Shape Recognition Via an a Contrario Model for Size Functions

Shape recognition methods are often based on feature comparison. When features are of different natures, combining the value of distances or (dis-)similarity measures is not easy since each feature has its own amount of variability. Statistical models are therefore needed. This article proposes a statistical method, namely an a contrario method, to merge features derived from several families of size functions. This merging is usually achieved through a touchy normalizing of the distances. The proposed model consists in building a probability measure. It leads to a global shape recognition method dedicated to perceptual similarities. 1 Global Shape Recognition 1.1 A Brief State of the Art Shape recognition is a central problem in computer vision. Geometrical shapes can be defined as outlines of objects and are thus formally connected bounded domains of the plane. Many shape recognition methods are global in the sense that the extracted features are computed over the whole solid shape. Since they mix global and local information, they are sensitive to occlusions (a part of the shape is hidden) or insertion (a part is added to the shape). There are however a large number of applications in which the shapes that are to be identified are not occluded, for which global methods are undoubtly useful. The present method actually enables global shape recognition. The global features are in general scalar numbers computed over the whole shape. Two classical global features are based on Fourier descriptors (after Zahn and Roskies [1]) or invariant moments [2,3] (following a founding work by Hu [4]). Affine invariant scalars for global shape representation can also be derived from A. Campilho and M. Kamel (Eds.): ICIAR 2006, LNCS 4142, pp. 410–421, 2006. c © Springer-Verlag Berlin Heidelberg 2006 Shape Recognition Via an a Contrario Model for Size Functions 411 wavelet coefficients (see for instance [5]). Scale-space representations can also be used to derive invariant representations. In this class of methods the Curvature Scale Space by Mokhtarian and Mackworth [6] is certainly the most popular approach. It consists in smoothing the shape boundary by curvature motion, while tracking the position of its inflexion points across the scales. This method yields similarity invariant representations and it has a certain amount of robustness to noise. Another invariant shape representation based on scale spaces can be found in Alvarez et al. [7] where shape invariants are built from the evolution of area and perimeter of the shapes undergoing the affine scale space. All these methods deal with similar shape recognition where similar is understood with respect to geometrical invariance (similar shapes are sought up to an affine transformation for example). On the contrary, a well-known, moment related, global method which deals with perceptual similarity is modal matching, by Sclaroff and Pentland [8]. In this method, solid shapes are represented by eigenmodes associated to a physical elastic model. This method permits relatively realistic shape deformations where the thin parts of the shape can alter more than the bulk (for example two human silhouettes are retrieved as similar, whatever the pose could be). 1.2 Contribution: Mixing Size Functions Through a Statistical Model The descriptors that are considered in the present shape recognition method are based on Size Functions (SFs), which are geometrical-topological descriptors, conceived for formalizing qualitative aspects of shapes. Although SFs carry both local and global information and cannot be classified as a purely global method, in this paper we shall use them as global features. We are indeed interested in the comparison of shapes when occlusions are not involved. When applied to shape recognition, SFs theory leads to (quasi) invariant descriptions. These descriptions have proven to be particularly useful for perceptual matching, mainly when no standard geometric templates are available [9]. See for example [10] where SFs are applied to trademark retrieval. As a single SF cannot provide a complete shape description (two dissimilar shapes could share similar SFs) several SFs have to be extracted from each shape. Each one of them enables to capture geometrical-topological information of different natures. The question is: how to merge these informations? The aim of this article is precisely to propose an appropriate statistical framework. The shape recognition algorithm derived from this model is dedicated to retrieve global shapes that have not undergone occlusions. 1.3 Plan of the Article We first give guidelines on Size Theory and explain how to extract geometrical information from shapes via several measuring functions (Sec. 2). In Sec. 3 we address the problem of retrieving shapes from a database that are similar to a query shape bymerging the extracted information through a statistical framework,