of Willmore-flow

This paper is concerned with diffuse-interface approximations of the Willmore flow. We first present numerical results of standard diffuse-interface models for colliding one dimensional interfaces. In such a scenario evolutions towards interfaces with corners can occur that do not necessarily describe the adequate sharp-interface dynamics. We therefore propose and investigate alternative diffuse-interface approximations that lead to a different and more regular behavior if interfaces collide. These dynamics are derived from approximate energies that converge to the $L^1$-lower-semicontinuous envelope of the Willmore energy, which is in general not true for the more standard Willmore approximation

Superquadric representation of scenes from multi-view range data

Object representation denotes representing three-dimensional (3D) real-world objects with known graphic or mathematic primitives recognizable to computers. This research has numerous applications for object-related tasks in areas including computer vision, computer graphics, reverse engineering, etc. Superquadrics, as volumetric and parametric models, have been selected to be the representation primitives throughout this research. Superquadrics are able to represent a large family of solid shapes by a single equation with only a few parameters. This dissertation addresses superquadric representation of multi-part objects and multiobject scenes. Two issues motivate this research. First, superquadric representation of multipart objects or multi-object scenes has been an unsolved problem due to the complex geometry of objects. Second, superquadrics recovered from single-view range data tend to have low confidence and accuracy due to partially scanned object surfaces caused by inherent occlusions. To address these two problems, this dissertation proposes a multi-view superquadric representation algorithm. By incorporating both part decomposition and multi-view range data, the proposed algorithm is able to not only represent multi-part objects or multi-object scenes, but also achieve high confidence and accuracy of recovered superquadrics. The multi-view superquadric representation algorithm consists of (i) initial superquadric model recovery from single-view range data, (ii) pairwise view registration based on recovered superquadric models, (iii) view integration, (iv) part decomposition, and (v) final superquadric fitting for each decomposed part. Within the multi-view superquadric representation framework, this dissertation proposes a 3D part decomposition algorithm to automatically decompose multi-part objects or multiobject scenes into their constituent single parts consistent with human visual perception. Superquadrics can then be recovered for each decomposed single-part object. The proposed part decomposition algorithm is based on curvature analysis, and includes (i) Gaussian curvature estimation, (ii) boundary labeling, (iii) part growing and labeling, and (iv) post-processing. In addition, this dissertation proposes an extended view registration algorithm based on superquadrics. The proposed view registration algorithm is able to handle deformable superquadrics as well as 3D unstructured data sets. For superquadric fitting, two objective functions primarily used in the literature have been comprehensively investigated with respect to noise, viewpoints, sample resolutions, etc. The objective function proved to have better performance has been used throughout this dissertation. In summary, the three algorithms (contributions) proposed in this dissertation are generic and flexible in the sense of handling triangle meshes, which are standard surface primitives in computer vision and graphics. For each proposed algorithm, the dissertation presents both theory and experimental results. The results demonstrate the efficiency of the algorithms using both synthetic and real range data of a large variety of objects and scenes. In addition, the experimental results include comparisons with previous methods from the literature. Finally, the dissertation concludes with a summary of the contributions to the state of the art in superquadric representation, and presents possible future extensions to this research

Stereological techniques for synthesizing solid textures from images of aggregate materials

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, February 2005.Includes bibliographical references (leaves 121-130).When creating photorealistic digital scenes, textures are commonly used to depict complex variation in surface appearance. For materials that have spatial variation in three dimensions, such as wood or marble, solid textures offer a natural representation. Unlike 2D textures, which can be easily captured with a photograph, it can be difficult to obtain a 3D material volume. This thesis addresses the challenge of extrapolating tileable 3D solid textures from images of aggregate materials, such as concrete, asphalt, terrazzo or granite. The approach introduced here is inspired by and builds on prior work in stereology--the study of 3D properties of a material based on 2D observations. Unlike ad hoc methods for texture synthesis, this approach has rigorous mathematical foundations that allow for reliable, accurate material synthesis with well-defined assumptions. The algorithm is also driven by psychophysical constraints to insure that slices through the synthesized volume have a perceptually similar appearance to the input image. The texture synthesis algorithm uses a variety of techniques to independently solve for the shape, distribution, and color of the embedded particles, as well as the residual noise. To approximate particle shape, I consider four methods-including two algorithms of my own contribution. I compare these methods under a variety of input conditions using automated, perceptually-motivated metrics as well as a carefully controlled psychophysical experiment. In addition to assessing the relative performance of the four algorithms, I also evaluate the reliability of the automated metrics in predicting the results of the user study. To solve for the particle distribution, I apply traditional stereological methods.(cont.) I first illustrate this approach for aggregate materials of spherical particles and then extend the technique to apply to particles of arbitrary shapes. The particle shape and distribution are used in conjunction to create an explicit 3D material volume using simulated annealing. Particle colors are assigned using a stochastic method, and high-frequency noise is replicated with the assistance of existing algorithms. The data representation is suitable for high-fidelity rendering and physical simulation. I demonstrate the effectiveness of the approach with side-by-side comparisons of real materials and their synthetic counterparts derived from the application of these techniques.by Robert Carl Jagnow.Ph.D

