Optimal histogram matching by monotone gray level transformation

This paper investigates the problem of optimal histogram matching using monotone gray level transformation, which always assigns all picture points of a given gray level i to another gray level T(i) such that if i ≥ j, then T(i) ≥ T(j). The objective is to find a transformed digital picture of a given picture such that the sum of absolute errors between the gray level histogram of the transformed picture and that of a reference picture is minimized. This is equivalent to placing k1 linearly ordered objects of different sizes one by one into k2 linearly ordered boxes of assorted sizes, such that the accumulated error of space underpacked or overpacked in the boxes is minimized; the placement function is monotonic, which ensures a polynomial time solution to this problem. A tree search algorithm for optimal histogram matching is presented which has time complexity O(k1 × k2). If the monotone property is dropped, then the problem becomes NP-complete, even if it is restricted to k2 = 2.

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