Investigating the Influence of Box-Constraints on the Solution of a Total Variation Model via an Efficient Primal-Dual Method

Abstract

In this paper, we investigate the usefulness of adding a box-constraint to the minimization of functionals consisting of a data-fidelity term and a total variation regularization term. In particular, we show that in certain applications an additional box-constraint does not effect the solution at all, i.e., the solution is the same whether a box-constraint is used or not. On the contrary, i.e., for applications where a box-constraint may have influence on the solution, we investigate how much it effects the quality of the restoration, especially when the regularization parameter, which weights the importance of the data term and the regularizer, is chosen suitable. In particular, for such applications, we consider the case of a squared $L^2$ data-fidelity term. For computing a minimizer of the respective box-constrained optimization problems a primal-dual semi-smooth Newton method is presented, which guarantees superlinear convergence.

1. Introduction

An observed image g, which contains additive Gaussian noise with zero mean and standard deviation $\sigma$, may be modeled as

$$g=K\widehat{u}+n$$

where $\widehat{u}$ is the original image, K is a linear bounded operator and n represents the noise. With the aim of preserving edges in images in [1] total variation regularization in image restoration was proposed. Based on this approach and assuming that $g\in L^2\left(\mathsf{\Omega }\right)$ and $K\in \mathcal{L}\left(L^2\left(\mathsf{\Omega }\right)\right)$, a good approximation of $\widehat{u}$ is usually obtained by solving

$$\underset{u\in BV\left(\mathsf{\Omega }\right)}{min}{\int }_{\mathsf{\Omega }}{\left|Du\right|\mathrm{such}\mathrm{that}(\mathrm{s}.\mathrm{t}.)\parallel Ku-g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2\le {\sigma }^2\left|\mathsf{\Omega }\right|$$

(1)

where $\mathsf{\Omega }\subset {\mathbb{R}}^2$ is a simply connected domain with Lipschitz boundary and $\left|\mathsf{\Omega }\right|$ its volume. Here ${\int }_{\mathsf{\Omega }}\left|Du\right|$ denotes the total variation of u in $\mathsf{\Omega }$ and $BV\left(\mathsf{\Omega }\right)$ is the space of functions with bounded variation, i.e., $u\in BV\left(\mathsf{\Omega }\right)$ if and only if $u\in L^1\left(\mathsf{\Omega }\right)$ and ${\int }_{\mathsf{\Omega }}\left|Du\right|<\infty$; see [2,3] for more details. We recall, that $BV\left(\mathsf{\Omega }\right)\subset L^2\left(\mathsf{\Omega }\right)$, if $\mathsf{\Omega }\subset {\mathbb{R}}^2$.

Instead of considering (1), we may solve the penalized minimization problem

$$\underset{u\in BV\left(\mathsf{\Omega }\right)}{min}{\parallel Ku-g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\alpha {\int }_{\mathsf{\Omega }}\left|Du\right|$$

(2)

for a given constant $\alpha >0$, which we refer to the $L^2$-TV model. In particular, there exists a constant $\alpha \ge 0$ such that the constrained problem (1) is equivalent to the penalized problem (2), if $g\in K\left(BV\right(\mathsf{\Omega }\left)\right)$ and K does not annihilate constant functions [4]. Moreover, under the latter condition also the existence of a minimizer of problem (1) and (2) is guaranteed [4]. There exist many algorithms that solve problem (1) or problem (2), see for example [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23] and references therein.

If in problem (2) instead of the quadratic $L^2$-norm an $L^1$-norm is used, we refer to it as the $L^1$-TV model. The quadratic $L^2$-norm is usually used when Gaussian noise is contained in the image, while the $L^1$-norm is more suitable for impulse noise [24,25,26].

If we additionally know (or assume) that the original image lies in the dynamic range $[c_{min},c_{max}]$, i.e., $c_{min}\le u\left(x\right)\le c_{max}$ for almost every (a.e.) $x\in \mathsf{\Omega }$, we incorporate this information into our problems (1) and (2) leading to

$$\underset{u\in BV\left(\mathsf{\Omega }\right)}{min}{\int }_{\mathsf{\Omega }}{\left|Du\right|\mathrm{s}.\mathrm{t}.\parallel Ku-g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2\le {\sigma }^2\left|\mathsf{\Omega }\right|\mathrm{and}u\in C$$

(3)

and

$$\underset{u\in BV\left(\mathsf{\Omega }\right)\cap C}{min}{\parallel Ku-g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\alpha {\int }_{\mathsf{\Omega }}\left|Du\right|,$$

(4)

respectively, where $C:=\{u\in L^2\left(\mathsf{\Omega }\right):c_{min}\le u\left(x\right)\le c_{max}\mathrm{for}\mathrm{a}.\mathrm{e}.x\in \mathsf{\Omega }\}$. In order to guarantee the existence of a minimizer of problems (3) and (4) we assume in the sequel that K does not annihilate constant functions. By noting that the characteristic function ${\chi }_C$ is lower semicontinuous this follows by the same arguments as in [4]. If additionally $g\in K\left(BV\right(\mathsf{\Omega })\cap C)$, then by [4] (Prop. 2.1) it follows that there exists a constant $\alpha \ge 0$ such that problem (3) is equivalent to problem (4).

For image restoration box-constraints have been considered for example in [5,27,28,29]. In [29] a functional consisting of an $L^2$-data term and a Tikhonov-like regularization term (i.e., $L^2$-norm of some derivative of u) in connection with box-constrained is presented together with a Newton-like numerical scheme. For box-constrained total variation minimization in [5] a fast gradient-based algorithm, called monoton fast iterative shrinkage/thresholding algorithm (MFISTA), is proposed and a rate of convergence is proven. Based on the alternating direction method of multipliers (ADMM) [30] in [27] a solver for the box-constrained $L^2$-TV and $L^1$-TV model is derived and shown to be faster than MFISTA. In [28] a primal-dual algorithm for the box-constrained $L^1$-TV model and for box-constrained non-local total variation is presented. In order to achieve a constrained solution, which is positive and bounded from above by some intensity value, in [31] an exponential type transform is applied to the $L^2$-TV model. Recently, in [32] a box-constraint is also incorporated in a total variation model with a combined $L^1$-$L^2$ data fidelity, proposed in [33], for removing simultaneously Gaussian and impulse noise in images.

Setting the upper bound in the set C to infinity and the lower bound to 0, i.e., $c_{min}=0$ and $c_{max}=+\infty$, leads to a non-negativity constraint. Total variation minimization with a non-negativity constraint is a well-known technique to improve the quality of reconstructions in image processing; see for example [34,35] and references therein.

In this paper we are concerned with the problems (3) and (4) when the lower bound $c_{min}$ and the upper bound $c_{max}$ in C are finite. However, the analysis and the presented algorithms are easily adjustable to the situation when one of the bounds is set to $-\infty$ or $+\infty$ respectively. Note, that a solution of problem (1) and problem (2) is in general not an element of the set C. However, since g is an observation containing Gaussian noise with zero mean, a minimizer of problem (2) lies indeed in C, if $\alpha$ in problem (2) is sufficiently large and the original image $\widehat{u}\in C$. This observation however rises the question whether an optimal parameter $\alpha$ would lead to a minimizer that lies in C. If this would be the case then incorporating the box-constraint into the minimization problem does not gain any improvement of the solution. In particular, there are situations in which a box-constraint is not effecting the solution at all (see Section 3 below). Additionally, we expect that the box-constrained problems are more difficult to handle and numerically more costly to solve than problem (1) and problem (2).

In order to answer the above raised question, we numerically compute optimal values of $\alpha$ for the box-constrained total variation and the non-box-constrained total variation and compare the resulting reconstructions with respect to quality measures. By optimal values we mean here parameters $\alpha$ such that the solutions of problem (1) and problem (2) or problem (3) and problem (4) coincide. Note, that there exists several methods for computing the regularization parameter; see for example [36] for an overview of parameter selection algorithms for image restoration. Here we use the pAPS-algorithm proposed in [36] to compute reasonable $\alpha$ in problem (2) and problem (4). For minimizing problem (4) we derive a semi-smooth Newton method, which should serve us as a good method for quickly computing rather exact solutions. Second order methods have been already proposed and used in image reconstruction; see [21,36,37,38,39]. However, to the best of our knowledge till now semi-smooth Newton methods have not been presented for box-constrained total variation minimization. In this setting, differently to the before mentioned approaches, the box-constraint adds some additional difficulties in deriving the dual problems, which have to be calculated to obtain the desired method; see Section 4 for more details. The superlinear convergence of our method is guaranteed by the theory of semi-smooth Newton methods; see for example [21]. Note, that our approach differs significantly from the Newton-like scheme presented in [29], where a smooth objective functional with a box-constraint is considered. This allows in [29] to derive a Newton method without dualization. Here, our Newton method is based on dualization and may be viewed as a primal-dual (Newton) approach.

We remark, that a scalar regularization parameter might not be the best choice for every image restoration problem, since images usually consist of large uniform areas and parts with fine details, see for example [36,38]. It has been demonstrated, for example in [36,38,40,41] and references therein, that with the help of spatially varying regularization parameters one might be able to restore images visually better than with scalar parameters. In this vein we also consider

$$\underset{u\in BV\left(\mathsf{\Omega }\right)}{min}{\parallel Ku-g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+{\int }_{\mathsf{\Omega }}\alpha \left|Du\right|$$

(5)

and

$$\underset{u\in BV\left(\mathsf{\Omega }\right)\cap C}{min}{\parallel Ku-g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+{\int }_{\mathsf{\Omega }}\alpha \left|Du\right|,$$

(6)

where $\alpha :\mathsf{\Omega }\to {\mathbb{R}}^{+}$ is a bounded continuous function [42]. We adapt our semi-smooth Newton method to approximately solve these two optimization problems and utilize the pLATV-algorithm of [36] to compute a locally varying $\alpha$.

Our numerical results show, see Section 6, that in a lot of applications the quality of the restoration is more a question of how to choose the regularization parameter then including a box-constraint. However, the solutions obtained by solving the box-constrained versions (3), (4) and (6) are improving the restorations slightly, but not drastically. Nevertheless, we also report on a medical applications where a non-negativity constraint significantly improves the restoration.

We realize that if the noise-level of the corrupted image is unknown, then we may use the information of the image intensity range (if known) to calculate a suitable parameter for problem (2). Note, that in this situation the optimization problems (1) and (3) cannot be considered since $\sigma$ is not at hand. We present a method which automatically computes the regularization parameter $\alpha$ in problem (2) provided the information that the original image $\widehat{u}\in [c_{min},c_{max}]$.

Hence the contribution of the paper is three-sided: (i) We present a semi-smooth Newton method for the box-constrained total variation minimization problems (3) and (6). (ii) We investigate the influence of the box-constraint on the solution of the total variation minimization models with respect to the regularization parameter. (iii) In case the noise-level is not at hand, we propose a new automatic regularization parameter selection algorithm based on the box-constraint information.

The outline of the rest of the paper is organized as follows: In Section 2 we recall useful definitions and the Fenchel-duality theorem which will be used later. Section 3 is devoted to the analysis of the box-constrained total variation minimization. In particular, we state that in certain cases adding a box-constraint to the considered problem does not change the solution at all. The semi-smooth Newton method for the box-constrained $L^2$-TV model (4) and its multiscale version (6) is derived in Section 4 and its numerical implementation is presented in Section 5. Numerical experiments investigating the usefulness of a box-constraint are shown in Section 6. In Section 7 we propose an automatic parameter selection algorithm by using the box-constraint. Finally, in Section 8 conclusions are drawn.

2. Basic Terminology

Let X be a Banach space. Its topological dual is denoted by $X^{*}$ and $\langle ·,·\rangle$ describes the bilinear canonical pairing over $X\times X^{*}$. A convex functional $J:X\to \overline{\mathbb{R}}$ is called proper, if $\{v\in X:J(v)\ne +\infty \}\ne \varnothing$ and $J\left(v\right)>-\infty$ for all $v\in X$. A functional $J:X\to \overline{\mathbb{R}}$ is called lower semicontinuous, if for every weakly convergent sequence $v^{\left(n\right)}\rightharpoonup \widehat{v}$ we have

$$\underset{v^{\left(n\right)}\rightharpoonup \widehat{v}}{lim\; inf}J\left(v\left(n\right)\right)\ge J\left(\widehat{v}\right).$$

For a convex functional $J:X\to \overline{\mathbb{R}}$ we define the subdifferential of J at $v\in X$ as the set valued function

$$\partial J\left(v\right):=\left\{\begin{array}{cc}\varnothing \hfill & \mathrm{if}J\left(v\right)=\infty ,\hfill \\ \{v^{*}\in X^{*}:\langle v^{*},u-v\rangle +J\left(v\right)\le J\left(u\right)\forall u\in X\}\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

It is clear from this definition, that $0\in \partial J\left(v\right)$ if and only if v is a minimizer of J.

