Variational Model with Nonstandard Growth Condition in Image Restoration and Contrast Enhancement

Abstract

We propose a new variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to the simultaneous contrast enhancement and denoising of color images. The characteristic feature of the proposed model is that we deal with a constrained non-convex minimization problem that lives in variable Sobolev-Orlicz spaces where the variable exponent is unknown a priori and it depends on a particular function that belongs to the domain of the objective functional. In contrast to the standard approach, we do not apply any spatial regularization to the image gradient. We discuss the consistency of the variational model, give the scheme for its regularization, derive the corresponding optimality system, and propose an iterative algorithm for practical implementations.

1 Introduction

A very promising approach to the image quality enhancement is to reduce the influence of the noise and improve the perceptibility of objects in the scene by increasing the brightness difference between objects and their background. In recent years, many contrast enhancement techniques have been proposed for digital images. Some approaches allow to improve the image contrast just in low light conditions [39, 44]. Other methods, called sharpening, focus on enforcing strong contours to remove the obtained blur, e.g., by the Gaussian convolution [38]. However, this kind of enhancement concerns only strong image contours while the contrast enhancement attempts to modify the gray level of objects not only in the contours neighborhood. In recent years, many different techniques have been proposed for the reconstruction of noise-affected digital images and their contrast enhancement. We refer to Ref. [40], where the authors focused on the problem of contrast enhancement of natural images captured with a digital camera, and gave a sufficiently complete overview of the existing methods with a detailed analysis of all pros and cons.

In this paper, we mainly focus on the development of a variational approach for simultaneous contrast enhancement of color images and their denoising. With that in mind, we propose a new variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to the simultaneous fusion and denoising of each spectral channel for input color images. In contrast to Ref. [24], we do not provide the color image restoration using the saturation-value Total Variation (TV), but instead, we are working just with the RGB representation of color images. However, as follows from the results of numerical simulations, the proposed approach does not strongly modify the histogram of the original image. This enables the model to preserve the global lighting sensation and to show that the hue of the main objects does not drastically change with the illumination. One of the most important advantages of this approach is the fact that the proposed model allows to synthesize at a high level of accuracy noise- and blur-free color images, that were captured in extremely low light conditions. This situation is typical for most remote sensing problems. Indeed, real-life satellite images frequently suffer from different types of noise, blur, and other atmosphere artifacts that can affect the radiation recovered by the sensors. As a result, such images lose their efficiency for the crop field monitoring problems and their utilization can lead to erroneous results and inferences.

The characteristic feature of the proposed model is that we deal with a constrained minimization problem with a special objective functional that lives in variable Sobolev-Orlicz spaces. This functional contains a spatially variable exponent characterizing the growth conditions and it can be seen as a replacement for the 1-norm in the TV regularization. Moreover, the variable exponent, which is associated with the non-standard growth, is unknown a priori and it depends on a particular function that belongs to the domain of the objective functional.

The idea of using a spatially varying exponent in a TV-like regularization method for image denoising dates back as early as 1997 [7] and it was put into practice in 2006 [11]. Both papers as well as some subsequent articles try to tackle variants of the problem

$$\begin{aligned} J(u)={\mathcal {D}}(u)+\lambda \int _{\Omega } |\nabla u(x)|^{p(\nabla u(x))}\,{\textrm{d}}x\longrightarrow \inf , \end{aligned}$$

(1)

where the exponent depends directly on the image u,  e.g.,

$$\begin{aligned} p(\nabla u)=1+ \frac{a^2}{a^2+|\nabla G_\sigma *u|^2}. \end{aligned}$$

(2)

Here, $\left( G_\sigma *v\right) (x)$ determines the convolution of the function v with the two-dimensional Gaussian filter kernel $G_\sigma .$

It has been demonstrated that this model possesses some favorable properties, particularly, when the edge preservation and the effective noise suppression are primary goals in image reconstruction. Furthermore, this model has been introduced specifically to address the issue of staircasing [42], which refers to the regularizer’s inclination towards piecewise constant functions. The appearance of the staircase effect is a notable drawback of the classical TV model. However, the non-convex model (1) did not gain significant attention for a long period due to its high numerical complexity and the absence of a rigorous mathematical substantiation of its consistency. Only particular solutions to this problem have been derived for a smoothed version of the integrand, using a weak notion of solution (see, for instance, Ref. [45]).

A recently developed alternative variant is the TV-like method [33] (see also Refs. [1, 12]), which computes the variable exponent p in an offline step and keeps it as a fixed parameter in the final optimization problem. This approach allows the exponent to vary based on the spatial location, enabling users to locally select whether to preserve edges or smooth intensity variations. However, there are only two natural types of imaging problems where this approach can be applied:

• single-channel imaging where first the exponent is computed from the given data and then is applied as prior in the subsequent minimization problem;

• dual-channel imaging where the secondary channel provides the exponent map that is used for regularization of the primary channel.

Thus, this circumstance imposes significant limitations from a practical point of view, especially in the case of multispectral satellite noisy images, where different channels can differ drastically.

The main purpose of this paper is to describe a robust approach for the simultaneous contrast enhancement and denoising of non-smooth multispectral images using an energy functional with non-standard growth, in particular a special form of anisotropic diffusion tensor for the regularization term and a term which is inspired by the variational model of Bertalmío et al. [5]. Following this approach, we aim to increase the perceptibility of objects in the scene and the noise robustness of the proposed model albeit it makes such variational problems completely non-smooth, non-convex, and, hence, significantly more difficult from a minimization point of view.

We consider a variational problem for the energy functional with non-standard growth p(x),  where the principle edge information for the contrast enhancement is mainly accumulated. Namely, for the simultaneous denoising and contrast enhancement of color images, we propose to solve the following constrained minimization problems:

$$\begin{aligned} J_i(f^0_i)=\inf _{v\in \Xi _i} J_i(v),\quad i=1,2,3 \end{aligned}$$

(3)

for each spectral channel of an input image separately, where the objective functional is non-convex and takes the form

$$\begin{aligned} J_i(v)=\int _{\Omega } |R_{\eta }\nabla v(x)|^{p(|\nabla v|)}\,{\textrm{d}}x +Q_i(v). \end{aligned}$$

(4)

Here, $Q_i(v)$ stands for the fidelity term and its specific form is described in detail in Sect. 3 together with the operator $R_{\eta }.$ The principle point that should be emphasized is the fact that we do not predefine the exponent p(x) a priori using for that the original image, but instead, we associate this characteristic with the current state of their spectral channels. We take it as follows:

$$\begin{aligned} p(\nabla u)=1+ \frac{a^2}{a^2+|\nabla u|^2}. \end{aligned}$$

(5)

Therefore, in contrast to the well-known approach, coming from the pioneering papers [2, 10], the principle difference of the models (5) and (2) is that we do not apply in (5) any spatial regularization of gradient $\nabla u.$ Because of this, the model (3)–(4) becomes an ill-posed problem from the mathematical point of view and can produce many unexpected phenomena. In particular, to our best knowledge, we have no results of the existence and consistency of the optimization problem (3)–(4). To overcome this problem, we could apply some regularization of the variable exponent p(x) in the form like (2).

However, it is well known that optimization problem (3)–(4) with the spatially regularized gradient has several serious practical and theoretical difficulties. The first one is that the spatial regularization of the gradient in the form (2) leads to the loss of accuracy in the case when the signal is noisy, with white noise (see, for instance, Ref. [10]). Then, the noise introduces very large, in theory unbounded, oscillations of the gradient $\nabla u.$ As a result, the conditional smoothing introduced by the model will not help, since all these noise edges will be kept.

The second drawback of the model with the regularized gradient is the fact that the space-invariant Gaussian smoothing inside the divergent term tends to push the edges in u away from their original locations. We refer to Ref. [41] where this issue was studied in detail. This effect, known as the edge dislocation, can be detrimental, especially in the context of the boundary detection problem and its application to remote the sensing and monitoring.

In view of this, our prime interest in this paper is to study the optimization problem (3)–(4) without the space-invariant Gaussian smoothing of the variable exponent p(x). In summary, the main contributions of our paper can be enumerated as follows.

• The variational statement for the simultaneous contrast enhancement and denoising of multispectral images in the form of the minimization problem in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional.

• Rigorous substantiation of the well-posedness of the variational problem with the non-standard growth functional.

• The proof of existence results to the approximation variational problems.

• The iterative algorithm for numerical implementations.

• Derivation of the first order necessary conditions for the original problem and their substantiation.

• Numerical experiments to study the performance of the new approach.

The remainder of the paper is organized as follows. In Sect. 2, we give preliminaries and some auxiliaries results. In Sect. 3, we present a novel variational problem for the denoising and contrast enhancement of non-smooth RGB images which can be viewed as an improved version of the variational model that has been recently proposed in Ref. [17]. Section 4 is devoted to the derivation of optimality conditions to the original problem and their substantiation. In Sect. 5, we discuss the possible ways for the relaxation of the minimization problem and its approximation. Specifically, we introduce a family of special minimization problems and show that each of these problems is solvable and their solutions are compact in an appropriate topology. We also discuss their approximating properties and give a convergence criterion of such sequence to an optimal solution of the original problem. Finally, for illustration, we give in Sect. 6 some results of numerical experiments.

2 Preliminaries

Let us recall some useful notations. For vectors $\xi \in {\mathbb {R}}^2$ and $\eta \in {\mathbb {R}}^2,$ $\left( \xi ,\eta \right) =\xi ^\textrm{t}\eta$ denotes the standard vector inner product in ${\mathbb {R}}^2,$ where $^\textrm{t}$ stands for the transpose operator. The norm $|\xi |$ is the Euclidean norm given by $|\xi |=\sqrt{(\xi ,\xi )}.$ Let $\Omega \subset {\mathbb {R}}^2$ be a bounded open set with a Lipschitz boundary $\partial \Omega$ and nonzero Lebesgue measure. For any subset $E\subset \Omega$ we denote by |E| its two-dimensional Lebesgue measure ${\mathcal {L}}^2(E).$ Let ${\overline{E}}$ denote the closure of E,  and $\partial E$ stands for its boundary. Let $\Omega \subset {\mathbb {R}}^2$ be a bounded connected open set with a sufficiently smooth boundary $\partial \Omega .$

2.1 Functional Spaces

For the convenience of the reader, we collect here the basic facts on functional spaces that will be used in the sequel. Let X denote a real Banach space with norm $\Vert \cdot \Vert _X,$ and let $X^\prime$ be its dual. Let $\left<\cdot ,\cdot \right>_{X^\prime ;X}$ be the duality form on $X^\prime \times X.$ By $\rightharpoonup$ and ${\mathop {\rightharpoonup }\limits ^{*}}$ we denote the weak and weak$^*$ convergence in normed spaces, respectively. For given $1 \leqslant p \leqslant +\infty ,$ the space $L^p(\Omega ;{\mathbb {R}}^2)$ is defined by

$$\begin{aligned} L^p(\Omega ;{\mathbb {R}}^2)=\left\{ f\!\!:\Omega \rightarrow {\mathbb {R}}^2\!\! :\ \Vert f\Vert _{L^p(\Omega ;{\mathbb {R}}^2)}<+\infty \right\} , \end{aligned}$$

where $\Vert f\Vert _{L^p(\Omega ;{\mathbb {R}}^2)}=\left( \int _{\Omega } |f(x)|^p\,{\textrm{d}}x\right) ^{1/p}$ for $1 \leqslant p<+\infty .$ The inner product of two functions f and g in $L^p(\Omega ;{\mathbb {R}}^2)$ with $p\in [1,\infty )$ is given by

$$\begin{aligned} \left( f,g\right) _{L^p(\Omega ;{\mathbb {R}}^2)}=\int _{\Omega } \left( f(x),g(x)\right) \!{\textrm{d}}x=\int _{\Omega } \sum _{k=1}^2 f_k(x) g_k(x){\textrm{d}}x. \end{aligned}$$

We denote by $C_c^\infty ({\mathbb {R}}^2)$ a locally convex space of all infinitely differentiable functions with compact support in ${\mathbb {R}}^2.$ We recall here some functional spaces that will be used throughout this paper. We define the Banach space $H^{1}(\Omega )$ as the closure of $C^\infty _c({\mathbb {R}}^2)$ with respect to the norm

$$\begin{aligned} \Vert y\Vert _{H^{1}(\Omega )}=\left( \int _{\Omega } \left( y^2+|\nabla y|^2\right) {\textrm{d}}x\right) ^{1/2}. \end{aligned}$$

We denote by $\left( H^{1}(\Omega )\right) ^\prime$ the dual space of $H^{1}(\Omega ).$ Hereinafter, $W^{1,1}(\Omega )$ stands for the Banach space of all functions $u\in L^1(\Omega )$ with respect to the norm

$$\begin{aligned} \Vert u\Vert _{W^{1,1}(\Omega )}=\Vert u\Vert _{L^1(\Omega )}+\Vert \nabla u\Vert _{L^1(\Omega )^2}. \end{aligned}$$

Given a real Banach space X,  we will denote by C([0, T]; X) the space of all continuous functions from [0, T] into X. We recall that a function $u\!:[0,T]\rightarrow X$ is said to be Lebesgue measurable if there exists a sequence $\left\{ u_k\right\} _{k\in {\mathbb {N}}}$ of step functions (i.e., $u_k=\sum _{j=1}^{n_k} a_j^k \chi _{A_j^k}$ for a finite number $n_k$ of Borel subsets $A_j^k\subset [0,T]$ and with $a_j^k\in X$) converging to u almost everywhere with respect to the Lebesgue measure in [0, T].

Then, for $1\leqslant p<\infty ,$ $L^p(0,T;X)$ is the space of all measurable functions $u\!:[0,T]\rightarrow X$ such that

$$\begin{aligned} \Vert u\Vert _{L^p(0,T;X)}=\left( \int _0^T \Vert u(t)\Vert ^p_X\,{\textrm{d}}t\right) ^{\frac{1}{p}}<\infty , \end{aligned}$$

while $L^\infty (0,T;X)$ is the space of measurable functions such that

$$\begin{aligned} \Vert u\Vert _{L^\infty (0,T;X)}=\sup _{t\in [0,T]}\Vert u(t)\Vert _X<\infty . \end{aligned}$$

A full presentation of this topic can be found in Ref. [21].

