Identifying multiple deformable parts on meshes and establishing dense correspondences between them are tasks of fundamental importance to computer graphics, with applications to e.g. geometric edit propagation and texture transfer. Much research has considered establishing correspondences between non-rigid surfaces, but little work can both identify similar multiple deformable parts and handle partial shape correspondences. This paper addresses two related problems, treating them as a whole: (i) identifying similar deformable parts on a mesh, related by a non-rigid transformation to a given query part, and (ii) establishing dense point correspondences automatically between such parts. We show that simple and efficient techniques can be developed if we make the assumption that these parts locally undergo isometric deformation. Our insight is that similar deformable parts are suggested by large clusters of point correspondences that are isometrically consistent. Once such parts are identified, dense point correspondences can be obtained by an iterative propagation process. Our techniques are applicable to models with arbitrary topology. Various examples demonstrate the effectiveness of our techniques.

We present a new method for interpolating both boundary values and gradients over a 2D polygonal domain. Despite various previous efforts, it remains challenging to define a closed-form interpolant that produces natural-looking functions while allowing flexible control of boundary constraints. Our method builds on an existing transfinite interpolant over a continuous domain, which in turn extends the classical mean value interpolant. We re-derive the interpolant from the mean value property of biharmonic functions, and prove that the interpolant indeed matches the gradient constraints when the boundary is piece-wise linear. We then give closed-form formula (as generalized barycentric coordinates) for boundary constraints represented as polynomials up to degree 3 (for values) and 1 (for normal derivatives) over each polygon edge. We demonstrate the flexibility and efficiency of our coordinates in two novel applications, smooth image deformation using curved cage networks and adaptive simplification of gradient meshes.

We investigate the propagation of errors through geometric transformations, such as re°ections, rotations, similarity transformations, and projections, and also the scalar product of vectors. This means computing tolerance zones of points which undergo such transformations, if these points and the transformations themselves are given by toleranced input data.

In this paper, we introduce a novel method that can generate a sequence of physical transformations between 3D models with different shape and topology. Feasible transformations are realized on a chain structure with connected components that are 3D printed. Collision-free motions are computed to transform between different configurations of the 3D printed chain structure. To realize the transformation between different 3D models, we first voxelize these input models into similar number of voxels. The challenging part of our approach is to generate a simple path --- as a chain configuration to connect most voxels. A layer-based algorithm is developed with theoretical guarantee of the existence and the path length. We find that collision-free motion sequence can always be generated when using a straight line as the intermediate configuration of transformation. The effectiveness of our method is demonstrated by both the simulation and the experimental tests taken on 3D printed chains.

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The polygonal approximation problem is a primary problem in computer graphics, pattern recognition, CAD/CAM, etc. In $R^2$, the cone intersection method (CIM) is one of the most eÆcient algorithms for approximating polygonal curves. With CIM Eu and Toussaint, by imposing an additional constraint and changing the given error criteria, resolve the three-dimensional weighted minimum number polygonal approximation problem with the parallel-strip error criterion (PS-WMN) under $L_2$ norm. In this paper, without any additional constraint and change of the error criteria, a CIM solution to the same problem with the line segment error criterion (LS-WMN) is presented, which is more frequently encountered than the PS-WMN is. Its time complexity is $O(n^3)$, and the space complexity is $O(n^2)$. An approximation algorithm is also presented, which takes $O(n^2)$ time and $O(n)$ space. Results of some examples are given to illustrate the eÆciency of these algorithms.

Finding correspondences between two surfaces is a fundamental operation in various applications in computer graphics and related fields. Candidate correspondences can be found by matching local signatures, but as they only consider local geometry, many are globally inconsistent. We provide a novel algorithm to prune a set of candidate correspondences to those most likely to be globally consistent. Our approach can handle articulated surfaces, and ones related by a deformation which is globally non-isometric, provided that the deformation is locally approximately isometric. Our approach uses an efficient diffusion framework, and only requires geodesic distance calculations in small neighbourhoods, unlike many existing techniques which require computation of global geodesic distances. We demonstrate that, for typical examples, our approach provides significant improvements in accuracy, yet also reduces time and memory costs by a factor of several hundred compared to existing pruning techniques. Our method is furthermore insensitive to holes, unlike many other methods.

Sharp edges are important shape features and their extraction has been extensively studied both on point clouds and surfaces. We consider the problem of extracting sharp edges from a sparse set of colour-and-depth (RGB-D) images. The noise-ridden depth measurements are challenging for existing feature extraction methods that work solely in the geometric domain (e.g. points or meshes). By utilizing both colour and depth information, we propose a novel feature extraction method that produces much cleaner and more coherent feature lines. We make two technical contributions. First, we show that intensity edges can augment the depth map to improve normal estimation and feature localization from a single RGB-D image. Second, we designed a novel algorithm for consolidating feature points obtained from multiple RGB-D images. By utilizing normals and ridge/valley types associated with the feature points, our algorithm is effective in suppressing noise without smearing nearby features.

Solving geometric constraint systems in 3-D is much more complicated than that in 2-D because the number of variables is larger and some of the results valid in 2-D cannot be extended for 3-D. In this paper, we propose a new DOF-based graph constructive method to geometric constraint systems solving that can efficiently handle well-, over- and under-constrained systems based on the dependence analysis. The basic idea is that the solutions of some geometric elements depend on some others because of the constraints between them. If some geometric elements depend on each other, they must be solved together. In our approach, we first identify all structurally redundant constraints, then we add some constraints to well constrain the system. And we prove that the order of a constraint system after processing under-constrained cases is not more than that of the original system multiplied by 5. After that, we apply a recursive searching process to identify all the clusters, which is shown to be capable of getting the minimum order-reduction result of a well-constrained system. We also briefly describe the constraint evaluation phase and show the implementation results of our method.

This work introduces a rigorous and practical approach for automatic N-RoSy field design on arbitrary surfaces with user defined field topologies. The user has full control of the number, positions and indices of the singularities, the turning numbers of the loops, and is able to edit the field interactively. We formulate N-RoSy field construction as designing a Riemannian metric, such that the holonomy along any loop is compatible with the local symmetry of N-RoSy fields. We prove the compatibility condition using discrete parallel transport. The complexity of N-RoSy field design is caused by curvatures. In our work, we propose to simplify the Riemannian metric to make it flat almost everywhere. This approach greatly simplifies the process and improves the flexibility, such that, it can design N-RoSy fields with single singularity, and mixed-RoSy fields. To demonstrate the effectiveness of our approach, we apply our design system to pen-and-ink sketching and geometry remeshing.