The conjugate function (or Legendre transform) of a convex function $J:X\to \overline{\mathbb{R}}$ is defined as $J^{*}:X^{*}\to \overline{\mathbb{R}}$ with

$$J^{*}\left(v^{*}\right)=\underset{v\in X}{sup}\{\langle v,v^{*}\rangle -J\left(v\right)\}.$$

From this definition we see that $J^{*}$ is the pointwise supremum of continuous affine functions and thus, according to [43] (Proposition 3.1, p. 14), convex, lower semicontinuous, and proper.

For an arbitrary set S we denote by ${\chi }_S$ its characteristic function defined by

$${\chi }_S\left(u\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}u\in S,\hfill \\ \infty \hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

We recall the Fenchel duality theorem; see, e.g., [43] for details.

Theorem 1 (Fenchel duality theorem).

Let X and Y be two Banach spaces with topological duals $X^{*}$ and $Y^{*}$, respectively, and $\Lambda :X\to Y$ a bounded linear operator with adjoint ${\Lambda }^{*}\in \mathcal{L}(Y^{*},X^{*})$. Further let $\mathcal{F}:X\to \mathbb{R}\cup \{\infty \}$, $\mathcal{G}:Y\to \mathbb{R}\cup \{\infty \}$ be convex, lower semicontinuous, and proper functionals. Assume there exists $u_0\in X$ such that $\mathcal{F}\left(u_0\right)<\infty$, $\mathcal{G}(\Lambda u_0)<\infty$ and $\mathcal{G}$ is continuous at $\Lambda u_0$. Then we have

$$\underset{u\in X}{inf}\mathcal{F}\left(u\right)+\mathcal{G}(\Lambda u)=\underset{p\in Y^{*}}{sup}-{\mathcal{F}}^{*}\left({\Lambda }^{*}p\right)-{\mathcal{G}}^{*}(-p)$$

(7)

and the problem on the right hand side of (7) admits a solution $\overline{p}$. Moreover, $\overline{u}$ and $\overline{p}$ are solutions of the two optimization problems in (7), respectively, if and only if

$$\begin{array}{c}\hfill {\Lambda }^{*}\overline{p}\in \partial \mathcal{F}\left(\overline{u}\right),\\ \hfill -\overline{p}\in \partial \mathcal{G}(\Lambda \overline{u}).\end{array}$$

3. Limitation of Box-Constrained Total Variation Minimization

In this section we investigate the difference between the box-constrained problem (4) and the non-box-constrained problem (1). For the case when the operator K is the identity I, which is the relevant case in image denoising, we have the following obvious result:

Proposition 1.

Let $K=I$ and $g\in C$, then the minimizer $u^{*}\in BV\left(\mathsf{\Omega }\right)$ of problem (2) lies also in the dynamic range $[c_{min},c_{max}]$, i.e., $u^{*}\in BV\left(\mathsf{\Omega }\right)\cap C$.

Proof of Proposition 1.

Assume $u^{*}\in BV\left(\mathsf{\Omega }\right)\backslash C$ is a minimizer of problem (2). Define a function $\tilde{u}$ such that

$$\tilde{u}\left(x\right):=\left\{\begin{array}{cc}{u}^{*}\left(x\right)\hfill & \mathrm{if}{c}_{min}\le u^{*}\left(x\right)\le c_{max},\hfill \\ c_{max}\hfill & \mathrm{if}{u}^{*}\left(x\right)>c_{max},\hfill \\ c_{min}\hfill & \mathrm{if}{u}^{*}\left(x\right)<c_{min},\hfill \end{array}\right.$$

for a.e. $x\in \mathsf{\Omega }$. Then we have that

$$\parallel u^{*}{-g\parallel }_{L^2\left(\mathsf{\Omega }\right)}>\parallel \tilde{u}{-g\parallel }_{L^2\left(\mathsf{\Omega }\right)},\mathrm{and}{\int }_{\mathsf{\Omega }}|D{u}^{*}|>{\int }_{\mathsf{\Omega }}\left|D\tilde{u}\right|.$$

(8)

This implies that $u^{*}$ is not a minimizer of problem (2), which is a contradiction and hence $u^{*}\in BV\left(\mathsf{\Omega }\right)\cap C$. ☐

This result is easily extendable to optimization problems of the type

$$\underset{u\in BV\left(\mathsf{\Omega }\right)}{min}{\alpha }_1{\parallel u-g\parallel }_{L^1\left(\mathsf{\Omega }\right)}+{\alpha }_2{\parallel u-g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+{\int }_{\mathsf{\Omega }}\left|Du\right|,$$

(9)

with ${\alpha }_1,{\alpha }_2\ge 0$ and ${\alpha }_1+{\alpha }_2>0$, since for $\tilde{u}$, defined as in the above proof, and a minimizer $u^{*}\in BV\left(\mathsf{\Omega }\right)\cap C$ of problem (9) the inequalities in (8) hold as well as $\parallel u^{*}{-g\parallel }_{L^1\left(\mathsf{\Omega }\right)}>{\parallel \tilde{u}-g\parallel }_{L^1\left(\mathsf{\Omega }\right)}$. Problem (9) has been already considered in [33,44,45] and can be viewed as a generalization of the $L^2$-TV model, since ${\alpha }_1=0$ in (9) yields the $L^2$-TV model and ${\alpha }_2=0$ in (9) yields the $L^1$-TV model.

Note, that if an image is only corrupted by impulse noise, then the observed image g is in the dynamic range of the original image. For example, salt-and-pepper noise contained images may be written as

$$g\left(x\right)=\left\{\begin{array}{cc}{c}_{min}\hfill & \mathrm{with}\mathrm{probability}{s}_1\in [0,1),\hfill \\ c_{max}\hfill & \mathrm{with}\mathrm{probability}{s}_2\in [0,1),\hfill \\ \widehat{u}\left(x\right)\hfill & \mathrm{with}\mathrm{probability}1-s_1-s_2,\hfill \end{array}\right.$$

with $1-s_1-s_2>0$ [46] and for random-valued impulse noise g is described as

$$g\left(x\right)=\left\{\begin{array}{cc}d\hfill & \mathrm{with}\mathrm{probability}s\in [0,1),\hfill \\ \widehat{u}\left(x\right)\hfill & \mathrm{with}\mathrm{probability}1-s,\hfill \end{array}\right.$$

with d being a uniformly distributed random variable in the image intensity range $[c_{min},c_{max}]$. Hence, following Proposition 1, in such cases considering constrained total variation minimization would not change the minimizer and no improvement in the restoration quality can be expected.

This is the reason why we restrict ourselves in the rest of the paper to Gaussian white noise contaminated images and consider solely the $L^2$-TV model.

It is clear that if a solution of the non box-constrained optimization problem already fulfills the box-constraint, then it is of course equivalent to a minimizer of the box-constraint problem. However, note that the minimizer is not unique in general.

In the following we compare the solution of the box-constrained optimization problem (4) with the solution of the unconstrained minimization problem (2).

Proposition 2.

Let $u\in C\cap BV\left(\mathsf{\Omega }\right)$ be a minimizer of

$$J^C\left(u\right):=\frac{1}{2}{\parallel u-g\parallel }_2^2+\alpha {\int }_{\mathsf{\Omega }}\left|Du\right|+{\chi }_C\left(u\right)$$

and $w\in BV\left(\mathsf{\Omega }\right)$ be a minimizer of

$$J\left(w\right):=\frac{1}{2}{\parallel w-g\parallel }_2^2+\alpha \left|Dw\right|\left(\mathsf{\Omega }\right).$$

Then we have that

1. 

$J^C\left(w\right)\ge J^C\left(u\right)=J\left(u\right)\ge J\left(w\right)$.

2. 

$\frac{1}{2}{\parallel u-w\parallel }_2^2\le J\left(u\right)-J\left(w\right)\le J^C\left(w\right)-J^C\left(u\right)$.

3. 

${\parallel u-w\parallel }_2^2\le {4\parallel \xi -g\parallel }_2^2+8\alpha \left|D\xi \right|\left(\mathsf{\Omega }\right)$ for any $\xi \in C\cap BV\left(\mathsf{\Omega }\right)$.

Proof of Proposition 2.

• Follows directly from the optimality of u and w.
• From [47] (Lemma 10.2) it follows that $\frac{1}{2}{\parallel u-w\parallel }_2^2\le J\left(u\right)-J\left(w\right).$ For the second inequality we make the observation that

  $$J^C\left(w\right)-J^C\left(u\right)=\left\{\begin{array}{cc}\infty \hfill & \mathrm{if}w\notin C\hfill \\ 0\hfill & \mathrm{if}w\in C,\hfill \end{array}\right.$$

  where we used the fact that $w=u$ if $w\in C$. This implies, that $J\left(u\right)-J\left(w\right)\le J^C\left(w\right)-J^C\left(u\right)$.
• For all $v\in C\cap BV\left(\mathsf{\Omega }\right)$ we have that

  $$\begin{array}{cc}\hfill {\parallel u-w\parallel }_2^2& \le 2\left({\parallel u-v\parallel }_2^2+{\parallel v-w\parallel }_2^2\right)\hfill \\ & \le 4\left(J\left(v\right)-J\left(u\right)+J\left(v\right)-J\left(w\right)\right)=8J\left(v\right)-4J\left(u\right)-4J\left(w\right),\hfill \end{array}$$

  where we used $2.$ and that ${(a+b)}^2\le 2(a^2+b^2)$. For any arbitrary $\xi \in C\cap BV\left(\mathsf{\Omega }\right)$, let $v=\xi$ and since $J\left(\xi \right)=\frac{1}{2}{\parallel \xi -g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\alpha \left|D\xi \right|\left(\mathsf{\Omega }\right)$ we get ${\parallel u-w\parallel }_2^2\le {4\parallel \xi -g\parallel }_2^2+8\alpha \left|D\xi \right|\left(\mathsf{\Omega }\right)$.

☐

If in Proposition 2 $\xi \in C\cap BV\left(\mathsf{\Omega }\right)$ is constant, then $\left|D\xi \right|\left(\mathsf{\Omega }\right)=0$ which implies that ${\parallel u-w\parallel }_2^2\le 4{\parallel \xi -g\parallel }_2^2$.

4. A Semi-Smooth Newton Method

4.1. The Model Problem

In general $K^{*}K$ is not invertible, which causes difficulties in deriving the dual problem of (4). In order to overcome this difficulties we penalize the $L^2$-TV model by considering the following neighboring problem

$$\underset{u\in C\cap H_0^1\left(\mathsf{\Omega }\right)}{min}\frac{1}{2}{\parallel Ku-g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\frac{\mu }{2}{\int }_{\mathsf{\Omega }}{|\nabla u|}^{2}dx+\alpha {\int }_{\mathsf{\Omega }}{|\nabla u|}_{{\ell }^2}dx,$$

(10)

where $\mu >0$ is a very small constant such that problem (10) is a close approximation of the total variation regularized problem (4). Note, that for $u\in H_0^1\left(\mathsf{\Omega }\right)$ the total variation of u in $\mathsf{\Omega }$ is equivalent to ${\int }_{\mathsf{\Omega }}{|\nabla u|}_{{\ell }^2}dx$ [3]. A typical example for which $K^{*}K$ is indeed invertible is $K=I$, which is used for image denoising. In this case, we may even set $\mu =0$, see Section 6. The objective functional in problem (10) has been already considered for example in [21,48] for image restoration. In particular in [21], a primal-dual semi-smooth Newton algorithm is introduced. Here, we actually adopt this approach to our box-constrained problem (10).

In the sequel we assume for simplicity that $-c_{min}=c_{max}=:c>0$, which changes the set C to $C:=\{u\in L^2\left(\mathsf{\Omega }\right):|u|\le c\}$. Note, that any bounded image $\widehat{u}$, i.e., which lies in the dynamic range $[a,b]$, can be easily transformed to an image $\tilde{u}\in [-c,c]$. Since this transform and K are linear, the observation g is also easily transformed to $\tilde{g}=K\tilde{u}+n$.

Example 1.

Let $\widehat{u}$ such that $a\le \widehat{u}\left(x\right)\le b$ for all $x\in \mathsf{\Omega }$. Then $|\widehat{u}\left(x\right)-\frac{b+a}{2}|\le \frac{b-a}{2}=:c$ for all $x\in \mathsf{\Omega }$ and we set $\tilde{u}=\widehat{u}-\frac{b+a}{2}$. Hence, $\tilde{g}=K\widehat{u}-K\frac{b+a}{2}+n=g-K\frac{b+a}{2}$.

