Reeb Graph Research Articles

Partial 3D Shape Retrieval by Reeb Pattern Unfolding

Abstract This paper presents a novel approach for fast and efficient partial shape retrieval on a collection of 3D shapes. Each shape is represented by a Reeb graph associated with geometrical signatures. Partial similarity between two shapes is evaluated by computing a variant of their maximum common sub‐graph. By investigating Reeb graph theory, we take advantage of its intrinsic properties at two levels. First, we show that the segmentation of a shape by a Reeb graph provides charts with disk or annulus topology only. This topology control enables the computation of concise and efficient sub‐part geometrical signatures based on parameterisation techniques. Secondly, we introduce the notion of Reeb pattern on a Reeb graph along with its structural signature. We show this information discards Reeb graph structural distortion and still depicts the topology of the related sub‐parts. The number of combinations to evaluate in the matching process is then dramatically reduced by only considering the combinations of topology equivalent Reeb patterns. The proposed framework is invariant against rigid transformations and robust against non‐rigid transformations and surface noise. It queries the collection in interactive time (from 4 to 30 seconds for the largest queries). It outperforms the competing methods of the SHREC 2007 contest in term of NDCG vector and provides, respectively, a gain of 14.1% and 40.9% on the approaches by Biasotti et al.[BMSF06]and Cornea et al.[CDS*05]. As an application, we present an intelligent modelling‐by‐example system which enables a novice user to rapidly create new 3D shapes by composing shapes of a collection having similar sub‐parts.

3D Mesh decomposition using Reeb graphs

Decomposition of complex 3D objects into simpler sub-parts is a challenging research subject with relevant outcomes for several application contexts. In this paper, an approach is proposed for decomposition of 3D objects based on Reeb graphs. The approach is motivated by perceptual principles and supports identification of salient object protrusions. Experimental results are presented to demonstrate the effectiveness of the proposed approach with respect to different solutions appeared in the literature, and with reference to ground-truth data obtained by manual decomposition of 3D objects.

Harmonic 1-form based skeleton extraction from examples

This paper presents a method to extract skeletons using examples. Our method is based on the observation that many deformations in real-world applications are isometric or near isometric. By taking advantage of the intrinsic property of harmonic 1-form, i.e., it is determined by the metric and independent of the resolution and embedding, our method can easily find a consistent mapping between the reference and example poses which can be in different resolutions and triangulations. We first construct the skeleton-like Reeb graph of a harmonic function defined on the given poses. Then by examining the changes of mean curvatures, we identify the initial locations of joints. Finally we refine the joint locations by solving a constrained optimization problem. We demonstrate the efficacy of the proposed framework by pose space deformation, skeleton transfer, shape segmentation and pose-invariant shape signature.

Extending Persistence Using Poincaré and Lefschetz Duality

Persistent homology has proven to be a useful tool in a variety of contexts, including the recognition and measurement of shape characteristics of surfaces in ℝ3. Persistence pairs homology classes that are born and die in a filtration of a topological space, but does not pair its actual homology classes. For the sublevelset filtration of a surface in ℝ3, persistence has been extended to a pairing of essential classes using Reeb graphs. In this paper, we give an algebraic formulation that extends persistence to essential homology for any filtered space, present an algorithm to calculate it, and describe how it aids our ability to recognize shape features for codimension 1 submanifolds of Euclidean space. The extension derives from Poincare duality but generalizes to nonmanifold spaces. We prove stability for general triangulated spaces and duality as well as symmetry for triangulated manifolds.

Efficient algorithms for computing Reeb graphs

The Reeb graph tracks topology changes in level sets of a scalar function and finds applications in scientific visualization and geometric modeling. We describe an algorithm that constructs the Reeb graph of a Morse function defined on a 3-manifold. Our algorithm maintains connected components of the two dimensional levels sets as a dynamic graph and constructs the Reeb graph in O ( n log n + n log g ( log log g ) 3 ) time, where n is the number of triangles in the tetrahedral mesh representing the 3-manifold and g is the maximum genus over all level sets of the function. We extend this algorithm to construct Reeb graphs of d-manifolds in O ( n log n ( log log n ) 3 ) time, where n is the number of triangles in the simplicial complex that represents the d-manifold. Our result is a significant improvement over the previously known O ( n 2 ) algorithm. Finally, we present experimental results of our implementation and demonstrate that our algorithm for 3-manifolds performs efficiently in practice.

Interactive Comparison of Scalar Fields Based on Largest Contours with Applications to Flow Visualization

Understanding fluid flow data, especially vortices, is still a challenging task. Sophisticated visualization tools help to gain insight. In this paper, we present a novel approach for the interactive comparison of scalar fields using isosurfaces, and its application to fluid flow datasets. Features in two scalar fields are defined by largest contour segmentation after topological simplification. These features are matched using a volumetric similarity measure based on spatial overlap of individual features. The relationships defined by this similarity measure are ranked and presented in a thumbnail gallery of feature pairs and a graph representation showing all relationships between individual contours. Additionally, linked views of the contour trees are provided to ease navigation. The main render view shows the selected features overlapping each other. Thus, by displaying individual features and their relationships in a structured fashion, we enable exploratory visualization of correlations between similar structures in two scalar fields. We demonstrate the utility of our approach by applying it to a number of complex fluid flow datasets, where the emphasis is put on the comparison of vortex related scalar quantities.