Let us recall that, for $1\leqslant p\leqslant \infty ,$ $L^p(0,T;X)$ is a Banach space. Moreover, if X is separable and $1\leqslant p<\infty ,$ then the dual space of $L^p(0,T;X)$ can be identified with $L^{p^\prime }(0,T;X^\prime ).$

2.2 Basic Facts on the Lebesgue and Sobolev Spaces with Variable Exponents

Let $p{:}~\Omega \rightarrow [p^{-},p^{+}]\subset (1,+\infty ),$ with $p^\pm =\text {const},$ be a given measurable function. Denote by $L^{p(\cdot )}(\Omega )$ the set of all measurable functions f(x) on $\Omega$ such that $\int _\Omega |f(x)|^{p(x)}\,{\textrm{d}}x<\infty .$ Then, $L^{p(\cdot )}(\Omega )$ is a reflexive separable Banach space with respect to the Luxemburg norm

$$\begin{aligned} \Vert f\Vert _{L^{p(\cdot )}(\Omega )}=\inf \left\{ \lambda >0\! :\ \int _\Omega \left| \frac{f(x)}{\lambda }\right| ^{p(x)}\,{\textrm{d}}x \leqslant 1\right\} . \end{aligned}$$

Moreover, in this case, the set $C^\infty _0(\Omega )$ is dense in $L^{p(\cdot )}(\Omega ).$ The relation between the modular $\int _\Omega |f(x)|^{p(x)}\,{\textrm{d}}x$ and the norm follows from the definition

$$\begin{aligned} \min \left\{ \Vert f\Vert ^{p^{-}}_{L^{p(\cdot )}(\Omega )},\Vert f\Vert ^{p^{+}}_{L^{p(\cdot )}(\Omega )}\right\} \leqslant \int _\Omega |f(x)|^{p(x)}\,{\textrm{d}}x \leqslant \max \left\{ \Vert f\Vert ^{p^{-}}_{L^{p(\cdot )}(\Omega )},\Vert f\Vert ^{p^{+}}_{L^{p(\cdot )}(\Omega )}\right\} . \nonumber \\ \end{aligned}$$

(6)

If $p(\cdot )=\text {const}>1$, then these inequalities transform into equalities. The following estimates are also well known (see, for instance, Refs. [22, 46]): if $f\in L^{p(\cdot )}(\Omega )$, then

$$\begin{aligned} \Vert f\Vert _{L^{p^{-}}(\Omega )}&\leqslant \left( 1+|\Omega |\right) ^{1/{p^{-}}} \Vert f\Vert _{L^{p(\cdot )}(\Omega )}, \end{aligned}$$

(7)

$$\begin{aligned} \Vert f\Vert _{L^{p(\cdot )}(\Omega )}&\leqslant \left( 1+|\Omega |\right) ^{1/{(p^{+})}^\prime }\Vert f\Vert _{L^{p^{+}}(\Omega )},\quad {(p^{+})}^\prime =\frac{{p^{+}}}{{p^{+}}-1},\quad \forall \,f\in L^{p^{+}}(\Omega ),\nonumber \\ \end{aligned}$$

(8)

$$\begin{aligned} \Vert f\Vert ^{p^{-}}_{L^{p(\cdot )}(\Omega )}-1&\leqslant \int _\Omega |f(x)|^{p(x)}\,{\textrm{d}}x\leqslant \Vert f\Vert ^{p^{+}}_{L^{p(\cdot )}(\Omega )}+1,\quad \forall \, f\in L^{p(\cdot )}(\Omega ). \end{aligned}$$

(9)

The next result can be viewed as an analogous of the Hölder inequality in Lebesgue spaces with variable exponents. If $f\in L^{p(\cdot )}(\Omega )$ and $g\in L^{p^\prime (\cdot )}(\Omega )$ with

$$\begin{aligned} p(x)\in [p^{-},p^{+}]\subset (1,+\infty ),\quad p^\prime (x)=\frac{p(x)}{p(x)-1}, \end{aligned}$$

then $\left( f,g\right) \in L^1(\Omega )$ and

$$\begin{aligned} \int _\Omega \left( f,g\right) \,{\textrm{d}}x\leqslant \left( \frac{1}{p^{-}}+\frac{1}{(p^\prime )^{-}}\right) \Vert f\Vert _{L^{p(\cdot )}(\Omega )} \Vert g\Vert _{L^{p^\prime (\cdot )}(\Omega )}\leqslant 2\Vert f\Vert _{L^{p(\cdot )}(\Omega )} \Vert g\Vert _{L^{p^\prime (\cdot )}(\Omega )}.\nonumber \\ \end{aligned}$$

(10)

Let $p_1(\cdot )$ and $p_2(\cdot )$ be measurable functions on $\Omega$ such that $p_i(x)\in [p_i^{-},p_i^{+}]\subset (1,+\infty )$ a.e. in $\Omega .$ In case $p_1(x)\geqslant p_2(x)$ a.e. in $\Omega ,$ the inclusion $L^{p_1(\cdot )}(\Omega )\subset L^{p_2(\cdot )}(\Omega )$ is continuous and

$$\begin{aligned} \Vert u\Vert _{L^{p_2(\cdot )}(\Omega )}\leqslant C\Vert u\Vert _{L^{p_1(\cdot )}(\Omega )},\quad \forall \, u\in L^{p_1(\cdot )}(\Omega ) \end{aligned}$$

(11)

with a constant $C=C(|\Omega |,p_1^{\pm },p_2^\pm ).$

The variable Sobolev space $W^{1,p(\cdot )}(\Omega )$ is defined as the set of functions

$$\begin{aligned} W^{1,p(\cdot )}(\Omega ):=\left\{ u\in W^{1,1}(\Omega )\cap L^{p(\cdot )}(\Omega )\!:\ |\nabla u(x)|^{p(x)}\in L^1(\Omega )\right\} \end{aligned}$$

equipped with the norm

$$\begin{aligned} \Vert u\Vert _{W^{1,p(\cdot )}(\Omega )}=\Vert u\Vert _{L^{p(\cdot )}(\Omega )}+\Vert \nabla u\Vert _{L^{p(\cdot )}(\Omega ;{\mathbb {R}}^N)}. \end{aligned}$$

(12)

Unlike classical Sobolev spaces, smooth functions are not necessarily dense in $W^{1,p(\cdot )}(\Omega ).$ Therefore, we define $H^{1,p(\cdot )}(\Omega )$ as the closure of the set $C^\infty ({\overline{\Omega }})$ in $W^{1,p(\cdot )}(\Omega )$-norm.

Let $C^{\textrm{log}}({\overline{\Omega }})$ be the set of functions continuous on ${\overline{\Omega }}$ with the logarithmic modulus of continuity, i.e.,

$$\begin{aligned} |p(x_1)-p(x_2)|\leqslant \omega (|x_1-x_2|), \end{aligned}$$

(13)

where $\omega \geqslant 0$ satisfies the condition: $\limsup _{\tau \rightarrow 0^{+}} \omega (\tau )\log \frac{1}{\tau }=C<+\infty ,$ $C=\text {const}.$ It is well known that for $p\in C^{\textrm{log}}({\overline{\Omega }})$ the set $C^\infty ({\overline{\Omega }})$ is dense in $W^{1,p(\cdot )}(\Omega )$ and the space $W^{1,p(\cdot )}(\Omega )$ coincides with the closure of $C^\infty ({\overline{\Omega }})$ with respect to the norm (12), i.e., in this case $W^{1,p(\cdot )}(\Omega )=H^{1,p(\cdot )}(\Omega ).$ In particular, if there exists $\delta \in (0,1]$ such that $p\in C^{0,\delta }({\overline{\Omega }}),$ then the set $C^\infty ({\overline{\Omega }})$ is dense in $W^{1,p(\cdot )}(\Omega ).$ Indeed, since

$$\begin{aligned} \lim \limits _{t\rightarrow 0} |t|^\delta \log \left( \frac{1}{|t|}\right) =0\quad \text {with } \delta \in (0,1], \end{aligned}$$

it follows from the Hölder continuity of $p(\cdot )$ that

$$\begin{aligned} |p(x)-p(y)|\leqslant C|x-y|^\delta\leqslant & {} \omega (|x-y|)\sup _{x,y\in \Omega }\left[ |x-y|^\delta \log (|x-y|^{-1})\right] \\\leqslant & {} C^\prime \omega (|x-y|) ,\quad \forall \,x,y\in \Omega , \end{aligned}$$

with $\omega (t)=C/\log (|t|^{-1}).$

Let $p(\cdot ), q(\cdot )\in C({\overline{\Omega }})$ be such that $p(x)\in [p^{-},p^{+}]\subset (1,2]$ and $q(x)<\frac{2p(x)}{2-p(x)}$ in ${\overline{\Omega }}.$ Then, the embedding $W^{1,p(\cdot )}(\Omega )\subset L^{q(\cdot )}(\Omega )$ is continuous and compact. Moreover, according to (11), we have a continuous embedding $W^{1,p(\cdot )}(\Omega )\subset W^{1,p^{-}}(\Omega ).$

For a more detailed presentation of the theory of these spaces, we refer to the monograph [22].

2.3 On the Dual Sobolev Space $H^{-1}(\Omega )$

Let $H^1_0(\Omega )$ be the standard Sobolev space, i.e., $H^1_0(\Omega )$ is the closure of $C^1_0(\Omega )$ with respect to the norm

$$\begin{aligned} \Vert u\Vert _{H^1_0(\Omega )}=\left( \int _\Omega |\nabla u(x)|^2\,{\textrm{d}}x\right) ^\frac{1}{2}. \end{aligned}$$

It is well known that for any $u^*\in H^{-1}(\Omega )$ there can be found a vector-function $g=\left[ g_1,g_2\right]$ in $L^2(\Omega ;{\mathbb {R}}^2)$ such that

$$\begin{aligned} \left<u^*,u\right>_{H^{-1}(\Omega );H^1_0(\Omega )}=\int _\Omega \left( g,\nabla u\right) _{{\mathbb {R}}^2}\,{\textrm{d}}x = \int _\Omega \left( g_1 \frac{\partial u}{\partial x_1}+g_2 \frac{\partial u}{\partial x_2}\right) \,{\textrm{d}}x. \end{aligned}$$

Therefore, it is clear now that

$$\begin{aligned} \Vert u^*\Vert _{H^{-1}(\Omega )}\leqslant \sqrt{\int _\Omega \left( g^2_1(x)+g_2^2(x)\right) \!{\textrm{d}}x}. \end{aligned}$$

(14)

On the other hand, due to the Lax-Milgram Theorem, the Dirichlet boundary value problem

$$\begin{aligned} -\Delta y=u^*\quad \text {in }\Omega ,\qquad y=0\quad \text {on } \partial \Omega \end{aligned}$$

(15)

has a unique solution $y=(-\Delta )^{-1}u^*\in H^1_0(\Omega )$ for each $u^*\in H^{-1}(\Omega ).$ Moreover, in view of the energy equality

$$\begin{aligned} \int _\Omega \left( \nabla y,\nabla y\right) _{{\mathbb {R}}^2}\!{\textrm{d}}x= \Vert \nabla y\Vert ^2_{L^2(\Omega ;{\mathbb {R}}^2)}=\Vert y\Vert ^2_{H^1_0(\Omega )}= \left<u^*,y\right>_{H^{-1}(\Omega );H^1_0(\Omega )}, \end{aligned}$$

(16)

which holds for the weak solution of the Dirichlet problem (15), we can deduce the following a priori estimate for the weak solution of the Dirichlet problem (15):

$$\begin{aligned} \Vert y\Vert _{H^1_0(\Omega )}\equiv \Vert (-\Delta )^{-1}u^*\Vert _{H^1_0(\Omega )}\equiv \Vert \nabla (-\Delta )^{-1}u^*\Vert _{L^2(\Omega ;{\mathbb {R}}^2)}\leqslant \Vert u^*\Vert _{H^{-1}(\Omega )}. \end{aligned}$$

Combining this result with (14), we get

$$\begin{aligned}&\Vert \nabla (-\Delta )^{-1}u^*\Vert _{L^2(\Omega ;{\mathbb {R}}^2)}\leqslant \Vert u^*\Vert _{H^{-1}(\Omega )}\leqslant \sqrt{\int _\Omega \left( g^2_1(x)+g_2^2(x)\right) {\textrm{d}}x} \nonumber \\&\quad {\mathop {=}\limits ^{\text {by}\,\, (16)}}\sqrt{\int _\Omega |\nabla y|^2_{{\mathbb {R}}^2}\,{\textrm{d}}x}=\Vert \nabla y\Vert _{L^2(\Omega ;{\mathbb {R}}^2)}=\Vert y\Vert _{H^1_0(\Omega )}= \Vert \nabla (-\Delta )^{-1}u^*\Vert _{L^2(\Omega ;{\mathbb {R}}^2)}. \end{aligned}$$

(17)

Hence, the norm in $H^{-1}(\Omega )$ can be defined as follows:

$$\begin{aligned} \Vert u^*\Vert _{H^{-1}(\Omega )}=\Vert \nabla (-\Delta )^{-1}u^*\Vert _{L^2(\Omega ;{\mathbb {R}}^2)}. \end{aligned}$$

(18)

2.4 Level Sets, Directional Gradients, and Texture Indexes

Let $u{:}~\Omega \rightarrow \overline{{\mathbb {R}}}$ be a given function. Then, for each $\lambda \in {\mathbb {R}}$, we can define the upper level set of u as follows:

$$\begin{aligned} Z_\lambda (u)=\left\{ u\geqslant \lambda \right\} :=\left\{ x\in \Omega \! :\ u(x)\geqslant \lambda \right\} . \end{aligned}$$

To describe this set, we assume that $u\in W^{1,1}(\Omega ).$ It was proven in Ref. [3] that if $u\in W^{1,1}(\Omega )$, then its upper level sets $Z_\lambda (u)$ are sets of finite perimeter. Therefore, the boundaries of level sets can be described by a countable family of Jordan curves with finite length, i.e., by continuous maps from the circle into the plane ${\mathbb {R}}^2$ without crossing points. As a result, at almost all points of almost all level sets of $u\in W^{1,1}(\Omega )$ we may define a unit normal vector $\theta (x).$ This vector field formally satisfies the following relations:

$$\begin{aligned} \left( \theta , \nabla u\right) = |\nabla u|\quad \text {and}\quad |\theta |\leqslant 1 \ \text {a.e. in } \Omega . \end{aligned}$$

In the sequel, we will refer to $\theta$ as the vector field of unit normals to the topographic map of a function u.

If $\theta \in L^\infty (\Omega ,{\mathbb {R}}^2)$ is a vector field of unit normals to the topographic map of some function $u(\cdot ),$ then for any function $v\in W^{1,1}(\Omega )$, we can define the so-called directional gradient of v following the rule (see Refs. [8, 9])

$$\begin{aligned} R_\eta \nabla v:=\nabla v-\eta ^2 \left( \theta ,\nabla v\right) \theta , \end{aligned}$$

(19)

where $\eta \in (0,1)$ is a given threshold. Since, for each function $v\in W^{1,1}(\Omega ),$ the expression $R_\eta \nabla v$ can be reduced to $(1-\eta ^2)\nabla v$ in those places of $\Omega$ where $\nabla v$ is collinear to the unit normal $\theta ,$ and to $\nabla v$ if $\nabla v$ is orthogonal to $\theta ,$ we have the following estimate:

$$\begin{aligned} (1-\eta ^2)|\nabla v|\leqslant |R_\eta \nabla v|\leqslant |\nabla v|\quad \text {in } \Omega . \end{aligned}$$

(20)

In what follows, with each function $u\in W^{1,1}(\Omega ),$ we associate the so-called texture index $p(|\nabla u|)$ using the rule (see Refs. [18,19,20] for comparison)

$$\begin{aligned} p(s)=1+\delta +\frac{a^2(1-\delta )}{a^2+s^2},\quad \forall \, s\in [0,+\infty ), \end{aligned}$$

(21)

where $0<\delta \ll 1$ is a given threshold. It is clear now that

$$\begin{aligned} p(|\nabla u|)\in [p^{-},p^{+}]\subset (1,2]\ \text {a.e. in } \Omega \text { with } p^{-}=1+\delta \text { and } p^{+}=2\nonumber \\ \end{aligned}$$

(22)

for each $u\in W^{1,1}(\Omega ).$

3 Statement of the Problem

In this section, we present a novel variational problem for the denoising and contrast enhancement of non-smooth RGB images which can be viewed as an improved version of the variational model recently proposed in Ref. [17].

