Theoretically-based algorithms for robustly tracking intersection curves of deforming surfaces

This paper applies singularity theory of mappings of surfaces to 3-space and the generic transitions occurring in their deformations to develop algorithms for continuously and robustly tracking the intersection curves of two deforming parametric spline surfaces, when the deformation is represented as a family of generalized offset surfaces. The set of intersection curves of two deforming surfaces over all time is formulated as an implicit 2-manifold I in an augmented (by time domain) parametric space R⁵. Hyperplanes corresponding to some fixed time instants may touchI at some isolated transition points, which delineate transition events, i.e. the topological changes to the intersection curves. These transition points are the 0-dimensional solution to a rational system of five constraints in five variables, and can be computed efficiently and robustly with a rational constraint solver using subdivision and hyper-tangent bounding cones. The actual transition events are computed by contouring the local osculating paraboloids. Away from any transition points, the intersection curves do not change topology and evolve according to a simple evolution vector field that is constructed in the Euclidean space in which the surfaces are embedded.     

带法向约束的隐式T样条曲线重构

目的 隐式曲线能够描述复杂的几何形状和拓扑结构,而传统的隐式B样条曲线的控制网格需要大量多余的控制点满足拓扑约束。有些情况下,获取的数据点不仅包含坐标信息,还包含相应的法向约束条件。针对这个问题,提出了一种带法向约束的隐式T样条曲线重建算法。方法 结合曲率自适应地调整采样点的疏密,利用二叉树及其细分过程从散乱数据点集构造2维T网格;基于隐式T样条函数提出了一种有效的曲线拟合模型。通过加入偏移数据点和光滑项消除额外零水平集,同时加入法向项减小曲线的法向误差,并依据最优化原理将问题转化为线性方程组求解得到控制系数,从而实现隐式曲线的重构。在误差较大的区域进行T网格局部细分,提高重建隐式曲线的精度。结果 实验在3个数据集上与两种方法进行比较,实验结果表明,本文算法的法向误差显著减小,法向平均误差由10^-3数量级缩小为10^-4数量级,法向最大误差由10^-2数量级缩小为10^-3数量级。在重构曲线质量上,消除了额外零水平集。与隐式B样条控制网格相比,3个数据集的T网格的控制点数量只有B样条网格的55.88%、39.80%和47.06%。结论 本文算法能在保证数据点精度的前提下,有效降低法向误差,消除了额外的零水平集。与隐式B样条曲线相比,本文方法减少了控制系数的数量,提高了运算速度。     

Topology preserving top-down compression of 2D vector fields using bintree and triangular quadtrees

We present a hierarchical top-down refinement algorithm for compressing 2D vector fields that preserves topology. Our approach is to reconstruct the data set using adaptive refinement that considers topology. The algorithms start with little data and subdivide regions that are most likely to reconstruct the original topology of the given data set. We use two different refinement techniques. The first technique uses bintree subdivision and linear interpolation. The second algorithm is driven by triangular quadtree subdivision with Coons patch quadratic interpolation. We employ local error metrics to measure the quality of compression and as a global metric we compute Earth Mover's Distance (EMD) to measure the deviation from the original topology. Experiments with both analytic and simulated data sets are presented. Results indicate that one can obtain significant compression with low errors without losing topological information. Advantages and disadvantages of different topology preserving compression algorithms are also discussed in the paper.     

Geometric Flows of Curves in Shape Space for Processing Motion of Deformable Objects

We introduce techniques for the processing of motion and animations of non‐rigid shapes. The idea is to regard animations of deformable objects as curves in shape space. Then, we use the geometric structure on shape space to transfer concepts from curve processing in ?ⁿ to the processing of motion of non‐rigid shapes. Following this principle, we introduce a discrete geometric flow for curves in shape space. The flow iteratively replaces every shape with a weighted average shape of a local neighborhood and thereby globally decreases an energy whose minimizers are discrete geodesics in shape space. Based on the flow, we devise a novel smoothing filter for motions and animations of deformable shapes. By shortening the length in shape space of an animation, it systematically regularizes the deformations between consecutive frames of the animation. The scheme can be used for smoothing and noise removal, e.g., for reducing jittering artifacts in motion capture data. We introduce a reduced‐order method for the computation of the flow. In addition to being efficient for the smoothing of curves, it is a novel scheme for computing geodesics in shape space. We use the scheme to construct non‐linear “Bézier curves” by executing de Casteljau's algorithm in shape space.     

