Shape Representation and Registration using Vector Distance Functions Hossam Abd EL Munim and Aly A. Farag Computer Vision and Image Processing Laboratory (CVIP) University of Louisville, Louisville, KY 40292 hossam,farag @cvip.uofl.edu Abstract This paper introduces a new method for shape registra- tion by matching vector distance functions. The vector dis- tance function representation is more flexible than the con- ventional signed distance map since it enables us to better control the shapes registration process by using more gen- eral transformations. Based on this model, a variational frame work is proposed for the global and local registration of shapes which does not need any point correspondences. The optimization criterion can handle efficiently the estima- tion of the global registration parameters. A closed form so- lution is provided to handle an incremental free form defor- mation model for covering the local deformations. This is an advantage over the gradient descent optimization which is biased towards the initialization and is more time con- suming. Results of real shapes registration will be demon- strated to show the efficiency of the proposed approach with small and large global/local deformations. 1. Introduction The shape registration aims to build a point correspon- dence between a given shape (source) boundary and a tem- plate (target) [1, 2]. It is a very important process in com- puter vision and medical imaging. The registration depends on the : 1) method how to represent shapes, 2) nature of the transformation to move the points from the source towards the target, and 3) dissimilarity measure. The latter can be defined according to either the shape boundary or its entire region. The iterative closest point algorithm was proposed in [3]. The approach is based on finding the correspondence based on the minimum distance criterion. Different shape regis- tration approaches based on this technique are provided in the literature (e.g. [4]). Shape registration is handled in [5] by matching signed distance functions. The dissimilarity measure allowed only the use of homogeneous scales which limits the efficiency of the process. Practically inhomogeneous scaling is nec- essary since data is gathered from different sources or sub- jects. The more general the transformation is, the better the results are. A variational approach to top-down image segmentation was proposed in [6]. A projective transformation is used with a single prior image is embedded into the image to be segmented without using any point correspondences. The prior shape contour is represented by a cone. Perspective distortion and scaling of the visible contour are allowed us- ing unlevel sections. Cremers et.al. investigated the dissimilarity measures for shapes represented by the signed distance function [7]. A symmetric pseudo distance, which is not biased to small ar- eas, was constituted as a dissimilarity measure. A shape- based segmentation technique was proposed which is pose invariant. Tracking of 2D and 3D objects’ examples were demonstrated in [8] as well. Different shapes registration approaches were proposed in the literature for example [9, 10, 11]. These approaches suffer from various problems, including scale variations and dependence on initialization. Also local deformations are not addressed efficiently. Vector distance functions (VDF’s) are used in [12] to evolve smooth manifolds. This representation defines a vec- tor that connects any point in space to the nearest point on the curve or surface. This representation can deal with shapes of different dimensions. We proposed shape representation by vector compo- nents in a different manner in our shape-based segmentation framework [13]. The vector components represent the vec- tor projections from any point in space to the nearest point on the shape boundary. We give a positive sign to the points inside the shape and negative to those outside the shape to mark these regions. We used a simple dissimilarity measure to handle the problem of inhomogeneous scaling. Also the vector map was designed to handle the segmentation prob- lem with the adaptive region model. In this paper, we use the VDF shape representation as a similarity measure in the shape registration process. More general transformations with different scaling , 1-4244-1180-7/07/$25.00 ©2007 IEEE