Representation and Self-Similarity of Shapes

Representing shapes is a signi cant problem for vi sion systems that must recognize or classify objects We derive a representation for a given shape by inves tigating its self similarities and constructing its shape axis SA and shape axis tree SA tree We start with a shape its boundary contour and two di erent parameterizations for the contour To measure its self similarity we consider matching pairs of points and their tangents along the boundary con tour i e matching the two parameterizations The matching or self similarity criteria may vary e g co circularity parallelism distance region homogene ity The loci of middle points of the pairing contour points are the shape axis and they can be grouped into a unique tree graph the SA tree The shape axis for the co circularity criteria is compared to the symmetry axis An interpretation in terms of object parts is also presented Introduction With the goal to compare shapes this paper aims to develop and compute a shape representation Motivation Shape Similarity Methods to compare two shapes contours based on evaluating global deformations tend to be sensi tive to occlusion and fail to account for local deforma tions such as articulations since these deformations may change the global appearance of objects consid erably while the entire deformation is concentrated in speci c points A class of methods compares objects by deforming one object into another and evaluating the amount of deformation applied in this process including Guaranteed methods typically uses dynamic pro gramming time warping to register two contours These are all string contour matching algorithms e g The problem with these approaches as pointed out in is that they do not account for re gion information and for symmetries see Figure Figure A shape contour followed by two shapes obtained by di erent deformations stretching into the rst one Local string deformation methods will fail to distinguish the dissimilarity between them and the rst one Our aim is to develop a shape representation of ob jects that allow similarity measures that account for local deformations symmetries and region informa tion Our representation gives a shape tree axis where every pair of consecutive nodes edge correspond to an object part The similarity measure between two objects is de ned by a tree matching scheme where the cost of matching edges is the cost of comparing two object parts local deformations and region informa tion can be accounted Tree matching uses the partial order induced by the trees but is rather more com plex than simple string matching algorithms More over node deletions account for occlusions and moving nodes account for articulations Figure shows result of using this representation for matching objects Shape Representation Our formulation is a variational one We start with a shape its boundary contour We seek a self similarity measure to structure the representation of shapes Our insight is to generate two di erent pa rameterizations for the contour and to measure its self similarity by matching the two parameterizations i e by matching pairs of points and their tangents along the boundary contour The matching or self similarity criteria may vary e g co circularity par allelism distance region homogeneity The middle