In this paper we analyze an algorithm which solves the point projection and the “inversion” problems for parametric curves and surfaces. It consists of a geometric second order iteration which converges faster than existing first order methods, and whose sensitivity to the choice of initial values is small. Applications include the ICP algorithm for shape registration.

We study error propagation through implicit geometric problems by linearizing and estimating the linearization error. The method is particularly useful for quadratic constraints, which turns out to be no big restriction for many geometric problems in applications.

This paper presents a marching method for computing intersection curves between two solids represented by subdivision surfaces of Catmull-Clark or Loop type. It can be used in trimming and boolean operations for subdivision surfaces. The main idea is to apply a marching method with geometric interpretation to trace the intersection curves. We first determine all intersecting regions, then find pairs of initial intersection points, and trace the intersection curves from the initial intersection points. Various examples are given to demonstrate the robustness and efficiency of our algorithm.

This paper considers the problem of placing mosaic tiles on a surface to produce a surface mosaic. We assume that the user specifies a mesh model, the size of the tiles and the amount of grout, and optionally, a few control vectors at key locations on the surface indicating the preferred tile orientation at these points. From these inputs, we place equal-sized rectangular tiles over the mesh such as to almost cover it, with controlled orientation. The alignment of the tiles follows a vector field which is interpolated over the surface from the control vectors, and also forced into alignment with any sharp creases, open boundaries, and boundaries between regions of different colors. Our method efficiently solves the problem by posing it as one of globally optimizing a spring-like energy in the Manhattan metric, using overlapping local parameterizations.We demonstrate the effectiveness of our algorithm with various examples.

Most approaches to least squares fitting of a B-spline surface to measurement data require a parametrization of the data point set and the choice of suitable knot vectors. We propose to use uniform knots in connection with a feature sensitive parametrization. This parametrization allocates more parameter space to highly curved feature regions and thus automatically provides more control points where they are needed.

Geometry processing algorithms often require the robust extraction of curvature information. We propose to achieve this with principal component analysis (PCA) of local neighborhoods, defined via spherical kernels centered on the given surface $\Phi$. Intersection of a kernel ball $B_r$ or its boundary sphere $S_r$ with the volume bounded by $\Phi$ leads to the so-called ball and sphere neighborhoods. Information obtained by PCA of these neighborhoods turns out to be more robust than PCA of the patch neighborhood $B_r \cap \Phi$ previously used. The relation of the quantities computed by PCA with the principal curvatures of $\Phi$ is revealed by an asymptotic analysis as the kernel radius $r$ tends to zero. This also allows us to define principal curvatures 'at scale $r$ in a way which is consistent with the classical setting. The advantages of the new approach are discussed in a comparison with results obtained by normal cycles and local fitting; whereas the former method somewhat lacks in robustness, the latter does not achieve a consistent behavior at features on coarse scales. As to applications, we address computing principal curves and feature extraction on multiple scales.

The computation of a rigid body transformation which optimally aligns a set of measurement points with a surface and related registration problems are studied from the viewpoint of geometry and optimization.We provide a convergence analysis for widely used registration algorithms such as ICP, using either closest points (Besl and McKay [2]) or tangent planes at closest points (Chen and Medioni [4]), and for a recently developed approach based on quadratic approximants of the squared distance function [24]. ICP based on closest points exhibits local linear convergence only. Its counterpart which minimizes squared distances to the tangent planes at closest points is a Gauss-Newton iteration; it achieves local quadratic convergence for a zero residual problem and - if enhanced by regularization and step size control - comes close to quadratic convergence in many realistic scenarios. Quadratically convergent algorithms are based on the approach in [24]. The theoretical results are supported by a number of experiments; there, we also compare the algorithms with respect to global convergence behavior, stability and running time.

Sharp edges, ridges, valleys and prongs are critical for the appearance and an accurate representation of a 3D model. In this paper, we propose a novel approach that deals with the global shape of features in a robust way. Based on a remeshing algorithm which delivers an isotropic mesh in a feature sensitive metric, features are recognized on multiple scales via integral invariants of local neighborhoods. Morphological and smoothing operations are then used for feature region extraction and classification into basic types such as ridges, valleys and prongs. The resulting representation of feature regions is further used for feature-specific editing operations.

We present a method for repairing topological errors on solid models in the form of small surface handles, which often arise from surface reconstruction algorithms. We utilize a skeleton representation that offers a new mechanism for identifying and measuring handles. Our method presents two unique advantages over previous approaches. First, handle removal is guaranteed not to introduce invalid geometry or additional handles. Second, by using an adaptive grid structure, our method is capable of processing huge models efficiently at high resolutions.

We present a method for modifying the topology of a 3D model with user control. The heart of our method is a guided topology editing algorithm. Given a source model and a user-provided target shape, the algorithm modifies the source so that the resulting model is topologically consistent with the target. Our algorithm permits removing or adding various topological features (e.g., handles, cavities and islands) in a common framework and ensures that each topological change is made by minimal modification to the source model. To create the target shape, we have also designed a convenient 2D sketching interface for drawing 3D line skeletons. As demonstrated in a suite of examples, the use of sketching allows more accurate removal of topological artifacts than previous methods, and enables creative designs with specific topological goals.