A Wavelet Perspective on Variational Perceptually-Inspired Color Enhancement

Abstract

The issue of perceptually-inspired correction of color and contrast in digital images has been recently analyzed with the help of variational principles. These techniques allowed building a general framework in which the action of many already existing algorithms can be more easily understood and compared in terms of intensification of local contrast and control of dispersion around the average intensity value. In this paper we analyze this issue from the dual perspective of wavelet theory, showing that it is possible to build energy functionals of wavelet coefficients that lead to a multilevel perceptually-inspired color correction. By computing the Euler–Lagrange equations associated to the wavelet-based functionals we were able to find an analytical formula for the modification of wavelet detail coefficients that overcomes the problem of an ad-hoc selection based on empirical considerations. Besides these theoretical results, the wavelet perspective provides the computational advantage of generating much faster algorithms in comparison with the spatial variational framework.

Appendix: Proof of proposition 1

Let us begin by proving the existence of a minimum of $\mathcal{E}_{w_j,\varphi ,\{a_{j,k}\},d^0_{j,k}}$. Let $\{d_{j,k}^n\}$ a minimizing sequence of detail coefficients. Since $\varphi \ge 0$, the sequence $d^0_{j,k} \log \frac{d^0_{j,k}}{d_{j,k}^n} - \left( d^0_{j,k} - d_{j,k}\right) \ge 0$ is bounded. Thus, modulo a subsequence, we may assume that $\{d_{j,k}^n\}$ converges to, say, $\{d_{j,k}\}$. Observe that since $\varphi (r)\rightarrow \infty$ as $r\rightarrow \infty$, the terms of the sequence $\{d_{j,k}^n\}$ are bounded away from $0$. Using this and the continuity of $\varphi$, we have that $\{d_{j,k}\}$ is a minimum of the energy.

Let us now pass to the computation of the Euler–Lagrange equations. We have to compute the first variation of the energy $\mathcal{E}_{w_j,\varphi ,\{a_{j,k}\},d^0_{j,k}} = \mathcal{C}_{w_j,\varphi ,\{a_{j,k}\}} + \mathcal{D}_{d^0_{j,k}}$ with respect to each coefficient $d_{j,k}$. The computation of the first variation of $\mathcal{D}_{d^0_{j,k}}$ with respect to $\{d_{j,k}\}$ is straightforward and gives:

$$\begin{aligned} \frac{\partial \mathcal{D}_{d^0_{j,k}}}{\partial \{d_{j,k}\}} = 1 - \frac{d^0_{j,k}}{d_{j,k}}. \end{aligned}$$

(13)

Let us consider now $\mathcal{C}_{w_j,\varphi ,\{a_{j,k}\}}$: if we denote with $(\overline{j},\overline{k})$ an auxiliary couple of indices, then we have

$$\begin{aligned} \frac{\partial \mathcal{C}_{w_j,\varphi ,\{a_{j,k}\}}}{\partial \{d_{\overline{j}, \overline{k}}\}}&= \sum _{k\in \varOmega } w_j \, \varphi '\left( \frac{a_{j,k}}{d_{j,k}}\right) \frac{-a_{j,k}\delta _{j-\overline{j},k-\overline{k}}}{(d_{j,k})^2}\nonumber \\&= - w_{\overline{j}} \varphi '\left( \frac{a_{\overline{j},\overline{k}}}{d_{\overline{j},\overline{k}}}\right) \frac{a_{\overline{j},\overline{k}}}{(d_{\overline{j},\overline{k}})^2}, \end{aligned}$$

(14)

where $\varphi ' \equiv \dfrac{d\varphi }{d\left( \frac{a_{\overline{j},\overline{k}}}{d_{\overline{j},\overline{k}}}\right) }$ and the Kronecker symbol

$$\begin{aligned} \delta _{j-\overline{j},k-\overline{k}} = \left\{ \begin{array}{ll} 1 &{} \text {if } j=\overline{j} \text { and } k=\overline{k};\\ 0 &{} \text {otherwise}, \end{array}\right. \end{aligned}$$

appears because of the derivative $\frac{\partial d_{j,k}}{\partial d_{\overline{j}, \overline{k}}}$. Adding (13) and (14), this last expressed in terms the original couple of indices $(j,k)$, and setting the expression to 0 we get

$$\begin{aligned} 1 - \frac{d^0_{j,k}}{d_{j,k}} - w_j \varphi '\left( \frac{a_{j,k}}{d_{j,k}}\right) \frac{a_{j,k}}{(d_{j,k})^2} = 0. \end{aligned}$$

(15)

Multiplying by $d_{j,k}$ and simplifying the algebraic expression, we arrive to the result stated in Proposition 1 for the generic function $\varphi$. When $\varphi \equiv \text {id}$ we obtain (12).