The original silhouettes of the Puma logo and a boy appear at the top, and their random affine transformations sampled to low resolution 64 × 64 patches at the second row in black. The normalized shapes with second-order moments appears in dark gray (bottom row) while the alternative method proposed by Cao et al. is presented in light gray (third row). 

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... the acquisition process. In [4], Cao et al. argue that normalization (canonization) of a planar shape that compensates for affine transformations and is based on second-order moments can be unstable. The authors propose alternatives based on the detection of flat intervals along the boundary. The next steps applied by Cao et al. involve center of mass estimation for the two regions created by a line parallel to the flat boundary line that goes through the center of mass. Parallel lines, area ratio, and center of mass are indeed robust measures preserved by affine transformations. On the other hand, a definition of flatness that is based on Euclidean distance and angles is not invariant to affine transformations. Moreover, if we limit our discussion to the analysis of simple closed contours there is a simple alternative for the first step propose in [4]. Experimenting with second order moments based normalization [13] we did not experience the instabilities reported by Cao et al. In fact, the moments based normalization proved to be equally stable as the centers of mass based alternative as can be seen in Figure 2. The method we propose in this section could be used to either initialize the Cao et al. canonization method or as compensation for the rotation ambiguity in moments based normalization. Let us assume that the contours we would like to normalize are interesting and therefor non-convex. In fact, convex contours could be classified by the simplest regular polygons that approximate the shape. A rough affine invariant canonical approximation for convex shapes could be triangles, squares, and circles that represent the rest of the regular polygons. Relying on area ratios and centers of mass, and based on [4], we define a robust affine invariant method for mapping a given contour into its canonical normalized shape. The steps of the method are as ...