Let $f=[f_1,f_2,f_3]^\textrm{t}\in L^2(\Omega ;{\mathbb {R}}^3)$ be an original color image. Let $\theta =\theta (f_i)=\left[ \begin{array}{ll}\theta _{1}\\ \theta _{2}\end{array}\right] \in L^\infty (\Omega ,{\mathbb {R}}^2)$ be a vector field of unit normals to the topographic map of the spectral channel $f_i,$

$$\begin{aligned} |\theta (x)|\leqslant 1\text { and }\left( \theta (x), \nabla f_i(x)\right) =| \nabla f_i(x)| \quad \text {a.e. in } \Omega . \end{aligned}$$

(23)

This vector field can be defined by the rule $\theta (x)=\frac{\nabla U(t,x)}{|\nabla U(t,x)|}$ with $t>0$ small enough, where U(t, x) is a solution of the following initial-boundary value problem:

$$\begin{aligned}&\frac{\partial U}{\partial t}=\text {div}\left( \frac{\nabla U}{|\nabla U|+ \kappa }\right) ,\quad t\in (0,+\infty ),\ x\in \Omega , \end{aligned}$$

(24)

$$\begin{aligned}&U(0,x)= f_i(x),\quad x\in \Omega , \end{aligned}$$

(25)

$$\begin{aligned}&\frac{\partial U(t,x)}{\partial \nu }=0,\quad t\in (0,+\infty ),\ x\in \partial \Omega \end{aligned}$$

(26)

with a relaxed version of the one-dimensional-Laplace operator in the principle part of (24). Here, $\kappa >0$ is a sufficiently small positive value.

Let $\eta \in (0,1)$ be a given threshold. Taking into account the definition of the Directional Total Variation (see Ref. [8]), we define the linear operator $R_{\eta }\!:{\mathbb {R}}^2\rightarrow {\mathbb {R}}^2$ as follows:

$$\begin{aligned} R_{\eta }\nabla v:=\nabla v-\eta ^2 \left( \theta ,\nabla v\right) \theta ,\quad \forall \,v\in W^{1,1}(\Omega ). \end{aligned}$$

(27)

Since $R_{\eta }\nabla v$ reduces to $(1-\eta ^2)\nabla v$ in those regions where the gradient $\nabla v$ is co-linear to $\theta ,$ and to $\nabla v,$ where $\nabla v$ is orthogonal to $\theta ,$ this operator does not enforce gradients in the direction $\theta .$

Remark 1

In the sequel, to reduce the number of parameters in the proposed model, we will set $\delta =\kappa$ in (24) and (21), and $\eta =1-\kappa$ in (19).

We also introduce the following set:

$$\begin{aligned} \Xi _i=\left\{ I\in H^{1,p(|\nabla I|)}(\Omega )\cap L^\infty (\Omega )\left| \begin{array}{c} \gamma _{i,0}\leqslant I(x)\leqslant \gamma _{i,1}\quad \text{ a.e. } \text{ in }\ \Omega ,\\ \gamma _{i,0}=\inf _{x\in \Omega } {f_i}(x),\\ \gamma _{i,1}=\sup _{x\in \Omega } {f_i}(x), \end{array} \right. \right\} \end{aligned}$$

(28)

where $p(\cdot )$ is given by (21), and $H^{1,p(|\nabla I|)}(\Omega )$ is the variable Sobolev space that can be defined as follows:

$$\begin{aligned} H^{1,p(|\nabla I|)}(\Omega ):=\textrm{cl}_{\Vert \cdot \Vert _{W^{1,p(|\nabla I|)}(\Omega )}}C^\infty _0({\mathbb {R}}^2). \end{aligned}$$

(29)

Further, for a given gray-scale image $I\in L^2(\Omega ),$ we define its average local contrast measure D(I) as follows (for comparison, we refer to Ref. [40]):

$$\begin{aligned} D(I)=\int _{\Omega }\int _{\Omega } W(x,y)\sqrt{\kappa ^2+\left| I(x)-I(y)\right| ^2}\,{\textrm{d}}x {\textrm{d}}y, \end{aligned}$$

(30)

where $\kappa >0$ is the same parameter as in (24), and $W\in L^2(\Omega \times \Omega )$ is a symmetric non-negative kernel such that

$$\begin{aligned} \int _{\Omega } W(x,y){\textrm{d}}x =1,\quad \forall \,y\in \Omega . \end{aligned}$$

A typical example of this function is the Gaussian kernel,

$$\begin{aligned} W(x,y)=\frac{1}{\sqrt{2{\uppi }} \sigma }\exp \left( {-\frac{|x-y|^2}{2\sigma ^2}}\right) ,\quad \sigma >0. \end{aligned}$$

As a result, the proposed variational approach for the contrast enhancement and denoising of color images can be stated as follows.

For each spectral channel $f_i$ $(i=1,2,3)$ of a given image $f=[f_1,f_2,f_3]^\textrm{t}\in L^2(\Omega ;{\mathbb {R}}^3),$ we generate a new one $f^0_i\in L^2(\Omega )$ as a solution of the following constrained minimization problem:

$$\begin{aligned} J_i(f^0_i)=\inf _{v\in \Xi _i} J_i(v), \end{aligned}$$

(31)

where

$$\begin{aligned} J_i(v)=\int _{\Omega } |R_{\eta }\nabla v(x)|^{p(|\nabla v|)}\,{\textrm{d}}x +\frac{\mu }{2}\left\| v -f_i\right\| ^2_{H^{-1}(\Omega )}+\frac{\lambda }{4} \left[ D(v)-cD(f_i)\right] ^2.\nonumber \\ \end{aligned}$$

(32)

Here, $\lambda >0$ and $\mu \in (0,1)$ are tuning parameters. The parameter $\lambda$ manages the trade-off between the fidelity term $\frac{\mu }{2}\left\| v -f_i\right\| ^2_{H^{-1}(\Omega )}$ and the contrast term $\frac{\lambda }{4}\left[ D(v)-cD(f_i)\right] ^2.$ As for the multiplier $c>0,$ we always suppose that $c>1$ and it provides a control of the contrast level expected for the result.

Before proceeding further, we provide some qualitative analysis of the variational problem (31)–(32). To begin with, we notice that, for each feasible solution $v\in \Xi ,$ the following two-side estimate:

$$\begin{aligned} p^{+}=2\geqslant p(|\nabla v|)> 1+\delta =:p^{-},\quad \text { for a.a.}\ \,x\in \Omega \end{aligned}$$

(33)

holds with $0<\delta \ll 1.$ Moreover, since $\eta \in (0,1)$ and $\eta \gg 0,$ we see that

$$\begin{aligned} (1-\eta ^2)|\nabla v|&\leqslant |R_{\eta }\nabla v|\leqslant |\nabla v|\quad \text {in}\,\, \Omega ,\\ \int _{\Omega }|R_{\eta }\nabla v(x)|^{p(|\nabla v|)}\,{\textrm{d}}x&\geqslant \int _{\Omega } (1-\eta ^2)^{p(|\nabla v|)}|\nabla v(x)|^{p(|\nabla v|)}\,{\textrm{d}}x \nonumber \end{aligned}$$

(34)

$$\begin{aligned}&\geqslant (1-\eta ^2)^2\int _{\Omega }|\nabla v(x)|^{p(|\nabla v|)}\,{\textrm{d}}x,\quad \forall \,v\in W^{1,p(|\nabla v|)}(\Omega ). \end{aligned}$$

(35)

As a result, for any $v\in \Xi _i,$ we have

$$\begin{aligned} \Vert v\Vert ^{p^{-}}_{W^{1,p(|\nabla v|)}(\Omega )}&=\left( \Vert v\Vert _{L^{p(|\nabla v|)}(\Omega )}+\Vert \nabla v\Vert _{L^{p(|\nabla v|)}(\Omega ;{\mathbb {R}}^2)}\right) ^{p^{-}}\nonumber \\&\leqslant C\left( \Vert v\Vert ^{p^{-}}_{L^{p(|\nabla v|)}(\Omega )}+\Vert \nabla v\Vert ^{p^{-}}_{L^{p(|\nabla v|)}(\Omega ;{\mathbb {R}}^2)}\right) \nonumber \\&{\mathop {\leqslant }\limits ^{\text {by}\,\, (9)}} C\left( \int _{\Omega } |v(x)|^{p(|\nabla v|)}\,{\textrm{d}}x+\int _{\Omega } |\nabla v(x)|^{p(|\nabla v|)}\,{\textrm{d}}x+2\right) \nonumber \\&\leqslant C\left( |\Omega |\gamma _{i,1}^{2}+\int _{\Omega } |\nabla v(x)|^{p(|\nabla v|)}\,{\textrm{d}}x+2\right) \nonumber \\&{\mathop {\leqslant }\limits ^{\text {by} \,\,(35)}} C\left( |\Omega |\gamma _{i,1}^{2}+2+\frac{1}{(1-\eta ^2)^2}\int _{\Omega }|R_{\eta }\nabla v(x)|^{p(|\nabla v|)}\,{\textrm{d}}x\right) \nonumber \\&\leqslant C\left( |\Omega |\gamma _{i,1}^{2}+2+\frac{1}{(1-\eta ^2)^2}J_i(v)\right) . \end{aligned}$$

(36)

Thus, the first term in the cost functional (32) can be considered as a regularizing term. As for the second term in (32), we make use of the following observation.

Remark 2

The model (32) is aimed not only at the contrast enhancement, but also to remove the additive noise in the so-called structured images, i.e., in images where the portion of high oscillatory edges is rather significant. In most cases, the satellite images with crop fields typically contain many high oscillatory edges (boundaries of the crop locations). Moreover, “the portion of noise” in such images can be different from channel to channel. Because of this, an important question is to separate pure noise from high oscillatory edges in each spectral channel. To handle this problem, Meyer [37] suggested to replace the standard $L^2$-fidelity term $\frac{\mu }{2}\Vert v -f_i\Vert ^2_{L^2(\Omega )},$ which is a typical component in the standard denoising models, by a weaker norm. As a plausible option of such weakening, Lieu and Vese [34] (see also Schönlieb [43]) have proposed to involve $H^{-1}(\Omega )$-norm instead of $\Vert \cdot \Vert _{L^2(\Omega )}.$ Thus, from this point of view, it is plausible to interpret the second term in (32) as the fidelity term.

Before we move on to the existence issues, we make use of the following result concerning the lower semicontinuity property of the modular $\int _\Omega |f(x)|^{p(x)}\,{\textrm{d}}x$ with respect to the weak convergence in $L^{p_k(\cdot )}(\Omega ).$ The proof of this assertion has been mainly inspired by the elegant proof of Lemma 1 in Ref. [13] (see also Ref. [47, Lemma 13.3] for comparison).

Proposition 1

Let $\left\{ p_k\right\} _{k\in {\mathbb {N}}}\subset [p^{-},p^{+}]$ be a given sequence such that

$$\begin{aligned} p_k(x)\rightarrow p(x)\quad \text {a.e. in } \Omega \text { as } k\rightarrow \infty . \end{aligned}$$

(37)

Let $\left\{ v_k\in W^{1,p_k(\cdot )}(\Omega )\right\} _{k\in {\mathbb {N}}}$ be a sequence such that

$$\begin{aligned}{} & {} \nabla v_k\rightharpoonup \nabla v\quad \text {weakly in } L^1(\Omega ;{\mathbb {R}}^2), \end{aligned}$$

(38)

$$\begin{aligned} \left\| |\nabla v_k(\cdot )|^{p_k(\cdot )}\right\| _{L^1(\Omega )}\leqslant & {} C\quad \text {for some positive constant } C \text { not depending on } k, \end{aligned}$$

(39)

and let $R_\eta \!\!:{\mathbb {R}}^2\rightarrow {\mathbb {R}}^2$ be the operator defined in (27) with some $\theta \in L^\infty (\Omega ;{\mathbb {R}}^2).$ Then, $\nabla v\in L^{p(\cdot )}(\Omega ;{\mathbb {R}}^2)$ and

$$\begin{aligned} \liminf _{k\rightarrow \infty } \int _{\Omega } |R_\eta \nabla v_k(x)|^{p_k(x)}\,{\textrm{d}}x\geqslant \int _{\Omega } |R_\eta \nabla v(x)|^{p(x)}\,{\textrm{d}}x. \end{aligned}$$

(40)

Proof

By Young’s inequality we have for $\xi ,\zeta \in {\mathbb {R}}^2$ and $1<p<\infty ,$

$$\begin{aligned} (\xi ,\zeta )\leqslant |\xi | |\zeta |\leqslant |\xi |^p+\frac{|\zeta |^{p^\prime }}{p^\prime p^{p^\prime /p}},\quad \frac{1}{p}+\frac{1}{p^\prime }=1. \end{aligned}$$

(41)

If now $\zeta$ is a function in $L^\infty (\Omega ;{\mathbb {R}}^2)$ and we make $p=p_k$ in (40) and use the assumption $p^{-}\leqslant p_k(x)\leqslant p^{+}$ for all $k\in {\mathbb {N}},$ then we derive

$$\begin{aligned} \int _{\Omega }\left( (R_\eta \nabla v_k,\zeta )-\frac{|\zeta |^{p_k^\prime (x)}}{p^\prime _k(x)p_k(x)^{p_k^\prime (x)/p_k(x)}}\right) \,{\textrm{d}}x \leqslant \int _{\Omega } |R_\eta \nabla v_k|^{p_k(x)}\,{\textrm{d}}x. \end{aligned}$$

(42)

Using (27) and assumptions (37) and (38), we can pass to the limit in (42) as $k\rightarrow \infty .$ As a result, we have

$$\begin{aligned} \int _{\Omega }\left( (R_\eta \nabla v,\zeta )-\frac{|\zeta |^{p^\prime (x)}}{p^\prime (x)p(x)^{p^\prime (x)/p(x)}}\right) \!{\textrm{d}}x \leqslant \liminf _{k\rightarrow \infty }\int _{\Omega } |R_\eta \nabla v_k|^{p_k(x)}\,{\textrm{d}}x:=L. \end{aligned}$$

(43)

Then, we consider the following function:

$$\begin{aligned} \zeta :=\frac{R_\eta \nabla v}{|R_\eta \nabla v|}p(x) |R_\eta \nabla v|_n^{\frac{1}{p^\prime (x)-1}}\quad \text {with }\ |R_\eta \nabla v|_n:=\max \left\{ |R_\eta \nabla v|,n\right\} ,\quad n>0. \end{aligned}$$

Inserting this function $\zeta$ into (43), we get

$$\begin{aligned} \int _{\Omega } \left( |R_\eta \nabla v| p(x) |R_\eta \nabla v|_n^{\frac{1}{p^\prime (x)-1}}-|R_\eta \nabla v|_n^{\frac{p^\prime (x)}{p^\prime (x)-1}}\frac{p(x)}{p^\prime (x)}\right) \,{\textrm{d}}x \leqslant L. \end{aligned}$$

This implies

$$\begin{aligned} \int _{\Omega } |R_\eta \nabla v|_n^{\frac{1}{p^\prime (x)-1}+1}\,{\textrm{d}}x\leqslant L. \end{aligned}$$

Since $\frac{1}{p^\prime (x)-1}+1=p(x),$ it follows that

$$\begin{aligned} \int _{\Omega } |R_\eta \nabla v|_n^{p(x)}\,{\textrm{d}}x\leqslant L. \end{aligned}$$

(44)

To conclude the proof, it remains to notice that the announced inequality (40) follows by letting $n\rightarrow \infty$ in (44). As for the inclusion $\nabla v\in L^{p(\cdot )}(\Omega ;{\mathbb {R}}^2)$ it is a direct consequence of assumption (39) and estimate (34).