Local and singularity-free G 1 triangular spline surfaces using a minimum degree scheme

We develop a scheme for constructing G ¹ triangular spline surfaces of arbitrary topological type. To assure that the scheme is local and singularity-free, we analyze the selection of scalar weight functions and the construction of the boundary curve network in detail. With the further requirements of interpolating positions, normals, and surface curvatures, we show that the minimum degree of such a triangular spline surface is 6. And we present a method for constructing boundary curves network, which consists of cubic Bézier curves. To deal with certain singular cases, the base mesh must be locally subdivided and we proposed an adaptive subdivision strategy for it. An application of our G ¹ triangular spline surfaces to the approximation of implicit surfaces is described. The visual quality of this scheme is demonstrated by some examples.     

A topology preserving level set method for geometric deformable models

Active contour and surface models, also known as deformable models, are powerful image segmentation techniques. Geometric deformable models implemented using level set methods have advantages over parametric models due to their intrinsic behavior, parameterization independence, and ease of implementation. However, a long claimed advantage of geometric deformable models-the ability to automatically handle topology changes-turns out to be a liability in applications where the object to be segmented has a known topology that must be preserved. We present a new class of geometric deformable models designed using a novel topology-preserving level set method, which achieves topology preservation by applying the simple point concept from digital topology. These new models maintain the other advantages of standard geometric deformable models including subpixel accuracy and production of nonintersecting curves or surfaces. Moreover, since the topology-preserving constraint is enforced efficiently through local computations, the resulting algorithm incurs only nominal computational overhead over standard geometric deformable models. Several experiments on simulated and real data are provided to demonstrate the performance of this new deformable model algorithm.     

Tracking deforming objects using particle filtering for geometric active contours

Tracking deforming objects involves estimating the global motion of the object and its local deformations as a function of time. Tracking algorithms using Kalman filters or particle filters have been proposed for finite dimensional representations of shape, but these are dependent on the chosen parametrization and cannot handle changes in curve topology. Geometric active contours provide a framework which is parametrization independent and allow for changes in topology, in the present work, we formulate a particle filtering algorithm in the geometric active contour framework that can be used for tracking moving and deforming objects. To the best of our knowledge, this is the first attempt to implement an approximate particle filtering algorithm for tracking on a (theoretically) infinite dimensional state space.     

Adaptive physics based tetrahedral mesh generation using level sets

We present a tetrahedral mesh generation algorithm designed for the Lagrangian simulation of deformable bodies. The algorithm’s input is a level set (i.e., a signed distance function on a Cartesian grid or octree). First a bounding box of the object is covered with a uniform lattice of subdivision-invariant tetrahedra. The level set is then used to guide a red green adaptive subdivision procedure that is based on both the local curvature and the proximity to the object boundary. The final topology is carefully chosen so that the connectivity is suitable for large deformation and the mesh approximates the desired shape. Finally, this candidate mesh is compressed to match the object boundary. To maintain element quality during this compression phase we relax the positions of the nodes using finite elements, masses and springs, or an optimization procedure. The resulting mesh is well suited for simulation since it is highly structured, has topology chosen specifically for large deformations, and is readily refined if required during subsequent simulation. We then use this algorithm to generate meshes for the simulation of skeletal muscle from level set representations of the anatomy. The geometric complexity of biological materials makes it very difficult to generate these models procedurally and as a result we obtain most if not all data from an actual human subject. Our current method involves using voxelized data from the Visible Male [1] to create level set representations of muscle and bone geometries. Given this representation, we use simple level set operations to rebuild and repair errors in the segmented data as well as to smooth aliasing inherent in the voxelized data.     

Self-intersections of offsets of quadratic surfaces: Part II, implicit surfaces

The paper investigates self-intersections of offsets of implicit quadratic surfaces. The quadratic surfaces are the simplest curved objects, referred to as quadrics, and are widely used in mechanical design. In an earlier paper, we have investigated the self-intersections of offsets of explicit quadratic surfaces, such as elliptic paraboloid, hyperbolic paraboloid and parabolic cylinder, since not only are they used in mechanical design, but also any regular surface can be locally approximated by such explicit quadratic surfaces. In this paper, we investigate the rest of the quadrics whose offsets may degenerate, i.e. the implicit quadratic-surfaces (ellipsoid, hyperboloid, elliptic cone, elliptic cylinder and hyperbolic cylinder). We found that self-intersection curves of offsets of all the implicit quadratic surfaces are planar implicit conics and their corresponding curve on the progenitor surface can be expressed as the intersection curve between an ellipsoid, whose semi-axes are proportional to the offset distance, and the implicit quadratic surfaces themselves.     

