A Multi-parameter Regularization Model for Deblurring Images Corrupted by Impulsive Noise

Abstract

This article considers image deblurring in the presence of impulse noise and proposes a new multi-parameter regularization model for image deblurring based on total variation (TV) and wavelet frame (WF), along with an efficient and effective solving algorithm for this restoring model. On the one hand, it is well known that the TV regularization-based Rudin–Osher–Fatemi model is very effective in preserving sharp edges and object boundaries which are generally the most important features to recover. On the other hand, WF-based approaches for image restoration have proven to be very successful in adaptively exploiting the regularity of natural images. By combining TV regularization and WF regularization, a novel multi-parameter regularization model is proposed for deblurring images in the presence of impulse noise. Numerically, the alternative direction method of multiplier (ADMM) with an adaptive scheme for choosing regularization parameters is provided and applied to this multi-parameter regularization model. Moreover, the convergence analysis of the ADMM is shown in the “Appendix.” Furthermore, numerical experiments involving images corrupted by various types of blurring kernels and different levels of noises indicate that the proposed model and algorithm outperform several state-of-the-art approaches in terms of the restoration quality, especially the ability to mitigate staircasing effects while preserving important features in images.

Appendix: Convergence Analysis

In this section, we will give the global convergence analysis of the proposed ADMM algorithm for IR, i.e., Algorithm 1, in Theorem 1.

Before we show Theorem 1, we first change the constrained problem (15) into another equivalent form by firstly bringing in several notations. From here, we let $X \triangleq {\mathbb {R}}^{M\times N}, Y \triangleq X \times X, Z \triangleq W(X) \subset X$, and $A \triangleq [\nabla , W, K]^{\mathrm {T}}$ denotes an operator that maps X into $Y \times Z \times X$, where $\nabla$, W, and K map X into Y, Z, and X, respectively. After introducing two convex functionals $F_(x)=0 : X \rightarrow {\mathbb {R}}$ and $G_(\zeta ) \triangleq G([v, z, T]^{\mathrm {T}}) = \left\| T \right\| _1 + \alpha \sum \left\| v \right\| _2+\beta \left\| z \right\| _1 : Y \times Z \times X \rightarrow {\mathbb {R}}$, we give the equivalent formulation of problem (15) as follows

$$\begin{aligned} \left\{ \begin{array}{ll} \min _{x,\zeta }~ F(x)+G(\zeta ) \\ \mathrm {s.t.} ~~~~~~Ax-\zeta =\wp \end{array}\right. \ \end{aligned}$$

(41)

where $\wp = [0, 0, y] \in Y \times Z \times X$. Then (17), the AL function of (15) turns into

$$\begin{aligned} {\mathcal L}_1(x, \zeta ; p) :=F(x)+G(\zeta )-\langle p, Ax-\zeta -\wp \rangle + (\lambda )/2\left\| Ax-\zeta -\wp \right\| _2^2 \end{aligned}$$

(42)

where $p = [p_1, p_2, p_3]^{\mathrm {T}} \in Y \times Z \times X$ and $\lambda = [\lambda _1, \lambda _2, \lambda _3]^{\mathrm {T}} \in {\mathbb {R}} \times {\mathbb {R}} \times {\mathbb {R}}$. Note that $\langle p, Ax-\zeta -\wp \rangle$ and $\frac{\lambda }{2}\left\| Ax-\zeta -\wp \right\| _2^2$ mean doing operations per-component, i.e., $\langle p_1, v-\nabla x \rangle + \langle p_2, z-Wx\rangle + \langle p_3, T-Kx+y \rangle$ and $\frac{\lambda _1}{2}\left\| v-\nabla x \right\| _2^2$ $+\frac{\lambda _2}{2}\left\| z-Wx \right\| _2^2 + \frac{\lambda _3}{2}\left\| T-Kx+y \right\| _2^2$ and so do the other operations.

Now, we give the convergence theorem of Algorithm 1.

Theorem 1

Under the following two assumptions, the sequence $\left\{ (x^k, \zeta ^k, p^k)\right\}$ generated by Algorithm 1 converges to a KKT point $(x^*, \zeta ^*, p^*)$ of (41).

