This thesis describes a general method for automatic reconstruction of accurate, concise, piecewise smooth surfaces from unorganized 3D points. Instances of surface reconstruction arise in numerous scientific and engineering applications, including reverse-engineering--the automatic generation of CAD models from physical objects.
Previous surface reconstruction methods have typically required additional knowledge, such as structure in the data, known surface genus, or orientation information. In contrast, the method outlined in this thesis requires only the 3D coordinates of the data points. From the data, the method is able to automatically infer the topological type of the surface, its geometry, and the presence and location of features such as boundaries, creases, and corners.
The reconstruction method has three major phases: (1) initial surface estimation, (2) mesh optimization, and (3) piecewise smooth surface optimization. A key ingredient in phase 3, and another principal contribution of this thesis, is the introduction of a new class of piecewise smooth representations based on subdivision. The effectiveness of the three-phase reconstruction method is demonstrated on a number of examples using both simulated and real data.
Phases 2 and 3 of the surface reconstruction method can also be used to approximate existing surface models. By casting surface approximation as a global optimization problem with an energy function that directly measures deviation of the approximation from the original surface, models are obtained that exhibit excellent accuracy to conciseness trade-offs. Examples of piecewise linear and piecewise smooth approximations are generated for various surfaces, including meshes, NURBS surfaces, CSG models, and implicit surfaces.

/pdf/surface-reconstruction-from-unorganized-points-4377nlb52v.pdf

Surface reconstruction from unorganized points

In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the symmetric eigenvalue problem, nonlinear eigenvalue problems, electronic structures computations, and signal processing. In addition to the new algorithms, we show how the geometrical framework gives penetrating new insights allowing us to create, understand, and compare algorithms. The theory proposed here provides a taxonomy for numerical linear algebra algorithms that provide a top level mathematical view of previously unrelated algorithms. It is our hope that developers of new algorithms and perturbation theories will benefit from the theory, methods, and examples in this paper.

http://www-math.mit.edu/~edelman/publications/geometry_of_algorithms.pdf

The Geometry of Algorithms with Orthogonality Constraints

Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows

A common fixed point theorem of S.L. and S.P. Singh is generalized by weakening commutativity hypotheses and by increasing the number of functions involved.

/pdf/compatible-mappings-and-common-fixed-points-117iccel17.pdf

Compatible mappings and common fixed points

A general-purpose computer vision system must be capable of recognizing three-dimensional (3-D) objects. This paper proposes a precise definition of the 3-D object recognition problem, discusses basic concepts associated with this problem, and reviews the relevant literature. Because range images (or depth maps) are often used as sensor input instead of intensity images, techniques for obtaining, processing, and characterizing range data are also surveyed.

Three-dimensional object recognition

Manifold Theory. Tensors. Semi-Riemannian Manifolds. Semi-Riemannian Submanifolds. Riemannian and Lorenz Geometry. Special Relativity. Constructions. Symmetry and Constant Curvature. Isometries. Calculus of Variations. Homogeneous and Symmetric Spaces. General Relativity. Cosmology. Schwarzschild Geometry. Causality in Lorentz Manifolds. Fundamental Groups and Covering Manifolds. Lie Groups. Newtonian Gravitation.

Semi-Riemannian Geometry With Applications to Relativity

/pdf/the-fundamental-equations-of-a-submersion-9v6wcmlmf3.pdf

The fundamental equations of a submersion.

Elementary Differential Geometry, Second Edition provides an introduction to the geometry of curves and surfaces.

Elementary differential geometry

/pdf/manifolds-of-negative-curvature-1v9lg8sdmr.pdf

Manifolds of negative curvature

This book offers an up-to-date look at black holes from a mathematical perspective containing many results never before published. It can be read as a broadly based introduction to modern techniques in differential geometry.

/pdf/the-geometry-of-kerr-black-holes-1ndp9s285i.pdf

Barrett O'Neill

Papers

Semi-Riemannian Geometry With Applications to Relativity

The fundamental equations of a submersion.

Elementary differential geometry

Manifolds of negative curvature

The geometry of Kerr black holes