Total generalized variation-based Retinex image decomposition

Abstract

Human visual system (HVS) can perceive color under varying illumination conditions, and Retinex theory is precisely aimed to simulate and explain how the HVS perceives reflectance regardless of different illumination conditions. In this paper, we introduce a reflectance and illumination decomposition model for the Retinex problem via total generalized variation regularization and $H^{1}$ decomposition. The total generalized variation regularization ameliorates the staircasing artifacts that appear in the reflectance component of existing total variation-based models and $H^{1}$ norm guarantees smoother illumination. We analyze the existence and uniqueness of the proposed model and employ an alternating minimization scheme based on split Bregman iteration. We present numerous numerical experiments on both grayscale and color images to make comparisons with several state-of-the-art methods and demonstrate that our method is comparable both quantitatively and qualitatively.

Appendix

In this appendix, we introduce detailed discretization of first-order and second-order differential operators using finite difference scheme.

The first-order forward and backward difference schemes are first given. Let $\Omega \rightarrow {\mathbb {R}}^{M\times N}$ denote the two-dimensional grayscale image space with size M and N. The coordinates x and y are oriented along columns and rows, respectively. So the first-order forward differences of u at point (i, j) along x and y directions are, respectively,

$$\begin{aligned} \partial _{x}^{+} u_{i,j}=\left\{ \begin{array}{ll} u_{i,j+1} - u_{i,j}&{} \mathrm{if} \ 1\le i \le M, 1 \le j < N,\\ u_{i,1} - u_{i,j}&{} \mathrm{if} \ 1 \le i \le M, j= N. \end{array}\right. \end{aligned}$$

$$\begin{aligned} \partial _{y}^{+}u_{i,j}=\left\{ \begin{array}{ll} u_{i+1,j} - u_{i,j}&{} \mathrm{if} \ 1\le i < M, 1 \le j \le N,\\ u_{1,j} - u_{i,j}&{} \mathrm{if} \ i = M, 1 \le j \le N. \end{array}\right. \end{aligned}$$

The first-order backward differences are, respectively,

$$\begin{aligned} \partial _{x}^{-} u_{i,j}=\left\{ \begin{array}{ll} u_{i,j} - u_{i,j-1}&{} \mathrm{if} \ 1\le i \le M, 1 < j \le N,\\ u_{i,j} - u_{i,N}&{} \mathrm{if} \ 1 \le i \le M, j= 1. \end{array}\right. \end{aligned}$$

$$\begin{aligned}\partial _{y}^{-} u_{i,j}=\left\{ \begin{array}{ll} u_{i,j} - u_{i-1,j}&{} \mathrm{if} \ 1< i \le M, 1 \le j \le N,\\ u_{i,j} - u_{M,j}&{} \mathrm{if} \ i = 1, 1\le j \le N. \end{array}\right. \end{aligned}$$

And the discrete second-order derivatives $\partial _{x}^{-}\partial _{x}^{+}u, \partial _{x}^{+}\partial _{x}^{-}u, \partial _{y}^{-}\partial _{y}^{+}u$ and $\partial _{y}^{+}\partial _{y}^{-}u$ at point (i, j) can be written by the corresponding compositions of the discrete first-order derivative, as follows

$$\begin{aligned} \begin{aligned} \partial _{x}^{+}\partial _{x}^{-} u_{i,j}&=\left\{ \begin{array}{ll} u_{i,N} - 2u_{i,j} + u_{i,j+1}&{} \mathrm{if} \ 1\le i\le M, j = 1,\\ u_{i,j-1} - 2u_{i,j} + u_{i,j+1}&{} \mathrm{if} \ 1\le i\le M, 1< j< N,\\ u_{i,j-1} - 2u_{i,j} + u_{i,1}&{} \mathrm{if} \ 1 \le i\le M, j = N. \end{array}\right. \\&=\partial _{x}^{-}\partial _{x}^{+} u_{i,j} \\ \partial _{y}^{+}\partial _{y}^{-} u_{i,j}&=\left\{ \begin{array}{ll} u_{M,j} - 2u_{i,j} + u_{i+1,j}&{} \mathrm{if} \ i = 1, 1\le j\le N,\\ u_{i-1,j} - 2u_{i,j} + u_{i+1,j}&{} \mathrm{if} \ 1< i< M, 1 \le j \le N,\\ u_{i-1,j} - 2u_{i,j} + u_{1,j}&{} \mathrm{if} \ i = M, 1 \le j\le N. \end{array}\right. \\&=\partial _{y}^{-}\partial _{y}^{+} u_{i,j} \end{aligned} \end{aligned}$$

Thus, the gradient, symmetrized derivative, divergence and Laplace can be discretized as follows, respectively:

$$\begin{aligned} \nabla u = \left( \partial _{x}^{+}u, \partial _{y}^{+}u \right) , \end{aligned}$$

(30)

$$\begin{aligned} {\mathcal {E}}({\mathbf {p}}) = \begin{pmatrix} \partial _{x}^{-} p_{1}&{}\frac{\partial _{y}^{-}p_{1} + \partial _{x}^{-}p_{2}}{2}\\ \frac{\partial _{y}^{-}p_{1} + \partial _{x}^{-}p_{2}}{2}&{} \partial _{y}^{-}p_{2}, \end{pmatrix} \end{aligned}$$

(31)

$$\begin{aligned} \mathrm {div}({\mathbf {p}}) = \partial _{x}^{-}p_{1} + \partial _{y}^{-}p_{2}, \end{aligned}$$

(32)

$$\begin{aligned} \Delta u_{i,j} = \mathrm {div}(\partial {u_{i,j}})=\partial _{x}^{-}\partial _{x}^{+}u_{i,j} + \partial _{y}^{-}\partial _{y}^{+}u_{i,j}. \end{aligned}$$

(33)

So the ${\mathcal {F}}(G_r)$ and ${\xi }_{r}$ in Eq. (22) can be written as

$$\begin{aligned} \begin{aligned} {\mathcal {F}}(G_r)&= \beta {\mathcal {F}}(i-l^{k}) - \theta _{1}{\mathcal {F}}\{\partial _{x}^{-}({w}_{1i,j}-b_{1i,j}) \\&\quad + \partial _{y}^{-}(w_{2i,j}-b_{2i,j})\}. \end{aligned} \end{aligned}$$

(34)

and

$$\begin{aligned} \begin{aligned} {\xi }_{r}&= \beta - \theta _{1}{\mathcal {F}}(\partial _{x}^{-}\partial _{x}^{+} + \partial _{y}^{-}\partial _{y}^{+}),\\&=\beta - 2\theta _{1}\left( \cos {\frac{2\pi n}{N} + \cos {\frac{2\pi m}{M}}}-2\right) . \end{aligned} \end{aligned}$$

(35)