Before proceeding to the existence issues, we provide, in the next section, a formal analysis of the optimality system for the problem (31)–(32).

4 Optimality Conditions

Let $f^0_i\in \Xi _i,$ with $i=1,2,3,$ be a point of local minimum in the problem (31)–(32), i.e., there exists a positive value $\tau >0$ such that

$$\begin{aligned} J_i(f^0_i)-J_i(v)\leqslant 0,\quad \forall \, v\in \Xi \text { s.t. } \Vert v-f^0_i\Vert _{W^{1,p^{-}}(\Omega )}<\tau . \end{aligned}$$

(45)

For simplicity, we assume that the two-side inequality

$$\begin{aligned} \gamma _{i,1}<f^0_i(x)<\gamma _{i,2} \end{aligned}$$

holds almost everywhere in $\Omega .$ Then, condition (45) can be rewritten as follows: for any smooth function $\varphi \in C^\infty ({\overline{\Omega }}),$ the inequality

$$\begin{aligned} J_i(f^0_i)-J_i(f_i^0+\sigma \varphi )\leqslant 0\quad \text {for } \sigma \text { small enough} \end{aligned}$$

(46)

holds. Hence, the scalar function

$$\begin{aligned} \psi (\sigma ):=J_i(f_i^0+\sigma \varphi )&=\int _{\Omega } |R_{\eta }\left( \nabla f_i^0(x)+\sigma \nabla \varphi (x)\right) |^{p(|\nabla f_i^0+\sigma \nabla \varphi |)}\,{\textrm{d}}x\\&\quad +\frac{\mu }{2}\left\| f_i^0+\sigma \varphi -f_i\right\| ^2_{H^{-1}(\Omega )}+\frac{\lambda }{4} \left[ D(f_i^0+\sigma \varphi )-cD(f_i)\right] ^2 \end{aligned}$$

has a minimum at $\sigma =0.$

Thus, to characterize the given feasible solution $f_i^{0}\in \Xi _i$ to the optimization problem (31)–(32), we make use of the Ferma’s Theorem. To do so, we show that the objective functional $J_i(v)$ is Gâteaux differentiable at $v=f_i^{0},$ that is, there exists a linear bounded functional $D_G J_i(f_i^{0})\in \left[ H^{1,p[\nabla f_i^0]}(\Omega )\right] ^\prime ={\mathcal {L}}\left( H^{1,p[\nabla f_i^0]}(\Omega ),{\mathbb {R}}\right)$ such that

$$\begin{aligned} J_i\left( f_i^0+\sigma h\right) =J_i\left( f_i^0\right) +\sigma D_G J_i(f_i^{0})[h]+r_i(h,\sigma ),\quad \forall \,h\in H^{1,p[\nabla f_i^0]}(\Omega ), \end{aligned}$$

(47)

where $|r_i(h,\sigma )|=o(|\sigma |)$ as $\sigma \rightarrow 0.$ Then, the condition $0\in {\textrm{Argmin}} \psi (\sigma )$ can be interpreted as

$$\begin{aligned} D_G J_i(f_i^{0})[\varphi ]=0,\quad \forall \,\varphi \in C^\infty ({\overline{\Omega }}). \end{aligned}$$

(48)

Keeping in mind the fact that the set of feasible solutions $\Xi _i$ to the problem (31)–(32) has an empty topological interior, we begin with the following auxiliary results, where $F^\prime (u)[h]$ stands for the directional derivative of a functional $F\!\!:X\rightarrow {\mathbb {R}}$ at the point $u\in X$ along a vector $h\in X,$ i.e.,

$$\begin{aligned} F^\prime (u)[h]=\lim _{\sigma \rightarrow 0}\frac{F(u+\sigma h)-F(u)}{\sigma }. \end{aligned}$$

Proposition 2

Let $f\in L^2(\Omega )$ be a given distribution and let

$$\begin{aligned} F_1(u)=\frac{1}{2} \Vert u-f\Vert ^2_{H^{-1}(\Omega )},\quad \forall \,u\in L^2(\Omega ). \end{aligned}$$

Then,

$$\begin{aligned} F^\prime _1(u)[h]=\left( (-\Delta )^{-1}(u-f),h\right) _{L^2(\Omega )},\quad \forall \, h\in L^2(\Omega ). \end{aligned}$$

(49)

Proof

The announced result immediately follows from the definition of the directional derivative and the following chain of transformations:

$$\begin{aligned} F_1(u+\sigma h)-F_1(u)&{\mathop {=}\limits ^{\text {by}\,\, (18)}} \frac{1}{2} \Vert \nabla (-\Delta )^{-1}(u+\sigma h-f)\Vert _{L^2(\Omega ;{\mathbb {R}}^2)}-\frac{1}{2} \Vert \nabla (-\Delta )^{-1}(u-f)\Vert _{L^2(\Omega ;{\mathbb {R}}^2)}\\&=\sigma \left( \nabla (-\Delta )^{-1}(u-f), \nabla (-\Delta )^{-1}h\right) _{L^2(\Omega ;{\mathbb {R}}^2)}+\sigma ^2\frac{1}{2} \Vert \nabla (-\Delta )^{-1}h\Vert ^2_{L^2(\Omega ;{\mathbb {R}}^2)}\\&=-\sigma \int _\Omega \textrm{div}\left[ \nabla (-\Delta )^{-1}(u-f)\right] (-\Delta )^{-1}h\,{\textrm{d}}x+\sigma ^2\frac{1}{2} \Vert h\Vert ^2_{H^{-1}(\Omega )}\\&= \sigma \int _\Omega (-\Delta )(-\Delta )^{-1}(u-f)(-\Delta )^{-1}h\,{\textrm{d}}x+\sigma ^2\frac{1}{2} \Vert h\Vert ^2_{H^{-1}(\Omega )}\\&=\sigma \left( (-\Delta )^{-1}(u-f),h\right) _{L^2(\Omega )}+o(\sigma ),\quad \forall \, u\in L^2(\Omega ). \end{aligned}$$

Proposition 3

Let $p\!:\Omega \rightarrow [p^{-},p^{+}]\subset (1,2],$ with $p^\pm =\text {const},$ be a given exponent and let

$$\begin{aligned} {\widetilde{F}}_2(u)=\int _{\Omega } |\nabla u(x)|^{p(x)}\,{\textrm{d}}x,\quad \forall \,u\in W^{1,p(\cdot )}(\Omega ). \end{aligned}$$

Then, for each $u\in W^{1,p(\cdot )}(\Omega ),$ we have

$$\begin{aligned} {\widetilde{F}}^\prime _2(u)[h]=\int _\Omega p(x)\left( |\nabla u(x) |^{p(x)-2}\nabla u(x),\nabla v(x)\right) \!{\textrm{d}}x,\quad \forall \, h\in W^{1,p(\cdot )}(\Omega ). \end{aligned}$$

(50)

Proof

Let $u,h\in W^{1,p(\cdot )}(\Omega )$ be given functions. We notice that

$$\begin{aligned} \frac{|\nabla u+\sigma \nabla h |^{p}-|\nabla u |^{p}}{\sigma } \rightarrow p\left( |\nabla u |^{p-2}\nabla u,\nabla h\right) \end{aligned}$$

as $\sigma \rightarrow 0$ almost everywhere in $\Omega .$ Furthermore, by convexity,

$$\begin{aligned} |\xi |^p - |\eta |^p\leqslant 2p\left( |\xi |^{p-1}+|\eta |^{p-1}\right) |\xi -\eta |, \end{aligned}$$

we have

$$\begin{aligned}&\Big |\frac{1}{\sigma }\Big (|\nabla u(x)+\sigma \nabla h(x) |^{p(x)}- |\nabla u(x)|^{p(x)}\Big )\Big |\nonumber \\ \leqslant&2p(x)\left( |\nabla u(x)+\sigma \nabla h(x) |^{p(x)-1}+ |\nabla u(x) |^{p(x)-1}\right) |\nabla h(x) |\nonumber \\ \leqslant&\text {const}\, \left( |\nabla u(x) |^{p(x)-1}+|\nabla h(x)|^{p(x)-1}\right) |\nabla h(x)|. \end{aligned}$$

(51)

Taking into account that

$$\begin{aligned} \int _\Omega |\nabla u(x)|^{p(x)-1}|\nabla v(x)|\,{\textrm{d}}x&\leqslant 2\Vert |\nabla u(x)|^{p(x)-1}\Vert _{L^{p^\prime (\cdot )}(\Omega )} \Vert |\nabla h(x)|\Vert _{L^{p(\cdot )}(\Omega )}\\&\leqslant 2\Vert |\nabla u(x) |^{p(x)-1}\Vert _{L^{p^\prime (\cdot )}(\Omega }\Vert \nabla h(x) \Vert _{L^{p(\cdot )}(\Omega ;{\mathbb {R}}^2)}, \end{aligned}$$

and

$$\begin{aligned} \int _\Omega |\nabla h(x)|^{p(x)}\,{\textrm{d}}x{\mathop {\leqslant }\limits ^{\textrm{by} \ (9)}} \Vert \nabla h\Vert ^2_{L^{p(\cdot )}(\Omega ;{\mathbb {R}}^2)}+1, \end{aligned}$$

we see that the right-hand side of the inequality (51) is an $L^1(\Omega )$-function. Therefore,

$$\begin{aligned} \lim _{\sigma \rightarrow 0} \frac{{\widetilde{F}}_2(u+\sigma h)-{\widetilde{F}}_2(u)}{\sigma }&= \lim _{\sigma \rightarrow 0}\int _\Omega \frac{|\nabla u(x)+\sigma \nabla h(x) |^{p}-|\nabla u(x) |^{p}}{\sigma }\!{\textrm{d}}x\\&=\int _\Omega p(x)\left( |\nabla u(x) |^{p(x)-2}\nabla u(x),\nabla h(x)\right) {\textrm{d}}x \end{aligned}$$

by the Lebesgue-dominated convergence theorem. From this, the representation (50) follows.

Proposition 4

Let $p\!:\Omega \rightarrow [p^{-},p^{+}]\subset (1,2],$ with $p^\pm =\text {const},$ be a given exponent and let

$$\begin{aligned} F_2(u)=\int _{\Omega } |R_\eta \nabla u(x)|^{p(x)}\,{\textrm{d}}x,\quad \forall \,u\in W^{1,p(\cdot )}(\Omega ), \end{aligned}$$

where the linear operator $R_{\eta }\!\!:{\mathbb {R}}^2\rightarrow {\mathbb {R}}^2$ is defined by the rule (27). Then, for each $u\in W^{1,p(\cdot )}(\Omega ),$ we have

$$\begin{aligned} F^\prime _2(u)[h]= & {} \int _\Omega p(x)\left( |R_\eta \nabla u(x) |^{p(x)-2}R_\eta \nabla u(x),\nabla h(x)\right) \!{\textrm{d}}x\nonumber \\{} & {} \!-\eta ^2 \int _\Omega p(x)\left( |R_\eta \nabla u(x) |^{p(x)-2}R_\eta \nabla u(x),\theta (x)\right) \left( \theta (x),\nabla h(x)\right) \!{\textrm{d}}x,\quad \forall \, h\in W^{1,p(\cdot )}(\Omega ).\nonumber \\ \end{aligned}$$

(52)

Proof

The representation (52) immediately follows from the definition of the directional derivative and Proposition 3.

Proposition 5

Let $u\in \Xi _i$ be a given feasible solution, let

$$\begin{aligned} p[\nabla u]:=1+\delta +\frac{a^2(1-\delta )}{a^2+|\nabla u|^2}, \end{aligned}$$

and let the functional $F_3\!:W^{1,1+\delta }(\Omega )\rightarrow {\mathbb {R}}$ be defined as follows:

$$\begin{aligned} F_3(u)=\int _{\Omega } |R_\eta \nabla v(x)|^{p[\nabla u]}\,{\textrm{d}}x,\quad \forall \,u\in W^{1,1+\delta }(\Omega ), \end{aligned}$$

where $v\in W^{1,p[\nabla u]}(\Omega )$ is a given function. Then, for each $v\in W^{1,p[\nabla u]}(\Omega )$ and for all $h\in W^{1,1+\delta }(\Omega ),$ we have

$$\begin{aligned} F^\prime _3(u)[h]=-\int _\Omega |R_\eta \nabla v(x) |^{p[\nabla u]} \frac{2a^2(1-\delta )\log \left( |R_\eta \nabla v(x) |\right) }{\left( a^2+|\nabla u|^2\right) ^2}\left( \nabla u, \nabla h\right) \!{\textrm{d}}x. \end{aligned}$$

(53)

Proof

The representation (53) immediately follows from the definition of the directional derivative.

