DECOMPOSITION OF RANGE IMAGES USING MARKOV RANDOM FIELDS Andreas Pichler Profactor Produktionsforschung Gmbh. Im Stadtgut A2 A-4407 Steyr-Gleink, Austria Robert B. Fisher School of Informatics University of Edinburgh Edinburgh, UK EH9 3JZ Markus Vincze Automation and Control Institute Vienna University of Technology A-1040 Vienna, Austria ABSTRACT This paper describes a computational model for deriving a decomposition of objects from laser rangefinder data. The process aims to produce a set of parts defined by compact- ness and smoothness of surface connectivity. Relying on a general decomposition rule, any kind of objects made up of free-form surfaces are partitioned. A robust method to par- tition the object based on Markov Random Fields (MRF), which allows to incorporate prior knowledge, is presented. Shape index and curvedness descriptors along with discon- tinuity and concavity distributions are introduced to classify region labels correctly. In addition, a novel way to classify the shape of a surface is proposed resulting in a better dis- tinction of concave, convex and saddle shapes. To achieve a reliable classification a multi-scale method provides a stable estimation of the shape index. 1. INTRODUCTION Image segmentation is an important early vision task to group pixels with similar characteristic into homogeneous regions. Many high level processing tasks (object recognition, sur- face representation using volumetric models for example) are based on such a preprocessed image. Descriptions ex- tracted by a system must reflect the characteristics of the environment. This paper is about computing such a descrip- tion in 3D range data. Segmentation of range images has been addressed by many researchers e.g. [1, 2]. Measurements are used to seg- ment a depth map into surface patches which are either pla- nar or curved and which are described formally by a func- tion. These methods are restricted to process mainly CAD (Computer Aided Design) models. The recognition of com- plex objects in a cluttered scene without a feature extrac- tion or segmentation step is presented by Johnson [3] who proposed a template matching procedure using spin images. Charvis et. al. [4] proposed point signatures for describ- ing the local shape around a point. Both methods aim to extract local shape templates for every point on the model. Unfortunately, the methods described above cannot be used to extract a more generic description of an object. This is a significant step which needs be carried out focusing on object classification among other things. Psychological studies have shown an underlying regu- larity of part recognition [5]. A partitioning rule on the ba- sis of the Transversality Regularity divides an object into its constituent parts along all contours of concave discon- tinuity of the tangent plane. The process described in our paper aims to produce a set of parts generated by a generic partitioning rule. The extraction of discontinuities does not always deliver a continuous or even closed boundary. Thus, boundary organization is needed to link spare discontinu- ity points. A problem is whether a putative point is labeled as boundary or dismissed due to noise in the image which often leads to misclassification of region labels. The ap- proach proposed in this paper assigns a feature vector taking a shape index and curvedness estimation into account. The combination of both ensures compactness, smoothness and convexity of surface connectivity. A Markov Random Field (MRF) is used to incorporate spatial interaction as well as to to estimate slow spatial variations of the feature charac- teristic of the image. The paper is structured along the com- putational steps that define the procedure for decomposing an object from sensor data. In Section 2, a mathematical formulation is defined to decompose a scene. A novel com- putation method is presented for consistent shape index es- timation. Furthermore, the spatial interaction of shape in- dex and curvedness components of the feature vector is de- cribed. 2. PROBLEM FORMULATION The decomposition scheme extracts convex constituent parts of an object based on the local shape information. A quan- titative measurement of the shape of a surface at a point s is the shape index S I (s) and the curvedness C(s). Accord- ing to Koenderink’s definition of the shape index [6] and its adopted version [7] nine shape categories S = {S 1 , ..., S 9 } are distinguished. The shape index provides a continous gradation between salient shapes such as convex, saddle and concave types. The curvedness is a measure of the scale or