Optimal histogram matching by monotone gray level transformation

Abstract

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 k 1 linearly ordered objects of different sizes one by one into k 2 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 ( k 1 × k 2). If the monotone property is dropped, then the problem becomes NP -complete, even if it is restricted to k 2 = 2.