Computational morphology : a computational geometric by G. T. Toussaint

By G. T. Toussaint

Computational Geometry is a brand new self-discipline of machine technological know-how that offers with the layout and research of algorithms for fixing geometric difficulties. there are numerous parts of research in several disciplines which, whereas being of a geometrical nature, have as their major part the extraction of an outline of the form or type of the enter facts. This concept is extra vague and subjective than natural geometry. Such fields comprise cluster research in information, laptop imaginative and prescient and development acceptance, and the size of shape and form-change in such components as stereology and developmental biology. This quantity is anxious with a brand new method of the learn of form and shape in those parts. Computational morphology is therefore excited by the therapy of morphology from the computational geometry viewpoint. This perspective is extra formal, dependent, procedure-oriented, and transparent than many past techniques to the matter and infrequently yields algorithms which are more straightforward to software and feature decrease complexity

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This problem can be solved in a way that is analogous to the two dimensional algorithms. Atkinson [4] considers a finite set D of points. Translate the origin to the centroid of D, and rotate the axes so that / coincides with the z axis. Represent each point in cylindrical coordinates (r ,ζ ,θ), then construct the list L (D ) as for points but where each entry of L (D ) has an extra coordinate z. The "polar sort" can be done on Θ, r, then z. The Knuth-Morris-Pratt algorithm can be used in the same way as in Section 2 tofindall rotational and reflective symmetries.

A similar result can be applied to reflections. Hence a maximal symmetric subset can be found in time O (n4\ogn). It would be interesting to know if there is a faster algorithm than this naive approach. , Dk} of symmetric subsets of D such that the union of the Dt· is D and k is minimal. , (

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