Exact histogram specification

TLDR

Exact histogram specification is ill‑posed for discrete images because cumulative distribution functions are not exactly invertible. The authors transform the problem into a K‑dimensional space and impose a strict pixel ordering to achieve invertible cumulative distribution functions. The new ordering improves upon the natural ordering, and experiments demonstrate its effectiveness for image enhancement, normalization, watermarking, and related tasks.

Abstract

While in the continuous case, statistical models of histogram equalization/specification would yield exact results, their discrete counterparts fail. This is due to the fact that the cumulative distribution functions one deals with are not exactly invertible. Otherwise stated, exact histogram specification for discrete images is an ill-posed problem. Invertible cumulative distribution functions are obtained by translating the problem in a K-dimensional space and further inducing a strict ordering among image pixels. The proposed ordering refines the natural one. Experimental results and statistical models of the induced ordering are presented and several applications are discussed: image enhancement, normalization, watermarking, etc.