A Fast Operator-splitting Method for Beltrami Color Image Denoising

Abstract

The Beltrami framework is a successful technique for color image denosing by regarding color images as manifolds embedded in a five dimensional spatial-chromatic space. It can ideally model the coupling between the color channels rather than treating them as if they were independent. However, the resulting model with high nonlinearity makes the related optimization problems difficult to solve numerically. In this paper, we propose an operator-splitting method for a variant of the Beltrami regularization model. From the optimality conditions associated with the minimization of the Beltrami regularized functional, we derive an initial value problem (gradient flow). We solve the gradient flow problem by an operator-splitting scheme involving three fractional steps. All three subproblem solutions can be obtained in closed form or computed by one-step Newton’s method. We demonstrate the efficiency and robustness of the proposed algorithm by conducting a series of experiments on real image denoising problems, where more than half of the computational time is saved compared to the existing augmented Lagrangian method (ALM) based algorithm for solving the Beltrami minimization model.

Appendix

Now we provide a brief introduction on the Lie and Marchuk-Yanenko schemes for the time-discretization of initial value problems. Consider the following steady-state problem

$$\begin{aligned} A(\varvec{X})=0, \end{aligned}$$

(40)

where the operator A maps the vector space $\varvec{V}$ into itself. The classical method to solve (40) is to formulated it as the following initial value problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial \varvec{X}}{\partial t}+A(\varvec{X})\ni 0~~\mathrm {on}~~(0,T)~(\mathrm {with}~0<T\le +\infty ) ,\\ \varvec{X}(0)=\varvec{X}_0. \end{array}\right. } \end{aligned}$$

(41)

Suppose that the problem (41) has steady-state solutions. These solutions are necessarily solutions of problem (40). We further assume the operator A has a nontrivial decomposition, namely

$$\begin{aligned} A=\sum _{j=1}^J A_j, \end{aligned}$$

(42)

where $J\ge 2$ and all $A_j, 0\le j\le J$, are individually simpler than A. Let $\tau >0$ be a time-discretization step, and denote $n\tau$ by $t^n$. Assuming that $\varvec{X}^n$ is the approximation of $\varvec{X}(t^n)$, the Lie scheme for solving (41) is given by (see Chapter 6 of [55] for its derivation)

$$\begin{aligned} \varvec{X}^0=\varvec{X}_0, \end{aligned}$$

(43)

then, for $n\ge 0$, $\varvec{X}^n\rightarrow \varvec{X}^{n+1}$ are updated as follows

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial \varvec{X}_j}{\partial t}+A_j(\varvec{X}_j)= 0~~\mathrm {on}~~(t^n,t^{n+1}),\\ \varvec{X}_{j}(t^n)=\varvec{X}^{n+(j-1)/J},~\varvec{X}^{n+j/J}=\varvec{X}_j(t^{n+1}), \end{array}\right. }\quad \text{ for }~~ j=1,\cdots ,J. \end{aligned}$$

(44)

The Lie scheme (43) and (44) is only semiconstructive since it requires to solve a series of initial value problems. Therefore, by discretizing the subproblems (44) using one step of backward Euler scheme, we obtain the following Marchuk-Yanenko scheme

$$\begin{aligned} \frac{\varvec{X}^{n+j/J}-\varvec{X}^{n+(j-1)/J}}{\tau }+A_j(\varvec{X}^{n+j/J})=0, \quad \text{ for } ~~ j=1,\cdots ,J. \end{aligned}$$

(45)

Due to asymptotic properties of the Lie and Marchuk-Yanenko schemes, the following phenomena generically holds [39]

• If converging for $j=1,\ldots ,J$, the sequences $(\varvec{X}^{n+j/J})_{n\ge 0}$ converge to different limits than the solution of (40) with the distances between them being ${\mathcal {O}}(\tau )$ at best.

• None of the above limits is a steady-state solution, but their distance to a steady-state solution converges to 0 as $\tau \rightarrow 0$ (if a steady-state solution does exist).

The above convergence theory has been proved in Chapter 6 of [55] supposing the space $\varvec{V}$ being finite dimensional and the operators $A_j$ being affine. Since the properties for $A_j$ do not hold in our Beltrami minimization model, the existing convergence theory cannot be applied to our algorithm, which need to be studied separately as our future works. Moreover, when the Lie or Marchuk-Yanenko scheme applies to multivalued operators such as the subdifferential of proper, lower semicontinuous, convex functionals, the first order accuracy is not guaranteed as well (our case in Sect. 3).