An image pixel based variational model for histogram equalization

Highlights

• We develop an image pixel based histogram equalization model for image contrast enhancement.

• A mean brightness term and a geometry constraint are incorporated in the proposed model.

• We propose to employ the alternating direction method of multipliers (ADMM) to solve the proposed model.

• The existence of the minimizer of the proposed model, and the convergence of the proposed algorithm are discussed.

Abstract

In this paper, we develop an image pixel based histogram equalization model for image contrast enhancement. The approach is to propose a variational model containing an energy functional to adjust the pixel values of an input image directly so that the resulting histogram can be redistributed to be uniform. This idea is different from existing histogram equalization algorithms where a histogram based on the input image is constructed, a mapping is determined to output a uniform histogram and then the pixel values of the input image are adjusted based on the mapping. In the variational model, a mean brightness term is incorporated to preserve the brightness of the input image, and a geometry constraint can also be added to keep the geometry structure of the input image. Theoretically, the existence of the minimizer of the proposed model, and the convergence of the proposed algorithm are given. Experimental results are reported to demonstrate that the performance of the proposed model are competitive with the other testing histogram equalization methods for several testing images.

Introduction

Contrast enhancement is to increase the visibility of images and to make them more pleasing or more appropriate for other image processing applications. The underlying principle in histogram equalization method is that the perceived distribution (histogram) of gray-levels in an image should be uniform. The most popular method for this purpose is to apply a transformation function to an input image such that the contrast of the transformed image is enhanced. It has been shown that this objective can be easily achieved by selecting the cumulative distribution function of the normalized histogram of the image gray-levels as the transformation function (GHE) [1].

However, GHE is not commonly used in consumer electronics such as TV because it does not preserve the brightness of the input image in the transformed image, and may cause undesirable artifacts. The brightness preserving bi-histogram equalization (BBHE) [2] has been proposed and developed to overcome this problem. BBHE first separates the histogram of an input image into two histograms based on the mean brightness of the input image: one is with the range from minimum gray level to the mean gray level and the other is with the range from the mean gray level to the maximum gray level. BBHE then performs equalization independently in these two histograms. It has been shown both mathematically and experimentally that BBHE is capable to preserve the brightness. BBHE is further extended to dualistic sub-image histogram equalization (DSIHE) [3] by maximizing the Shannon entropy of the transformed image. Minimum mean brightness error bi-histogram equalization (MMBEBHE) [4] is another extension of BBHE, which performs the separation by using the threshold level that yields the minimal difference of the mean brightness between the input image and the enhanced image. Another famous improvement of BBHE is recursive mean-separate histogram equalization (RMSHE) [5] which may be difficult to determine the optimal value of the recursive mean. Minimum within-class variance multi-histogram equalization (MWCVMHE) [6] automatically partitions the input histogram into multiple sub-histograms by minimizing within-class variance and then applies histogram equalization in each sub-histogram separately. MWCVMHE can also handle the brightness preservation problem, however, it may generate low-contrast output images.

Variational formulations of histogram equalization have been proposed and studied in the literature, see Jafar and Ying [7], Altas et al. [8], Wang and Ye [9], Sapiro and Caselles [10], Wang et al. [11], Arici et al. [12], and Hashemi et al. [13]. A popular variational histogram equalization model is constrained variational histogram equalization (CVHE) [7] which basically extends the variational definition of the GHE algorithm by adding a mean brightness constraint to formulate a functional optimization problem, the solution of which defines a new gray level transformation function for contrast enhancement. In Altas et al. [8], the minimization of the cumulative spacing between histogram bars in the least squares sense is considered. In Wang and Ye [9], brightness preserving histogram equalization (BPHEME) is formulated based on the entropy maximization. A Lyapounov functional for the histogram modification flow is proposed in Sapiro and Caselles [10], which is the sum of a global measure of contrast and the quadratic dispersion around the middle gray level. Convex optimization is used in Wang et al. [11] to transform the input image histogram into the flattest histogram subject to a mean brightness constraint (FHSABP). Histogram modification framework (HMF) [12] is treated as an optimization problem that minimizes a cost function [12], which can also handle contrast enhancement. To design a parameter free contrast enhancement algorithm based on HMF, a genetic algorithm (GA) is employed in Hashemi et al. [13], which finds a target histogram by maximizing a contrast measure based on edge information (CEBGA).

Contrast enhancement based on local information by using histogram equalization has been proposed in Wang and Ng [14], Kwok et al. [15], Liu et al. [16], Celik [17] recently. In Wang and Ng [14], the authors proposed 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 (LHE). In order to minimize the differences among the local transformation at the nearby pixel locations, the spatial regularization of the transformation is also incorporated into the functional for the equalization process. In Kwok et al. [15], histogram equalization is performed in several divided sectors of the input image, and the output image is derived by a weighted-sum aggregation on the basis of an intensity gradient measure. A non-overlapped sub-blocks and local histogram projection based contrast enhancement (NOSHP) is presented in Liu et al. [16] which executes histogram projection (HP) in numbers of non-overlapped sub-blocks. In Celik [17], a two-dimensional histogram equalization (2DHE) algorithm is developed by utilizing contextual information around each pixel to enhance the contrast of an input image. We remark that a mean brightness constraint is not considered in these local enhancement methods except that in Wang and Ng [14].

On the other hand, histogram transfer method has been studied and proposed for color matching of two or more images, see Papadakis et al. [18], Grundland and Dodgson [19], Pichon et al. [20], Delon [21], Pitié and Kokaram [22]. The problem is to find a mapping in the space of colors that transforms the source into an image with a color distribution as similar as possible to that of the target. In Papadakis et al. [18], the authors proposed a variational formulation for histogram transfer. They studied an energy functional composed by three terms: one tends to approach the cumulative histograms of the transformed images, the other two tend to maintain the colors and geometry of the original images. In Grundland and Dodgson [19], the matching of color histograms is proposed on the three channels independently. A color histogram equalization is introduced in Pichon et al. [20] by computing a transfer map using mesh deformations to fit the existing histogram to a uniform histogram. In Delon [21] and Pitié and Kokaram [22], the formulation of color histogram matching in terms of Monge–Kantorovich’s mass transportation theory has been considered.

In this paper, we develop an image pixel based histogram equalization model for image contrast enhancement. Our idea is different from the above histogram equalization algorithms like GHE, BBHE, MWCVMHE, CVHE, FHSABP, and LHE, which are based on intensity transfer by using a transformation function. Here we propose a variational model containing an energy functional to adjust the pixel values of an input image directly so that the resulting histogram can be redistributed to be uniform. Our target is an enhanced output image instead of a mapping computed by histogram equalization algorithms. In the proposed model, a mean brightness term is incorporated to preserve the brightness of the input image, and a geometry constraint can also be added to keep the geometry structure of the input image. Theoretically, the existence of the minimizer of the proposed model is given. Experimental results are reported to demonstrate that the performance of the proposed model are competitive with the other testing histogram equalization methods for several testing images.

The outline of this paper is organized as follows. In Section 2, we will describe the proposed model and present some theoretical results. In Section 3, We will present the algorithm to solve the proposed model numerically, and also give some convergence analysis about the proposed algorithm. In Section 4, we will show the numerical examples to illustrate the effectiveness of the proposed algorithm. The concluding remarks will be given in Section 5.