VS: A surface-based system for topological analysis, quantization and visualization of voxel data

VS is a simple system consisting of several techniques for various volumetric problems. Based on the marching cubes algorithm, it operates with one space sweep through the voxels and extracts all topological information: detection of all isosurfaces, partitioning the data into connected components on the basis of surface connectivity, and association of surfaces with any internal surfaces to arbitrary levels of nesting. VS extends Baker's "Weaving Wall" method by associating topological cavities with their outer surface, and by using efficient data structures for the voxel traversal and for the connected component detection. Its runtime, on average, is only about 2% more than the speeded-up marching cubes algorithm. VS operates on the original voxels without using the contour tree required by other approaches. Using linear-time preprocessing, VS constructs a data structure which can be utilized for any isovalue or interval of isovalues. An accurate estimate of each component's volume is based on the volume enclosed by the outer surface, minus the volume of any internal cavities. VS enables noise reduction by eliminating components (and cavities) with a small volume. Different components can be rendered with different visual attributes, and cutaway views by arbitrary cut-planes are displayed as if the objects were solid, without adding any new surface patches to match the intersection.

自己隠蔽の回避を目的としたRegion-based Contour Treeによる3次元等値面の分割

自己隠蔽の回避を目的としたRegion-based Contour Treeによる3次元等値面の分割

Markerless Motion Capture with Structure Estimation Capability

We present a markerless motion capture system able to determine the kinematic structure while measuring joint movement. In addition to volume data, we also use texture data to precisely measure the degrees of freedom that do not affect the shape, e.g., pronation/supination angles of the forearm and shank. We first obtain topology using a Reeb graph and independently build a tentative articulated-body chain model of the subject for each frame. We then extract a common optimized chain model by comparing joint angles of tentative models of all frames to identify which joints are related to describing the movement of the subject. Our system thus measures movement without prior knowledge of the structure. The system identifies the link length of objects with known structures based on measured data.

スケルトンを用いたＴｉｍｅ‐Ｖａｒｙｉｎｇ Ｍｅｓｈからの動き抽出と類似動作検索

We propose an algorithm for motion extraction from Time-Varying Mesh (TVM). TVM is a sequence of 3D models made for real-world dynamic objects. TVM store detailed information about 3D objects such as their shape, color, and motion. Therefore, TVM is an attractive technology for next-generation multimedia. However, the management of a TVM database, which is used for archiving and retrieving, is a significant problem because it is difficult to locate and track feature points in TVM. For efficient and effective archive and retrieval systems, motion extraction and tracking is essential. In our method, skeletons are extracted from TVM using a Reeb graph, and motion tracking is achieved by fitting a reference skeleton to the other skeletons. We developed a geodesic function using principal component analysis (PCA) for fast and noiseless skeleton extraction. Robust fitting was achieved by implementing an end-node tracking strategy. In addition, we have developed an efficient motion retrieval system compatible with conventional motion capture data using the extracted skeletons. As a result, we have developed a low-cost query generation technique using motion-capture systems and cross search between TVM and motion capture data.

Time-varying Reeb graphs for continuous space–time data

Time-varying Reeb graphs for continuous space–time data

Time-varying Reeb graphs for continuous space–time data

The Reeb graph is a useful tool in visualizing real-valued data obtained from computational simulations of physical processes. We characterize the evolution of the Reeb graph of a time-varying continuous function defined in three-dimensional space. We show how to maintain the Reeb graph over time and compress the entire sequence of Reeb graphs into a single, partially persistent data structure, and augment this data structure with Betti numbers to describe the topology of level sets and with path seeds to assist in the fast extraction of level sets for visualization.

Differential topology methods for shape description

AbstractDifferential topology, and specifically Morse theory, provides a suitable setting for formalizing and solving several problems related to shape analysis. In this field, we discuss how a shape can be analyzed according to the properties of a real function defined on it (e.g., harmonic fields or laplacian eigenfunctions), and how these properties can be stored in compact and informative descriptors. We refer to Reeb graphs, that encode the configuration of level sets and critical points of the function

Enhancing 3D mesh topological skeletons with discrete contour constrictions

This paper describes a unified and fully automatic algorithm for Reeb graph construction and simplification as well as constriction approximation on triangulated surfaces. The key idea of the algorithm is that discrete contours – curves carried by the edges of the mesh and approximating the continuous contours of a mapping function – encode both topological and geometrical shape characteristics. Therefore, a new concise shape representation, enhanced topological skeletons, is proposed, encoding the contours’ topological and geometrical evolution. First, mesh feature points are computed. Then they are used as geodesic origins for the computation of an invariant mapping function that reveals the shape most significant features. Next, for each vertex in the mesh, its discrete contour is computed. As the set of discrete contours recovers the whole surface, each of them can be analyzed, both to detect topological changes and constrictions. Constriction approximations enable Reeb graphs refinement into more visually meaningful skeletons, which we refer to as enhanced topological skeletons. Extensive experiments showed that, without any preprocessing stage, proposed algorithms are fast in practice, affine-invariant and robust to a variety of surface degradations (surface noise, mesh sampling and model pose variations). These properties make enhanced topological skeletons interesting shape abstractions for many computer graphics applications.

