By Brezis H.
Este texto recoge en una forma sensiblemente mas elaborada un curso de Maitrise impartido en los angeles Universidad Pierre y Marie Curie (Paris VI). Supone conocidos los elementos basicos de los angeles Topologia normal, de l. a. Integraci6n y del Calculo Diferencial.
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Reproduced with permission from D. D. M. A. Santiago “Feature extraction from a signature based on the turning angle function for the classification of breast tumors”, Journal of Digital Imaging, 21(2):129-144, 2008. ©Springer.
The TAF keeps track of the turning angle of the contour, increasing with convex regions and decreasing with concave regions. The turning angle of a segment si is the difference or step between TC (si ) and TC (si+1 ). The turning angle ranges in the interval (−180◦ , 180◦ ). Negative values represent concave regions and positive values represent convex regions. For a convex contour, TC (sn ) is a monotonic function, starting at an arbitrary value φ and increasing to φ + 2π . For a nonconvex polygon, TC (sn ) can become arbitrarily large, because it accumulates the total amount of turning angles, obeying the range of 2π between the starting point and the final point .
This may not be a desirable step, depending on the application. Rangayyan et al.  proposed a polygonal modeling procedure that eliminates this limitation of the method of Ventura and Chen . The procedure proposed by Rangayyan et al.  begins by segmenting the given closed contour into a set of piecewise-continuous curved parts; this is achieved by locating the points of inflection on the contour, based on its first, second, and third derivatives. The algorithm retains the initial points of inflection in the final polygonal model, thereby constraining the fit of the model to the contour provided.