
Research
Abstracts  2007 
Robust Modeling of TwoDimensional ShapesJared Glover, Nicholas Roy & Daniela RusThe ProblemWe wish to model the variability of 2D shapes (or a set of views of 3D shapes), in order to robustly perform inference tasks such as object recognition and object boundary estimation in the presence of sensor noise, occlusions, and rigid transformations. MotivationIn computer vision, most work on 2D shape recognition has focussed on discriminative models of sensing. Shape dissimilarity metrics (or shape distances) are formed, and one compares a new shape to a database of exemplars in order to find the best match. Much of the work has gone into finding the right shape distance, and typically a transformation is taken on the raw geometric data (e.g. an object silhouette contour) in order to perform the classification in the more discriminative shape space. In addition, care must be taken in choosing the exemplar shapes so that the size of the classification problem does not blow up with the number of object classes. Such techniques typically work well when the ability to discriminate between classes (i.e. classification) is the most important problem to be solved. However, if the geometry of objects within the same shape class varies in a significant way (such as in figure 1), or one wishes to perform additional inference tasks such as boundary estimation (such as in figure 2), it can be very difficult to construct an appropriate shape distance. Discriminative techniques can be difficult to us for robust shape modeling of both the object class and boundary estimation. Some methods require a transformation which makes it very difficult or even impossible to recover the original geometry (e.g. fourier descriptors [6], curvature scale space [7]). Other methods require many hard decisions to be made in the modeling process (e.g. skeletons [8], partbased models [5,9]), making it very difficult to develop statistical variants of the complete shape of the model. Thus, we seek a model for shape which is generative, and is as close to the raw geometric data (e.g. pixel or range measurement locations) as possible.
ApproachWe will use a seemingly simple, yet powerful model representing object shape as the complete set of boundary points. The drawbacks to this model are obviouswe are (for now) restricted to closed contours, we cannot represent continuous curves, and we must solve a correspondence problem in order to compare shapes. However, the advantages are (i) it can be made entirely nonparametric (i.e. we can incorporate the pixel or range measurement locations in the model), (ii) we have a natural cyclic ordering of points around the contour, and (iii) most importantly, we can draw upon a vast amount of established literature on the statistical shape modeling of sets of ordered landmarks [1,4]. Preliminary ResultsWe have demonstrated the ability to perform boundary estimation in the presence of occlusions [3], and we have also worked on methods for finding point correspondences between two shapes [2].
Future WorkIn future work, we plan to apply statistical shape modeling to image segmentation, robotic navigation under uncertainty, and robotic manipulation. References:[1] I. Dryden and K. Mardia. Statistical Shape Analysis. John Wiley and Sons, 1998. [2] Jared Glover, Christian Uldall Pedersen, and Erik Taarnhøj. Solving the cyclic order preserving assignment problem. Advanced Algorithms (6.854) Final Project, December 2006. [3] Jared Glover, Daniela Rus, Nicholas Roy, and Geoff Gordon. Robust models of object geometry. In Proceedings of the IROS Workshop on From Sensors to Human Spatial Concepts, Beijing, China, 2006. [4] D.G. Kendall, D. Barden, T.K. Carne, and H. Le. Shape and Shape Theory. John Wiley and Sons, 1999. [5] L. J. Latecki and R. Lakamper. Contourbased shape similarity. In Proc. of Int. Conf. on Visual Information Systems, volume LNCS 1614, pages 617624, June 1999. [6] C. C. Lin and R. Chellappa. Classification of partial 2d shapes using fourier descriptors. IEEE Trans. Pattern Anal. Mach. Intell., 9(5):686 690, 1987. [7] F. Mokhtarian and A. K. Mackworth. A theory of multiscale curvaturebased shape representation for planar curves. In IEEE Trans. Pattern Analysis and Machine Intelligence, volume 14, 1992. [8] Kaleem Siddiqi, Ali Shokoufandeh, Sven J. Dickinson, and Steven W. Zucker. Shock graphs and shape matching. In ICCV, pages 222229, 1998. [9] Mirela Tanase and Remco C. Veltkamp. Partbased shape retrieval. In MULTIMEDIA '05: Proceedings of the 13th annual ACM international conference on Multimedia, pages 543546, New York, NY, USA, 2005. 

