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

https://www.cs.ucla.edu/~dt/papers/mia96/mia96.pdf

Deformable models in medical image analysis: a survey

The polygon remains a popular graphics primitive for computer graphics application. Besides having a simple representation, computer rendering of polygons is widely supported by commercial graphics hardware and software. However, because the polygon is linear, often thousands or millions of primitives are required to capture the details of complex geometry. Models of this size are generally not practical since rendering speeds and memory requirements are proportional to the number of polygons. Consequently applications that generate large polygonal meshes often use domain-specific knowledge to reduce model size. There remain algorithms, however, where domainspecific reduction techniques are not generally available or appropriate. One algorithm that generates many polygons is marching cubes. Marching cubes is a brute force surface construction algorithm that extracts isodensity surfaces from volume data, producing from one to five triangles within voxels that contain the surface. Although originally developed for medical applications, marching cubes has found more frequent use in scientific visualization where the size of the volume data sets are much smaller than those found in medical applications. A large computational fluid dynamics volume could have a finite difference grid size of order 100 by 100 by 100, while a typical medical computed tomography or magnetic resonance scanner produces over 100 slices at a resolution of 256 by 256 or 512 by 512 pixels each. Industrial computed tomography, used for inspection and analysis, has even greater resolution, varying from 512 by 512 to 1024 by 1024 pixels. For these sampled data sets, isosurface extraction using marching cubes can produce from 500k to 2,000k triangles. Even today’s graphics workstations have trouble storing and rendering models of this size. Other sampling devices can produce large polygonal models: range cameras, digital elevation data, and satellite data. The sampling resolution of these devices is also improving, resulting in model sizes that rival those obtained from medical scanners. This paper describes an application independent algorithm that uses local operations on geometry and topology to reduce the number of triangles in a triangle mesh. Although our implementation is for the triangle mesh, it can be directly applied to the more general polygon mesh. After describing other work related to model creation from sampled data, we describe the triangle decimation process and its implementation. Results from two different geometric modeling applications illustrate the strengths of the algorithm.

/pdf/decimation-of-triangle-meshes-3fvv4shbtq.pdf

Decimation of triangle meshes

We present a method for solving the following problem: Given a set of data points scattered in three dimensions and an initial triangular mesh M0, produce a mesh M, of the same topological type as M0, that fits the data well and has a small number of vertices. Our approach is to minimize an energy function that explicitly models the competing desires of conciseness of representation and fidelity to the data. We show that mesh optimization can be effectively used in at least two applications: surface reconstruction from unorganized points, and mesh simplification (the reduction of the number of vertices in an initially dense mesh of triangles).

Mesh optimization

This paper presents a new reflectance model for rendering computer synthesized images. The model accounts for the relative brightness of different materials and light sources in the same scene. It describes the directional distribution of the reflected light and a color shift that occurs as the reflectance changes with incidence angle. The paper presents a method for obtaining the spectral energy distribution of the light reflected from an object made of a specific real material and discusses a procedure for accurately reproducing the color associated with the spectral energy distribution. The model is applied to the simulation of a metal and a plastic.

A reflectance model for computer graphics

We propose a new approach to the problem of generating a simple topologically-closed geometric model from a point-sampled volume data set. We call such a model a Geometrically Deformed Model or GDM. A GDM is created by placing a 'seed' model in the volume data set. The model is then deformed by a relaxation process that minimizes a set of constraints that provides a measure of how well the model fits the features in the data. Constraints are associated with each vertex in the model that control local deformation, interaction between the model and the data set, and the shape and topology of the model. Once generated, a GDM can be used for visualization, shape recognition, geometric measurements, or subjected to a series of geometric operations. This technique is of special importance because of the advent of nondestructive sensing equipment (CT, MRI) that generates point samples of true three-dimensional objects.

Geometrically deformed models: a method for extracting closed geometric models form volume data

An overview of automatic feature recognition techniques for computer-aided process planning

The problem of fitting a 2D broadcloth composite ply (modelled as a piece of deformable woven cloth) onto a 3D curved surface (modelled with a NURBS surface) is considered. ‘Fitting’ means that a ply is deformed and applied to a surface so that it is everywhere in contact with the surface. A fitting method is presented that incorporates a flexible method for specifying initial conditions, and allows for the simulation of a wide range of fitting configurations.

Fitting a woven-cloth model to a curved surface: mapping algorithms

With the advent of lighter and stronger composite materials over the last two decades, more and more metals have been replaced by composite materials in aircraft and other vehicles. It is not suprising that the airframes of certain aircraft (e.g. Lear Fan 2100) are totally fabricated from composite materials. Composites consist of a reinforcing material suspended in a “matrix” material (e.g. epoxy) that bonds it to adjacent reinforcing materials. The three major composite forms are chopped fiber, unidirectional tape, and (bidirectional) broadcloth. Today’s aircraft mostly employ broadcloth composites partly because they have outstanding strength-to-weight ratio, and partly because their structural properties may be tailored to the expected load in different directions [16]. Broadcloth composites have both vertical and horizontal threads (weft and warp) interwoven to form a sheet of cloth as seen in Fig 1. They strongly resist stretching along thread directions, but can be deformed flexibly along thread diagonals by changing the angle between vertical and horizontal threads. This allows a composite broadcloth to be “formed” into virtually any curved surface. In order to reinforce aircraft structures, multiple sheets (plies) of broadcloth composites are usually laid one on top of the other, a lamination process. With this trend of replacing metals with composite materials, new lamination technologies have been developed to manufacture composite materials in order to form an airframe. During the lamination process, plies of composite sheets must be cut to the required size, with the required fiber orientation, and laid up onto the surface of a die or molding tool prior to curing. It is this process of deforming a broadcloth composite sheet into a 3D shape by pressing it onto a mold that we will refer to as a “fitting.” Unfortunately, the cutting and orientation process has not yet been fully automated, and it has been mostly implemented with manual procedures. It is therefore a time-consuming, non-reproducable, and inaccurate process in need of design automation tools. In this paper we present a number of modeling methods that may be used to automate the design of 3D broadcloth composite parts. First, we describe a model (a Tchebychev net) which allows us to simulate the deformation of woven materials into a specific 3D shape. Two algorithms are described for performing the actual fitting. The first algorithm simulates the fit by solving the Tchebychev net formula using a finite difference technique. The second algorithm simulates the fit by reducing the problem to a surface-surface intersection problem. Once we establish the techniques for simulating a fit, we can discuss the quality and acceptibility of the fit. In general, a good fit is the one that consumes the smallest area of the material, that has the smallest deformation energy, and that is free of manufacturing anomalies such as wrinkles and breaks. We will present mathematical tools that allow us to measure the “goodness” of a fit with a Tchebychev net, and that allow us to visually identify where a possible anomalous event may occur. Specifically, we will introduce a path-dependent Gaussian curvature integral that is defined at an arbitrary point in a surface region. With a path-dependent Gaussian curvature integral, we will show that it is possible not only to predict anomalous events, or wrinkles in particular, but to provide a solution to preventing them. Finally, we will propose three methods for preventing anomalous events: (1) automatic generation of a good initial condition, (2) dart insertion, and (3) surface shape modification. Providing different methods for preventing

/pdf/modeling-methods-for-the-design-of-3d-broadcloth-composite-3q14xppj7x.pdf

Michael J. Wozny

Papers

Geometrically deformed models: a method for extracting closed geometric models form volume data

An overview of automatic feature recognition techniques for computer-aided process planning

Fitting a woven-cloth model to a curved surface: mapping algorithms

Modeling methods for the design of 3D broadcloth composite parts

The M-Space Theory of Tolerances