A full Eulerian finite difference approach for solving fluid-structure coupling problems

A new simulation method for solving fluid-structure coupling problems has been developed. All the basic equations are numerically solved on a fixed Cartesian grid using a finite difference scheme. A volume-of-fluid formulation (Hirt and Nichols (1981, J. Comput. Phys., 39, 201)), which has been widely used for multiphase flow simulations, is applied to describing the multi-component geometry. The temporal change in the solid deformation is described in the Eulerian frame by updating a left Cauchy-Green deformation tensor, which is used to express constitutive equations for nonlinear Mooney-Rivlin materials. In this paper, various verifications and validations of the present full Eulerian method, which solves the fluid and solid motions on a fixed grid, are demonstrated, and the numerical accuracy involved in the fluid-structure coupling problems is examined.Comment: 38 pages, 27 figures, accepted for publication in J. Comput. Phy

Complex scene modeling and segmentation with deformable simplex meshes

In this thesis we present a system for 3D reconstruction and segmentation of complex real world scenes. The input to the system is an unstructured cloud of 3D points. The output is a 3D model for each object in the scene. The system starts with a model that encloses the input point cloud. A deformation process is applied to the initial model so it gets close to the point cloud in terms of distance, geometry and topology. Once the deformation stops the model is analyzed to check if more than one object is present in the point cloud. If necessary a segmentation process splits the model into several parts that correspond to each object in the scene. Using this segmented model the point cloud is also segmented. Each resulting sub-cloud is treated as a new input to the system. If, after the deformation process, the model is not segmented a refinement process improves the objective and subjective quality of the model by concentrating vertices around high curvature areas. The simplex mesh reconstruction algorithm was modified and extended to suit our application. A novel segmentation algorithm was designed to be applied on the simplex mesh. We test the system with synthetic and real data obtained from single objects, simple. and complex scenes. In the case of the synthetic data different levels of noise are added to examine the performance of the system. The results show that the systems performs well for either of the three cases and also in the presence of low levels of noise

Mathematics and Algorithms in Tomography

This is the eighth Oberwolfach conference on the mathematics of tomography. Modalities represented at the workshop included X-ray tomography, sonar, radar, seismic imaging, ultrasound, electron microscopy, impedance imaging, photoacoustic tomography, elastography, vector tomography, and texture analysis

Simulation of pore-scale flow using finite element-methods

I present a new finite element (FE) simulation method to simulate pore-scale flow. Within the pore-space, I solve a simplified form of the incompressible Navier-Stoke’s equation, yielding the velocity field in a two-step solution approach. First, Poisson’s equation is solved with homogeneous boundary conditions, and then the pore pressure is computed and the velocity field obtained for no slip conditions at the grain boundaries. From the computed velocity field I estimate the effective permeability of porous media samples characterized by thin section micrographs, micro-CT scans and synthetically generated grain packings. This two-step process is much simpler than solving the full Navier Stokes equation and therefore provides the opportunity to study pore geometries with hundreds of thousands of pores in a computationally more cost effective manner than solving the full Navier-Stoke’s equation. My numerical model is verified with an analytical solution and validated on samples whose permeabilities and porosities had been measured in laboratory experiments (Akanji and Matthai, 2010). Comparisons were also made with Stokes solver, published experimental, approximate and exact permeability data. Starting with a numerically constructed synthetic grain packings, I also investigated the extent to which the details of pore micro-structure affect the hydraulic permeability (Garcia et al., 2009). I then estimate the hydraulic anisotropy of unconsolidated granular packings. With the future aim to simulate multiphase flow within the pore-space, I also compute the radii and derive capillary pressure from the Young-Laplace equation (Akanji and Matthai,2010

Interactive Segmentation of 3D Medical Images with Implicit Surfaces

To cope with a variety of clinical applications, research in medical image processing has led to a large spectrum of segmentation techniques that extract anatomical structures from volumetric data acquired with 3D imaging modalities. Despite continuing advances in mathematical models for automatic segmentation, many medical practitioners still rely on 2D manual delineation, due to the lack of intuitive semi-automatic tools in 3D. In this thesis, we propose a methodology and associated numerical schemes enabling the development of 3D image segmentation tools that are reliable, fast and interactive. These properties are key factors for clinical acceptance. Our approach derives from the framework of variational methods: segmentation is obtained by solving an optimization problem that translates the expected properties of target objects in mathematical terms. Such variational methods involve three essential components that constitute our main research axes: an objective criterion, a shape representation and an optional set of constraints. As objective criterion, we propose a unified formulation that extends existing homogeneity measures in order to model the spatial variations of statistical properties that are frequently encountered in medical images, without compromising efficiency. Within this formulation, we explore several shape representations based on implicit surfaces with the objective to cover a broad range of typical anatomical structures. Firstly, to model tubular shapes in vascular imaging, we introduce convolution surfaces in the variational context of image segmentation. Secondly, compact shapes such as lesions are described with a new representation that generalizes Radial Basis Functions with non-Euclidean distances, which enables the design of basis functions that naturally align with salient image features. Finally, we estimate geometric non-rigid deformations of prior templates to recover structures that have a predictable shape such as whole organs. Interactivity is ensured by restricting admissible solutions with additional constraints. Translating user input into constraints on the sign of the implicit representation at prescribed points in the image leads us to consider inequality-constrained optimization

