In this paper we describe a hierarchical face clustering algorithm for triangle meshes based on fitting primitives belonging to an arbitrary set. The method proposed is completely automatic, and generates a binary tree of clusters, each of which fitted by one of the primitives employed. Initially, each triangle represents a single cluster; at every iteration, all the pairs of adjacent clusters are considered, and the one that can be better approximated by one of the primitives forms a new single cluster. The approximation error is evaluated using the same metric for all the primitives, so that it makes sense to choose which is the most suitable primitive to approximate the set of triangles in a cluster. Based on this approach, we implemented a prototype which uses planes, spheres and cylinders, and have experimented that for meshes made of 100k faces, the whole binary tree of clusters can be built in about 8 seconds on a standard PC. The framework here described has natural application in reverse engineering processes, but it has been also tested for surface de-nosing, feature recovery and character skinning.

Curve-skeletons are thinned 1D representations of 3D objects useful for many visualization tasks including virtual navigation, reduced-model formulation, visualization improvement, animation, etc. There are many algorithms in the literature describing extraction methodologies for different applications; however, it is unclear how general and robust they are. In this paper, we provide an overview of many curve-skeleton applications and compile a set of desired properties of such representations. We also give a taxonomy of methods and analyze the advantages and drawbacks of each class of algorithms.

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Understanding shapes has been a challenging issue for many years, firstly motivated by computer vision and more recently by many complex applications in diverse fields, such as medical imaging, animation, or product modeling. Moreover, the results achieved so far prompted a significant amount of work in very innovative research fields such as semantic-based knowledge systems dealing with multi-dimensional media. This paper describes the historical evolvement of the research done at IMATI-GE / CNR in the field of shape understanding. The most significant methods developed are classified and described along with some results, and discussed with respect to applications. Open issues are outlined along with future research plans.

Given a continuous scalar field f: X → IR where X is a topological space, a level set of f is a set {x ∈ X: f(x) = α} for some value α ∈ IR. The level sets of f can be subdivided into connected components. As α changes continuously, the connected components in the level sets appear, disappear, split and merge. The Reeb graph of f encodes these changes in connected components of level sets. It provides a simple yet meaningful abstraction of the input domain. As such, it has been used in a range of applications in fields such as graphics and scientific visualization. In this paper, we present the first sub-quadratic algorithm to compute the Reeb graph for a function on an arbitrary simplicial complex K. Our algorithm is randomized with an expected running time O(m log n), where m is the size of the 2-skeleton of K (i.e, total number of vertices, edges and triangles), and n is the number of vertices. This presents a significant improvement over the previous Θ(mn) time complexity for arbitrary complex, matches (although in expectation only) the best known result for the special case of 2-manifolds, and is faster than current algorithms for any other special cases (e.g, 3-manifolds). Our algorithm is also very simple to implement. Preliminary experimental results show that it performs well in practice.

Abstract—We present an algorithm for curve skeleton extraction via Laplacian-based contraction. Our algorithm can be applied to surfaces with boundaries, polygon soups, and point clouds. We develop a contraction operation that is designed to work on generalized discrete geometry data, particularly point clouds, via local Delaunay triangulation and topological thinning. Our approach is robust to noise and can handle moderate amounts of missing data, allowing skeleton-based manipulation of point clouds without explicit surface reconstruction. By avoiding explicit reconstruction, we are able to perform skeleton-driven topology repair of acquired point clouds in the presence of large amounts of missing data. In such cases, automatic surface reconstruction schemes tend to produce incorrect surface topology. We show that the curve skeletons we extract provide an intuitive and easy-to-manipulate structure for effective topology modification, leading to more faithful surface reconstruction. Keywords-curve skeleton; point cloud; Laplacian; contraction; topology repair; surface reconstruction I.

Assessing the similarity among 3D shapes is a challenging research topic, and effective shape descriptions have to be devised in order to support the matching process. There is a growing consensus that shapes are recognized and coded mentally in terms of relevant parts and their spatial configuration, or structure. The presentation will discuss the definition and use of structural descriptions for assessing shape similarity. The idea is to define a shape description framework based on results of differential topology which deal with the description of shapes by means of the properties of one, or more, real-valued functions defined over the shape. Studying these properties, several topological descriptions of the shape can be defined, which may also encode different geometric and morphological attributes that globally and locally describe the shape. Examples and results will be discussed and ongoing work outlined. 1

Abstract. In this paper, 3D shape retrieval methodology suited for search in special category of 3D shape is presented. The proposed approach employs a fully unsupervised segmentation algorithm to decompose 3D models into components. Shape distribution vectors describing the resulting components are extracted and together with connectivity relations identify a 3D model. The 3D-shapes we are interested in this paper are models of furniture. Ontology of furniture that we started building will be used in annotation and then key word based retrieval of furniture models. A mapping between low level features extracted by the above mentioned algorithm and ontology concepts is performed. The proposed approach bridges the gap between keyword-based approaches and query-by-example approaches by using not only the low-level features but also a domain ontology. 1

In dip-coating processes a three-dimensional object, e.g. an entire car body, is dipped into a liquid bath. In order to simulate such processes, the space surrounding the object is decomposed into the so-called flow volumes, for which each intersection with a horizontal plane is connected. At any time the liquid’s surface then has a unique level within such a flow volume, which greatly simplifies the simulation of the liquid. The decomposition into flow volumes corresponds to the Reeb graph of the object’s exterior (considered as 3-manifold with boundary) with respect to the height function. This article presents an algorithm which computes this decomposition for an object represented as oriented triangular boundary mesh. First critical vertices of the surface are identified, which include the upper and lower ends of flow volumes. Using local information about horizontal intersection planes near the critical points, a sweep plane algorithm then constructs the volume decomposition in a second step. It is shown that the method can deal with realistic data. 1

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