Problem (10) can be equivalently written as

$$\underset{u\in H_0^1\left(\mathsf{\Omega }\right)}{min}\frac{1}{2}{\parallel Ku-g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\frac{\mu }{2}{\int }_{\mathsf{\Omega }}{|\nabla u|}^{2}dx+\alpha {\int }_{\mathsf{\Omega }}{|\nabla u|}_{{\ell }^2}dx+{\chi }_C\left(u\right).$$

(11)

If $u^{*}\in H_0^1\left(\mathsf{\Omega }\right)$ is a solution of problem (10) (and equivalently problem (11)), then there exists ${\lambda }^{*}\in H_0^1{\left(\mathsf{\Omega }\right)}^{*}$ and ${\sigma }^{*}\in \partial R\left(u\right)$, where $R\left(u\right):={\int }_{\mathsf{\Omega }}{|\nabla u|}_{{\ell }^2}dx$, such that

$$\begin{array}{c}\hfill K^{*}K{u}^{*}-K^{*}g-\mu \Delta u^{*}+\alpha {\sigma }^{*}+{\lambda }^{*}=0\\ \hfill \langle {\lambda }^{*},u-u^{*}\rangle \le 0\end{array}$$

for all $u\in C\cap H_0^1\left(\mathsf{\Omega }\right)$.

For implementation reasons (actually for obtaining a fast, second-order algorithm) we approximate the non-smooth characteristic function ${\chi }_C$ by a smooth function in the following way

$$\begin{array}{cc}\hfill {\chi }_C\left(u\right)& \approx \frac{\eta }{2}\left(\parallel max\{u-c_{max},0\}{\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+{\parallel max\{c_{min}-u,0\}\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2\right)\hfill \\ & =\frac{\eta }{2}\left(\parallel max\left\{\right|u|-c_{},0{\}\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2\right),\hfill \end{array}$$

where $\eta >0$ is large. This leads to the following optimization problem

$$\underset{u\in H_0^1\left(\mathsf{\Omega }\right)}{min}\frac{1}{2}{\parallel Ku-g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\frac{\mu }{2}{\int }_{\mathsf{\Omega }}{|\nabla u|}^{2}dx+\alpha {\int }_{\mathsf{\Omega }}{|\nabla u|}_{{\ell }^2}dx+\frac{\eta }{2}\left(\parallel max\left\{\right|u|-c_{},0{\}\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2\right).$$

(12)

Remark 1.

By the assumption $-c_{min}=c_{max}$ we actually excluded the cases (i) $c_{min}=0$, $c_{max}=+\infty$ and (ii) $c_{min}=-\infty$, $c_{max}=0$. In these situations we just need to approximate ${\chi }_C$ by (i) $\frac{\eta }{2}\left({\parallel max\{-u,0\}\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2\right)$ and (ii) $\frac{\eta }{2}\left({\parallel max\{u,0\}\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2\right)$. By noting this, in a similar fashion as done below for problem (12), a primal-dual semi-smooth Newton method can be derived for these two cases.

4.2. Dualization

By a standard calculation one obtains that the dual of problem (12) is given by

$$\underset{\overrightarrow{p}=({\overrightarrow{p}}_1,p_2)\in {\mathbb{L}}^2\left(\mathsf{\Omega }\right)\times L^2\left(\mathsf{\Omega }\right)}{sup}-\frac{1}{2}\left|\right||{\Lambda }^{*}\overrightarrow{p}+K^{*}{g\left|\right||}_B^2+\frac{1}{2}{\parallel g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2-{\chi }_A(-{\overrightarrow{p}}_1)-\frac{1}{2\eta }\parallel -p_2{\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2-{\parallel -c{p}_2\parallel }_{L^1\left(\mathsf{\Omega }\right)}$$

(13)

with ${\Lambda }^{*}\overrightarrow{p}=-div{\overrightarrow{p}}_1+p_2$ and $A:=\{v\in {\mathbb{L}}^2\left(\mathsf{\Omega }\right):|v{|}_{{\ell }^2}\le \alpha \}$. As the divergence operator does not have a trivial kernel, the solution of the optimization problem (13) is not unique. In order to render the problem (13) strictly concave we add an additional term yielding the following problem

$$\begin{array}{cc}\hfill \underset{\overrightarrow{p}\in {\mathbb{L}}^2\left(\mathsf{\Omega }\right)\times L^2\left(\mathsf{\Omega }\right)}{min}& \frac{1}{2}\left|\right||{\Lambda }^{*}\overrightarrow{p}+K^{*}{g\left|\right||}_B^2-\frac{1}{2}{\parallel g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+{\chi }_A(-{\overrightarrow{p}}_1)+\frac{1}{2\eta }{\parallel -p_2\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & +\parallel -c{p}_2{\parallel }_{L^1\left(\mathsf{\Omega }\right)}+\frac{\gamma }{2\alpha }{\parallel p_1\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2,\hfill \end{array}$$

(14)

where $\gamma >0$ is a fixed parameter.

Proposition 3.

The dual problem of problem (14) is given by

$$\underset{u\in H_0^1\left(\mathsf{\Omega }\right)}{min}\frac{1}{2}{\parallel Ku-g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\frac{\mu }{2}{\parallel \nabla u\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\alpha {\int }_{\mathsf{\Omega }}{\varphi }_{\gamma }(\nabla u)\left(x\right)dx+\frac{\eta }{2}\parallel max\left\{\right|u|-{c,0\}\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2$$

(15)

with

$${\varphi }_{\gamma }\left(\overrightarrow{q}\right)\left(x\right)=\left\{\begin{array}{cc}\frac{1}{2\gamma }{\left|\overrightarrow{q}\left(x\right)\right|}_{{\ell }^2}^2\hfill & if|\overrightarrow{q}{\left(x\right)|}_{{\ell }^2}<\gamma \hfill \\ |\overrightarrow{q}{\left(x\right)|}_{{\ell }^2}-\frac{\gamma }{2}\hfill & if|\overrightarrow{q}{\left(x\right)|}_{{\ell }^2}\ge \gamma .\hfill \end{array}\right.$$

The proof of this statement is a bit technical and therefore deferred to Appendix A.

Similar as in [21] one can show that the solution of problem (15) converges to the minimizer of (12) as $\gamma \to 0$.

From the Fenchel duality theorem we obtain the following characterization of solutions u and $\overrightarrow{p}$ of problem (15) and problem (14) (note that $p=-q$)

$$\begin{array}{cc}\hfill div{\overrightarrow{p}}_1-p_2& =K^{*}Ku-K^{*}g-\mu \Delta u\mathrm{in}{H}^{-1}\left(\mathsf{\Omega }\right)\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\overrightarrow{p}}_1& =\frac{\alpha }{\gamma }\nabla u\mathrm{if}{\left|{\overrightarrow{p}}_1\right|}_{{\ell }^2}<\alpha \mathrm{in}{\mathbb{L}}^2\left(\mathsf{\Omega }\right)\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\overrightarrow{p}}_1& =\alpha \frac{\nabla u}{|\nabla u|}\mathrm{if}{\left|{\overrightarrow{p}}_1\right|}_{{\ell }^2}={\alpha }_1\mathrm{in}{\mathbb{L}}^2\left(\mathsf{\Omega }\right)\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill p_2& =\eta max\left\{\right|u|-c,0\}sign\left(u\right)\mathrm{in}{L}^2\left(\mathsf{\Omega }\right).\hfill \end{array}$$

(19)

This system can be solved efficiently by a semi-smooth Newton algorithm. Moreover, equations (17) and (18) can be condensed into ${\overrightarrow{p}}_1=\frac{\alpha \nabla u}{max\{\gamma ,|\nabla u{|}_{{\ell }^2}\}}$.

4.3. Adaptation to Non-Scalar $\alpha$

For locally adaptive $\alpha$, i.e., $\alpha :\mathsf{\Omega }\to {\mathbb{R}}^{+}$ is a function, the minimization problem (12) changes to

$$\underset{u\in H_0^1\left(\mathsf{\Omega }\right)}{min}\frac{1}{2}{\parallel Ku-g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\frac{\mu }{2}{\int }_{\mathsf{\Omega }}{|\nabla u|}^{2}dx+{\int }_{\mathsf{\Omega }}\alpha \left(x\right){|\nabla u|}_{{\ell }^2}dx+\frac{\eta }{2}\left(\parallel max\left\{\right|u|-c_{},0{\}\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2\right).$$

(20)

Its dual problem is given by

$$\underset{\overrightarrow{p}\in {\mathbb{L}}^2\left(\mathsf{\Omega }\right)\times L^2\left(\mathsf{\Omega }\right)}{min}\frac{1}{2}\left|\right||{\Lambda }^{*}\overrightarrow{p}+K^{*}{g\left|\right||}_B^2-\frac{1}{2}{\parallel g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+{\chi }_{\tilde{A}}(-{\overrightarrow{p}}_1)+\frac{1}{2\eta }\parallel -p_2{\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+{\parallel -c{p}_2\parallel }_{L^1\left(\mathsf{\Omega }\right)},$$

where $\tilde{A}:=\{v\in {\mathbb{L}}^2{\left(\mathsf{\Omega }\right):\left|v\left(x\right)\right|}_{{\ell }^2}\le \alpha \left(x\right)\}$. Similarly but slightly different as above, cf. problem (14), we penalize by

$$\begin{array}{cc}\hfill \underset{\overrightarrow{p}\in {\mathbb{L}}^2\left(\mathsf{\Omega }\right)\times L^2\left(\mathsf{\Omega }\right)}{min}& \frac{1}{2}\left|\right||{\Lambda }^{*}p+K^{*}{g\left|\right||}_B^2-\frac{1}{2}{\parallel g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+{\chi }_A(-{\overrightarrow{p}}_1)+\frac{1}{2\eta }{\parallel -p_2\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & +\parallel -c{p}_2{\parallel }_{L^1\left(\mathsf{\Omega }\right)}+\frac{\gamma }{2}{\parallel p_1\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2.\hfill \end{array}$$

Then the dual of this problem turns out to be

$$\underset{u\in H_0^1\left(\mathsf{\Omega }\right)}{min}\frac{1}{2}{\parallel Ku-g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\frac{\mu }{2}{\parallel \nabla u\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+{\int }_{\mathsf{\Omega }}{\varphi }_{\gamma ,\alpha }(\nabla u)\left(x\right)dx+\frac{\eta }{2}\parallel max\left\{\right|u|-{c,0\}\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2$$

(21)

with

$${\varphi }_{\gamma ,\alpha }\left(\overrightarrow{q}\right)\left(x\right)=\left\{\begin{array}{cc}\frac{1}{2\gamma }{\left|\overrightarrow{q}\left(x\right)\right|}_{{\ell }^2}^2\hfill & \mathrm{if}|\overrightarrow{q}{\left(x\right)|}_{{\ell }^2}<\gamma \alpha \left(x\right)\hfill \\ \alpha \left(x\right)|\overrightarrow{q}{\left(x\right)|}_{{\ell }^2}-\frac{\gamma }{2}{\left|\alpha \left(x\right)\right|}^2\hfill & \mathrm{if}|\overrightarrow{q}{\left(x\right)|}_{{\ell }^2}\ge \gamma \alpha \left(x\right).\hfill \end{array}\right.$$

Denoting by u a solution of problem (21) and $\overrightarrow{p}$ the solution of the associated pre-dual problem, the optimality conditions due to the Fenchel theorem [43] are given by

$$\begin{array}{cc}\hfill div{\overrightarrow{p}}_1-p_2& =K^{*}Ku-K^{*}g-\mu \Delta u\hfill \\ \hfill {\overrightarrow{p}}_1& =\frac{\alpha \nabla u}{max\{\gamma \alpha ,|\nabla u\left|\right\}}\hfill \\ \hfill p_2& =\eta max\left\{\right|u|-c,0\}sign\left(u\right).\hfill \end{array}$$

5. Numerical Implementation

Similar as in the works [21,36,37,38,39], where semi-smooth Newton methods for non-smooth systems emerging from image restoration models have been derived, we can solve the discrete version of the system (16)–(19), using finite differences, efficiently by a primal-dual algorithm. Therefore let $u^h\in {\mathbb{R}}^N$, $p_1^h\in {\mathbb{R}}^{2N}$, $p_2^h\in {\mathbb{R}}^N$, $g^h\in {\mathbb{R}}^N$, denote the discrete image intensity, the dual variables, and the observed data vector, respectively, where $N\in \mathbb{N}$ is the number of elements (pixels) in the discrete image ${\mathsf{\Omega }}^h$. Moreover, we denote by ${\alpha }^h>0$ the regularization parameter. Correspondingly we define ${\nabla }^h\in {\mathbb{R}}^{2N\times N}$ as the discrete gradient operator, ${\Delta }^h\in {\mathbb{R}}^{N\times N}$ as the discrete Laplace operator, $K^h\in {\mathbb{R}}^{N\times N}$ as a discrete operator, and ${\left(K^h\right)}^t$ its transpose. Moreover, ${div}^h=-{\left({\nabla }^h\right)}^t$. Here $|·|$, $max\{·,·\}$, and $sign(·)$ are understood for vectors in a component-wise sense. Moreover, we use the function $\left[\right|·\left|\right]:{\mathbb{R}}^{2N}\to {\mathbb{R}}^{2N}$ with $\left[\right|v^h{\left|\right]}_i=\left[\right|v^h{\left|\right]}_{i+N}=\sqrt{{\left(v_i^h\right)}^2+{\left(v_{i+N}^h\right)}^2}$ for $1\le i\le N$.