Proposition 6

Let $u\in \Xi$ be a feasible solution, and let

$$\begin{aligned} F_4(u)=\frac{\lambda }{4} \left[ D(u)-cD(f_i)\right] ^2, \end{aligned}$$

where $f_i\in L^2(\Omega )$ is a given spectral channel, $c=\text {const}>1,$ and

$$\begin{aligned} D(u)=\int _{\Omega }\int _{\Omega } W(x,y)\sqrt{\kappa ^2+\left| u(x)-u(y)\right| ^2}\,{\textrm{d}}x {\textrm{d}}y. \end{aligned}$$

Then, the directional derivative of $F_4{:}~L^2(\Omega )\rightarrow {\mathbb {R}}$ at the given point u along a vector $h\in L^2(\Omega )$ takes the form

$$\begin{aligned} F^\prime _4(u)[h]=\lambda \left( D(u)-cD(I_f)\right) \int _\Omega \left( \int _\Omega W(x,y)\frac{u(x)-u(y)}{\sqrt{\kappa ^2+\left| u(x)-u(y)\right| ^2}}\!{\textrm{d}}y\right) h(x)\,{\textrm{d}}x.\nonumber \\ \end{aligned}$$

(54)

Proof

The representation (54) immediately follows from the definition of the directional derivative and the following chain of transformations:

$$\begin{aligned}{} & {} D(u+\sigma h)-D(u)\\= & {} \int _{\Omega }\int _{\Omega } W(x,y)\left( \sqrt{\kappa ^2+\left| u(x)-u(y)+\sigma (h(x)-h(y))\right| ^2}-\sqrt{\kappa ^2+\left| u(x)-u(y)\right| ^2}\right) \!{\textrm{d}}x {\textrm{d}}y\\= & {} \int _{\Omega }\int _{\Omega } W(x,y) \frac{\left| u(x)-u(y)+\sigma (h(x)-h(y))\right| ^2-\left| u(x)-u(y)\right| ^2}{\sqrt{\kappa ^2+\left| u(x)-u(y)+\sigma (h(x)-h(y))\right| ^2}+\sqrt{\kappa ^2+\left| u(x)-u(y)\right| ^2}} \,{\textrm{d}}x {\textrm{d}}y\\= & {} \sigma \int _{\Omega }\left( \int _{\Omega } \left[ W(x,y) +W(y,x)\right] \frac{u(x)-u(y)}{\sqrt{\kappa ^2+\left| u(x)-u(y)\right| ^2}}\,{\textrm{d}}y\right) h(x)\, {\textrm{d}}y+o(\sigma ^2). \end{aligned}$$

We are now able to show that the objective functional $J_i(v)$ is Gâteaux differentiable at $v=f_i^{0}.$ With that in mind, we utilize the representation (32). As a result, we see that

$$\begin{aligned} J^\prime _i(f_i^{0})[\varphi ]&= \int _\Omega p(|\nabla f_i^{0}|)\left( |R_\eta \nabla f_i^{0}(x) |^{p(|\nabla f_i^{0}|)-2}R_\eta \nabla f_i^{0}(x),\nabla \varphi (x)\right) \!{\textrm{d}}x\nonumber \\&\quad -\eta ^2\int _\Omega p(|\nabla f_i^{0}|)\left( |R_\eta \nabla f_i^{0}(x) |^{p(|\nabla f_i^{0}|)-2}R_\eta \nabla f_i^{0}(x),\theta (x)\right) \left( \theta (x),\nabla \varphi (x)\right) \!{\textrm{d}}x\nonumber \\&\quad -\int _\Omega |R_\eta \nabla f_i^{0}(x) |^{p(|\nabla f_i^{0}|)} \frac{2a^2(1-\delta )\log \left( |R_\eta \nabla f_i^{0}(x) |\right) }{\left( a^2+|\nabla f_i^{0}(x))|^2\right) ^2}\left( \nabla f_i^{0}, \nabla \varphi \right) \!{\textrm{d}}x\nonumber \\&\quad +\lambda \left( D(f_i^{0})-cD(f_i)\right) \int _\Omega \left( \int _\Omega W(x,y)\frac{f_i^{0}(x)-f_i^{0}(y)}{\sqrt{\kappa ^2+\left| f_i^{0}(x)-f_i^{0}(y)\right| ^2}}\,{\textrm{d}}y\right) \varphi (x)\!{\textrm{d}}x\nonumber \\&\quad + \int _{\Omega } \Big [(-\Delta )^{-1}(f_i^{0}-f_i)\Big ]\varphi (x)\!{\textrm{d}}x=0,\quad \,\forall \,\varphi \in C^\infty ({\overline{\Omega }}). \end{aligned}$$

(55)

Thus, $J_i^\prime (f_i^{0}){:}~C^\infty ({\overline{\Omega }})\rightarrow {\mathbb {R}}$ is a linear functional.

Let us show that each term in (55) can be extended by the continuity to the entire Sobolev space $H^{1,p(|\nabla f_i^{0}|)}(\Omega ).$ To this end, it is enough to establish the existence of a constant $M>0$ such that

$$\begin{aligned} \left| J_i^\prime (f_i^{0})[\varphi ]\right| \leqslant M \Vert \varphi \Vert _{W^{1,p(|\nabla f_i^{0}|)}(\Omega )},\quad \forall \, \varphi \in C^\infty ({\overline{\Omega }}). \end{aligned}$$

(56)

Indeed, rewriting (55) in the form

$$\begin{aligned} J_i^\prime (f_i^{0})[\varphi ]=S_1+S_2+S_3+S_4+S_5, \end{aligned}$$

where the one-to-one correspondence to (55) is preserved, we see that

$$\begin{aligned} |S_1|&\leqslant \Vert p\Vert _{L^\infty (\Omega )}\int _\Omega \left( |R_\eta \nabla f_i^{0} |^{p(|\nabla f_i^{0}|)-2}R_\eta \nabla f_i^{0},\nabla \varphi \right) \!{\textrm{d}}x\\&{\mathop {\leqslant }\limits ^{\text {by}\,\, (10)}} 2p^{+}\Vert |R_\eta \nabla f_i^{0} |^{p(|\nabla f_i^{0}|)-2}R_\eta \nabla f_i^{0}\Vert _{L^{p^\prime (|\nabla f_i^{0}|)}(\Omega ;{\mathbb {R}}^2)} \Vert \nabla \varphi \Vert _{L^{p(|\nabla I^{0}|)}(\Omega ;{\mathbb {R}}^2)} \\&{\mathop {\leqslant }\limits ^{\text {by}\,\, (9)}} 2p^{+} \left( 1+\int _{\Omega }|R_\eta \nabla f_i^{0}|^{p(|\nabla f_i^{0}|)}\,{\textrm{d}}x\right) ^{1/(p^\prime )^{-}} \Vert \varphi \Vert _{W^{1,p(|\nabla f_i^{0}|)}(\Omega )}, \end{aligned}$$

where $p^{+}=2,$ $(p^\prime )^{-}=p^{+}/(p^{+}-1)=2,$ and

$$\begin{aligned} \int _{\Omega }|R_\eta \nabla f_i^{0}|^{p(|\nabla f_i^{0}|)}\,{\textrm{d}}x {\mathop {\leqslant }\limits ^{\text {by}\,\, (9)}}1+\Vert \nabla f_i^{0}\Vert ^{p^{+}}_{L^{p(|\nabla f_i^{0}|)}(\Omega ;{\mathbb {R}}^2)}\leqslant 1+\Vert f_i^0\Vert ^2_{W^{1,p(|\nabla f_i^{0}|)}(\Omega )}<+\infty \end{aligned}$$

by the assumption $f_i^0\in \Xi _i.$ Thus, there exists a constant $M_1>0$ such that

$$\begin{aligned} |S_1|\leqslant M_1\Vert \varphi \Vert _{W^{1,p(|\nabla f_i^{0}|)}(\Omega )}. \end{aligned}$$

(57)

Arguing similarly, it can be shown that a constant $M_2>0$ exists such that

$$\begin{aligned} |S_2|\leqslant M_2\Vert \varphi \Vert _{W^{1,p(|\nabla f_i^{0}|)}(\Omega )}. \end{aligned}$$

(58)

As for the third term in (55), we notice that

$$\begin{aligned} |\nabla f_i^{0} |^{2} \frac{|\log \left( |\nabla f_i^{0} |\right) |}{\left( a^2+|\nabla f_i^{0}|^2\right) ^2}\leqslant \frac{|\log \left( |\nabla f_i^{0} |\right) |}{a^2+|\nabla f_i^{0}|^2}<+\infty \quad \text {as}\,\, \nabla f_i^{0}|\rightarrow \infty \end{aligned}$$

by the L’Hôpital’s rule. Using similar arguments, we see that

$$\begin{aligned} |\nabla f_i^{0} |^{2} \frac{|\log \left( |\nabla f_i^{0} |\right) |}{\left( a^2+|\nabla f_i^{0}|^2\right) ^2}\leqslant \frac{1}{a^4}|\nabla f_i^{0} |^{2} |\log \left( |\nabla f_i^{0} |\right) |<+\infty \quad \text {as}\ |\nabla f_i^{0}|\rightarrow 0. \end{aligned}$$

Thus, we can deduce the existence of a constant $M_2>0$ such that

$$\begin{aligned} |S_2|&\leqslant 2a^2(1-\delta )\left\| |R_\eta \nabla f_i^{0} |^{2} \frac{|\log \left( |R_\eta \nabla f_i^{0} |\right) |}{\left( a^2+|\nabla f_i^{0}|^2\right) ^2}\right\| _{L^\infty (\Omega )}\nonumber \\&\quad \times \int _\Omega \left( |R_\eta \nabla f_i^{0} |^{p(|\nabla f_i^{0}|)-2}R_\eta \nabla f_i^{0},\nabla \varphi \right) \!{\textrm{d}}x\leqslant M_3\Vert \varphi \Vert _{W^{1,p(|\nabla f_i^{0}|)}(\Omega )}. \end{aligned}$$

(59)

It remains to notice that in view of the obvious inclusions

$$\begin{aligned}{} & {} \int _\Omega W(\cdot ,y)\frac{f_i^{0}(\cdot )-f_i^{0}(y)}{\sqrt{\kappa ^2+\left| f_i^{0}(\cdot )-f_i^{0}(y)\right| ^2}}\,{\textrm{d}}y\in L^2(\Omega ),\\{} & {} \quad (-\Delta )^{-1}(f_i^{0}-f_i)\in L^2(\Omega ), \end{aligned}$$

the existence of constants $M_3$ and $M_4$ such that

$$\begin{aligned} |S_j|\leqslant M_j \Vert \varphi \Vert _{L^2(\Omega )}\leqslant M_j\Vert \varphi \Vert _{W^{1,p(|\nabla f_i^{0}|)}(\Omega )},\quad j=4,5, \end{aligned}$$

(60)

immediately follows from (55) and the Cauchy inequality.

Utilizing the estimates (57), (59), and (60), we finally arrive at the inequality (56) with $M=\max \{M_1,M_2,M_3,M_4,M_5\}.$ Thus, the mapping $\varphi \mapsto J_i^\prime (f_i^{0})[\varphi ]$ can be defined for all $\varphi \in H^{1,p(|\nabla f_i^{0}|)}(\Omega )$ using the density of $C^\infty ({\overline{\Omega }})$ in $H^{1,p(|\nabla f_i^{0}|)}(\Omega )$ (see (29)) and the standard rule

$$\begin{aligned} D_G J_i(f_i^{0})[\varphi ]=\lim _{k\rightarrow \infty }D_G J_i(f_i^{0})[\varphi _k], \end{aligned}$$

where $\left\{ \varphi _k\right\} _{k\in {\mathbb {N}}}\subset C^\infty _c({\mathbb {R}}^2)$ and $\varphi _k\rightarrow \varphi$ strongly in $H^{1,p(|\nabla f_i^{0}|)}(\Omega ).$ Hence, the objective functional $J_i(v)$ is Gâteaux differentiable at $v=f_i^{0},$ and

$$\begin{aligned} D_G J_i(f_i^{0})[h]=J^\prime (f_i^{0})[h],\quad \forall \, h\in H^{1,p(|\nabla f_i^{0}|)}(\Omega ). \end{aligned}$$

To get the final relations for optimality conditions, it remains to observe that identity (55) implies the following equalities in the sense of distributions:

$$\begin{aligned}{} & {} -\mathop {\textrm{div}}\Big [p(x)|R_\eta \nabla f_i^{0} |^{p(|\nabla f_i^{0}|)-2}R_\eta \nabla f_i^{0}\Big ]\nonumber \\{} & {} +\eta ^2\mathop {\textrm{div}}\Big [ p(x)\left( |R_\eta \nabla f_i^{0} |^{p(|\nabla f_i^{0}|)-2}R_\eta \nabla f_i^{0},\theta \right) \theta \Big ]\nonumber \\{} & {} +2a^2(1-\delta )\mathop {\textrm{div}}\left[ |R_\eta \nabla f_i^{0} |^{p(|\nabla f_i^{0}|)} \frac{\log \left( |R_\eta \nabla f_i^{0} |\right) }{\left( a^2+|\nabla f_i^{0}|^2\right) ^2}\nabla f_i^{0}\right] \nonumber \\{} & {} +\lambda \left( D(f_i^{0})-cD(f_i)\right) \int _\Omega W(x,y)\frac{f_i^{0}(x)-f_i^{0}(y)}{\sqrt{\kappa ^2 +\left| f_i^{0}(x)-f_i^{0}(y)\right| ^2}}\,{\textrm{d}}y\nonumber \\{} & {} +\mu (-\Delta )^{-1}(f_i^{0}-f_i)=0\quad \text {in }\ \Omega , \end{aligned}$$

(61)

$$\begin{aligned}{} & {} \Big (|\nabla f_i^{0} |^{p(|\nabla f_i^{0}|)-2}\nabla f_i^{0},\nu \Big )=0\quad \text {on }\ \partial \Omega , \end{aligned}$$

(62)

where $\nu$ denotes the unit outward normal to the boundary $\partial \Omega .$

5 Existence Issues and Regularization of the Original Optimization Problem

The main question we are going to discuss in this section is to find out whether the problem (31)–(32) admits at least one solution. With that in mind, we make use of the so-called indirect approach [16, 29]. The main idea of this approach is to show that the original minimization problem (31)–(32) can be efficiently approximated by a special family of optimization problems of a similar structure but with the spatial regularization of the exponent $p(|\nabla u|)$ in the form

$$\begin{aligned} p_{\varepsilon }(|\nabla u|)=1+\delta +\frac{a^2(1-\delta )}{a^2+\left| (\nabla K_{\varepsilon }*u)(x)\right| ^2}, \end{aligned}$$

(63)

where $(\nabla K_{\varepsilon }*u)$ stands for the Steklov smoothing operator.

