A Variational Histogram Equalization Method for Image Contrast Enhancement

Contrast enhancement is one of the major tasks in image processing. The purpose of image contrast enhancement is to increase the visibility of images and to make them more suitable for other image processing applications and analysis. Histogram equalization is one of the well-known methods for enhancing the contrast of images due to its simplicity and comparatively better performance on different kinds of images. The idea is that a transformation function is applied to an input image such that the contrast of the transformed image is enhanced. Existing histogram equalization methods do not distinguish the image pixel values in different pixel locations, their performance may cause some unexpected distortions in practice. Several variational formulations of histogram equalization have been proposed and studied in the literature. In these formulations, either a mean brightness constraint is not considered or the differences among pixel transformations at the nearby pixel locations are not studied.

The main aim of this proposal is to develop a histogram equalization algorithm for image contrast enhancement. Our idea is to propose a variational approach containing an energy functional to determine a local transformation such that the histogram can be redistributed locally, and the brightness of the transformed image can be preserved. In order to minimize the differences among the local transformation at the nearby pixel locations, the spatial regularization of the transformation is also incorporated in the functional for the equalization process. In the variational problem, we consider both the H1-norm regularization and the total variation regularization of the transformation in the model. The existence and uniqueness of their minimizers of the two proposed models will be studied and analyzed. Moreover, the alternating direction method is developed to solve the two proposed models. Its convergence and performance will be analyzed and evaluated respectively by testing for a wide range of image types in different image processing applications.

Applications of variational methods to color images and high dynamic range images will be considered, studied and analyzed. We extend and develop our variational histogram equalization model (i) in the CIE LCH color space to enhance color image contrast, and remove unnatural appearances including color missing and visually disturbing artifacts; and (ii) in the generation of a high dynamic range image from a low dynamic range image through spatial regularization and range regularization.