5.1. Scalar $\alpha$

The discrete version of (16)–(19) reads as

$$\begin{array}{cc}\hfill 0& =-{div}^h{p}_1^h+\eta D\left(m_0\right)+{\left(K^h\right)}^t{K}^h{u}^h-{\left(K^h\right)}^t{g}^h-\mu {\Delta }^h{u}^h\hfill \\ \hfill 0& =D^h\left(m_{\gamma }\right)p_1^h-{\alpha }^h{\nabla }^h{u}^h\hfill \end{array}$$

(22)

where $D^h\left(v\right)$ is a diagonal matrix with vector v in its diagonal, $m_0:=sign\left(u\right)max\left\{\right|u|-c,0\}$, and $m_{\gamma }:=max\{\gamma ,[|{\nabla }^h{u}^h\left|\right]\}$. We define

$$\begin{array}{c}{\chi }_{{\mathcal{A}}_{\gamma }}=D^h\left(t_{\gamma }\right)\mathrm{with}{\left(t_{\gamma }\right)}_i=\left\{\begin{array}{cc}0\hfill & \mathrm{if}{\left(m_{\gamma }\right)}_i=\gamma ,\hfill \\ 1\hfill & \mathrm{else};\hfill \end{array}\right.\hfill \\ {\chi }_{{\mathcal{A}}_c^{max}}=D^h\left(t_c^{max}\right)\mathrm{with}{\left(t_c^{max}\right)}_i=\left\{\begin{array}{cc}0\hfill & \mathrm{if}{\left(m_c^{max}\right)}_i=0,\hfill \\ 1\hfill & \mathrm{else};\hfill \end{array}\right.\hfill \\ {\chi }_{{\mathcal{A}}_c^{min}}=D^h\left(t_c^{min}\right)\mathrm{with}{\left(t_c^{min}\right)}_i=\left\{\begin{array}{cc}0\hfill & \mathrm{if}{\left(m_c^{min}\right)}_i=0,\hfill \\ 1\hfill & \mathrm{else},\hfill \end{array}\right.\hfill \end{array}$$

where $m_c^{max}:=max\{u-c,0\}$ and $m_c^{min}:=max\{u+c,0\}$. Further, we set

$$\begin{array}{c}{M}^h\left(v\right)=\left(\begin{array}{cc}{D}^h\left(v_x\right)& D^h\left(v_y\right)\\ D^h\left(v_x\right)& D^h\left(v_y\right)\end{array}\right)\mathrm{with}v={(v_x,v_y)}^t\in {\mathbb{R}}^{2N}.\hfill \end{array}$$

Applying a generalized Newton step to solve (22) at $(u_k^h,p_{1,k}^h)$ yields

$$\begin{array}{c}\hfill \left(\begin{array}{cc}\eta ({\chi }_{{\mathcal{A}}_c^{max}}+{\chi }_{{\mathcal{A}}_c^{min}})+{\left(K^h\right)}^t{K}^h-\mu {\Delta }^h& -{div}^h\\ C_k^h\nabla & D^h\left(m_{\gamma }\right)\end{array}\right)\left(\begin{array}{c}{\delta }_u\\ {\delta }_{p_1}\end{array}\right)=\left(\begin{array}{c}-{\mathfrak{F}}_1^k\\ -{\mathfrak{F}}_2^k\end{array}\right)\end{array}$$

(23)

where

$$\begin{array}{c}{C}_k^h=D^h\left(p_{1,k}^h\right){\chi }_{{\mathcal{A}}_{\gamma }}{D}^h{\left(m_{\gamma }\right)}^{-1}{M}^h(\nabla u_k^h)-{\alpha }^h{D}^h\left(e_{2N}\right)\hfill \\ {\mathfrak{F}}_1^k=-{div}^h{p}_{1,k}^h+\eta D^h\left(m_0\right)+{\left(K^h\right)}^t{K}^h{u}_k^h-{\left(K^h\right)}^{t}g-\mu {\Delta }^h{u}_k^h\hfill \\ {\mathfrak{F}}_2^k=D^h\left(m_{\gamma }\right)p_{1,k}^h-{\alpha }^h{\nabla }^h{u}_k^h\hfill \end{array}$$

and $e_N\in {\mathbb{R}}^N$ is the identity vector. The diagonal matrix $D\left(m_{\gamma }\right)$ is invertible, i.e.,

$$\begin{array}{c}\hfill {\delta }_{p_1}=D^h{\left(m_{\gamma }\right)}^{-1}(-{\mathfrak{F}}_2^k-C_k^h\nabla {\delta }_u)\end{array}$$

and hence we can eliminate ${\delta }_{p_1}$ from the Newton system resulting in

$$H_k{\delta }_u=f_k$$

(24)

where

$$\begin{array}{c}{H}_k:=\eta ({\chi }_{{\mathcal{A}}_c^{max}}+{\chi }_{{\mathcal{A}}_c^{min}})+{\left(K^h\right)}^t{K}^h-\mu {\Delta }^h+{div}^h{D}^h{\left(m_{\gamma }\right)}^{-1}{C}_k^h\nabla ,\hfill \\ f_k:=-{\mathfrak{F}}_1^k-{div}^h{D}^h{\left(m_{\gamma }\right)}^{-1}{\mathfrak{F}}_2^k.\hfill \end{array}$$

If $H_k$ is positive definite, then the solution ${\delta }_u$ of (24) exists and is a descent direction of (15). However, in general we cannot expect the positive definiteness of $H_k$. In order to ensure that $H_k$ is positive definite, we project $p_{1,k}^h$ onto its feasible set by setting $({\left(p_{1,k}^h\right)}_i,{\left(p_{1,k}^h\right)}_{i+N})$ to ${\alpha }^{h}max\{{\alpha }^h,\left[\right|p_{1,k}^h{\left|\right]}_i{\}}^{-1}({\left(p_{1,k}^h\right)}_i,{\left(p_{1,k}^h\right)}_{i+N})$ for $i=1,\dots ,N$ which guarantees

$$\left[\right|p_{1,k}^h{\left|\right]}_i\le {\alpha }^h$$

(25)

for $i=1,\dots ,2N$. The modified system matrix, denoted by $H_k^{+}$, is then positive definite. Then our semi-smooth Newton solver may be written as:

Primal-dual Newton method (pdN): Initialize $(u_0^h,p_{1,0}^h)\in {\mathbb{R}}^N\times {\mathbb{R}}^{2N}$ and set $k:=0$.  1. Determine the active sets ${\chi }_{{\mathcal{A}}_c^{max}}\in {\mathbb{R}}^{N\times N}$, ${\chi }_{{\mathcal{A}}_c^{min}}\in {\mathbb{R}}^{N\times N}$, ${\chi }_{{\mathcal{A}}_{\gamma }}\in {\mathbb{R}}^{N\times N}$,  2. If (25) is not satisfied, then compute $H_k^{+}$; otherwise set $H_k^{+}:=H_k$.  3. Solve $H_k^{+}{\delta }_u=f_k$ for ${\delta }_u$.  4. Compute ${\delta }_{p_1}$ by using ${\delta }_u$.  5. Update $u_{k+1}^h:=u_k^h+{\delta }_u$ and $p_{1,k+1}^h:=p_{1,k}^h+{\delta }_{p_1}$.  6. Stop or set $k:=k+1$ and continue with step 1).

This algorithm converges at a superlinear rate, which follows from standard theory; see [20,21]. The Newton method is terminated as soon as the initial residual is reduced by a factor of ${10}^{-4}$.

Note, that, since $\eta =0$ implies $p_2=0$, in this case the proposed primal-dual Newton method becomes the method in [21].

5.2. Non-Scalar $\alpha$

A similar semi-smooth Newton method might be derived for the locally adaptive case by noting that then ${\alpha }^h\in {\mathbb{R}}^N$, and hence the second equation in (22) changes to

$$0=D^h\left(m_{\gamma }\right)p_1^h-D^h\left({({\alpha }^h,{\alpha }^h)}^t\right){\nabla }^h{u}^h,$$

where $m_{\gamma }:=max\{\gamma {\alpha }^h,\left[\right|{\nabla }^h{u}^h\left|\right]\}$ leading to (23) with

$$C_k^h=D^h\left(p_{1,k}^h\right){\chi }_{{\mathcal{A}}_{\gamma }}{D}^h{\left(m_{\gamma }\right)}^{-1}{M}^h(\nabla u_k^h)-D^h\left({({\alpha }^h,{\alpha }^h)}^t\right)$$

and

$${\mathfrak{F}}_2^k=D^h\left(m_{\gamma }\right)p_{1,k}^h-D^h\left({({\alpha }^h,{\alpha }^h)}^t\right){\nabla }^h{u}_k^h.$$

The positive definite modified matrix $H_k^{+}$ is then obtained by setting $({(p_{1,k}^h)}_i,{(p_{1,k}^h)}_{i+N})$ to ${\alpha }_i^{h}max\{{\alpha }_i^h,\left[\right|p_{1,k}^h{\left|\right]}_i{\}}^{-1}({\left(p_{1,k}^h\right)}_i,{\left(p_{1,k}^h\right)}_{i+N})$ for $i=1,\dots ,N$.

6. Numerical Experiments

For our numerical studies we consider the images shown in Figure 1 of size 256 × 256 pixels and in Figure 2. The image intensity range of all original images considered in this paper is $[0,1]$, i.e., $c_{min}=0$ and $c_{max}=1$. Our proposed algorithms automatically transform this images into the dynamic range $[-c,c]$, here with $c=1/2$. That is, let $\widehat{u}\in [0,1]$ be the original image before any corruption, then $\widehat{u}\left(x\right)-\frac{1}{2}\in [-\frac{1}{2},\frac{1}{2}]$. Moreover, the solution generated by the semi-smooth Newton method is afterwards back-transformed, i.e., the generated solution $\tilde{u}$ is transformed to $\tilde{u}+\frac{1}{2}$. Note that ${max}_x\tilde{u}\left(x\right)+\frac{1}{2}$ is not necessarily in $[0,1]$, except $\tilde{u}\in [-\frac{1}{2},\frac{1}{2}]$.

As a comparison for the different restoration qualities of the restored image we use the PSNR [49] (peak signal-to-noise ratio) given by

$$PSNR=20log\frac{1}{\parallel \widehat{u}-u^{*}\parallel },$$

where $\widehat{u}$ denotes the original image before any corruption and $u^{*}$ the restored image, which is widely used as an image quality assessment measure, and the MSSIM [50] (mean structural similarity), which usually relates to perceived visual quality better than PSNR. In general, when comparing PSNR and MSSIM, large values indicate better reconstruction than small values.

In our experiments we also report on the computational time (in seconds) and the number of iterations (it) needed until the considered algorithms are terminated.

In all the following experiments the parameter $\mu$ is chosen to be 0 for image denoising (i.e., $K=I$), since then no additional smoothing is needed, and $\mu ={10}^{-6}$ if $K\ne I$ (i.e., for image deblurring, image inpainting, for reconstructing from partial Fourier-data, and for reconstructing from sampled Radon-transform).

6.1. Dependency on the Parameter $\eta$

We start by investigating the influence of the parameter $\eta$ on the behavior of the semi-smooth Newton algorithm and its generated solution. Let us recall, that $\eta$ is responsible how strictly the box-constraint is adhered. In order to visualize how good the box-constraint is fulfilled for a chosen $\eta$ in Figure 3 we depict ${max}_{x}u\left(x\right)-c_{max}$ and ${min}_{x}u\left(x\right)-c_{min}$ with $c_{max}=1$, $c_{min}=0$, and u being the back-transformed solution, i.e., $u=\tilde{u}+\frac{1}{2}$, where $\tilde{u}$ is obtained via the semi-smooth Newton method. As long as ${max}_{x}u\left(x\right)-c_{max}$ and ${min}_{x}u\left(x\right)-c_{min}$ are positive and negative, respectively, the box-constraint is not perfectly adhered. From our experiments for image denoising and image deblurring, see Figure 3, we clearly see that the larger $\eta$ the more strictly the box-constraint is adhered. In the rest of our experiments we choose $\eta ={10}^6$, which seems sufficiently large to us and the box-constraint seems to hold accurately enough.