Let $K{:}~{\mathbb {R}}^2\rightarrow {\mathbb {R}}$ be a positive compactly supported function such that

$$\begin{aligned} K\in C^\infty _0({\mathbb {R}}^2),\quad \int _{{\mathbb {R}}^2} K(x){\textrm{d}}x=1,\quad \text {and}\ K(x)=K(-x),\ \forall \,x\in {\mathbb {R}}^2. \end{aligned}$$

For any ${\varepsilon }>0,$ we set $K_{\varepsilon }(x)={\varepsilon }^{-2}K\left( \frac{x}{{\varepsilon }}\right) .$ Then, the following properties of the convolution:

$$\begin{aligned} u_{\varepsilon }(x):=(K_{\varepsilon }*u)(x)=\int _{\Omega } K_{\varepsilon }(x-y) u(y){\textrm{d}}y,\quad \forall \,u\in L^1(\Omega ) \end{aligned}$$

are well known [23]:

• (i) $u_{\varepsilon }\in C^\infty (\Omega )$ for all ${\varepsilon }>0;$

• (ii) $u_{\varepsilon }(x)\rightarrow u(x)$ almost everywhere in $\Omega ;$

• (iii) if $u\in L^p(\Omega )$ with $1\leqslant p<\infty ,$ then $u_{\varepsilon }\rightarrow u$ in $L^p(\Omega ).$

We introduce the following family of approximating problems to the problem (31)–(32):

$$\begin{aligned} J_{i,{\varepsilon }}(f^0_{i,{\varepsilon }})=\inf _{v\in \Xi _{i,{\varepsilon }}} J_{i,{\varepsilon }}(v),\quad i=1,2,3, \end{aligned}$$

(64)

where ${\varepsilon }$ is a small parameter which varies within a strictly decreasing sequence of positive numbers converging to 0, 

$$\begin{aligned} J_{i,{\varepsilon }}(v)&=\int _{\Omega } |R_{\eta }\nabla v(x)|^{p_{\varepsilon }(|\nabla v|)}\,{\textrm{d}}x +\frac{\mu }{2}\left\| v -f_i\right\| ^2_{H^{-1}(\Omega )}+\frac{\lambda }{4} \left[ D(v)-cD(f_i)\right] ^2, \end{aligned}$$

(65)

$$\begin{aligned} \Xi _{i,{\varepsilon }}&=\left\{ I\in H^{1,p_{\varepsilon }(|\nabla I|)}(\Omega )\cap L^\infty (\Omega )\left| \begin{array}{c} \gamma _{i,0}\leqslant I(x)\leqslant \gamma _{i,1}\quad \text{ a.e. } \text{ in }\ \Omega ,\\ \gamma _{i,0}=\inf _{x\in \Omega } f_{i}(x),\\ \gamma _{i,1}=\sup _{x\in \Omega } f_{i}(x) \end{array} \right. \right\} \end{aligned}$$

(66)

and $p_{\varepsilon }(|\nabla v|)$ is defined in (63).

Before proceeding further, we make use of a few technical results.

Lemma 1

[17, Lemma 1] Let $\left\{ v_k\right\} _{k\in {\mathbb {N}}}\subset L^\infty (\Omega )$ be a sequence of measurable functions such that $v_k(x)\rightarrow v(x)$ weakly-$*$ in $L^\infty (\Omega )$ for some $v\in L^\infty (\Omega ).$ Let

$$\begin{aligned} \left\{ p_k=1+\delta +\frac{a^2(1-\delta )}{a^2+\left| (\nabla K_{\varepsilon }*v_k)(x)\right| ^2}\right\} _{k\in {\mathbb {N}}} \end{aligned}$$

be the corresponding sequence of exponents. Then,

$$\begin{aligned}{} & {} p_{k,{\varepsilon }} \rightarrow p_{\varepsilon }=1+\delta +\frac{a^2(1-\delta )}{a^2+\left| (\nabla K_{\varepsilon }*u)\right| ^2}\quad \text {uniformly in }{\overline{\Omega }}\text { as } k\rightarrow \infty ,\nonumber \\{} & {} 1+\delta +\frac{a^2(1-\delta )}{{a^2+\Vert K_{\varepsilon }\Vert ^2_{C^1(\overline{\Omega -\Omega })} \sup _{k\in {\mathbb {N}}}\Vert v_k\Vert ^2_{L^1(\Omega )}}}\leqslant p_{k,{\varepsilon }}(x) \leqslant 2,\quad \forall \,x\in \Omega ,\ \forall \,k\in {\mathbb {N}},\nonumber \\ \end{aligned}$$

(67)

where

$$\begin{aligned} \Vert K_{\varepsilon }\Vert _{C^1(\overline{\Omega -\Omega })}=\max \limits _{\begin{array}{c} z=x-y\\ x\in {\overline{\Omega }}, y\in {\overline{\Omega }} \end{array}} \Big [|K_{\varepsilon }(z)|+|\nabla K_{\varepsilon }(z)|\Big ]. \end{aligned}$$

Lemma 2

[40, Proposition B.2] The mapping $v\mapsto \frac{\lambda }{4} \left[ D(v)-cD(f_i)\right] ^2$ is continuous from $L^2(\Omega )$ endowed with the strong topology to ${\mathbb {R}}$ with pointwise convergence.

Proposition 7

[17] Let $\left\{ p_{k,{\varepsilon }}\right\} _{k\in {\mathbb {N}}}$ be a sequence of exponents that satisfies all preconditions of Lemma 1. If a bounded sequence $\left\{ f_k\in L^{p_{k,{\varepsilon }}(\cdot )}(\Omega )\right\} _{k\in {\mathbb {N}}}$ converges weakly in $L^{1+{\delta }}(\Omega )$ to f,  then $f\in L^{p_{\varepsilon }(\cdot )}(\Omega ),$ $f_k\rightharpoonup f$ in variable $L^{p_{k,{\varepsilon }}(\cdot )}(\Omega ).$

We are now in a position to prove the existence of minimizers for the proposed approximating problem (64)–(66).

Theorem 1

Let $\Omega$ be an open bounded and connected sub-domain of ${\mathbb {R}}^2$ with a Lipschitz boundary. Let $f_i\in L^2(\Omega )$ be a given spectral channel of an image arguably contaminated by additive Gaussian noise with zero mean. Then, for each ${\varepsilon }>0,$ the minimization problem (64)–(66) admits at least one solution $f^0_{i,{\varepsilon }}$ in $W^{1,p^{-}}(\Omega )\cap L^\infty (\Omega )$ such that $I^0_{i,{\varepsilon }}\in H^{1,p[\nabla f^0_{i,{\varepsilon }}]}(\Omega ).$

Proof

To begin with, let us notice that, for each ${\varepsilon }>0,$ the indicated minimization problem is consistent, i.e., $J_{i,{\varepsilon }}(u)<+\infty$ for any $u\in \Xi _{i,{\varepsilon }}.$ Since $\Xi _{i,{\varepsilon }}\ne \varnothing$ and $0\leqslant J_{i,{\varepsilon }}(v)<+\infty$ for all $v\in \Xi _{i,{\varepsilon }},$ it follows that there exists a non-negative value $\zeta \geqslant 0$ such that $\zeta _{\varepsilon }=\inf \limits _{v\in \Xi _{i,{\varepsilon }}}J_{i,{\varepsilon }}(v).$ Let $\left\{ v^{\varepsilon }_k\right\} _{k\in {\mathbb {N}}}$ be a minimizing sequence for (64)–(66), i.e.,

$$\begin{aligned} \left\{ v^{\varepsilon }_k\right\} _{k\in {\mathbb {N}}}\subset \Xi _{i,{\varepsilon }}\ \text { and } \lim \limits _{k\rightarrow \infty } J_{i,{\varepsilon }}\left( v^{\varepsilon }_k\right) =\zeta _{\varepsilon }. \end{aligned}$$

Without lost of generality, we can suppose that $J_{i,{\varepsilon }}\left( v^{\varepsilon }_k\right) \leqslant \zeta _{\varepsilon }+1$ for all $k\in {\mathbb {N}}.$ From this and estimate (36), we deduce

$$\begin{aligned}{} & {} \Vert v^{\varepsilon }_k\Vert ^{p^{-}}_{W^{1,p_{\varepsilon }(|\nabla v^{\varepsilon }_k|)}}\leqslant C\left( |\Omega |\gamma _{i,1}^{2}+2+\frac{\zeta _{\varepsilon }+1}{(1-\eta ^2)^2}\right) ,\quad \forall \,k\in {\mathbb {N}}, \end{aligned}$$

(68)

$$\begin{aligned}{} & {} \Vert v^{\varepsilon }_k\Vert _{L^\infty (\Omega )}\leqslant \gamma _{i,1},\quad \forall \,k\in {\mathbb {N}}. \end{aligned}$$

(69)

Hence, in view of (7) and (33), the sequence $\left\{ v^{\varepsilon }_k\right\} _{k\in {\mathbb {N}}}$ is bounded in $W^{1,p^{-}}(\Omega ).$ Therefore, there exists a subsequence of $\left\{ v^{\varepsilon }_{k}\right\} _{k\in {\mathbb {N}}},$ still denoted by the same index, and a vector-function $f_{i,{\varepsilon }}^0\in W^{1,p^{-}}(\Omega )$ such that

$$\begin{aligned}{} & {} v^{\varepsilon }_{k}\rightarrow f_{i,{\varepsilon }}^0\ \text {strongly in } L^q(\Omega ) \text { for all } q\in [1,(p^{-})^*) \text { as } k\rightarrow \infty , \end{aligned}$$

(70)

$$\begin{aligned}{} & {} v^{\varepsilon }_{k}{\mathop {\rightharpoonup }\limits ^{*}}f_{i,{\varepsilon }}^0\ \text {weakly -}*\text { in } L^\infty (\Omega ) \text { as } k\rightarrow \infty , \end{aligned}$$

(71)

$$\begin{aligned}{} & {} v^{\varepsilon }_{k}\rightharpoonup f_{i,{\varepsilon }}^0\ \text {weakly in } W^{1,p^{-}}(\Omega ) \text { as } k\rightarrow \infty , \end{aligned}$$

(72)

where, by the Sobolev embedding theorem, $(p^{-})^*=\frac{2p^{-}}{2-p^{-}}=\frac{2+2\delta }{1-\delta }>2.$

In view of this and the smoothness of the kernel $K_{\varepsilon },$ we see that the operator

$$\begin{aligned} L^{p^{-}}(\Omega ;{\mathbb {R}}^2)\ni \nabla v\mapsto p_{\varepsilon }(|\nabla v|)\in C({\overline{\Omega }}) \end{aligned}$$

is compact (see Lemma 1). Therefore, (71)–(72) imply that $p_{\varepsilon }(|\nabla v^{\varepsilon }_{k}|) \rightarrow p_{\varepsilon }(|\nabla f^0_{i,{\varepsilon }}|)$ in $C({\overline{\Omega }}).$ Passing then to a subsequence if necessary, we have (see Propositions 1 and 7):

$$\begin{aligned}{} & {} v^{\varepsilon }_k(x)\rightarrow f^0_{i,{\varepsilon }}(x)\ \text { a.e. in } \Omega .\\{} & {} v^{\varepsilon }_k\rightharpoonup f^0_{i,{\varepsilon }}\text { weakly in variable } L^{p_{\varepsilon }(|\nabla v^{\varepsilon }_k|)}(\Omega ),\nonumber \\{} & {} \nabla v^{\varepsilon }_k\rightharpoonup \nabla f_{i,{\varepsilon }}^0\ \text { weakly in variable } L^{p_{\varepsilon }(|\nabla v^{\varepsilon }_k|)}(\Omega ;{\mathbb {R}}^{2}).\nonumber \end{aligned}$$

(73)

Hence, $f^0_{i,{\varepsilon }}\in W^{1,p_{\varepsilon }(|\nabla f^0_{i,{\varepsilon }}|)}(\Omega ).$

Further we notice that, for each $k\in {\mathbb {N}},$ $\gamma _{i,0}\leqslant v^{\varepsilon }_k(x)\leqslant \gamma _{i,1}$ a.a. in $\Omega .$ Then, it follows from (73) that the limit function $f_{i,{\varepsilon }}^0(x)$ is also subjected to the same restriction. Thus, $f^0_{i,{\varepsilon }}\in W^{1,p_{\varepsilon }(|\nabla f^0_{i,{\varepsilon }}|)}(\Omega )\cap L^\infty (\Omega )$ is a feasible solution to the minimization problem (64)–(66).

It remains to show that $f^0_{i,{\varepsilon }}$ is a minimizer of this problem. Indeed, taking into account the properties (68), (72), and the fact that $\theta \in L^\infty (\Omega ,{\mathbb {R}}^2),$ we see that the sequence

$$\begin{aligned} \left\{ \left| R_{\eta }\nabla v^{\varepsilon }_{k}:=\nabla v^{\varepsilon }_{k}-\eta ^2 \left( \theta ,\nabla v^{\varepsilon }_{k}\right) \theta \right| \in L^{p_{\varepsilon }(|\nabla v^{\varepsilon }_k|)}(\Omega ;{\mathbb {R}}^2)\right\} _{k\in {\mathbb {N}}} \end{aligned}$$

is bounded in the variable space $L^{p_{\varepsilon }(|\nabla v^{\varepsilon }_k|)}(\Omega ;{\mathbb {R}}^2)$ and weakly convergent to $|R_{\eta }\nabla f^0_{i,{\varepsilon }}|$ in $L^{p^{-}}(\Omega ;{\mathbb {R}}^2).$ Hence, by Proposition 1, the following lower semicontinuous property:

$$\begin{aligned} \liminf _{k\rightarrow \infty } \int _{\Omega } |R_{\eta }\nabla v^{\varepsilon }_{k}(x)|^{p_{\varepsilon }(|\nabla v^{\varepsilon }_k|)}\,{\textrm{d}}x\geqslant \int _{\Omega } |R_{\eta }\nabla f^0_{i,{\varepsilon }}(x)|^{p_{\varepsilon }(|\nabla f^0_{i,{\varepsilon }}|)}\,{\textrm{d}}x \end{aligned}$$

holds. Combining this relation with the following ones:

$$\begin{aligned}{} & {} \lim _{k\rightarrow \infty }\left\| v^{\varepsilon }_k -f_i\right\| ^2_{H^{-1}(\Omega )}=\left\| f^0_{i,{\varepsilon }} -f_i\right\| ^2_{H^{-1}(\Omega )},\\{} & {} \lim _{k\rightarrow \infty }\left[ D(v^{\varepsilon }_k)-cD(f_i)\right] ^2=\left[ D(f^0_{i,{\varepsilon }})-cD(f_i)\right] ^2, \end{aligned}$$

which are direct consequences of Lemma 2 and compactness of the embedding $L^2(\Omega )\subset H^{-1}(\Omega ),$ we finally obtain

$$\begin{aligned} \zeta _{\varepsilon }&=\inf \limits _{v\in \Xi _{i,{\varepsilon }}}J_{i,{\varepsilon }}(v)=\lim _{k\rightarrow \infty } J_{i,{\varepsilon }}\left( v^{\varepsilon }_k\right) =\liminf _{k\rightarrow \infty }J_{i,{\varepsilon }}\left( v^{\varepsilon }_k\right) \\&\geqslant \int _{\Omega } |R_{\eta }\nabla f_{i,{\varepsilon }}^0(x)|^{p_{\varepsilon }(|\nabla f^0_{i,{\varepsilon }}|)}\,{\textrm{d}}x +\frac{\mu }{2}\left\| f_{i,{\varepsilon }}^0 -f_i\right\| ^2_{H^{-1}(\Omega )}\\&\quad +\frac{\lambda }{4} \left[ D(f_{i,{\varepsilon }}^0)-cD(f_i)\right] ^2 = J_{i,{\varepsilon }}(f_{i,{\varepsilon }}^0). \end{aligned}$$

Thus, $f_{i,{\varepsilon }}^0$ is a minimizer to the problem (64)–(66).

Taking this existence result into account, we pass to the study of approximation properties of the problem (64)–(66). Namely, the main question we are going to discuss further is whether we can establish the convergence of minima of (64)–(66) to minima of (31)–(32) as ${\varepsilon }$ tends to zero. In other words, we aim to show that some optimal solutions to (31)–(32) can be approximated by the solutions of (64)–(66). To this end, we make use of some results of the variational convergence of minimization problems [26, 29,30,31] and begin with some auxiliaries (see also Refs. [14, 15, 30, 35, 36] for other aspects of this concept).