Active haptic exploration for 3D shape reconstruction.

by Fung Wai Keung.Thesis (M.Phil.)--Chinese University of Hong Kong, 1996.Includes bibliographical references (leaves 146-151).Acknowledgements --- p.viiiAbstract --- p.1Chapter 1 --- Overview --- p.3Chapter 1.1 --- Tactile Sensing in Human and Robot --- p.4Chapter 1.1.1 --- Human Hands and Robotic Hands --- p.4Chapter 1.1.2 --- Mechanoreceptors in skin and Tactile Sensor Arrays --- p.7Chapter 1.2 --- Motivation --- p.12Chapter 1.3 --- Objectives --- p.13Chapter 1.4 --- Related Work --- p.14Chapter 1.4.1 --- Using Vision Alone --- p.15Chapter 1.4.2 --- Integration of Vision and Touch --- p.15Chapter 1.4.3 --- Using Touch Sensing Alone --- p.17Chapter 1.4.3.1 --- Ronald S. Fearing's Work --- p.18Chapter 1.4.3.2 --- Peter K. Allen's Work --- p.22Chapter 1.5 --- Outline --- p.26Chapter 2 --- Geometric Models --- p.27Chapter 2.1 --- Introduction --- p.27Chapter 2.2 --- Superquadrics --- p.27Chapter 2.2.1 --- 2D Superquadrics --- p.27Chapter 2.2.2 --- 3D Superquadrics --- p.29Chapter 2.3 --- Model Recovery of Superquadric Models --- p.31Chapter 2.3.1 --- Problem Formulation --- p.31Chapter 2.3.2 --- Least Squares Optimization --- p.33Chapter 2.4 --- Free-Form Deformations --- p.34Chapter 2.4.1 --- Bernstein Basis --- p.36Chapter 2.4.2 --- B-Spline Basis --- p.38Chapter 2.5 --- Other Geometric Models --- p.41Chapter 2.5.1 --- Generalized Cylinders --- p.41Chapter 2.5.2 --- Hyperquadrics --- p.42Chapter 2.5.3 --- Polyhedral Models --- p.44Chapter 2.5.4 --- Function Representation --- p.45Chapter 3 --- Sensing Strategy --- p.54Chapter 3.1 --- Introduction --- p.54Chapter 3.2 --- Sensing Algorithm --- p.55Chapter 3.2.1 --- Assumption of objects --- p.55Chapter 3.2.2 --- Haptic Exploration Procedures --- p.56Chapter 3.3 --- Contour Tracing --- p.58Chapter 3.4 --- Tactile Sensor Data Preprocessing --- p.59Chapter 3.4.1 --- Data Transformation and Sensor Calibration --- p.60Chapter 3.4.2 --- Noise Filtering --- p.61Chapter 3.5 --- Curvature Determination --- p.64Chapter 3.6 --- Step Size Determination --- p.73Chapter 4 --- 3D Shape Reconstruction --- p.80Chapter 4.1 --- Introduction --- p.80Chapter 4.2 --- Correspondence Problem --- p.81Chapter 4.2.1 --- Affine Invariance Property of B-splines --- p.84Chapter 4.2.2 --- Point Inversion Problem --- p.87Chapter 4.3 --- Parameter Triple Interpolation --- p.91Chapter 4.4 --- 3D Object Shape Reconstruction --- p.94Chapter 4.4.1 --- Heuristic Approach --- p.94Chapter 4.4.2 --- Closed Contour Recovery --- p.97Chapter 4.4.3 --- Control Lattice Recovery --- p.102Chapter 5 --- Implementation --- p.105Chapter 5.1 --- Introduction --- p.105Chapter 5.2 --- Implementation Tool - MATLAB --- p.105Chapter 5.2.1 --- Optimization Toolbox --- p.107Chapter 5.2.2 --- Splines Toolbox --- p.108Chapter 5.3 --- Geometric Model Implementation --- p.109Chapter 5.3.1 --- FFD Examples --- p.111Chapter 5.4 --- Shape Reconstruction Implementation --- p.112Chapter 5.5 --- 3D Model Reconstruction Examples --- p.120Chapter 5.5.1 --- Example 1 --- p.120Chapter 5.5.2 --- Example 2 --- p.121Chapter 6 --- Conclusion --- p.128Chapter 6.1 --- Future Work --- p.129Appendix --- p.133Bibliography --- p.14