6.2. Box-Constrained Versus Non-Box-Constrained

In the rest of this section we are going to investigate how much the solution (and its restoration quality) depends on the box-constraint and if this is a matter on how the regularization parameter is chosen. We start by comparing for different values of $\alpha$ the solutions obtained by the semi-smooth Newton method without a box-constraint (i.e., $\eta =0$) with the ones generated by the same algorithm with $\eta ={10}^6$ (i.e., a box-constraint is incorporated). Our obtained results are shown in

Table 1. Reconstruction of the cameraman-image corrupted by Gaussian white noise with $\sigma =0.1$ for different regularization parameters $\alpha$.

$\mathit{\alpha }$	pdN with $\mathit{\eta }=0$				Box-Constrained pdN
	PSNR	MSSIM	Time	It	PSNR	MSSIM	Time	It
0.001	20.165	0.27649	183.29	365	20.635	0.28384	178.27	365
0.01	21.464	0.29712	55.598	117	21.905	0.30472	55.245	117
0.096029	27.134	0.35214	14.615	33	27.135	0.35221	14.465	33
0.096108	27.132	0.35201	14.91	33	27.133	0.35207	14.317	33
0.4	22.079	0.16816	14.779	34	22.079	0.16816	14.982	34
⌀	23.5947	0.28919	56.6388	116.4	23.7773	0.2922	55.4557	116.4

for image denoising and in

Table 2. Reconstruction of the cameraman-image corrupted by Gaussian blur and Gaussian white noise with $\sigma =0.1$ for different regularization parameters $\alpha$.

$\mathit{\alpha }$	pdN with $\mathit{\eta }=0$				Box-Constrained pdN
	PSNR	MSSIM	Time	It	PSNR	MSSIM	Time	It
0.001	5.9375	0.039866	1197	143	11.568	0.083341	1946	145
0.01	21.908	0.18938	863	67	21.964	0.1946	2011	67
0.051871	21.815	0.18115	759	37	21.814	0.18113	1299	37
0.051868	21.814	0.18114	752	36	21.814	0.18114	1255	37
0.4	19.823	0.090709	1431	61	19.823	0.090709	1454	61
⌀	18.2593	0.13645	1000	68.8	19.3966	0.14618	1593	69.4

for image deblurring. We obtain, that for small $\alpha$ we gain “much” better results with respect to PSNR and MSSIM with a box-constraint than without. The reason for this is that if no box-constraint is used and $\alpha$ is small then nearly no regularization is performed and hence noise, which is violating the box-constraint, is still present. Therefore incorporating a box-constraint is reasonable for these choices of parameters. However, if $\alpha$ is sufficiently large then we numerically observe that the solution of the box-constrained and non-box-constrained problem are the same. This is not surprising, because there exists $\overline{\alpha }>0$ such that for all $\alpha >\overline{\alpha }$ the solution of problem (2) is $\frac{1}{\left|\mathsf{\Omega }\right|}{\int }_{\mathsf{\Omega }}g$, see [4] (Lemma 2.3). That is, for such $\alpha$ the minimizer of problem (2) is the average of the observation which lies in the image intensity range of the original image, as long as the mean of Gaussian noise is 0 (or sufficiently small). This implies that in such a case the minimizer of problem (2) and problem (4) are equivalent. Actually this equivalency already holds if $\alpha$ is sufficiently large such that the respective solution of problem (2) lies in the dynamic range of the original image, which is the case in our experiments for $\alpha =0.4$. Hence, whether it makes sense or not to incorporate a box-constraint into the considered model depends on the choice of parameters. The third and fourth value of $\alpha$ in Table 1 and Table 2 refer to the ones which equalize problem (2) and problem (1), and respectively problem (4) and problem (3). In the sequel we call such parameters optimal, since a solution of the penalized problem also solves the related constrained problem. However, we note that these $\alpha$-values are in general not giving the best results with respect to PSNR and MSSIM, but they are usually close to the results with the largest PSNR and MSSIM. For both type of applications, i.e., image denoising and image deblurring, these optimal $\alpha$-values are nearly the same for problem (2) and problem (1), and respectively problem (4) and problem (3) and hence also the PSNR and MSSIM of the respective results are nearly the same. Nevertheless, we mention that for image deblurring the largest PSNR and MSSIM in these experiments is obtained for $\alpha =0.01$ with a box-constraint.

In

Table 3. Reconstruction of the cameraman-image corrupted by Gaussian white noise with $\sigma =0.1$ for different regularization parameters $\alpha$ using the pdN with thresholded g.

$\mathit{\alpha }$	PSNR	MSSIM	Time	It
0.001	20.64	0.28388	107.05	349
0.01	21.949	0.30504	32.336	113
0.096029	26.528	0.33078	8.9479	33
0.096108	26.526	0.33066	8.8966	33
0.4	21.666	0.15794	9.4218	35
⌀	23.4617	0.28166	33.3304	112.6

and

Table 4. Reconstruction of the cameraman-image corrupted by Gaussian blur and Gaussian white noise with $\sigma =0.1$ for different regularization parameters $\alpha$ using the pdN with thresholded g.

$\mathit{\alpha }$	PSNR	MSSIM	Time	It
0.001	6.6758	0.046454	1091	140
0.01	21.929	0.19231	743	65
0.051871	21.709	0.1722	659	35
0.051868	21.709	0.1722	651	35
0.4	19.683	0.087447	1438	61
⌀	18.3413	0.13412	916	67.2

we also report on an additional strategy. In this approach we threshold (or project) the observation g such that the box-constraint holds in any pixel and use then the proposed Newton method with $\eta =0$. For large $\alpha$ this is an inferior approach, but for small $\alpha$ this seems to work similar to incorporating a box-constraint, at least for image denoising. However, it is outperformed by the other approaches.

6.3. Comparison with Optimal Regularization Parameters

In order to determine the optimal parameters $\alpha$ for a range of different examples we assume that the noise-level $\sigma$ is at hand and utilize the pAPS-algorithm presented in [36]. Alternatively, instead of computing a suitable $\alpha$, we may solve the constrained optimization problems (1) and (3) directly by using the alternating direction methods of multipliers (ADMM). An implementation of the ADMM for solving problem (1) is presented in [51], which we refer to as the ADMM in the sequel. For solving problem (3) a possible implementation is suggested in [27]. However, for comparison purposes we use a slightly different version, which uses the same succession of updates as the ADMM in [51], see Appendix B for a description of this version. In the sequel we refer to this algorithm as the box-contrained ADMM. We do not expect the same results for the pAPS-algorithm and the (box-contrained) ADMM, since in the pAPS-algorithm we use the semi-smooth Newton method which generates an approximate solution of problem (15), that is not equivalent to problem (1) and problem (3). In all the experiments in the pAPS-algorithm we set the initial regularization parameter to be ${10}^{-3}$.

6.3.1. pdN versus ADMM

We start by comparing the performance of the proposed primal-dual semi-smooth Newton method (pdN) and the ADMM. In these experiments we assume that we know the optimal parameters $\alpha$, which are then used in the pdN. Note, that a fair comparison of these two methods is difficult, since they are solving different optimization problems, as already mentioned above. However, we still compare them in order to understand better the performance of the algorithms in the sequel section.

The comparison is performed for image denoising and image deblurring and the respective findings are collected in

Table 5. Reconstruction of the cameraman-image corrupted by Gaussian white noise with standard deviation $\sigma$.

$\mathit{\sigma }$	$\mathit{\alpha }$	Box-Constrained pdN				Box-Constrained ADMM
		PSNR	MSSIM	Time	It	PSNR	MSSIM	Time	It
0.3	0.34586	22.485	0.18619	18.844	41	22.2	0.17829	897.82	127
0.2	0.21408	24.054	0.24497	14.847	33	23.852	0.23917	808.67	100
0.1	0.096108	27.132	0.35201	14.91	33	26.963	0.35118	716.37	70
0.05	0.043393	30.567	0.47437	22.902	51	30.488	0.47503	656.66	48
0.01	0.0071847	40.417	0.75235	59.533	133	40.542	0.75864	454.45	24
0.005	0.0032996	45.164	0.8686	89.674	199	45.423	0.87718	501.59	24
⌀		31.6363	0.47975	36.785	81.667	31.5781	0.47991	672.5961	65.5

and

Table 6. Reconstruction of the cameraman-image corrupted by Gaussian blur and Gaussian white noise with standard deviation $\sigma$.

$\mathit{\sigma }$	$\mathit{\alpha }$	Box-Constrained pdN				Box-Constrained ADMM
		PSNR	MSSIM	Time	It	PSNR	MSSIM	Time	It
0.3	0.2342	20.382	0.10678	1551.5	55	20.361	0.099691	2217.3	256
0.2	0.13169	20.981	0.13262	1434	41	20.978	0.12702	2593.1	265
0.1	0.051871	21.814	0.18113	1407.4	37	21.825	0.17536	3404.5	292
0.05	0.01951	22.484	0.22905	2691.4	51	22.501	0.22423	4065.4	305
0.01	0.0012674	24.293	0.34618	2440.5	126	24.237	0.34082	7185.3	358
0.005	0.00031869	25.451	0.40405	1903.4	149	25.377	0.3987	26,985	1081
⌀		22.5674	0.2333	1904.6943	76.5	22.5464	0.22764	7741.7188	426.17

. From there we clearly observe, that the proposed pdN with $\eta ={10}^6$ reaches in all experiments the desired reconstruction significantly faster than the box-constrained ADMM. While the number of iterations for image denoising is approximately the same for both methods, for image deblurring the box-constrained pdN needs significantly less iterations than the other method. In particular, the pdN needs nearly the same amount of iterations independently of the application. However, more iterations for small $\sigma$ are needed. Note, that the pdN converges at a superlinear rate and hence a faster convergence than the box-constrained ADMM is not surprising but supports the theory.

6.3.2. Image Denoising

In

Table 7. PSNR- and MSSIM-values of the reconstruction of different images corrupted by Gaussian white noise via pAPS using the primal-dual Newton method.