Reeb graphs for shape analysis and applications

Reeb graphs are compact shape descriptors which convey topological information related to the level sets of a function defined on the shape. Their definition dates back to 1946, and finds its root in Morse theory. Reeb graphs as shape descriptors have been proposed to solve different problems arising in Computer Graphics, and nowadays they play a fundamental role in the field of computational topology for shape analysis. This paper provides an overview of the mathematical properties of Reeb graphs and reconstructs its history in the Computer Graphics context, with an eye towards directions of future research.

Robust on-line computation of Reeb graphs

Reeb graphs are a fundamental data structure for understanding and representing the topology of shapes. They are used in computer graphics, solid modeling, and visualization for applications ranging from the computation of similarities and finding defects in complex models to the automatic selection of visualization parameters. We introduce an on-line algorithm that reads a stream of elements (vertices, triangles, tetrahedra, etc.) and continuously maintains the Reeb graph of all elements already reed. The algorithm is robust in handling non-manifold meshes and general in its applicability to input models of any dimension. Optionally, we construct a skeleton-like embedding of the Reeb graph, and/or remove topological noise to reduce the output size. For interactive multi-resolution navigation we also build a hierarchical data structure which allows real-time extraction of approximated Reeb graphs containing all topological features above a given error threshold. Our extensive experiments show both high performance and practical linear scalability for meshes ranging from thousands to hundreds of millions of triangles. We apply our algorithm to the largest, most general, triangulated surfaces available to us, including 3D, 4D and 5D simplicial meshes. To demonstrate one important application we use Reeb graphs to find and highlight topological defects in meshes, including some widely believed to be "clean."

Robust on-line computation of Reeb graphs

Reeb graphs are a fundamental data structure for understanding and representing the topology of shapes. They are used in computer graphics, solid modeling, and visualization for applications rangin...

The Genus of a Digital Image Boundary Is Determined by Its Foreground, Background, and Reeb Graphs

We prove that the genus of the boundary of a digital image is precisely half of the sum of the cycle ranks of three particular graphs: the foreground and background which capture topological information about the digital image and its complement, respectively, and the Reeb graph, relative to the natural height function, associated with the digital image's boundary. We prove several additional results, including a characterization of when the cycle rank of the Reeb graph fails to equal the genus of the digital image's boundary (which can happen by virtue of the failure of the natural height function on the boundary of the digital image to be a Morse function).

Topology-Controlled Volume Rendering

Topology provides a foundation for the development of mathematically sound tools for processing and exploration of scalar fields. Existing topology-based methods can be used to identify interesting features in volumetric data sets, to find seed sets for accelerated isosurface extraction, or to treat individual connected components as distinct entities for isosurfacing or interval volume rendering. We describe a framework for direct volume rendering based on segmenting a volume into regions of equivalent contour topology and applying separate transfer functions to each region. Each region corresponds to a branch of a hierarchical contour tree decomposition, and a separate transfer function can be defined for it. The novel contributions of our work are 1) a volume rendering framework and interface where a unique transfer function can be assigned to each subvolume corresponding to a branch of the contour tree, 2) a runtime method for adjusting data values to reflect contour tree simplifications, 3) an efficient way of mapping a spatial location into the contour tree to determine the applicable transfer function, and 4) an algorithm for hardware-accelerated direct volume rendering that visualizes the contour tree-based segmentation at interactive frame rates using graphics processing units (GPUs) that support loops and conditional branches in fragment programs.

Systoles of 2-complexes, Reeb graph, and Grushko decomposition

Let X be a finite 2-complex with unfree fundamental group. We prove lower bounds for the area of a metric on X, in terms of the square of the least length of a noncontractible loop in X. We thus establish a uniform systolic inequality for all unfree 2-complexes. Our inequality improves the constant in M. Gromov’s inequality in this dimension. The argument relies on the Reeb graph and the coarea formula, combined with an induction on the number of freely indecomposable factors in Grushko’s decomposition of the fundamental group. More specifically, we construct a kind of a Reeb space “minimal model” for X, reminiscent of the “chopping off long fingers” construction used by Gromov in the context of surfaces. As a consequence, we prove the agreement of the Lusternik-Schnirelmann and systolic categories of a 2-complex.