Assumption 1

There exists a saddle point $\theta ^* \triangleq (x^*,\zeta ^*,p^*)$ to problem (41), namely $x^*$, $\zeta ^*$, and $p^*$ satisfy the KKT conditions

$$\begin{aligned} p^* \in \partial G(\zeta ^*),~A^{\mathrm {T}}p^* \in \partial F(x^*),~Ax^*-\zeta ^*=\wp . \end{aligned}$$

(43)

where F(x) and $G(\zeta )$ are both convex functionals, hence, there exist two scalars $\tau _F$ and $\tau _G$ satisfy the following two inequalities, respectively.

$$\begin{aligned} \langle t_1-t_2,x_1-x_2 \rangle \ge \tau _F \left\| x_1-x_2 \right\| _2^2, \forall ~ x_1,x_2,t_1\in \partial F(x_1),t_2\in \partial F(x_2) \end{aligned}$$

(44)

$$\begin{aligned} \langle s_1-s_2,\zeta _1-\zeta _2 \rangle \ge \tau _G \left\| \zeta _1-\zeta _2 \right\| _2^2, \forall ~ \zeta _1,\zeta _2,s_1\in \partial G(\zeta _1),s_2\in \partial G(\zeta _2). \end{aligned}$$

(45)

Assumption 2

$2-\xi >(1-\xi )^2.$

Before giving the proof of Theorem 1, for simplicity, we straighten the image $x \in {\mathbb {R}}^{M\times N}$ into a vector sill denoted as $x \in {\mathbb {R}}^{MN}$ by stacking the columns of x and so do the other variables such as $\zeta$ and p. Therefore, we have $\zeta ,~p \in {\mathbb {R}}^{3MN}$, A is an operator that maps ${\mathbb {R}}^{MN}$ into ${\mathbb {R}}^{3MN}$ and $A^{\mathrm {T}}A \in {\mathbb {R}}^{MN \times MN}$. Moreover, we let $\left\| x \right\| _\Xi ^2 \triangleq x^{\mathrm {T}}\Xi x$, where $\Xi \in {\mathbb {R}}^{MN \times MN}$. Furthermore, we introduce the following symbols

$$\begin{aligned}&{{\widehat{p}}} := p^k-\lambda (Ax^{k+1}-\zeta ^{k+1}-\wp ) \end{aligned}$$

(46)

$$\begin{aligned}&\theta ^*:= \left( \begin{array}{c} x^*\\ \zeta ^*\\ p^*\\ \end{array} \right) , \theta ^k:= \left( \begin{array}{c} x^k\\ \zeta ^k\\ p^k\\ \end{array} \right) , \widehat{\theta }:= \left( \begin{array}{c} x^{k+1}\\ \zeta ^{k+1}\\ \widehat{p}\\ \end{array} \right) ,~\forall ~ k=0,1,2,\ldots \end{aligned}$$

(47)

where $\theta ^*$ is a KKT point, $\theta ^k$ is the current point, $\widehat{\theta }$ is the next point as if $\xi =1$, and

$$\begin{aligned}&G_0:=\left( \begin{array}{ccc} {\mathbf {I}}_{N^2} &{} &{} \\ &{} {\mathbf {I}}_{3N^2} &{} \\ &{} &{} \xi {\mathbf {I}}_{3N^2} \\ \end{array}\right) , ~G_1:=\left( \begin{array}{ccc} \lambda A^{\mathrm {T}}A &{} &{} \\ &{} \mathbf {0}_{3N^2} &{} \\ &{} &{} \frac{1}{\lambda } {\mathbf {I}}_{3N^2} \\ \end{array}\right) \end{aligned}$$

(48)

$$\begin{aligned}&G:=G_0^{-1}G_1:=\left( \begin{array}{ccc} \lambda A^{\mathrm {T}}A &{} &{} \\ &{} \mathbf {0}_{3N^2} &{} \\ &{} &{} \frac{1}{\xi \lambda } {\mathbf {I}}_{3N^2} \\ \end{array}\right) \end{aligned}$$

(49)

where ${\mathbf {I}}$ and $\mathbf {0}$ refer to the identity matrix and the zero matrix with proper size, respectively. From now on, we also denoted $\lambda \triangleq \lambda _1=\lambda _2$ for simplicity.

From these definitions, it follows

$$\begin{aligned} \theta ^{k+1}=\theta ^k-G_0(\theta ^k-\widehat{\theta }) \end{aligned}$$

(50)

and $G \ge 0$; hence, G is a semi-norm.

Proof

We divide this process into the following three parts. $\square$

Step 1 To show the boundedness of the sequence $\left\{ \theta ^k\right\}$.