Image	$\mathit{\sigma }$	pAPS with pdN with $\mathit{\eta }=0$					pAPS with Box-Constrained pdN
		PSNR	MSSIM	Time	$\mathit{\alpha }$	It	PSNR	MSSIM	Time	$\mathit{\alpha }$	It
phantom	0.3	19.274	0.30365	693.59	0.28721	40	19.302	0.30456	486.9	0.28544	20
	0.2	22.024	0.37461	553.44	0.19055	37	22.06	0.37521	393.67	0.18921	17
	0.1	27.471	0.44124	565.80	0.09621	37	27.518	0.4412	389.39	0.095365	17
	0.05	33.173	0.46878	858.33	0.048744	37	33.228	0.46847	421.18	0.04827	16
	0.01	46.409	0.50613	1605.4	0.010597	37	46.46	0.50559	791.34	0.010509	17
	0.005	51.805	0.52587	2245.3	0.0057362	35	51.846	0.52522	925.46	0.0056958	15
cameraman	0.3	22.485	0.18619	1591	0.34586	102	22.485	0.18621	847.53	0.34579	50
	0.2	24.054	0.24497	919.63	0.21408	66	24.056	0.24508	528.85	0.21398	32
	0.1	27.132	0.35201	580.91	0.096108	39	27.135	0.35221	646.44	0.096029	41
	0.05	30.567	0.47437	549.23	0.043393	24	30.571	0.47458	561.43	0.04336	25
	0.01	40.417	0.75235	677.79	0.0071847	12	40.418	0.75238	645.32	0.0071837	12
	0.005	45.164	0.8686	745.67	0.0032996	9	45.164	0.8686	701.1	0.0032995	9
barbara	0.3	20.618	0.30666	1419.7	0.34118	84	20.618	0.30666	735.83	0.34118	39
	0.2	21.649	0.39964	788.49	0.20109	47	21.649	0.39971	358.6	0.20105	18
	0.1	24.241	0.58555	336.17	0.089758	23	24.241	0.58564	319.09	0.089746	23
	0.05	27.884	0.75178	326.59	0.04286	15	27.885	0.7518	286.61	0.042858	16
	0.01	38.781	0.9322	345.64	0.0079133	9	38.781	0.9322	311.25	0.0079133	9
	0.005	44.056	0.9681	395.8	0.0037205	7	44.056	0.9681	330.97	0.0037205	7
house	0.3	23.827	0.1839	2750	0.41771	154	23.829	0.18392	1185.9	0.41763	75
	0.2	25.611	0.23397	1460.6	0.26795	116	25.611	0.23397	610.2	0.26796	45
	0.1	28.855	0.31916	690.33	0.11979	75	28.855	0.31916	348.92	0.11979	34
	0.05	32.04	0.4074	574.34	0.050505	39	32.041	0.40741	563.92	0.050502	40
	0.01	40.292	0.75174	493.84	0.0071206	11	40.292	0.75174	468.11	0.0071206	11
	0.005	44.989	0.86648	527	0.0033035	8	44.989	0.86648	502.38	0.0033035	8
lena	0.3	21.905	0.29155	1925	0.41731	120	21.905	0.29155	953.83	0.41731	57
	0.2	23.506	0.36317	1064.9	0.25937	85	23.506	0.36317	585.24	0.25937	39
	0.1	26.37	0.49351	537.69	0.11246	46	26.369	0.49351	302.32	0.11246	21
	0.05	29.615	0.62313	566.71	0.047771	27	29.615	0.62313	579.3	0.04777	28
	0.01	39.261	0.91371	526.89	0.0068096	97	39.262	0.91371	546.75	0.0068091	10
	0.005	44.672	0.97133	626.47	0.0032764	8	44.673	0.97133	652.97	0.0032762	8
bones	0.3	25.744	0.34395	5830.2	0.86048	310	25.743	0.34395	749.26	0.8605	35
	0.2	27.637	0.39821	2949	0.56086	238	27.637	0.39821	747.28	0.56086	45
	0.1	30.908	0.49398	1232.8	0.27216	141	30.908	0.49398	364.71	0.27216	32
	0.05	34.284	0.58386	612.74	0.12735	87	34.284	0.58386	362.62	0.12735	41
	0.01	43.174	0.7449	386.03	0.020815	33	43.174	0.7449	479.77	0.020814	33
	0.005	47.493	0.80124	340.02	0.0091423	23	47.493	0.80124	470.83	0.0091423	23
cookies	0.3	21.466	0.31394	1117.4	0.38254	87	21.466	0.31396	857.99	0.38252	42
	0.2	23.136	0.40787	709.8	0.25117	62	23.136	0.40787	320.84	0.25118	12
	0.1	26.498	0.55614	398.59	0.12598	42	26.498	0.55614	290.67	0.12598	18
	0.05	30.292	0.67741	382.29	0.06257	30	30.293	0.67741	523.47	0.06257	30
	0.01	40.482	0.85926	414.09	0.011324	15	40.482	0.85926	567.65	0.011324	15
	0.005	45.39	0.91128	470.54	0.0052653	12	45.39	0.91128	653.55	0.0052653	12
numbers	0.3	17.593	0.33171	520.53	0.27654	31	17.862	0.35167	358.5	0.26337	6
	0.2	20.658	0.39035	415.03	0.18442	30	20.936	0.40758	375.14	0.17526	9
	0.1	26.259	0.44549	365.01	0.092576	30	26.56	0.45925	318.58	0.087618	8
	0.05	32.061	0.47618	466.24	0.046674	30	32.382	0.48698	321.63	0.044044	8
	0.01	45.511	0.51733	1039	0.0099524	31	45.869	0.52176	467.07	0.0093273	7
	0.005	51.036	0.53022	1474.9	0.0053171	30	51.44	0.53149	463.78	0.0049401	5
⌀		32.0368	0.5343	9,597,189		54.5833	32.0828	0.5357	534.877		23.75

and

Table 8. PSNR- and MSSIM-values of the reconstruction of different images corrupted by Gaussian white noise via the ADMM by solving the constrained versions.

Image	$\mathit{\sigma }$	ADMM				Box-Constrained ADMM
		PSNR	MSSIM	Time	It	PSNR	MSSIM	Time	It
phantom	0.3	18.956	0.29789	1245	118	18.922	0.29225	999.39	137
	0.2	21.615	0.37299	1203.8	108	21.538	0.36632	1068.9	123
	0.1	27.021	0.45439	868.12	79	26.941	0.44525	995.99	91
	0.05	32.823	0.5007	576.35	47	32.731	0.4847	650.9	53
	0.01	46.446	0.60023	444.34	23	47.36	0.54521	524.88	27
	0.005	53.734	0.5907	566.53	24	53.59	0.51937	913.28	38
cameraman	0.3	22.33	0.18003	889.65	99	22.2	0.17829	897.82	127
	0.2	23.895	0.23961	836.26	80	23.852	0.23917	808.67	100
	0.1	27.002	0.35006	690.45	60	26.963	0.35118	716.37	70
	0.05	30.513	0.47359	564.73	43	30.488	0.47503	656.66	48
	0.01	40.486	0.75493	409.62	22	40.542	0.75864	454.45	24
	0.005	45.329	0.8728	494.94	23	45.423	0.87718	501.59	24
barbara	0.3	20.6	0.30878	891.68	94	20.604	0.31256	683.92	112
	0.2	21.655	0.40225	827.72	74	21.654	0.40468	628.45	91
	0.1	24.224	0.5883	583.96	50	24.215	0.58921	547.54	61
	0.05	27.889	0.75319	449.51	35	27.874	0.75405	470.8	41
	0.01	38.921	0.9338	441.87	21	38.978	0.9341	374.69	24
	0.005	44.413	0.97035	503.68	21	44.584	0.97166	425.92	23
house	0.3	23.761	0.19254	1072.7	108	23.689	0.19468	674.76	128
	0.2	25.659	0.24287	949.97	85	25.609	0.24367	709.26	108
	0.1	28.907	0.32034	678	54	28.875	0.32115	596.95	70
	0.05	32.054	0.40609	532.99	35	32.029	0.40732	478.67	42
	0.01	40.394	0.7555	422.07	19	40.438	0.75891	331.36	21
	0.005	45.201	0.87241	492.11	20	45.278	0.8758	408.17	22
lena	0.3	21.834	0.29247	950.81	102	21.85	0.2995	570.59	115
	0.2	23.455	0.36437	724	81	23.46	0.36828	670.96	97
	0.1	26.349	0.49495	584.08	55	26.333	0.49605	641.47	67
	0.05	29.604	0.62425	456.63	39	29.588	0.62541	554.51	44
	0.01	39.422	0.91694	377.76	22	39.486	0.91822	438.7	25
	0.005	45.021	0.97391	470.81	24	45.106	0.97436	517.97	26
bones	0.3	25.829	0.35686	750.12	110	25.051	0.36748	611.23	115
	0.2	27.689	0.40363	734.13	90	27.486	0.41281	652.2	94
	0.1	30.942	0.48823	624.02	58	31.02	0.49267	755.06	75
	0.05	34.319	0.57951	343.83	35	34.37	0.57521	624.7	49
	0.01	43.011	0.73032	193.87	14	42.9	0.72002	270.92	16
	0.005	47.446	0.79549	204.98	12	47.432	0.78519	285.72	14
cookies	0.3	21.436	0.31503	789.78	102	21.461	0.32264	788.24	118
	0.2	23.129	0.40742	713.25	82	23.128	0.41261	772.93	99
	0.1	26.527	0.55478	554.95	55	26.506	0.55871	659.5	70
	0.05	30.364	0.67447	422.84	37	30.338	0.67746	589.77	45
	0.01	40.513	0.85593	234.35	15	40.589	0.85815	316.64	19
	0.005	45.382	0.90918	255.79	14	45.307	0.90892	319.82	16
numbers	0.3	17.257	0.32157	1640.7	147	17.487	0.34599	1311.9	146
	0.2	20.279	0.38382	1594	133	20.49	0.42	1215.8	134
	0.1	25.901	0.44284	981.06	95	26.125	0.48656	874.87	85
	0.05	31.847	0.47604	719.72	52	32.412	0.51241	532.91	46
	0.01	45.726	0.52365	643.88	26	47.089	0.51102	1001.3	56
	0.005	52.512	0.53468	641.43	29	52.859	0.51942	1756.9	83
⌀		32.0754	0.5386	671.7274	57.7292	32.1302	0.5389	671.9570	67.8958

we summarize our findings for image denoising. We observe that adding a box-constraint to the considered optimization problem leads to a possibly slight improvement in PSNR and MSSIM. While in some cases there is some improvement (see for example the image “numbers”) in other examples no improvement is gained (see for example the image “barbara”). In order to make the overall improvement more visible, in the last row of Table 7 and Table 8 we add the average PSNR and MSSIM of all computed restorations. It shows that on average we may expect a gain of around 0.05 PSNR and around 0.001 MSSIM, which is nearly nothing. Moreover, we observe, that the pAPS-algorithm computes the optimal $\alpha$ for the box-constrained problem on average faster than the one for the non-box-constrained problem. We remark, that the box-constrained version needs less (or at maximum the same amount of) iterations as the version with $\eta =0$. The reason for this might be that in each iterations, due to the thresholding of the approximation by the box-constraint, a longer or better step towards the minimizer than by the non-box-constrained pAPS-algorithm is performed. At the same time also the reconstructions of the box-constrained pAPS-algorithm yield higher PSNR and MSSIM than the ones obtained by the pAPS-algorithm with $\eta =0$. The situation seems to be different for the ADMM. On average, the box-constrained ADMM and the (non-box-constrained) ADMM need approximately the same run-time.

For several examples (i.e., the images “phantom”, “cameraman”, “barbara”, “house”) the choice of the regularization parameter by the box-constrained pAPS-algorithm with respect to the noise-level is depicted in Figure 4. Clearly, the parameter is selected to be smaller the less noise is present in the image.

We are now wondering whether a box-constraint is more important when the regularization parameter is non-scalar, i.e., when $\alpha :\mathsf{\Omega }\to {\mathbb{R}}^{+}$ is a function. For computing suitable locally varying $\alpha$ we use the pLATV-algorithm proposed in [36], whereby we set in all considered examples the initial (non-scalar) regularization parameter to be constant ${10}^{-2}$. Note, that the continuity assumption on $\alpha$ in problem (5) and problem (6) is not needed in our discrete setting, since ${\sum }_{x\in {\mathsf{\Omega }}^h}\alpha \left(x\right)\left|{\nabla }^h{u}^h\left(x\right)\right|$ is well defined for any $\alpha \in {\mathbb{R}}^N$. We approximate such $\alpha$ for problem (20) with $\eta =0$ (unconstrained) and with $\eta ={10}^6$ (box-constrained) and obtain also here that the gain with respect to PSNR and MSSIM is of the same order as in the scalar case, see

Table 9. PSNR- and MSSIM-values of the reconstruction of different images corrupted by Gaussian white noise via pLATV using the primal-dual Newton method.