Lemma 3

Let $\left\{ {\varepsilon }_k\right\} _{k\in {\mathbb {N}}}$ be a sequence of positive numbers converging to zero as $k\rightarrow \infty .$ Let

$$\begin{aligned} \left\{ v_k\right\} _{k\in {\mathbb {N}}}\quad \text {and} \quad \left\{ p_k:=1+\delta +\frac{a^2(1-\delta )}{a^2+ \left| (\nabla K_{{\varepsilon }_k}*v_k)\right| ^2}\right\} _{k\in {\mathbb {N}}} \end{aligned}$$

be sequences such that

$$\begin{aligned}{} & {} v_k\in \Xi _{i,{\varepsilon }_k},\quad \forall \,k\in {\mathbb {N}}, \end{aligned}$$

(74)

$$\begin{aligned}{} & {} v_k(x)\rightarrow v(x)\quad \text {a.e. in } \Omega , \end{aligned}$$

(75)

$$\begin{aligned}{} & {} v_k\rightarrow v\quad \text {strongly in } L^2(\Omega ), \end{aligned}$$

(76)

$$\begin{aligned}{} & {} \nabla v_k\rightharpoonup \nabla v\quad \text {weakly in } L^{p^{-}}(\Omega ;{\mathbb {R}}^{2}), \end{aligned}$$

(77)

$$\begin{aligned}{} & {} \left\| |\nabla v_k(\cdot )|^{p_k(\cdot )}\right\| _{L^1(\Omega )}\leqslant C\quad \text {for some positive constant } C \text { not depending on } k, \end{aligned}$$

(78)

$$\begin{aligned}{} & {} p_k(x)\rightarrow p(x):=1+\delta +\frac{a^2(1-\delta )}{a^2+\left| \nabla v\right| ^2}\quad \text {a.e. in } \Omega . \end{aligned}$$

(79)

Then,

$$\begin{aligned} v\in \Xi _i\quad \text {and}\quad J_i(v)\leqslant \liminf _{k\rightarrow \infty } J_{i,{\varepsilon }_k}(v_k),\quad \forall \,i=1,2,3. \end{aligned}$$

(80)

Proof

The following relations:

$$\begin{aligned} \lim _{k\rightarrow \infty }\left\| v_k -f_i\right\| ^2_{H^{-1}(\Omega )}=\left\| v -f_i\right\| ^2_{H^{-1}(\Omega )}, \end{aligned}$$

(81)

$$\begin{aligned} \lim _{k\rightarrow \infty }\left[ D(v_k)-cD(f_i)\right] ^2=\left[ D(v)-cD(f_i)\right] ^2 \end{aligned}$$

(82)

are a direct consequence of Lemma 2, compactness of the embedding $L^2(\Omega )\subset H^{-1}(\Omega ),$ and condition (76). We also notice that, in view of representation

$$\begin{aligned} R_{\eta }\nabla v_k:=\nabla v_k-\eta ^2 \left( \theta ,\nabla v_k\right) \theta ,\quad \forall \,k\in {\mathbb {N}}, \end{aligned}$$

Proposition 1 and the initial assumptions (77)–(79) lead to the conclusion:

$$\begin{aligned} \nabla v\in L^{p(\cdot )}(\Omega ;{\mathbb {R}}^2)\quad \text {and}\quad \liminf _{k\rightarrow \infty } \int _{\Omega } |R_{\eta }\nabla v_k(x)|^{p_k(x)}\,{\textrm{d}}x \geqslant \int _{\Omega } |R_{\eta }\nabla v(x)|^{p(x)}\,{\textrm{d}}x. \end{aligned}$$

As a result, combining the last inequality with (81)–(82), we arrive at the announced relation (80)$_2.$

It remains to show that v is a feasible solution to the problem (31)–(32), i.e., $v\in \Xi _i.$ To this end, we take into account the inclusion $\nabla v\in L^{p(\cdot )}(\Omega ;{\mathbb {R}}^2)$ established above and the fact that $v_k\in \Xi _{i,{\varepsilon }_k}$ for each $k\in {\mathbb {N}}.$ Then, it follows from (75) that $\gamma _{i,0}\leqslant v(x)\leqslant \gamma _{i,1}$ almost everywhere in $\Omega ,$ and, therefore, $v\in \Xi _i.$

Lemma 4

For each feasible solution $v\in \Xi _i$ to the original problem (31)–(32), there can be found a sequence $\left\{ v_{\varepsilon }\right\} _{{\varepsilon }\rightarrow 0}$ such that

$$\begin{aligned}{} & {} v_{\varepsilon }\in \Xi _{i,{\varepsilon }},\quad \forall \,{\varepsilon }\in (0,{\varepsilon }_0)\ \text {with } {\varepsilon }_0>0 \text { sufficiently small},\end{aligned}$$

(83)

$$\begin{aligned}{} & {} v_e(x)\rightarrow v(x)\quad \text {a.e. in } \Omega \text { as } {\varepsilon }\rightarrow 0, \end{aligned}$$

(84)

$$\begin{aligned}{} & {} v_{\varepsilon }\rightarrow v\quad \text {strongly in } L^2(\Omega ), \end{aligned}$$

(85)

$$\begin{aligned}{} & {} \nabla v_{\varepsilon }\rightarrow \nabla v\quad \text {strongly in } L^{p^{-}}(\Omega ;{\mathbb {R}}^{2}), \end{aligned}$$

(86)

$$\begin{aligned}{} & {} \left\| |\nabla v_{\varepsilon }(\cdot )|^{p_{\varepsilon }(\cdot )}\right\| _{L^1(\Omega )}\leqslant C\quad \text {for some positive constant } C \text { not depending on } {\varepsilon }, \end{aligned}$$

(87)

$$\begin{aligned}{} & {} p_{\varepsilon }(x):=1+\delta +\frac{a^2(1-\delta )}{a^2+\left| (\nabla K_{{\varepsilon }}*v_{\varepsilon })\right| ^2}\rightarrow p(x):=1+\delta +\frac{a^2(1-\delta )}{a^2+\left| \nabla v\right| ^2}\quad \text {a.e. in } \Omega , \nonumber \\ \end{aligned}$$

(88)

$$\begin{aligned}{} & {} J_i(v)\geqslant \limsup _{{\varepsilon }\rightarrow 0} J_{i,{\varepsilon }}(v_{\varepsilon }). \end{aligned}$$

(89)

Proof

Let v be an arbitrary feasible solution to the problem (31)–(32). We define the sequence $\left\{ v_{\varepsilon }\right\} _{{\varepsilon }\rightarrow 0}$ as a smooth mollification of v with the kernel $K_{\varepsilon },$ i.e.,

$$\begin{aligned} v_{\varepsilon }(x):=(K_{\varepsilon }*v)(x)=\int _{\Omega } K_{\varepsilon }(x-y) v(y)\,{\textrm{d}}y,\quad \forall \,x\in \Omega . \end{aligned}$$

Then, properties (84)–(86) are direct consequences of the Steklov smoothing procedure (see (i)–(iii)). Moreover, in view of (84) and the fact that $\gamma _{i,0}\leqslant v(x)\leqslant \gamma _{i,1}$ a.e. in $\Omega ,$ we can suppose that the same restriction for $v_{\varepsilon }$

$$\begin{aligned} \gamma _{i,0}\leqslant v_{\varepsilon }(x)\leqslant \gamma _{i,1}\quad \text {a.e. in } \Omega \end{aligned}$$

(90)

holds true with ${\varepsilon }>0$ small enough.

Since $v_{\varepsilon }\rightarrow v$ strongly in $W^{1,p^{-}}(\Omega ),$ it follows without loss of generality that $\nabla v_{\varepsilon }(x)\rightarrow \nabla v(x)$ almost everywhere in $\Omega .$ As a result, this implies the pointwise convergence (88). Hence,

$$\begin{aligned} |R_\eta \nabla v_{\varepsilon }(x)|^{p_{\varepsilon }(x)}\rightarrow |R_\eta \nabla v(x)|^{p(x)}\quad \text {a.e. in }\Omega . \end{aligned}$$

From this and the fact that $|R_\eta \nabla v(x)|^{p(x)}\in L^1(\Omega ),$ we deduce

$$\begin{aligned} |R_\eta \nabla v_{\varepsilon }|^{p_{\varepsilon }(\cdot )}\rightarrow |R_\eta \nabla v|^{p(\cdot )}\quad \text {strongly in }L^1(\Omega ). \end{aligned}$$

(91)

Thus, $\left\| |R_\eta \nabla v_{\varepsilon }(\cdot )|^{p_{\varepsilon }(\cdot )}\right\| _{L^1(\Omega )}\leqslant C$ for some positive constant C not depending on ${\varepsilon }.$ Hence, in view of estimates (34), we get $\nabla v_{\varepsilon }\in L^{p_{\varepsilon }(\cdot )}(\Omega ;{\mathbb {R}}^2)$ for ${\varepsilon }$ small enough. From this and (90), the assertion (83) follows. Moreover, the following equality:

$$\begin{aligned} \lim _{{\varepsilon }\rightarrow 0}\int _{\Omega } |R_\eta \nabla v_{\varepsilon }|^{p_{\varepsilon }(\cdot )}\,{\textrm{d}}x= \int _{\Omega } |R_\eta \nabla v|^{p(\cdot )}\,{\textrm{d}}x \end{aligned}$$

(92)

immediately follows from (91).

It remains to observe that

$$\begin{aligned} \lim _{k\rightarrow \infty }\left\| v_k -f_i\right\| ^2_{H^{-1}(\Omega )}=\left\| v -f_i\right\| ^2_{H^{-1}(\Omega )}, \end{aligned}$$

(93)

$$\begin{aligned} \lim _{k\rightarrow \infty }\left[ D(v_k)-cD(f_i)\right] ^2=\left[ D(v)-cD(f_i)\right] ^2, \end{aligned}$$

(94)

by Lemma 2 and compactness of the embedding $L^2(\Omega )\subset H^{-1}(\Omega ).$ As a result, we conclude from (92), (93), and (94) that, in fact, instead of the announced inequality (89), we have $J_i(v)=\lim _{{\varepsilon }\rightarrow 0} J_{i,{\varepsilon }}(v_{\varepsilon }).$ The proof is complete.

We are now in a position to prove the main result of this section.

Theorem 2

Assume that the original minimization problem (31)–(32) has a non-empty set of minimizers. Let $\left\{ f^0_{i,{\varepsilon }}\in \Xi _{i,{\varepsilon }}\right\} _{{\varepsilon }>0}$ be a sequence of solutions to the corresponding minimization problems (64)–(66). Let $\left\{ p_{\varepsilon }:=1+\delta +\frac{a^2(1-\delta )}{a^2+\left| (\nabla K_{{\varepsilon }}*f^0_{i,{\varepsilon }})\right| ^2}\right\} _{{\varepsilon }>0}$ be the sequence of associated exponents. Assume that the sequence $\left\{ p_{\varepsilon }\right\} _{{\varepsilon }>0}$ is compact with respect to the pointwise convergence in $\Omega .$ Then, there exists an element $f_i^0\in \Xi$ such that, up to a subsequence,

$$\begin{aligned}{} & {} f^0_{i,{\varepsilon }}(x)\rightarrow f_i^0(x)\quad \text {a.e. in } \Omega \text { as } {\varepsilon }\rightarrow 0, \end{aligned}$$

(95)

$$\begin{aligned}{} & {} f^0_{i,{\varepsilon }}\rightarrow f_i^0\quad \text {strongly in } L^2(\Omega ), \end{aligned}$$

(96)

$$\begin{aligned}{} & {} \nabla f^0_{i,{\varepsilon }}\rightharpoonup \nabla f_i^0\quad \text {weakly in } L^{p^{-}}(\Omega ;{\mathbb {R}}^{2}), \end{aligned}$$

(97)

$$\begin{aligned}{} & {} \left\| |\nabla f^0_{i,{\varepsilon }}(\cdot )|^{p_{\varepsilon }(\cdot )}\right\| _{L^1(\Omega )}\leqslant C\quad \text {for some positive constant } C \text { not depending on } {\varepsilon }, \end{aligned}$$

(98)

$$\begin{aligned}{} & {} \inf _{v\in \Xi _{i,{\varepsilon }}}J_i(v)=J_i(f_i^0)=\lim _{{\varepsilon }\rightarrow 0} J_{i,{\varepsilon }}(f^0_{i,{\varepsilon }})=\lim _{{\varepsilon }\rightarrow 0}\inf _{v\in \Xi _{i,{\varepsilon }}}J_{i,{\varepsilon }}(v_{\varepsilon }). \end{aligned}$$

(99)

Proof

First, we observe that a given sequence of minimizers for approximating problems (64)–(66) is compact with respect to the convergence (95)–(97). Indeed, for an arbitrary test function $\varphi \in C^\infty _c({\mathbb {R}}^2),$ we have

$$\begin{aligned} \varphi \in H^{1,p_{\varepsilon }(\cdot )}(\Omega ),\quad \forall \,{\varepsilon }>0. \end{aligned}$$

Let us assume that this function satisfies the pointwise constraints $\gamma _{i,0} \leqslant \varphi (x) \leqslant \gamma _{i,1}$ in $\Omega .$ Then, $\varphi \in \Xi _{i,{\varepsilon }}$ for all ${\varepsilon }>0,$ and, therefore, we can suppose that

$$\begin{aligned} J_{i,{\varepsilon }}(f^0_{i,{\varepsilon }})=\inf _{v\in \Xi _{i,{\varepsilon }}}J_{i,{\varepsilon }}(v_{\varepsilon }) \leqslant J_{i,{\varepsilon }}(\varphi ) \leqslant \sup _{{\varepsilon }> 0}J_{i,{\varepsilon }}(\varphi ) \leqslant C<+\infty ,\quad \forall \,{\varepsilon }>0. \end{aligned}$$

Hence,

$$\begin{aligned} \sup _{{\varepsilon }> 0}\int _{\Omega } |R_\eta \nabla f^0_{i,{\varepsilon }}(\cdot )|^{p_{\varepsilon }(\cdot )}\,{\textrm{d}}x\leqslant C\quad \text {and}\quad \sup _{{\varepsilon }> 0}\Vert f^0_{i,{\varepsilon }}\Vert _{L^2(\Omega )}\leqslant \sqrt{\Vert \Omega \Vert } \gamma _{i,1}. \end{aligned}$$

(100)

Combining this issue with estimates (34), we see that the sequence $\Big \{\! f^0_{i,{\varepsilon }}\in \Xi _{i,{\varepsilon }}\!\Big \}_{{\varepsilon }>0}$ is bounded in $W^{1,p^{-}}(\Omega ).$ Hence, there exists a subsequence $\Big \{\! f^0_{i,k}\in \Xi _{i,{\varepsilon }_k}\!\Big \}_{k\in {\mathbb {N}}}$ of $\Big \{\! f^0_{i,{\varepsilon }}\in \Xi _{i,{\varepsilon }}\!\Big \}_{{\varepsilon }>0},$ and a function $f_i^0\in W^{1,p^{-}}(\Omega )$ such that

$$\begin{aligned}{} & {} f^0_{i,k}\rightarrow f_i^0\ \text{ strongly } \text{ in } \ L^q(\Omega ) \ \mathrm{for \ all} \ q\in [1,(p^{-})^*),\nonumber \\{} & {} f^0_{i,k} \rightharpoonup f_i^0\ \text{ weakly } \text{ in } \ W^{1,p^{-}}(\Omega ) \ \text {as} \ k\rightarrow \infty , \end{aligned}$$

(101)

where, by the Sobolev embedding theorem, $(p^{-})^*=\frac{2p^{-}}{2-p^{-}}=\frac{2+2\delta }{1-\delta }>2+\delta .$ From this, the conditions (95)–(97) follow, whereas (98) is a consequence of (34) and the boundedness property (100).