The optimality conditions of the subproblems of Algorithm 1 are

$$\begin{aligned}&-{{\widehat{p}}}+\lambda A(x^k-x^{k+1}) \in \partial G(\zeta ^{k+1}) \end{aligned}$$

(51)

$$\begin{aligned}&A^{\mathrm {T}}{\widehat{p}} \in \partial F(x^{k+1}). \end{aligned}$$

(52)

By the convexity of G and the first optimality condition in Theorem 1, and (51), it follows that

$$\begin{aligned} \langle \zeta ^{k+1}-\zeta ^*,-({\widehat{p}}-p^*-\lambda A(x^k-x^{k+1}))\rangle \ge \tau _G \left\| \zeta ^{k+1}-\zeta ^* \right\| _2^2. \end{aligned}$$

(53)

Similarly, by the convexity of g, the second optimality condition in Theorem 1, and (52), we have

$$\begin{aligned}&\langle x^{k+1}-x^*, A^{\mathrm {T}}({{\widehat{p}}}-p^*-\lambda A(x^k-x^{k+1})\rangle + \lambda A^{\mathrm {T}}A(x^k-x^{k+1}) \nonumber \\&\quad \ge \tau _F \left\| x^{k+1}-x^* \right\| _2. \end{aligned}$$

(54)

In addition, it follows the third optimality condition in Theorem 1 and (46) that

$$\begin{aligned} A(x^{k+1}-x^*)-(\zeta ^{k+1}-\zeta ^*)=\frac{1}{\lambda }(p^k-{\widehat{p}}). \end{aligned}$$

(55)

Then, adding (53) and (54) and using (55) give

$$\begin{aligned} \frac{1}{\lambda }< & {} p^k-{\widehat{p}},{\widehat{p}}-p^*-\lambda A(x^k-x^{k+1})>+<x^{k+1}-x^*,\lambda A^{\mathrm {T}} A(x^k-x^{k+1})> \nonumber \\\ge & {} ~ \tau _F \left\| x^{k+1}-x^* \right\| _2^2 +\tau _G \left\| \zeta ^{k+1}-\zeta ^* \right\| _2^2 \end{aligned}$$

(56)

which can be simplified as

$$\begin{aligned} (\widehat{\theta }-\theta ^*)^{\mathrm {T}}G_1(\theta ^k-\widehat{\theta })\ge & {} ~<A(x^k-x^{k+1}),p^k-{\widehat{p}}> \nonumber \\&+\,\tau _F \left\| x^{k+1}-x^* \right\| _2^2+~\tau _G \left\| \zeta ^{k+1}-\zeta ^* \right\| _2^2. \end{aligned}$$

(57)

By rearranging the terms, we have

$$\begin{aligned} (\theta ^k-\theta ^*)^{\mathrm {T}}G_1(\theta ^k-\widehat{\theta })\ge & {} ~ \left\| \theta ^k-\widehat{\theta } \right\| _{G_1}^2+<A(x^k-x^{k+1}),p^k-{\widehat{p}}> \nonumber \\&+~\tau _F \left\| x^{k+1}-x^* \right\| _2^2 +\tau _G \left\| \zeta ^{k+1}-\zeta ^* \right\| _2^2.~~~~ \end{aligned}$$

(58)

From the relationship (50), we get the following identity

$$\begin{aligned} \left\| \theta ^k-\theta ^* \right\| _G^2-\left\| \theta ^{k+1}-\theta ^* \right\| _G^2= 2(\theta ^k-\theta ^*)^{\mathrm {T}}G_1(\theta ^k-\widehat{\theta })-\left\| G_0(\theta ^k-\widehat{\theta }) \right\| _G^2.\nonumber \\ \end{aligned}$$

(59)

By (58), we have

$$\begin{aligned}&\left\| \theta ^k-\theta ^* \right\| _G^2-\left\| \theta ^{k+1}-\theta ^* \right\| _G^2\ge 2\left\| \theta ^k-\widehat{\theta } \right\| _{G_1}^2-\left\| \theta ^k-\widehat{\theta } \right\| _{G_1G_0}^2 \nonumber \\&\quad +~2<A(x^k-x^{k+1}),p^k-{\widehat{p}}>+2\tau _F \left\| x^{k+1}-x^* \right\| _2^2 + 2\tau _G \left\| \zeta ^{k+1}-\zeta ^* \right\| _2^2\nonumber \\ \end{aligned}$$

