Permanence-Based Shape Decomposition in Binary Pyramids Gunilla Borgefors 1 , Giuliana Ramella 2 , Gabriella Sanniti di Baja 2 1 Centre for Image Analysis, SLU, Uppsala, Sweden, gunilla@cb.uu.se 2 Istituto di Cibernetica, CNR, Arco Felice (Napoli), Italy, gsdb,gr @imagm.cib.na.cnr.it Abstract An algorithm to decompose hierarchically bidimensional patterns is introduced. The single-scale input pattern is first transformed into a multi-scale data set. The multi-resolution skeleton is then computed and its hierarchical decomposition is obtained by using the notion of permanence. A constrained reverse distance transformation is applied to the skeleton components to reconstruct the regions into which the pattern is decomposed. A merging process then reduces the number of components to the most significant ones and improves decomposition stability. 1. Introduction Region-based shape representation has become more and more popular in computer vision, as it allows tackling complex tasks as recognition in presence of occlusion or deformation of parts of the shape. Regions that are sufficiently far apart from the region where an occlusion or a deformation occurred are expected to remain unbiased; in most cases, the description of the unbiased regions together with information on their mutual spatial relations can be enough to perform recognition in a robust way. A number of algorithms can be found in the literature, focussing either on the decomposition of the interior of the shape, e.g. [1-4], or on the segmentation of its boundary, e.g. [5]. Other methods take somehow into account information from both the interior and the boundary, e.g. [6-7]. Naturally, whichever algorithm is used to decompose the shape, the obtained regions should be meaningful and simple, and their organisation should preferably be hierarchical. Multi-resolution representations are convenient for pattern recognition, especially when recognition implies the use of a matching phase. Shape complexity depends on resolution. Only the most significant regions are preserved in all scales, while other regions appear only at higher resolutions. Matching can be preliminarily performed using only lower resolution representations, where a small number of regions are generally involved. This process allows sorting out from the library of prototypes a smaller number of entries on which to perform a more detailed comparison at higher resolution. In this communication we use a binary AND-pyramid to transform a single scale representation of a binary image into a multi-resolution representation. The mechanism used to build the pyramid is implicitly based on width information. Only the thickest regions constituting the shape appear at all resolution levels, while narrower regions soon disappear when the resolution decreases. By introducing the notion of permanence through the pyramid levels, a hierarchical shape decomposition scheme becomes possible. The shape is decomposed at all levels of the pyramid so that region-based recognition techniques can be applied at any resolution. Our decomposition method is actually based on the decomposition of the interior of the shape. We compute the skeleton of the shape at all resolution levels and hierarchically decompose it. Hence, we follow Blum’s idea and interpret a shape as a collection of overlapping (maximal) disks, [1]. By suitably reversing skeletonization, we obtain at all pyramid levels a partition of the shape into constituting regions. Merging criteria are introduced to simplify the representation and get rid of marginally significant regions. At every pyramid level any region R, regarded as non significant, is merged to one of its adjacent regions by using information on the geometrical properties of the regions in the shape partition at that level, as well as by taking into account the history of the regions homologous of R in already inspected pyramid levels. The algorithm is general purpose and has been designed without referring to a particular class of images. It has been tested on a large number of differently structured images. The results show that the regions are as large as they can be, constrained by the homogeneous thickness criterion, and relatively stable. This makes our method promising to use for object recognition. In Section 2, we transform a single-scale image into a multi-scale one and compute the multi-resolution skeleton. Sections 3 and 4 briefly describe the procedures to obtain the hierarchical decomposition of the skeleton and shape reconstruction, respectively. Merging criteria aimed at simplifying the decomposition are presented in Section 5. Finally, Section 6 briefly discusses the obtained results. 2. Multi-Scale Shape Representation Let I be a black and white image having size 2 n ×2 n . For the sake of simplicity, we suppose that the set of the black pixels (pattern) consists of a single 8-connected component, while the set of the white pixels (background) may include any number of 4-connected components. We assume that all the pixels on the border of I are white. The 2×2 AND-pyramid of I is built as follows. The highest, original, resolution level in the pyramid coincides with I. The next, lower, resolution level 2 n-1 ×2 n-1 is built by partitioning I into 2×2 blocks of pixels, the children, and