Thus, we may suppose that for the subsequence $\left\{ f^0_{i,k}\in \Xi _{{\varepsilon }_k}\right\} _{k\in {\mathbb {N}}}$ all preconditions of Lemma 3 are fulfilled. Therefore, property (80) leads us to the conclusion that $f_i^0\in \Xi _i$ and

$$\begin{aligned} \liminf _{k\rightarrow \infty }\inf _{v\in \Xi _{i,e_k}} J_{i,{\varepsilon }_k}(v)= \liminf _{k\rightarrow \infty } J_{{i,{\varepsilon }_k}}(f^0_{i,k}) \geqslant J_i(f_i^0)\geqslant \inf _{v\in \,\Xi _i}J_i(v) = J_i(f_i^*), \end{aligned}$$

(102)

where $f_i^*\in \Xi$ is a minimizer for (31)–(32).

Fig. 1

Original image (left) and its smoothed version without contrasting $(\mu =0)$ (right)

Fig. 2

Variants of contrast enhancement with the corresponding histograms (from the left to the right): $c=2$ and $\text {window}=5,$ $c=2$ and $\text {window}=7$

Fig. 3

Variants of contrast enhancement with the corresponding histograms (from the left to the right): $c=10$ and $\text {window}=5,$ $c=10$ and $\text {window}=7$

Then, Lemma 4 implies the existence of a realizing sequence $\left\{ f^*_{i,{\varepsilon }}\in \Xi _{i,{\varepsilon }}\right\} _{{\varepsilon }>0}$ such that $f^*_{i,{\varepsilon }}\rightarrow f^*_i$ as ${\varepsilon }\rightarrow 0$ in the sense of relations (84)–(88), and

$$\begin{aligned} J_i(f_i^*)\geqslant \limsup _{{\varepsilon }\rightarrow 0} J_{i,{\varepsilon }}(f^*_{i,{\varepsilon }}). \end{aligned}$$

Utilizing this fact, we have

$$\begin{aligned} \inf _{v\in \,\Xi _i}J_i(v)&= J_i(f^*_i)\geqslant \limsup _{{\varepsilon }\rightarrow 0} J_{i,{\varepsilon }}(f^*_{i,{\varepsilon }})\geqslant \limsup _{{\varepsilon }\rightarrow 0}\inf _{v\in \,\Xi _{i,{\varepsilon }}} J_{i,{\varepsilon }}(v) \nonumber \\&\geqslant \limsup _{k\rightarrow \infty } \inf _{v\,\in \,\Xi _{i,{\varepsilon }_k}} J_{i,{\varepsilon }_k}(v)= \limsup _{k\rightarrow \infty }J_{i,{\varepsilon }_k}(f^0_{i,k}). \end{aligned}$$

(103)

From this and (102) we deduce that

$$\begin{aligned} \liminf _{k\rightarrow \infty } J_{i,{\varepsilon }_k}(f^0_{i,k})\geqslant \limsup _{k\rightarrow \infty }J_{i,{\varepsilon }_k}(f^0_{i,k}). \end{aligned}$$

Hence, we can combine (102) and (103) to get

$$\begin{aligned} J_i(f_i^0) =J_i(f_i^*)= \inf _{v\in \Xi _i}J_i(v)=\lim _{k\,\rightarrow \infty }\inf _{v\,\in \, \Xi _{i,{\varepsilon }_k}} J_{i,{\varepsilon }_k}(v). \end{aligned}$$

(104)

Using these relations and the fact that the problem (31)–(32) is solvable, we may suppose that $f_i^*=f_i^0.$ Since the equality (104) holds for all subsequences of $\left\{ f^0_{i,{\varepsilon }}\right\} _{{\varepsilon }>0},$ which are convergent in the sense of relations (95)–(97), it follows that these limits coincide and, therefore, $f_i^0$ is the limit of the whole sequence $\left\{ f^0_{i,{\varepsilon }}\right\} _{{\varepsilon }>0}.$ Then, using the same argument for the sequence of minimizers as we did for the subsequence $\left\{ f^0_{i,{\varepsilon }_k}\right\} _{\,k\in \,{\mathbb {N}}},$ we finally obtain

$$\begin{aligned} \liminf _{{\varepsilon }\,\rightarrow 0}\inf _{v\in \,\Xi _{i,{\varepsilon }}} J_{i,{\varepsilon }}(v)&= \liminf _{{\varepsilon }\,\rightarrow 0} J_{i,{\varepsilon }}(f^0_{i,{\varepsilon }}) \geqslant J_i(f_i^0)= \inf _{v\in \,\Xi _i}J_i(v)\\&\geqslant \limsup _{{\varepsilon }\rightarrow 0}J_{i,{\varepsilon }}(f^*_{i,{\varepsilon }}) \geqslant \limsup _{{\varepsilon }\rightarrow 0}\inf _{v\in \,\Xi _{i,{\varepsilon }}} J_{i,{\varepsilon }}(v)\\&=\limsup _{{\varepsilon }\,\rightarrow 0}J_{i,{\varepsilon }}(f^0_{i,{\varepsilon }}), \end{aligned}$$

and this concludes the proof.

Remark 3

It is worth to emphasize a few principle issues from Theorem 2. The first one is that, in practice, the assumption concerning solvability of the original optimization problem is not so restricted and, in principle, it can be omitted. Indeed, any digital color image $f=[f_1,f_2,f_3]^t$ is originally defined on some grid G. Therefore, each of its spectral channels $f_i\Big |_G$ can be associated with some real-valued matrix. Hence, we can always suppose that the exponent $p(|\nabla f_i|)\Big |_G$ is the restriction on the same grid of some Lipschitz-continuous function $p(\cdot )\!\!:\Omega \rightarrow {\mathbb {R}}.$ Then, arguing as in the proof of Theorem 1, the solvability of the problem (31)–(32) can be easily established.

The second point, that should be emphasized here, is the assumption about the compactness property of the sequence of associated exponents $\left\{ p_{\varepsilon }:=1+\delta +\frac{a^2(1-\delta )}{a^2+\left| (\nabla K_{{\varepsilon }}*f^0_{i,{\varepsilon }})\right| ^2}\right\} _{{\varepsilon }>0}$ with respect to the pointwise convergence in $\Omega .$ Since this property is crucial in Theorem 2, we propose to consider it as an easily realized in practice criterion for the verification of whether the approximating sequence $\left\{ f^0_{i,{\varepsilon }}\in \Xi _{i,{\varepsilon }}\right\} _{{\varepsilon }>0}$ leads to some optimal solution of the original problem.

6 Numerical Results

To illustrate the implementation of the proposed model (31)–(32) to the simultaneous denoising and contrast enhancement of color images, we make use of the optimality conditions in the form of (62). In other words, we have dropped the two-side constraints $\gamma _{i,0}\leqslant v(x)\leqslant \gamma _{i,1}$ from the sets $\Xi _i,$ and instead we control the fulfillment of this two-side constraint at each step of the numerical approximations.

Fig. 4

Variants of contrast enhancement with the corresponding histograms (from the left to the right): original image, restored image with $c=20$ and $\text {window}=15$

Fig. 5

Variants of contrast enhancement with the corresponding histograms (from the left to the right): original and restored with $c=10$ and $\text {window}=7$

Fig. 6

Variants of contrast enhancement with the corresponding histograms (from the left to the right): original image, restored image with $c=10$ and $\text {window}=5$

Since, in practical implementations, it is reasonable to define the solution of the problem (31)–(32) using a “gradient descent” strategy, we can start with some initial image $f=[f_1,f_2,f_3]^\textrm{t}\in L^2(\Omega ;{\mathbb {R}}^3)$ and pass to the following system of three initial-boundary value problems for quasi-linear parabolic equations with Neumann boundary conditions:

$$\begin{aligned} \frac{\partial f_i^{0}}{\partial t}&= \mathop {\textrm{div}}\Big [p(x)|R_\eta \nabla f_i^{0} |^{p(|\nabla f_i^{0}|)-2}R_\eta \nabla f_i^{0}\Big ]\nonumber \\&\quad -\eta ^2\mathop {\textrm{div}}\Big [ p(x)\left( |R_\eta \nabla f_i^{0} |^{p(|\nabla f_i^{0}|)-2}R_\eta \nabla f_i^{0},\theta \right) \theta \Big ]\nonumber \\&\quad -2a^2(1-\delta )\mathop {\textrm{div}}\left[ |R_\eta \nabla f_i^{0} |^{p(|\nabla f_i^{0}|)} \frac{\log \left( |R_\eta \nabla f_i^{0} |\right) }{\left( a^2+|\nabla f_i^{0}|^2\right) ^2}\nabla f_i^{0}\right] \nonumber \\&\quad -\lambda \left( D(f_i^{0})-cD(f_i)\right) \int _\Omega W(x,y)\frac{f_i^{0}(x)-f_i^{0}(y)}{\sqrt{\kappa ^2 +\left| f_i^{0}(x)-f_i^{0}(y)\right| ^2}}\,{\textrm{d}}y\nonumber \\&\quad -\mu (-\Delta )^{-1}(f_i^{0}-f_i)=0\quad \text {in }\ (0,T)\times \Omega , \end{aligned}$$

(105)

$$\begin{aligned}&\Big (|\nabla f_i^{0} |^{p(|\nabla f_i^{0}|)-2}\nabla f_i^{0},\nu \Big )=0\quad \text {on }\ (0,T)\times \partial \Omega , \end{aligned}$$

(106)

$$\begin{aligned}&f_i^{0}(0,\cdot )=f_{i}(\cdot ),\quad i=1,2,3\quad \text { in }\ \Omega . \end{aligned}$$

(107)

For numerical simulations, we set: $\delta =\kappa$ in (24) and (21), and $\eta =1-\kappa$ in (19), $\kappa =0.001,$ $\lambda =0.1,$ and $\mu =2.$ As for the noise estimator $a>0$ in (21), we use the choice of Black et al. [6], i.e.,

$$\begin{aligned} a=\frac{1.482\,6}{\sqrt{2}}\text {MAD}(\nabla f_i), \end{aligned}$$

where $\textrm{MAD}$ denotes the median absolute deviation of the corresponding spectral channel $f_i$ of the original image $f=[f_1,f_2,f_3]^\textrm{t}$ that can be computed as

$$\begin{aligned} \text {MAD}(\nabla f_i)={\textrm{median}}\left[ \Big | |\nabla f_i|-{\textrm{median}}\left( \left| \nabla f_i\right| \right) \Big |\right] \end{aligned}$$

and ${\textrm{median}}\left( \left| \nabla {\widetilde{S}}_i\right| \right)$ represents the median over the band $S_i\!\!\!:G_H\rightarrow {\mathbb {R}}$ to the gradient amplitude.

Fig. 7

Influence of the contrast enhancement scale on the result (from the left to the right): original, $c=10$ and $\text {window}=5,$ $c=20$ and $\text {window}=5$

Fig. 8

Variants of contrast enhancement with the corresponding histograms (from the left to the right): original image, restored image with $c=10$ and $\text {window}=5$

Fig. 9

Variants of contrast enhancement with the corresponding histograms (from the left to the right): original image, restored image with $c=10$ and $\text {window}=5$

To guarantee the stability of the proposed algorithm, we make use of the following condition:

$$\begin{aligned} 2\left( \frac{1}{\kappa }+\lambda +\mu \right) \Delta t<1. \end{aligned}$$

There are numerous approaches to solve quasi-linear partial differential equations (see Refs. [4, 25] for various techniques). Since we are dealing with pixels in image processing, finite-difference approaches and an explicit scheme of the forward Euler method are arguably the best options. The number of iterations for each spectral channel can be defined experimentally. We used $10^3$-iterations. As for the size of the kernel W(x, y) used for D,  this size manages the scale of the contrast enhancement. In our experiments, we used it equal to 3, 5, 7, 15,  albeit it can be related to the size of the input image.

The most expensive computation is the one of D and $\nabla D$ embedded in the computation of the right-hand side of the system (105). For acceleration of these computations, we can refer to Ref. [40], where the efficient Bernstein polynomials approximation has been proposed.

As follows from the result of numerical simulations (see Figs. 3, 5, 6, 7, 8, 9), parameters c,  $\mu ,$ $\lambda ,$ and the size of $\text {window}$ for the kernel W(x, y) are crucial for the contrast enhancement and these parameters have to be tuned in dependence on the desired result. In particular, we observe that at a large scale $(\text {window})$ and low contrast level c,  the proposed model can produce an image with more details, but with the same lighting sensation as the original one. To show how the choice of the parameters c,  $\mu ,$ $\lambda ,$ and $\text {window}$ affect the results of contrast enhancement, we supplied all images in Figs. 1, 2, 3, 5, 6, 7, 8, and 9 by the histograms of their luma components which represent the perceptual brightness of the color images $I\!\!:\Omega \rightarrow {\mathbb {R}}^3.$ To this end, we used the following representation for the luma: $Y_I(x)=\alpha _R I_R(x)+ \alpha _G I_G(x)+ \alpha _B I_B(x)$ with

$$\begin{aligned} \alpha _R=0.299,\quad \alpha _G=0.587,\quad \alpha _B=0.114. \end{aligned}$$

Here, $I_R,$ $I_G,$ and $I_B$ stand for the intensities of a given image in R, G, and B spectral channels, respectively.

In particular, as follows from the obtained histograms, the proposed variational model is sufficiently sensitive to the choice of the weight coefficient c,  whereas the size of $\text {window}$ for the kernel W(x, y) affects the contrast enhancement in rather a mild manner (see Figs. 2 and 3).

7 Conclusions

We introduce a variational model with the non-standard growth condition for the restoration and contrast enhancement of multi-band images. We show that increasing the average local contrast measure improves the perceived contrast of the image. We have obtained sufficient conditions for the convergence of the minimization algorithm. The contrast scale and level in our model are adjustable, so that the proposed approach can be considered as fully adaptive. Our enhancement method for color images works directly on the RGB images without any pre- and/or post-processing. The automatic adaptation of the parameters to the content of the considered image could be a future direction of research.