An Efficient Algorithm for Determining an Aesthetic Shape Connecting Unorganized 2D Points

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Abstract

We present anefficient algorithm for determining an aesthetically pleasing shape boundary connecting all the points in a given unorganized set of 2D points, with no other information than point coordinates. By posing shape construction as a minimisation problem which follows the Gestalt laws, our desired shape inline image is non-intersecting, interpolates all points and minimizes a criterion related to these laws. The basis for our algorithm is an initial graph, an extension of the Euclidean minimum spanning tree but with no leaf nodes, called as the minimum boundary complex inline image. inline image and inline image can be expressed similarly by parametrizing a topological constraint. A close approximation of inline image, termed inline image can be computed fast using a greedy algorithm. inline image is then transformed into a closed interpolating boundary inline image in two steps to satisfy inline image’s topological and minimization requirements. Computing inline image exactly is an NP (Non-Polynomial)-hard problem, whereas inline image is computed in linearithmic time. We present many examples showing considerable improvement over previous techniques, especially for shapes with sharp corners. Source code is available online.

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