A cartoon-plus-texture image decomposition model for blind deconvolution

Abstract

In this paper, we study a blind deconvolution problem by using an image decomposition technique. Our idea is to make use of a cartoon-plus-texture image decomposition procedure into the deconvolution problem. Because cartoon and texture components can be represented differently in images, we can adapt suitable regularization methods to restore their components. In particular, the total variational regularization is used to describe the cartoon component, and Meyer’s G-norm is employed to model the texture component. In order to obtain the restored image automatically, we also use the generalized cross validation method efficiently and effectively to estimate their corresponding regularization parameters. Experimental results are reported to demonstrate that the visual quality of restored images by using the proposed method is very good, and is competitive with the other testing methods.

Appendix

In this section, we first show the detailed algorithm for minimizing the main problem (2.2). Noting that the algorithm for solving u-subproblem (3.5) has been given in Sect. 3.1, we then summarize the proposed algorithms for solving h-subproblem (3.4) and v-subproblem (3.6).

The problems in (6.1), (3.7), and (6.3) can be solved by using fast Fourier transforms, see Huang et al. (2008). The solutions of the problems in (6.2) and (3.8) are given by,

$$\begin{aligned} \tilde{h}^{i+1}_1 = {{\mathrm{P_{ROF}}}}(h^{i+1}, \beta ), \ \ \tilde{u}^{i+1}_1 = {{\mathrm{P_{ROF}}}}(u^{i+1}, \beta ), \end{aligned}$$

where $\mathrm{P}_{\mathrm{ROF}}(g, \alpha )$ represents the solution of the ROF model (Rudin et al. 1992) applied to g with a coefficient $\alpha$, and it can be solved by the projection algorithm (Chambolle 2004; Ng et al. 2007) or the Split Bregman algorithm (Goldstein and Osher 2009). The solution of the problem (6.4) is given as follows,

$$\begin{aligned} \tilde{v}^{i+1}_1 = v^{i+1}-\frac{1}{\beta }\mathrm{P}_{\mathrm{ROF}}\left( \beta v^{i+1}, \frac{1}{\beta \mu }\right) \!, \end{aligned}$$

(6.5)

Actually, the Euler–Lagrange equation of problem (6.4) is given by,

$$\begin{aligned} 0\in \partial J^{*}\left( \frac{\tilde{v}}{\mu }\right) -\beta (v^{i+1}-\tilde{v}) \Longleftrightarrow \beta (v^{i+1}-\tilde{v})\in \partial J^{*}\left( \frac{\tilde{v}}{\mu }\right) \Longleftrightarrow \frac{\tilde{v}}{\mu }\in \partial J(\beta (v^{i+1}-\tilde{v})), \end{aligned}$$

Let $w = \beta (v^{i+1}-\tilde{v})$, and we have,

$$\begin{aligned} \tilde{v} = v^{i+1}-\frac{w}{\beta }. \end{aligned}$$

(6.6)

Then we derive the following equation,

$$\begin{aligned} \frac{v^{i+1}}{\mu }-\frac{w}{\beta \mu }\in \partial J(w)\Longleftrightarrow 0\in \partial J(w)-\frac{1}{\beta \mu }(\beta v^{i+1}-w). \end{aligned}$$

Consequently, we have $w = \mathrm{P}_{\mathrm{ROF}}(\beta v^{i+1}, \frac{1}{\beta \mu })$, and combining this with (6.6), we derive the solution (6.5). Again noting that (6.1), (3.7), and (6.3) are linear least squares problems with Tikhonov regularization, and therefore the associated regularization parameters $\frac{\lambda _{1}\beta }{2}$, $\frac{\lambda _{2}\beta }{2}$, and $\frac{\lambda _{3}\beta }{2}$ can be estimated by using GCV technique.