Image	$\mathit{\sigma }$	pLATV with pdN with $\mathit{\eta }=0$				pLATV with Box-Constrained pdN
		PSNR	MSSIM	Time	It	PSNR	MSSIM	Time	It
phantom	0.3	19.629	0.31986	517.78	13	19.744	0.32143	710.59	16
	0.2	22.405	0.38578	286.66	13	22.525	0.38706	607.27	15
	0.1	27.802	0.44701	389.34	13	27.936	0.44762	552.49	14
	0.05	33.48	0.47301	526.89	13	33.57	0.4733	571.06	14
	0.01	46.546	0.50749	139.72	4	46.625	0.50646	98.973	3
	0.005	52.038	0.52569	112.1	3	52.062	0.52512	105.89	3
cameraman	0.3	22.382	0.18185	775.5	15	22.393	0.18186	838.55	19
	0.2	24.032	0.24038	605.56	13	24.029	0.23937	593.9	16
	0.1	27.16	0.35301	304.83	12	27.175	0.35285	415.09	13
	0.05	30.702	0.4745	310.03	10	30.696	0.47315	262.59	10
	0.01	40.647	0.73386	135.88	4	40.647	0.73386	144.01	4
	0.005	45.292	0.86678	365.19	7	45.292	0.86678	357.42	7
barbara	0.3	20.527	0.30278	572.44	16	20.516	0.3023	652.83	18
	0.2	21.73	0.39872	411.5	13	21.729	0.39882	459.09	14
	0.1	24.503	0.58885	285.8	9	24.486	0.58693	302.46	10
	0.05	28.196	0.75135	309.53	7	28.198	0.75151	304.89	7
	0.01	38.898	0.92794	424.8	18	38.898	0.92794	425.12	18
	0.005	44.186	0.96871	147.57	5	44.186	0.96871	150.91	5
house	0.3	23.661	0.18529	593.22	18	23.704	0.18526	698.35	22
	0.2	25.507	0.23789	478.33	17	25.51	0.23741	522.86	18
	0.1	28.736	0.32695	304.43	13	28.741	0.32581	332.47	14
	0.05	31.94	0.4217	184.75	11	31.943	0.42182	320.22	11
	0.01	40.423	0.73752	451.47	13	40.423	0.73752	377.89	13
	0.005	45.118	0.85458	286.48	7	45.118	0.85458	222.54	7
lena	0.3	21.829	0.29245	624.96	19	21.828	0.29232	727.69	21
	0.2	23.442	0.3634	471.91	16	23.445	0.36326	538.81	17
	0.1	26.403	0.49438	347.04	13	26.406	0.49419	374.1	14
	0.05	29.703	0.62778	396.43	8	29.704	0.62784	403.49	8
	0.01	39.324	0.91256	652.2	16	39.324	0.91256	659.92	16
	0.005	44.736	0.97081	244.61	5	44.737	0.97081	237.28	5
bones	0.3	25.633	0.36194	869.22	37	25.518	0.35855	1122.8	38
	0.2	27.6	0.41895	665.85	33	27.506	0.41674	823.95	33
	0.1	30.964	0.51162	419.72	26	30.862	0.51085	435.6	24
	0.05	34.413	0.59991	291.13	19	34.408	0.59947	374.07	19
	0.01	43.521	0.76261	98.45	6	43.521	0.76261	93.765	6
	0.005	47.564	0.7997	103.29	6	47.564	0.7997	95.637	6
cookies	0.3	21.415	0.31722	597.85	17	21.371	0.3164	653.83	18
	0.2	23.126	0.40895	500.29	16	23.107	0.4087	485.2	15
	0.1	26.487	0.55758	237.87	12	26.501	0.55811	332.12	12
	0.05	30.301	0.68332	413.36	12	30.295	0.68292	175.11	9
	0.01	40.525	0.86052	261.76	8	40.525	0.86052	236.96	8
	0.005	45.499	0.90364	229.85	5	45.499	0.90364	174.58	5
numbers	0.3	17.627	0.33232	421.86	11	17.821	0.34957	654.22	20
	0.2	20.7	0.39072	313.32	10	20.888	0.40559	556.07	19
	0.1	26.296	0.4455	258.05	10	26.528	0.45766	440.13	17
	0.05	32.104	0.47624	339.03	10	32.369	0.48578	501.47	15
	0.01	45.527	0.51735	92.644	2	45.938	0.52169	59.513	2
	0.005	51.265	0.53078	207.86	5	51.536	0.53125	257.45	6
⌀		32.1155	0.5365	374.5490	12.2708	32.1531	0.5375	425.8590	13.4167

.

For $\sigma =0.1$ and the image “barbara” we show in Figure 5 the reconstructions generated by the considered algorithms. As indicated by the quality measures, all the reconstructions look nearly alike, whereby in the reconstructions produced by the pLATV-algorithm details, like the pattern of the scarf, are (slightly) better preserved. The spatially varying $\alpha$ of the pLATV-algorithm is depicted in Figure 6. There we clearly see, that at the scarf around the neck and shoulder the values of $\alpha$ are small, allowing to preserve the details better.

6.3.3. Image Deblurring

Now we consider the images in Figure 1a–c, convolve them first with a Gaussian kernel of size $9\times 9$ and standard deviation 3 and then add some Gaussian noise with mean 0 and standard deviation $\sigma$. Here we again compare the results obtained by the pAPS-algorithm, the ADMM, and the pLATV-algorithm for the box-constrained and non-box-constrained problems. Our findings are summarized in

Table 10. PSNR- and MSSIM-values of the reconstruction of different images corrupted by Gaussian blur and Gaussian white noise via pAPS or pLATV using the primal-dual Newton method or via the ADMM by solving the constrained versions.

Image	$\mathit{\sigma }$	pAPS with pdN with $\mathit{\eta }=\mathbf{0}$				pAPS with Box-Constrained pdN
		PSNR	MSSIM	Time	It	PSNR	MSSIM	Time	It
phantom	0.05	16.41	0.21695	11,603	11	16.801	0.23085	33,002	11
	0.01	18.624	0.30997	12,663	10	18.861	0.32574	18,744	8
	0.005	20.554	0.36713	5892.9	6	21.01	0.38225	17,245	9
cameraman	0.05	22.482	0.2288	8328.5	11	22.484	0.22905	24,162	11
	0.01	24.281	0.34554	4799.8	5	24.293	0.34618	13,102	5
	0.005	25.436	0.4035	5611.4	8	25.451	0.40405	12,033	7
barbara	0.05	21.363	0.37833	5422.8	8	21.363	0.37832	10,657	8
	0.01	22.052	0.49312	2820.6	5	22.052	0.49311	6188.1	5
	0.005	23.084	0.57223	4275.5	13	23.084	0.57223	11,121	13
⌀		21.5874	0.3684	6824	8.5556	21.7108	0.37353	16,250	8.5556
Image	$\mathit{\sigma }$	ADMM				Box-Constrained ADMM
		PSNR	MSSIM	Time	It	PSNR	MSSIM	Time	It
phantom	0.05	16.427	0.2691	6243.1	465	16.811	0.27613	8186.6	581
	0.01	18.581	0.3094	34,683	1424	18.809	0.32544	33,141	1327
	0.005	20.505	0.3677	40,801	1426	20.906	0.38285	43,753	1485
cameraman	0.05	22.51	0.2242	2870.1	214	22.501	0.22423	4065.4	305
	0.01	24.236	0.34073	8268.2	428	24.237	0.34082	7185.3	358
	0.005	25.373	0.3986	30,944	1200	25.377	0.3987	26985	1081
barbara	0.05	21.417	0.38039	1996.6	169	21.421	0.38113	2807.4	237
	0.01	22.036	0.48965	13,282	678	22.038	0.48964	12,012	627
	0.005	23.043	0.56871	40,949	1527	23.038	0.56841	37,430	1423
⌀		21.5699	0.37205	20,004	836.78	21.6821	0.37637	19,507	824.89
Image	$\mathit{\sigma }$	pLATV with pdN with $\mathit{\eta }=\mathbf{0}$				pLATV with Box-Constrained pdN
		PSNR	MSSIM	Time	It	PSNR	MSSIM	Time	It
phantom	0.05	16.43	0.21966	14,732	12	16.824	0.23305	29,567	10
	0.01	18.967	0.31788	39,014	49	19.602	0.33679	11,950	52
	0.005	20.729	0.36861	49,658	67	22.041	0.39052	11,940	71
cameraman	0.05	22.533	0.22964	26,380	22	22.534	0.22973	70,208	22
	0.01	24.599	0.34708	57,513	47	24.622	0.34769	113,175	47
	0.005	25.77	0.40389	39,662	60	25.796	0.40451	110,646	60
barbara	0.05	21.377	0.3808	26,779	24	21.377	0.3808	49,808	24
	0.01	22.432	0.50565	22,433	51	22.448	0.50625	78,720	52
	0.005	23.652	0.59532	16,168	61	23.658	0.59545	53,559	61
⌀		21.8319	0.37428	32,482	43.667	22.1003	0.38053	81,893	44.333

. Also here we observe a slight improvement with a box-constraint with respect to PSNR and MSSIM. The choice of the regularization parameters by the box-constrained pAPS-algorithm is depicted in Figure 7. In Figure 8 we present for the image “cameraman” and $\sigma =0.01$ the reconstructions produced by the respective methods. Also here, as indicated by the quality measures, all the restorations look nearly the same. The locally varying $\alpha$ generated by the pLATV-algorithm are depicted in Figure 9.

6.3.4. Image Inpainting

The problem of filling in and recovering missing parts in an image is called image inpainting. We call the missing parts inpainting domain and denote it by $D\subset \mathsf{\Omega }$. The linear bounded operator K is then a multiplier, i.e., $Ku=1_{\mathsf{\Omega }\backslash D}·u$, where $1_{\mathsf{\Omega }\backslash D}$ is the indicator function of $\mathsf{\Omega }\backslash D$. Note, that K is not injective and hence $K^{*}K$ is not invertible. Hence in this experiment we need to set $\mu >0$ so that we can use the proposed primal-dual semismooth Newton method. In particular, as mentioned above, we choose $\mu ={10}^{-6}$.

In the considered experiments the inpainting domain are gray bars as shown in Figure 10a, where additionally additive white Gaussian noise with $\sigma =0.1$ is present. In particular, we consider examples with $\sigma \in \{0.3,0.2,0.1,0.05,0.01,0.005\}$. The performance of the pAPS- and pLATV-algorithm with and without a box-constraint reconstructing the considered examples are summarized in

Table 11. PSNR- and MSSIM-values for the application inpainting via pAPS.

Image	$\mathit{\sigma }$	pAPS with pdN with $\mathit{\eta }=0$					pAPS with Box-Constrained pdN
		PSNR	MSSIM	Time	$\mathit{\alpha }$	It	PSNR	MSSIM	Time	$\mathit{\alpha }$	It
lena	0.3	21.151	0.26378	1709.9	0.37358	105	21.151	0.26378	958.59	0.37358	50
	0.2	22.555	0.33033	1075.8	0.23336	72	22.555	0.33032	578.35	0.23336	32
	0.1	24.922	0.44992	578.34	0.10369	41	24.922	0.44992	370.42	0.10369	18
	0.05	27.005	0.56734	513.1	0.044922	25	27.005	0.56735	507.37	0.044919	25
	0.01	29.618	0.82318	524.73	0.006614	9	29.618	0.82319	516.86	0.0066133	9
	0.005	29.912	0.87427	569.85	0.00319	8	29.912	0.87427	674.02	0.0031896	8
cookies	0.3	20.761	0.27956	1189.8	0.34456	74	20.763	0.27963	806.65	0.34448	35
	0.2	22.138	0.36599	761.34	0.22529	55	22.138	0.36599	228.76	0.22529	10
	0.1	24.624	0.50595	419.34	0.11088	36	24.624	0.50595	283.69	0.11088	15
	0.05	26.967	0.62721	359.67	0.05467	26	26.967	0.62721	409.44	0.05467	26
	0.01	30.05	0.80847	481.89	0.01008	14	30.05	0.80847	547.4	0.01008	14
	0.005	30.438	0.85701	491.23	0.0047745	11	30.438	0.85701	590.77	0.0047745	11
⌀		25.83	0.57403	617.22		36	25.83	0.57404	477.78		18.5

and

Table 12. PSNR- and MSSIM-values for the application inpainting via pLATV.

Image	$\mathit{\sigma }$	pLATV with pdN with $\mathit{\eta }=0$				pLATV with Box-Constrained pdN
		PSNR	MSSIM	Time	It	PSNR	MSSIM	Time	It
lena	0.3	21.027	0.26266	684.18	14	21.036	0.26239	836.18	16
	0.2	22.45	0.32986	555.66	11	22.457	0.32939	584.66	12
	0.1	24.893	0.4498	446.84	12	24.868	0.45004	400.3	9
	0.05	26.982	0.56904	474.61	9	26.983	0.56912	460.22	9
	0.01	29.621	0.82242	796.53	16	29.621	0.82242	775.88	16
	0.005	29.987	0.87461	285.24	5	29.987	0.87461	284.79	5
cookies	0.3	20.546	0.27499	660.05	11	20.548	0.27574	753.89	13
	0.2	21.965	0.36179	458.1	10	21.975	0.36259	578.94	11
	0.1	24.538	0.50627	193.2	10	24.547	0.50651	251.68	11
	0.05	26.862	0.63007	322.75	7	26.863	0.6301	381.66	7
	0.01	30.047	0.80869	83.563	2	30.047	0.80869	99.415	2
	0.005	30.254	0.84914	181.44	5	30.254	0.84914	213.15	5
⌀		25.7643	0.56161	428.51	9.3333	25.765	0.56173	468.3961	9.6667

. We observe, that adding a box-constraint does not seem to change the restoration considerably. However, as in the case of image denoising, the pAPS-algorithm with box-constrained pdN needs less iterations and hence less time than the same algorithm without a box-constraint to reach the stopping criterion. Figure 10 shows a particular example for image inpainting and denoising with $\sigma =0.1$. It demonstrates that visually there is nearly no difference between the restoration obtained by the considered approaches. Moreover, we observe that the pLATV-algorithm seems to be not suited to the task of image inpainting. A reason for this might be, that the pLATV-algorithm does not take the inpainting domain correctly into account. This is visible in Figure 11 where the spatially varying $\alpha$ seems to be chosen small in the inpainting domain, which not necessarily seems to be a suitable choice.

