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The distance transform algorithm is popular in computer vision and machinelearning domains. It is used to minimize quadratic functions over a grid ofpoints. Felzenszwalb and Huttenlocher (2004) describe an O(N) algorithm forcomputing the minimum distance transform for quadratic functions. Theiralgorithm works by computing the lower envelope of a set of parabolas definedon the domain of the function. In this work, we describe an average time O(N)algorithm for maximizing this function by computing the upper envelope of a setof parabolas. We study the duality of the minimum and maximum distancetransforms, give a correctness proof of the algorithm and its runtime, anddiscuss potential applications.

Proof of Correctness and Time Complexity Analysis of a Maximum Distance Transform Algorithm