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On Applying the Farey Sequence for Shape Representation in Z2
Abstract
Describing the shape of an object is a well-studied, yet ever-engrossing problem, because an appropriate description can improve the efficiency of a shape matching algorithm, thereby enriching subsequent applications. The authors propose a novel boundary-based shape description using the Farey sequence to capture an object shape represented as a sequence of discrete straight line segments. The straight edges are extracted directly from a gray-scale image without resorting to any edge map detection, and without using any thinning procedure. Then we merge the straight pieces, which are almost collinear but usually small in length, by employing the novel idea of an Augmented Farey Table (AFT). An AFT is a preprocessed data structure that provides us the Farey indices based on which the amount of linearity of three consecutive vertices of a polygon in the digital plane, is decided. Using the final straight pieces after AFT-based merging, the authors build a shape description using the Farey indices of the merged/larger pieces. In particular, the method would be computationally attractive for polygonal approximation and shape description of a large database of gray-scale images. Experimental results demonstrate its usefulness, efficiency, and elegance.
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