3D object comparison based on shape descriptors

S. Biasotti, S. Marini
2005 International journal of computer applications in technology  
Due to the recent improvements to 3D object acquisition, visualisation and modelling technologies, the number of 3D models available on the web is more and more growing, and there is an increasing demand for tools supporting the automatic search for 3D objects in digital archives. Traditional methods for 3D shape retrieval roughly filter shape information or perform a punctual comparison of models. In this paper, we discuss on the advantages of approaching the shape matching problem through 3D
more » ... raph-like descriptors, which decompose the shape into relevant subparts. In our approach, shapes are compared using a graph matching technique that includes a structured process, which identifies the most similar object portions. In particular, we investigate on the properties of these descriptors and of our matching method in the CAD context. Introduction The needs to extract knowledge from massive volumes of digital content rapidly increases and new forms of content are coming in evidence, such as 3D animations and virtual or augmented reality. Whilst it has become relatively easier to generate 3D information and to interact with the geometry of shapes, it is harder to structure, filter, organise and retrieve it. These considerations are changing the approach to 3D object modelling. Until now a primary challenge in computer graphics has been how to build and render complete and effective models, now the key issue is how to find and interpret them. In this sense, methods for automatically extracting the semantic content of digital shapes and generating shape models in which knowledge/semantics is explicitly represented will become more and more necessary. This will allow browsing the web or digital object repositories using enhanced search engines not simply based on text-searches but on shape and semantics (e.g., content and context based search engines capable of answering semantics-based queries) [53]. In our understanding the knowledge of a digital shape may be organised at three different levels of representation: geometric, structural and semantic level [15]. A first organisation of the shape data into a computational structure gives access to the geometric level of representation, where different types of geometric models can be used to represent the same object form. As examples, we can list polygons, surface models (splines, NURBS, implicit surfaces,...) [34], solid models (3D mesh, Brep, CSG) [22][28][36], clusters of pixels or voxels (shapes segmented within an image or volume) [36], etc. In a geometric model, topological and geometric information are coded explicitly or implicitly in a computer processable structure. Then, a structural level of representation is reached by organising the geometric information and/or shape data to reflect and/or make explicit the association between parts/components of shapes. Examples of structural models are: multi-resolution and multi-scale models [27], curvature based surface decompositions [52][14], topological decompositions [12][13], shape segmentations, etc.. At the highest level of segmentation, the semantic one, there is the association of a specific semantics to structured and/or geometric models through annotation of shapes, or shape parts, according to the concepts formalised by the semantic domain. Therefore, a semantic model is the representation of a shape embedded into a specific context. The method discussed in this paper approaches the problem of using a structural representation for shape comparison purposes. The first step to fulfil this task is to associate a signature, that is a so-called shape descriptor, to a geometric model. In particular, we would desire that the chosen shape descriptor concisely represents the shape features, is invariant to rigid and similarity transformations, is insensitive to noise and small extra feature, is robust with respect to topological deformations, is suitable for multi-scale analysis and is computationally efficient and simple to store. Moreover, it is our opinion that a signature that organises the object shape in a topologically consistent framework provides a relation between shape structure and semantics. Several methods have been proposed to extract the salient information stored in a model; such as descriptors based on shape distributions [33], spherical harmonics [16][39], statistical distribution of the shape points in the space [48] or the highcurvature regions [20], while others try to organise and interpret the shape features through a graph representation, such as skeletal curves [46] and Reeb graphs [5][21] [42] . In particular, we will discuss on the advantage of considering a skeleton based representation of the shape as the signature for approaching the 3D shape comparison problem and on the possible extension of the application domain from freeform models to mechanical ones. Furthermore, we will give a novel interpretation of our approach, showing that the heuristic method we presented in [6] may be deduced from the more general problem of approximating the maximum common sub-graph [30]; in addition, the heuristic choices we proposed in [6] can be modularly inserted in a general methodology and other approximations may be introduced. The reminder of this paper is organised as follows. First, an overview of the existing skeletal representations and their use on shape representation and retrieval is proposed, focusing, in particular, on the Reeb graph and the matching approaches based on that structure. Then, our approach to 3D shape comparison is described and arranged in the general graph isomorphism context. Examples and results are proposed in section 4 and compared with those obtained through the spherical harmonics approach in [16]; in addition, the suitability of our method for object matching is discussed. Conclusions and future developments end the paper. Related Work on shape representation and retrieval Existing techniques can represent the geometry of a shape with high detail, typically through a dense mesh of simple basic elements such as triangles, tetrahedra, etc. Such meshes can approximate the geometry of a shape arbitrarily well, but they fail in describing the morphological structure of the shape, which has a fundamental importance for shape classification and understanding. On the contrary, iconic models, intended as concise, part-based representations of a shape, provide more structured descriptions, even if sometimes less accurate. In this context shape distributions [33], which evaluate the distribution on the surface of a shape function that measures the geometric properties of the model, and spherical harmonics [16] are expressive tools. The latter descriptor, in particular, decomposing the model into a collection of functions defined on concentric spheres with respect the centre of mass, is invariant to object rotations. The original shape cannot be recovered from these shape descriptors but comparison may be efficiently accomplished using traditional distances between functions. Furthermore, these descriptors do not identify the correspondence between the most similar object sub-parts preventing any reasoning based on the shape structure (e.g.: reasoning about building differences between mechanical artefacts). In advanced fields, such as virtual human modelling, available modelling tools to represent structured geometry are focused on adding a skeleton to the 3D geometry in order to animate it and provide different degrees of realism (from segmented nondeformable bodies to anatomically accurate deformable meshes). In addition, there has been a considerable amount of work in the literature on extracting critical features (point, integral lines, etc.) from 2D scalar fields describing grey-level images and terrains, and more recently some work has been done on volume data, again on extracting a critical net or on representing the topological structure of the iso-surfaces through the so-called contour tree. Skeletal descriptions A skeletal structure should encode the decomposition of a shape into relevant parts, or features, which may have either a geometric or an application-dependent meaning. Therefore, it is important to detect salient features over non-significant ones and define a mapping between the skeleton and the full geometry, so that the two levels can be Reeb graph representations has to be taken into account during the similarity analysis: in fact each function emphasizes different aspects of the object shape. Figure 7 highlights how the choice of the function in the Reeb graph representation influences the matching results. In fact, the topology of the teapot has been modified and the graphs result much different. The graph obtained through the distance from the barycentre function is a representation of the spatial distribution of the object with respect to its centre of mass: even if a part of the handle has been removed, the remaining part folds on itself, generating a critical points in the Reeb function. On the contrary, the graph based on the integral geodesic does not take into account the spatial embedding, thus the broken handle of the teapot results in a maximum critical point with respect to the geodesic distance, neglecting the shape of the handle itself.
doi:10.1504/ijcat.2005.006465 fatcat:2gt44aoajfcs7apwbvgjflx3xm