(60)

and thus it follows

$$\begin{aligned}&\left\| \theta ^k-\theta ^* \right\| _G^2-\left\| \theta ^{k+1}-\theta ^* \right\| _G^2\nonumber \\&\quad \ge \varphi (\theta ^k-\widehat{\theta })+~2\tau _F \left\| x^{k+1}-x^* \right\| _2^2+~2\tau _G \left\| \zeta ^{k+1}-\zeta ^* \right\| _2^2 \end{aligned}$$

(61)

where

$$\begin{aligned} \varphi (\theta ^k-\widehat{\theta })= & {} \varphi (x^k-x^{k+1},\zeta ^k-\zeta ^{k+1},p^k-{\widehat{p}}) \nonumber \\= & {} \left\| x^k-x^{k+1} \right\| _{\lambda A^{\mathrm {T}}A}^2+\frac{2-\xi }{\lambda } \left\| p^k-{\widehat{p}} \right\| _2^2+2(p^k-{\widehat{p}})^{\mathrm {T}}A(x^k-x^{k+1}).\nonumber \\ \end{aligned}$$

(62)

Then, $\left\| \theta ^k-\theta ^* \right\| _G^2$ is bounded, as a result of which we have the boundedness of ${(x^k,p^k)}$ and that of ${\zeta ^k}$ by (55).

Step 2 To show that for a certain $\rho >0$, $\left\| \theta ^k-\theta ^* \right\| _G^2+\frac{\lambda }{\rho }\left\| \varUpsilon ^k \right\| _2^2$ is monotonically nonincreasing, where $\varUpsilon ^k = Ax^k-\zeta ^k-\wp$ is the residual at iteration k, instead of which, we turn to proof that there exist $\varpi >0$ such that

$$\begin{aligned}&\left( \left\| \theta ^k-\theta ^* \right\| _G^2+\frac{\lambda }{\rho } \left\| \varUpsilon ^k \right\| _2^2\right) -\left( \left\| \theta ^{k+1}-\theta ^* \right\| _G^2+\frac{\lambda }{\rho } \left\| \varUpsilon ^{k+1} \right\| _2^2\right) \nonumber \\&\quad \ge \varpi \left\| \theta ^k-\theta ^{k+1} \right\| _G^2+2\tau _F \left\| x^k-x^{k+1} \right\| _2^2+2\tau _F \left\| x^{k+1}-x^* \right\| _2^2\nonumber \\&\qquad +\,2\tau _G \left\| \zeta ^{k+1}-\zeta ^* \right\| _2^2. \end{aligned}$$

(63)

We first derive a lower bound for the cross-term $(p^k-{\widehat{p}})^{\mathrm {T}}A(x^k-x^{k+1})$. Combining (52) with the definition of ${\widehat{p}}$, i.e., (46), we have

$$\begin{aligned} \left\{ \begin{array}{ll} A^{\mathrm {T}}[p^{k-1}-\lambda (Ax^k-\zeta ^k-\wp )] \in \partial F(x^k) \\ A^{\mathrm {T}}{\widehat{p}} \in \partial F(x^{k+1}).\end{array}\right. \ \end{aligned}$$

(64)

The difference of the two terms on the left in (64) is

$$\begin{aligned} A^{\mathrm {T}}[p^{k-1}-{\widehat{p}}-\lambda (Ax^k-\zeta ^k-\wp )]=A^{\mathrm {T}}(p^k-{\widehat{p}})-\lambda (1-\xi )A^{\mathrm {T}}(Ax^k-\zeta ^k-\wp ).\nonumber \\ \end{aligned}$$

(65)

By (64), (65), and (46), we get

$$\begin{aligned}&<A^{\mathrm {T}}(p^k-{\widehat{p}}) , x^k-x^{k+1}> \nonumber \\&\quad \ge \tau _F \left\| x^k-x^{k+1} \right\| _2^2+<(1-\xi )\lambda A^{\mathrm {T}}(Ax^k-\zeta ^k-\wp ),x^k-x^{k+1}> \end{aligned}$$