Topology of real algebraic space curves

In this paper we give a new projection-based algorithm for computing the topology of a real algebraic space curve given implicitly by a set of equations. Under some genericity conditions, which may be reached through a linear change of coordinates, we show that a plane projection of the given curve, together with a special polynomial in the ideal of the curve contains all the information needed to compute its topological shape. Our method is also designed in such a way to exploit important particular cases such as complete intersection curves or curves contained in nonsingular surfaces.     

A framework for dynamic implicit curve approximation by an irregular discrete approach

The approximation of implicit planar curves by line segments is a very classical problem. Many algorithms use interval analysis to approximate this curve, and to handle the topology of the final reconstruction. In this article, we use discrete geometry tools to build an original geometrical and topological representation of the implicit curve. The polygonal approximation contains few segments, and the Reeb graph permits to sum up efficiently the shape and the topology of the curve. Furthermore, we propose two algorithms to process local cells refinement and local cells grouping schemes. We illustrate these schemes with a global system that efficiently handles manual or automatic fast updates on the global reconstruction, by considering topological or geometrical constraints. We also compare the speed and the quality of our approach with two classical methods.     

General Object Reconstruction Based on Simplex Meshes

In this paper, we propose a general tridimensional reconstruction algorithm of range and volumetric images, based on deformable simplex meshes. Simplex meshes are topologically dual of triangulations and have the advantage of permitting smooth deformations in a simple and efficient manner. Our reconstruction algorithm can handle surfaces without any restriction on their shape or topology. The different tasks performed during the reconstruction include the segmentation of given objects in the scene, the extrapolation of missing data, and the control of smoothness, density, and geometric quality of the reconstructed meshes. The reconstruction takes place in two stages. First, the initialization stage creates a simplex mesh in the vicinity of the data model either manually or using an automatic procedure. Then, after a few iterations, the mesh topology can be modified by creating holes or by increasing its genus. Finally, an iterative refinement algorithm decreases the distance of the mesh from the data while preserving high geometric and topological quality. Several reconstruction examples are provided with quantitative and qualitative results.     

Metamorphs: deformable shape and appearance models

This paper presents a new deformable modeling strategy aimed at integrating shape and appearance in a unified space. If we think traditional deformable models as active contours or evolving curve fronts, the new deformable shape and appearance models we propose are deforming disks or volumes. Each model has not only boundary shape but also interior appearance. The model shape is implicitly embedded in a higher dimensional space of distance transforms, thus represented by a distance map image. In this way, both shape and appearance of the model are defined in the pixel space. A common deformation scheme, the Free Form Deformations (FFD), parameterizes warping deformations of the volumetric space in which the model is embedded in, hence deforming both model boundary and interior simultaneously.     

Sketching freeform meshes using graph rotation functions

We present a new approach for sketching free form meshes with topology consistency. Firstly, we interpret the given 2D curve to be the projection of the 3D curve with the minimum curvature. Then we adopt a topology-consistent strategy based on the graph rotation system, to trace the simple faces on the interconnecting 3D curves. With the face tracing algorithm, our system can identify the 3D surfaces automatically. After obtaining the boundary curves for the faces, we apply Delaunay triangulation on these faces. Finally, the shape of the triangle mesh that follows the 3D boundary curves is computed by using harmonic interpolation. Meanwhile our system provides real-time algorithms for both control curve generation and the subsequent surface optimization. With the incorporation of topological manipulation into geometrical modeling, we show that automatically generated models are both beneficial and feasible.     

