Variational Contrast Enhancement of Gray-Scale and RGB Images

Abstract

The aim of this paper is twofold. First, we propose a new method for enhancing the contrast of gray-value images. We use the difference of the average local contrast measures between the original and the enhanced images within a variational framework. This enables the user to intuitively control the contrast level and the scale of the enhanced details. Moreover, our model avoids large modifications of the original image histogram. Thereby it preserves the global illumination of the scene and it can cope with large areas having similar gray values. The minimizer of the proposed functional is computed by a gradient descent algorithm in connection with a polynomial approximation of the average local contrast measure. The polynomial approximation is computed via Bernstein polynomials. In the second part, the approach is extended to a variational enhancement method for color images. The model approximately preserves the hue of the original image and additionally includes a total variation term to correct the possible noise. The method requires no post-  or preprocessing. The minimization problem is solved with a hybrid primal–dual algorithm. Experiments demonstrate the efficiency and the flexibility of the proposed approaches in comparison with state-of-the-art methods.

An Efficient Computation by Polynomial Approximation

First, we show how the sign function can be approximated by Bernstein polynomials. The Bernstein polynomial on [0, 1] of degree n is defined by

$$\begin{aligned} B^n_k(t) :={\left( {\begin{array}{c}n\\ k\end{array}}\right) } t^k (1-t)^{n-k}. \end{aligned}$$

By the Weierstraß’ theorem, any continuous function $f:[0,1] \rightarrow {\mathbb {R}}$ can be approximated by

$$\begin{aligned} b_n(f,t)=\displaystyle \sum _{k=0}^n f\left( \dfrac{k}{n}\right) B^n_k(t) \end{aligned}$$

(35)

and $\Vert f - b_n(f,\cdot )\Vert _\infty$ goes to zero as $n \rightarrow \infty$. We approximate the jump function $g:[0,1]\rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} g(t) := \left\{ \begin{array}{ll} 0&{}\mathrm{if} \; t \in [0,\tfrac{1}{2}),\\ \tfrac{1}{2} &{}\mathrm{if} \; t = \tfrac{1}{2},\\ 1&{}\mathrm{if} \; t \in (\tfrac{1}{2},1], \end{array} \right. \end{aligned}$$

and use $P(t) := 2b_n\left( g,\dfrac{t+1}{2} \right) -1$ as an approximation of the sign function. Next, we write $b_n\left( g,\dfrac{t+1}{2} \right)$ as the sum of monomials

$$\begin{aligned} \left( \dfrac{1}{2} \right) ^n\sum _{k=0}^n {\left( {\begin{array}{c}n\\ k\end{array}}\right) } g \left( \dfrac{k}{n} \right) \sum _{i=0}^k \sum _{j=0}^{n-k} {\left( {\begin{array}{c}k\\ i\end{array}}\right) }{\left( {\begin{array}{c}n-k \\ j\end{array}}\right) } (-1)^j t^{i+j}. \end{aligned}$$

(36)

Having a polynomial representation $P(t) = \sum _{m=0}^n c_m t^m$ available, we use [8] to efficiently compute $\nabla \tilde{C}(u)(x)$ if $w(x,y) = G(x-y)$:

$$\begin{aligned} \nabla \tilde{C}(u)(x)&=\sum _{y \in \Lambda } w(x,y)\sum _{m=0}^n c_{m}(u(x)-u(y))^m \\&=\sum _{y \in \Lambda }w(x,y)\sum _{m=0}^n c_{m} (-1)^{k} \sum _{k=0}^{m}\genfrac(){0.0pt}1{m}{k}u(y)^k u(x)^{m-k}\\&=\sum _{k=0}^n a_{k}(x) \sum _{y\in \Lambda } w(x,y)u(y)^k\\&=\sum _{k=0}^n a_{k}(x)(G * u^k)(x) \end{aligned}$$

with $a_{k}(x)$ a polynomial depending on u(x). The convolution can be computed in a fast way using the fast Fourier transform. Note that in [8], the sign function is not directly approximated, but a continuous approximation of it, namely a slope function. Even for this function, the Chebyshev approximation will not stay within $[-1,1]$.