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Chapter 2: Literature Review 15feature vector. The binary Hamming distance is used to measure the similaritybetween two shapes. To account for the invariance to Euclidean transformations theshape needs to be normalized. Chakrabarti et al. [Chakrabarti et al., 2000] improve thegrid descriptor by using an adaptive resolution (AR) representation acquired byapplying quad-tree decomposition on the bitmap representation of the shape.Typically, shape methods use rectangular-grid sampling to acquire shape information.The shape representation so derived is usually not translation, rotation and scalinginvariant. Extra normalization is therefore required. Goshtasby [Goshtasby, 1985]proposes the use of a shape matrix which is derived from a circular raster samplingtechnique. The idea is similar to normal raster sampling. However, rather than overlaythe normal square grid on a shape image, a polar raster of concentric circles and radiallines is overlaid in the center of the mass. The binary value of the shape is sampled atthe intersections of the circles and radial lines. The shape matrix is formed such thatthe circles correspond to the matrix columns and the radial lines correspond to thematrix rows. Prior to the sampling, the shape is scale normalized using the maximumradius of the shape. The resultant matrix representation is invariant to translation,rotation, and scaling. Since the sampling density is not constant with the polarsampling raster, Taza et al. represent shape using a weighed shape matrix, which givesmore weight to peripheral samples in [Taza et al., 1989]. However, since a shapematrix is a sparse sampling of shape, it is easily affected by noise. Besides, shapematching using a shape matrix is expensive. Parui et al. propose a shape descriptionbased on the relative areas of the shape contained in concentric rings located in theshape center of the mass in [Parui et al., 1986].Structural methods for region-based shape description usually involve the convexhulls and medial axis described in [Davies,1997], [Blum,1967] and [Morse,1994]. Aregion R is convex if and only if for any two points x 1 ; x 2 R, the whole linesegment x 1 x 2 is inside the region. The convex hull of a region is the smallest convexregion H which satisfies the condition R H. The difference H − R is called theconvex deficiency D of the region R. The extraction of the convex hull can beachieved either using the boundary-tracing method from [Sonka et al., 1993] or byusing morphological methods from [Gonzalez and Woods, 1992]. Since shapeboundaries tend to be irregular because of digitization noise and variations insegmentation result in a convex deficiency that has small, meaningless componentsscattered throughout the boundary. Common practice is to first smooth a boundaryprior to partitioning. The polygon approximation is particularly attractive because itcan reduce the computation time taken for extracting the convex hull from O (n 2 ) to O(n) (n being the number of points in the shape). The extraction of convex hull can be asingle process which finds significant convex deficiencies along the boundary. Afuller representation of the shape is obtained by a recursive process which results in aconcavity tree. Here the convex hull of an object is first obtained with its convexdeficiencies, then the convex hulls and deficiencies of the convex deficiencies are