A level-set approach for the metamorphosis of solid models

We present a new approach to 3D shape metamorphosis. We express the interpolation of two shapes as a process where one shape deforms to maximize its similarity with another shape. The process incrementally optimizes an objective function while deforming an implicit surface model. We represent the deformable surface as a level set (iso-surface) of a densely sampled scalar function of three dimensions. Such level-set models have been shown to mimic conventional parametric deformable surface models by encoding surface movements as changes in the grayscale values of a volume data set. Thus, a well-founded mathematical structure leads to a set of procedures that describes how voxel values can be manipulated to create deformations that are represented as a sequence of volumes. The result is a 3D morphing method that offers several advantages over previous methods, including minimal need for user input, no model parameterization, flexible topology, and subvoxel accuracy     

Multiresolution on spherical curves

In this paper, we present an approximating multiresolution framework of arbitrary degree for curves on the surface of a sphere. Multiresolution by subdivision and reverse subdivision allows one to decrease and restore the resolution of a curve, and is typically defined by affine combinations of points in Euclidean space. While translating such combinations to spherical space is possible, ensuring perfect reconstruction of the curve remains challenging. Hence, current spherical multiresolution schemes tend to be interpolating or midpoint-interpolating, as achieving perfect reconstruction in these cases is more straightforward. We use a simple geometric construction for a non-interpolating and non-midpoint-interpolating multiresolution scheme on the sphere, which is made up of easily generalized components and based on a modified Lane–Riesenfeld algorithm.     

Building 3D surface networks from 2D curve networks with application to anatomical modeling

Constructing 3D surfaces that interpolate 2D curves defined on parallel planes is a fundamental problem in computer graphics with wide applications including modeling anatomical structures. Typically the problem is simplified so that the 2D curves partition each plane into only two materials (e.g., air versus tissue). Here we consider the general problem where each plane is partitioned by a curve network into multiple materials (e.g., air, cortex, cerebellum, etc.). We present a novel method that automatically constructs a surface network from curve networks with arbitrary topology and partitions an arbitrary number of materials. The surface network exactly interpolates the curve network on each plane and is guaranteed to be free of gaps or self-intersections. In addition, our method provides a flexible framework for user interaction so that the surface topology can be modified conveniently when necessary. As an application, we applied the method to build a high-resolution 3D model of the mouse brain from 2D anatomical boundaries defined on 350 tissue sections. The surface network accurately models the partitioning of the brain into 17 abutting anatomical regions with complex topology.     

Topological models for boundary representation: a comparison with n-dimensional generalized maps

In boundary representation, a geometric object is represented by the union of a ‘topological’ model, which describes the topology of the modelled object, and an ‘embedding’ model, which describes the embedding of the object, for instance in three-dimensional Euclidean space. In recent years, numerous topological models have been developed for boundary representation, and there have been important developments with respect to dimension and orientability. In the main, two types of topological models can be distinguished. ‘Incidence graphs’ are graphs or hypergraphs, where the nodes generally represent the cells of the modelled subdivision (vertex, edge, face, etc.), and the edges represent the adjacency and incidence relations between these cells. ‘Ordered’ models use a single type of basic element (more or less explicitly defined), on which ‘element functions’ act; the cells of the modelled subdivision are implicitly defined in this type of model. In this paper some of the most representative ordered topological models are compared using the concepts of the n-dimensional generalized map and the n-dimensional map. The main result is that ordered topological models are (roughly speaking) equivalent with respect to the class of objects which can be modelled (i.e. with respect to dimension and orientability).     

Visualizing nonmanifold and singular implicit surfaces with point clouds

We use octree spatial subdivision to generate point clouds on complex nonmanifold implicit surfaces in order to visualize them. The new spatial subdivision scheme only uses point sampling and an interval exclusion test. The algorithm includes a test for pruning the resulting plotting nodes so that only points in the closest nodes to the surface are used in rendering. This algorithm results in improved image quality compared to the naive use of intervals or affine arithmetic when rendering implicit surfaces, particularly in regions of high curvature. We discuss and compare CPU and GPU versions of the algorithm. We can now render nonmanifold features such as rays, ray-like tubes, cusps, ridges, thin sections that are at arbitrary angles to the octree node edges, and singular points located within plot nodes, all without artifacts. Our previous algorithm could not render these without severe aliasing. The algorithm can render the self-intersection curves of implicit surfaces by exploiting the fact that surfaces are singular where they self-intersect. It can also render the intersection curves of two implicit surfaces. We present new image space and object space algorithms for rendering these intersection curves as contours on one of the surfaces. These algorithms are better at rendering high curvature contours than our previous algorithms. To demonstrate the robustness of the node pruning algorithm we render a number of complex implicit surfaces such as high order polynomial surfaces and Gaussian curvature surfaces. We also compare the algorithm with ray casting interms of speed and image quality. For the surfaces presented here, the point clouds can be computed in seconds to minutes on atypical Intel based PC. Once this is done, the surfaces can be rendered at much higher frame rates to allow some degree of interactive visualization.