(66)

to which applying the Cauchy–Schwarz inequality gives

$$\begin{aligned}&(p^k-{\widehat{p}})^{\mathrm {T}}A\left( x^k-x^{k+1}\right) \ge <\sqrt{\lambda }\left( Ax^k-\zeta ^k-\wp \right) ,(1-\xi )\sqrt{\lambda }A\left( x^k-x^{k+1}\right)> \nonumber \\&\quad +\,\tau _F \left\| x^k-x^{k+1} \right\| _2^2 \nonumber \\&\ge -\frac{\lambda }{2\rho } \left\| Ax^k-\zeta ^k-\wp \right\| _2^2-\frac{(1-\xi )^2 \lambda \rho }{2} \left\| A\left( x^k-x^{k+1}\right) \right\| _2^2\nonumber \\&\quad +\,\tau _F \left\| x^k-x^{k+1} \right\| _2^2,~\forall ~\rho >0. \end{aligned}$$

(67)

Substituting (67) into (61) and using the definition of ${\widehat{p}}$, we have

$$\begin{aligned}&\left\| \theta ^k-\theta ^* \right\| _G^2+\frac{\lambda }{\rho } \left\| \varUpsilon ^k \right\| _2^2\nonumber \\&\quad \ge \left\| \theta ^{k+1}-\theta ^* \right\| _G^2+\frac{\lambda }{\rho } \left\| \varUpsilon ^{k+1} \right\| _2^2+\lambda \left( 2-\xi -\frac{1}{\rho }\right) \left\| \varUpsilon ^{k+1} \right\| _2^2 \nonumber \\&\qquad +~\lambda \left( 1-(1-\xi )^2\rho \right) \left\| A\left( x^k-x^{k+1}\right) \right\| _2^2+2\tau _F \left\| x^k-x^{k+1} \right\| _2^2 \nonumber \\&\qquad +~2\tau _F \left\| x^{k+1}-x^* \right\| _2^2+2\tau _G \left\| \zeta ^{k+1}-\zeta ^* \right\| _2^2.~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$

(68)

To prove such $\varpi >0$ exists for (63), we only need the existence of $\rho >0$ such that $2-\xi -\frac{1}{\rho }$ and $1-(1-\xi )^2\rho >0$, which holds if and only if $2-\xi >(1-\xi )^2$, i.e., $\xi \in (0,\frac{1+\sqrt{5}}{2})$.

Step 3 To show the global convergence of Algorithm 1.

Being bounded, $\left\{ \theta ^k\right\}$ has a converging subsequence $\left\{ \theta ^{k_j}\right\}$. Let $\bar{\theta }=\lim _{j\rightharpoondown \infty }\theta ^{k_j}$. Next we will show $\bar{\theta }$ is a KKT point. Let $\theta ^*$ denote an arbitrary KKT point.

As we know that $\left\| \theta ^k-\theta ^* \right\| _G^2+\frac{\lambda }{\rho }\left\| \varUpsilon ^k \right\| _2^2$ is monotonically nonincreasing and thus converging. Due to $\varpi >0$ , $\left\| \theta ^k-\theta ^{k+1} \right\| _G^2= \lambda \left\| A(x^k-x^{k+1}) \right\| _2^2+\frac{1}{\lambda \xi } \left\| p^k-p^{k+1} \right\| _2^2 \rightarrow 0$, so $x^k-x^{k+1}\rightarrow 0$ and $p^k-p^{k+1}\rightarrow 0$ and the following formula holds

$$\begin{aligned} \varUpsilon ^k= (Ax^k-\zeta ^k-\wp ) \rightarrow 0,~~ as~~k\rightarrow \infty . \end{aligned}$$

(69)

Consequently, $\left\| \theta ^k-\theta ^* \right\| _G^2$ also converges.

By passing limit on (69) over the subsequence, we have

$$\begin{aligned} A{\bar{x}}-{\bar{\zeta }}=\wp . \end{aligned}$$

(70)

Now on both sides of (51) and (52) taking limit over the subsequence, we obtain $-\bar{p} \in \partial G(\bar{\zeta })$, $A^{\mathrm {T}}\bar{p} \in \partial F(\bar{x})$. Then, together with (70), $\bar{\theta }$ is a KKT point of problem (41) and now we can let $\theta ^*=\bar{\theta }$. From $\left\{ \theta ^{k_j}\right\} \rightarrow \bar{\theta }$ in j and the convergence of $\parallel \theta ^k-\theta ^*\parallel _G^2$ it follows $\left\| \theta ^k-\theta ^* \right\| _G^2\rightarrow 0$ in k, which implies that $p^k\rightarrow p^*$, $x^k\rightarrow x^*$ and $\zeta ^k\rightarrow \zeta ^*$ following from (69) and (70).

Then, we have finished the proof of Theorem 1. $\square$