A topological approach for segmenting human body shape

Segmentation of a 3D human body, is a very challenging problem in applications exploiting human scan data. To tackle this problem, the paper proposes a topological approach based on the discrete Reeb graph (DRG) which is an extension of the classical Reeb graph to handle unorganized clouds of 3D points. The essence of the approach concerns detecting critical nodes in the DRG, thereby permitting the extraction of branches that represent parts of the body. Because the human body shape representation is built upon global topological features that are preserved so long as the whole structure of the human body does not change, our approach is quite robust against noise, holes, irregular sampling, frame change and posture variation. Experimental results performed on real scan data demonstrate the validity of our method

A Combinatorial Parametric Engineering Model for Solid Freeform Fabrication

Fabricated parts are often represented as compact connected smooth 3-manifolds with boundary, where the boundaries consist of compact smooth 2-manifolds. This class of mathematical structures includes topological spaces with enclosed voids and tunnels. Useful information about these structures are coded into level functions (Morse functions) which map points in the 3-manifold onto their height above a fixed plane. By definition, Morse functions are smooth functions, all of whose critical points are nondegenerate. This information is presented by the Reeb graph construction that develops a topologically informative skeleton of the manifold whose nodes are the critical points of the Morse function and whose edges are associated with the connected components between critical slices. This approach accurately captures the SFF process: using a solid geometric model of the part, defining surface boundaries; selecting a part orientation; forming planar slices, decomposing the solid into a sequence of thin cross-sectional polyhedral layers; and then fabricating the part by producing the polyhedra by additive manufacturing. This note will define a qualitative and combinatorial parametric engineering model of the SFF part design process. The objects under study will be abstract simplicial complexes K with boundary ∂K. Systems of labeled 2-surfaces in K, called slices, will be associated with the cross-sectional polyhedral layers. The labeled slices are mapped into a family of digraph automata, which, unlike cellular automata, are defined not on regular lattices with simple connectivities (cells usually have either 4 or 8 cell neighborhoods) but on unrestricted digraphs whose connectivities are irregular and more complicated.Mechanical Engineerin

Image = Structure + Few Colors

Topology plays an important role in computer vision by capturing the structure of the objects. Nevertheless, its potential applications have not been sufficiently developed yet. In this paper, we combine the topological properties of an image with hierarchical approaches to build a topology preserving irregular image pyramid (TIIP). The TIIP algorithm uses combinatorial maps as data structure which implicitly capture the structure of the image in terms of the critical points. Thus, we can achieve a compact representation of an image, preserving the structure and topology of its critical points (maxima, the minima and the saddles). The parallel algorithmic complexity of building the pyramid is O(log d) where d is the diameter of the largest object.We achieve promising results for image reconstruction using only a few color values and the structure of the image, although preserving fine details including the texture of the image

Decision Making for Rapid Information Acquisition in the Reconnaissance of Random Fields

Research into several aspects of robot-enabled reconnaissance of random fields is reported. The work has two major components: the underlying theory of information acquisition in the exploration of unknown fields and the results of experiments on how humans use sensor-equipped robots to perform a simulated reconnaissance exercise. The theoretical framework reported herein extends work on robotic exploration that has been reported by ourselves and others. Several new figures of merit for evaluating exploration strategies are proposed and compared. Using concepts from differential topology and information theory, we develop the theoretical foundation of search strategies aimed at rapid discovery of topological features (locations of critical points and critical level sets) of a priori unknown differentiable random fields. The theory enables study of efficient reconnaissance strategies in which the tradeoff between speed and accuracy can be understood. The proposed approach to rapid discovery of topological features has led in a natural way to to the creation of parsimonious reconnaissance routines that do not rely on any prior knowledge of the environment. The design of topology-guided search protocols uses a mathematical framework that quantifies the relationship between what is discovered and what remains to be discovered. The quantification rests on an information theory inspired model whose properties allow us to treat search as a problem in optimal information acquisition. A central theme in this approach is that "conservative" and "aggressive" search strategies can be precisely defined, and search decisions regarding "exploration" vs. "exploitation" choices are informed by the rate at which the information metric is changing.Comment: 34 pages, 20 figure

Affine-invariant skeleton of 3D shapes

No abstract availabl

An edit distance for Reeb graphs

We consider the problem of assessing the similarity of 3D shapes using Reeb graphs from the standpoint of robustness under perturbations. For this purpose, 3D objects are viewed as spaces endowed with real-valued functions, while the similarity between the resulting Reeb graphs is addressed through a graph edit distance. The cases of smooth functions on manifolds and piecewise linear functions on polyhedra stand out as the most interesting ones. The main contribution of this paper is the introduction of a general edit distance suitable for comparing Reeb graphs in these settings. This edit distance promises to be useful for applications in 3D object retrieval because of its stability properties in the presence of noise