Exact Histogram Specification

Image enhancement is a fundamental but important topic in image processing field. The main aim of image enhancement is to improve visual effects such that the processed image is more applicable to specific use than the original one. Histogram correction technique is one of the commonly used methods of image enhancement. In usual, it has two classes of histogram correction technique, histogram equalization and specification. Histogram equalization is the earliest and the most commonly used technique in image processing. By means of histogram equalization yield more balanced and better contrasted images. Histogram specification is also a commonly used technique in image processing and it is used to obtain the image which has special histogram. Histogram specification popularizes histogram equalization.At first, this paper introduces the fundamental principles,properties and applications of histogram. In addition, the fundamental principles of histogram equalization and histogram specification. In the continuous case, statistical models of histogram equalization/ specification would yield exact results, their discrete counterparts fail. In the case of discrete, although it probably is obvious for histogram equalization by now, histogram specification is a trial-and-error process for the most part. In this paper, we introduce some commonly used methods of histogram specification. Although the introduced mapping laws can improve the image qualities in different aspects, the results are approximate. This is due to the fact that the number of pixels in an image is usually considerably larger than the number of graylevels. In the discrete case, the cumulative distribution functions are staircase function, hence they are not invertible except in the case when pixels take distinct values.For the above-mentioned topic, we introduce the exact histogram specification, a kind of arithmetic which based on ordering theory. Firstly, we present the principle of exact histogram specification , namely ordering relation defined on image pixel, which can induce almost strict ordering. Then, we discuss the method how to induce such a strict ordering. Speaking specifically, by using a vector operator, the problem is transferred from a scalar image to a K-dimensional space by associating a vector to each pixel by lexicographically ordering. Next, a strict ordering is induced among the image pixels and invertible cumulative distribution function is obtained. Furthermore, we provide the theoretical analysis of the ordering existence. Also, the experimental results,the statistical models of the induced ordering are presented. Once ordering is achieved, pixels are immediately separated into classes and assigned to the desired graylevel. The proposed strict ordering is consistent with the natural one and thus, the information content of images is generally preserved.Finally, several applications related to exact histogram specification are discussed: image enhancement, normalization, histogram specification inversion, watermarking, etc.