Digital geometric invariance and shape representation

In this paper, we present conditions which guarantee that a realistic digitization process preserves the qualitative differential geometry of the object boundary, such as convexity and inflection. This is possible since very few digital boundary patterns are shown to be realizable, and each such digital pattern has a well-defined geometric interpretation with respect to tangent direction. Using the set of realizable boundary patterns, we can recover geometric properties of the digitized object boundary, such as convexity and inflection. In addition, since all the realizable patterns are known, any other pattern can be labeled as either noise or a boundary discontinuity. Since each of these patterns has a well-defined tangent span, an adjacency graph can be generated from the patterns and this graph can be used to recursively generate the set of all possible digital boundary curves. The digitization process used in this paper is equivalent to setting a very low threshold value on the sensor output.