6.3.5. Reconstruction from Partial Fourier-Data

In magnetic resonance imaging one wishes to reconstruct an image which is only given by partial Fourier data and additionally distorted by some additive Gaussian noise with zero mean and standard deviation $\sigma$. Hence, the linear bounded operator is $K=S\circ \mathcal{F}$, where $\mathcal{F}$ is the $2D$ Fourier matrix and S is a downsampling operator which selects only a few output frequencies. The frequencies are usually sampled along radial lines in the frequency domain, in particular in our experiments along 32 radial lines, as visualized in Figure 12.

In our experiments we consider the images of Figure 2, transformed to its Fourier frequencies. As already mentioned, we sample the frequencies along 32 radial lines and add some Gaussian noise with zero mean and standard deviation $\sigma$. In particular, we consider different noise-levels, i.e., $\sigma =\{0.3,0.2,0.1,0.05,0.01,0.005\}$. We reconstruct the obtained data via the pAPS- and pLATV-algorithm by using the semi-smooth Newton method first with $\eta =0$ (no box-constraint) and then with $\eta ={10}^6$ (with box-constraint). In

Table 13. PSNR- and MSSIM-values of the reconstruction of sampled Fourier data corrupted by Gaussian white noise via the pAPS- and pLATV-algorithm using the primal-dual Newton method.

Image	$\mathit{\sigma }$	pAPS with pdN with $\mathit{\eta }=\mathbf{0}$			pAPS with Box-Constrained pdN
		PSNR	MSSIM	CPU-Time	PSNR	MSSIM	CPU-Time
Shepp-	0.3	18.888	0.16233	3787.2	19.000	0.1685	5509.2
Logan	0.2	20.524	0.21302	2844.9	20.696	0.22086	3673.5
phantom	0.1	24.256	0.2905	1884.7	24.496	0.29896	2582.3
	0.05	28.639	0.34972	2008.5	28.948	0.35833	2115
	0.01	40.168	0.42734	1993.3	40.711	0.43325	1349.5
	0.005	45.263	0.44714	2225.4	45.933	0.45199	951.49
knee	0.3	21.606	0.26553	22,466	21.606	0.26553	36,054
	0.2	22.985	0.30965	15,705	22.985	0.30965	31,072
	0.1	25.017	0.37061	11,561	25.017	0.37056	24,994
	0.05	26.443	0.41652	8803.4	26.445	0.41661	21,056
	0.01	27.912	0.47141	4996.9	27.959	0.47267	11,707
	0.005	28.035	0.47683	6076.9	28.089	0.47843	13,116
⌀		27.4781	0.35005	7029.4365	27.657	0.35378	12,848.2064
Image	$\mathit{\sigma }$	pLATV with pdN with $\mathit{\eta }=\mathbf{0}$			pLATV with Box-Constrained pdN
		PSNR	MSSIM	CPU-Time	PSNR	MSSIM	CPU-Time
Shepp-	0.3	18.99	0.16078	5445.3	17.148	0.11219	15,500
Logan	0.2	20.567	0.21006	3179.1	19.324	0.17395	11,719
phantom	0.1	24.376	0.29028	2491.5	23.51	0.27083	4623.8
	0.05	28.569	0.34645	1926.1	28.303	0.34392	7125.8
	0.01	39.475	0.41775	266.7	39.579	0.42053	695.74
	0.005	43.782	0.43085	465.09	43.627	0.43096	1373.9
knee	0.3	15.583	0.18089	17,413	16.011	0.186	17,750
	0.2	18.87	0.24419	11,640	19.227	0.25069	14,414
	0.1	23.525	0.34652	3663.5	23.64	0.34945	9220.4
	0.05	26.307	0.41393	1545.6	26.341	0.4159	4165.6
	0.01	27.044	0.46069	4091.1	27.055	0.4612	12,059
	0.005	24.773	0.41841	10,499	24.639	0.4172	34,409
⌀		25.9885	0.3267	5218.8	25.7003	0.3194	11,088

we collect our findings. We observe that the pLATV-algorithm seems not to be suitable for this task, since it is generating inferior results. For scalar $\alpha$ we observe as before, that a slight improvement with respect to PSNR and MSSIM is expectable when a box-constraint is used. In Figure 13 we present the reconstructions generated by the considered algorithms for a particular example, demonstrating the visual behavior of the methods.

6.3.6. Reconstruction from Sampled Radon-Data

In computerized tomography instead of a Fourier-transform a Radon-transform is used in order to obtain a visual image from the measured physical data. Also here the data is obtained along radial lines. Here we consider the Shepp-Logan phantom, see Figure 14a, and a slice of a body, see Figure 15a. The sinogram in Figure 14a and Figure 15b are obtained by sampling along 30 and 60 radial lines, respectively, Note, that the sinogram is in general noisy. Here the data is corrupted by Gaussian white noise with standard deviation $\sigma$, whereby $\sigma =0.1$ for the data of the Shepp-Logan phantom and $\sigma =0.05$ for the data of the slice of the head. Using the inverse Radon-transform we obtain Figure 16a,b, which is obviously a suboptimal reconstruction. A more sophisticated approach utilizes the $L^2$-TV model which yields the reconstruction depicted in Figure 16b,e, where we use the pAPS-algorithm and the proposed primal-dual algorithm with $\eta =0$. However, since an image can be assumed to have non-negative values, we may incorporate a non-negativity constraint via the box-constrained $L^2$-TV model yielding the result in Figure 16c,f, which is a much better reconstruction. Also here the parameter $\alpha$ is automatically computed by the pAPS-algorithm and the non-negativity constraint is incorporated by setting $\eta ={10}^6$ in the semi-smooth Newton method. In order to compute the Radon-matrix in our experiments we used the FlexBox [52].

Other applications where a box-constraint, and in particular a non-negativity improves the image reconstruction quality significantly include for example magnetic particle imaging, see for example [53] and references therein.

7. Automated Parameter Selection

We recall, that if the noise-level $\sigma$ is not known, then the problems (1) and (3) cannot be considered. Moreover, the selection of the parameter $\alpha$ in problem (2) cannot be achieved by using the pAPS-algorithm, since this algorithm is based on problem (1). Note, that also other methods, like the unbiased predictive risk estimator method (UPRE) [54,55] and approaches based on the Stein unbiased risk estimator method (SURE) [56,57,58,59,60] use knowledge of the noise-level and hence cannot be used for selecting a suitable parameter if $\sigma$ is unknown.

If we assume that $\sigma$ is unknown but the image intensity range of the original image $\widehat{u}$ is known, i.e., $\widehat{u}\in [c_{min},c_{max}]$, then we may use this information for choosing the parameter $\alpha$ in problem (2). This may be performed by applying the following algorithm:

Box-constrained automatic parameter selection (bcAPS): Initialize ${\alpha }_0>0$ (sufficiently small) and set $n:=0$    1. Solve $u_n\in arg{min}_{u\in BV\left(\mathsf{\Omega }\right)}{\parallel Ku-g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+{\alpha }_n{\int }_{\mathsf{\Omega }}\left|Du\right|$.  2. If $u_n\notin [c_{min},c_{max}]$ increase ${\alpha }_n$ (i.e., ${\alpha }_{n+1}:=\tau {\alpha }_n$ with $\tau >1$), else STOP.  3. Set $n:=n+1$ and continue with step 1.

Here $\tau >1$ is an arbitrary parameter chosen manually such that the generated restoration u is not over-smoothed, i.e., there exist $x\in \mathsf{\Omega }$ such that $u\left(x\right)\approx c_{min}$ and/or $u\left(x\right)\approx c_{max}$. In our experiments it turned out that $\tau =1.05$ seems to be a reasonable choice, so that the generated solution has the wished property.

Numerical Examples

In our experiments the minimization problem in step 1 of the bcAPS algorithm is approximately solved by the proposed primal-dual semi-smooth Newton method with $\eta =0$. We set the initial regularization parameter ${\alpha }_0={10}^{-4}$ for image denoising and ${\alpha }_0={10}^{-3}$ for image deblurring. Moreover, we set $\tau =1.05$ in the bcAPS-algorithm to increase the regularization parameter.

Experiments for image denoising, see

Table 14. PSNR- and MSSIM-values of the reconstruction of the cameraman-image corrupted by Gaussian white noise with standard deviation $\sigma$ via the bcAPS algorithm using the primal-dual Newton method with $\eta =0$.

$\mathit{\sigma }$	bcAPS				pAPS
	PSNR	MSSIM	Time	$\mathit{\alpha }$	PSNR	MSSIM	Time	$\mathit{\alpha }$
0.3	22.230	0.17478	1065.8	0.381058	22.485	0.18619	1591	0.34586
0.2	23.637	0.22552	1084.9	0.245634	24.054	0.24497	919.63	0.21408
0.1	26.621	0.32588	840.2	0.112528	27.132	0.35201	580.91	0.096108
0.05	29.388	0.41062	817.9	0.059676	30.567	0.47437	549.23	0.043393
0.01	39.332	0.70321	552.4	0.009346	40.417	0.75235	677.79	0.0071847
0.005	44.508	0.84591	415.6	0.003883	45.164	0.8686	745.67	0.0032996

, show that the bcAPS-algorithm finds suitable parameters in the sense that the PSNR and MSSIM of these reconstructions is similar to the ones obtained with the pAPS-algorithm (when $\sigma$ is known); also compare with Table 7 and Table 10. This is explained by the observation that also the regularization parameters $\alpha$ calculated by the bcAPS-algorithm do not differ much from the ones obtained via the pAPS-algorithm. For image deblurring, see

Table 15. PSNR- and MSSIM-values of the reconstruction of the cameraman-image corrupted Gaussian blur and Gaussian white noise with standard deviation $\sigma$ via the bcAPS algorithm using the primal-dual Newton method with $\eta =0$.

$\mathit{\sigma }$	bcAPS				pAPS
	PSNR	MSSIM	Time	$\mathit{\alpha }$	PSNR	MSSIM	Time	$\mathit{\alpha }$
0.05	22.304	0.21524	76,216.3	0.027598	22.482	0.2288	8328.5	0.019529
0.01	22.956	0.26523	86,068.9	0.010401	24.281	0.34554	4799.8	0.0012714
0.005	23.024	0.27018	96,837.0	0.009434	25.436	0.4035	5611.4	0.00031872

, the situation is not so persuasive. In particular, the obtained regularization parameter of the two considered methods differ more significantly than before, resulting in different PSNR and MSSIM. However, in the case $\sigma =0.05$ the considered quality measures of the generated reconstructions are nearly the same.

We also remark, that in all the experiments the pAPS-algorithm generated reconstructions, which have larger PSNR and MSSIM than the ones obtained by the bcAPS-algorithm. From this observation it seems more useful to know the noise-level than the image intensity range. However, if the noise-level is unknown but the image intensity is known, then the bcAPS-algorithm may be a suitable choice.

8. Conclusions

In this work we investigated the quality of restored images when the image intensity range of the original image is additionally incorporated into the $L^2$-TV model as a box-constraint. We observe that this box-constraint may indeed improve the quality of reconstructions. However, if the observation already fulfills the box-constraint, then it clearly does not change the solution at all. Moreover, in a lot of applications the proper choice of the regularization parameter seems much more important than an additional box-constraint. Nevertheless, also then a box-constraint may improve the quality of the restored image, although the improvement is then only very little. On the contrary the additional box-constraint may improve the computational time significantly. In particular, for image deblurring and in magnetic resonance imaging using the pAPS-algorithm the computational time is about doubled, while the quality of the restoration is basically not improved. This suggests, that for these applications an additional box-constraint may not be reasonable. Note, that the run-time of the ADMM is independent whether a box-constraint is used or not.

For certain applications, as in computerized tomography, a box-constraint (in particular a non-negativity constraint) improves the reconstruction considerably. Hence, the question rises under which conditions an additional box-constraint indeed has significant influence on the reconstruction when the present parameters are chosen in a nearly optimal way.

If the noise-level of an corrupted image is unknown but the image intensity range of the original image is at hand, then the image intensity range may be used to calculate a suitable regularization parameter $\alpha$. This can be done as explained in Section 7. Potential future research may consider different approaches, as for example in an optimal control setting. Then one may want to solve

$$\begin{array}{c}\underset{u,\alpha }{min}\parallel max\{u-c_{max},0\}{\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+{\parallel max\{c_{min}-u,0\}\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\kappa J\left(\alpha \right)\hfill \\ \mathrm{s}.\mathrm{t}.u\in arg\underset{u}{min}{\parallel Ku-g\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\alpha {\int }_{\mathsf{\Omega }}\left|Du\right|,\hfill \end{array}$$

where $\kappa >0$ and J is a suitable functional, cf. [61,62,63] for other optimal control approaches in image reconstruction.