1. Introduction and Some Preliminaries
The main goal of any image denoising problem is to restore the noise-free gray-scale image
from the observed one
. In this paper, we start from the assumption that the observed image can be represented as
where
n is the white Gaussian noise following the Gaussian distribution
and
v stands for the noise with a probably strong impulsive nature, which the Gaussian model fails to describe. We assume that both noises occur simultaneously and independently in the entire domain.
To eliminate both the Gaussian noise
n and impulse noise
v, we propose the following optimal control problem:
subject to the constraints
Here,
is a bounded, open, simply connected set, its boundary
is assumed to be sufficiently smooth,
is a positive value,
is a given positive parameter,
is the original noise-corrupted image,
is the pre-denoised image when applying a median filter to
f,
,
a.e. in
are given distributions, and
is the so-called directional sparsity term which, in fact, measures the
-norm in the space of the
-norm in time. Additionally,
is the matrix of anisotropy, and as for the variable exponent
, we define it with the rule
where
Here, stands for the zero extension of u to the entire space , and and are given small positive values. As for the parameters and , they act as regularization and smoothing parameters.
It is clear now that, for each function , the inclusion holds almost everywhere in with and .
The study of optimal control problems for PDEs with variable nonlinearity is motivated by various applications in the image enhancement, where some special cases of Equations (
2)–(
5) appear as the natural generalization of the classical Perona–Malik model [
1,
2,
3,
4]. We also refer to [
5], where the authors dealt with a special case of the model in Equations (
2)–(
5) and show the given class of optimal control problems is well posed.
The main benefit of the proposed model in Equations (
2)–(
5) is the manner in which this model accommodates the local image information. It is easy to see that if the gradient of the noisy image
f is sufficiently large (i.e., likely edges) at some places, then only total variation (or shortly
-based) diffusion will be used there. However, if at some points the gradient is sufficiently close to zero (i.e., it is a homogeneous region), then the model becomes isotropic. At the rest of the locations, the diffusion is somewhere between Gaussian and TV-based. However, as immediately follows from Equation (
7) and definition of the matrix
, the type of anisotropy is not completely predefined by the structure of the original noisy image
f. Moreover, the image
u after denoising may have other shapes of homogeneous regions and other structures with other locations for the edges.
In spite of the fact that there are many other different variants for the choice of the diffusivity term in Equation (
2) using, for instance, the so-called directional total variation [
6] and flexible space-variant anisotropic regularization [
7], to the best of our knowledge, the effective choice of this operator for general image denoising problem with different noise distributions remains an open problem.
In recent years, many different techniques have been proposed for the reconstruction of noise-affected digital images. In particular, the following nonlinear hybrid diffusion model, which is a symbiosis of the mean curvature diffusion with the Gaussian heat diffusion, has been proposed for image denoising (see [
8]):
where
Here, f is an input image, and are fixed constants, is a bounded open domain of with a sufficiently smooth boundary, and is the unit outwardly normal to the boundary .
The important characteristic of this model is the fact that it has a hybrid diffusion type which combines the mean curvature diffusion with the heat diffusion such that inside those regions, where the gradient of
u is small enough, the new model acts like a heat equation and results in isotropic smoothing, whereas near the region’s contours where the magnitude of the gradient is large, this model acts like a mean curvature equation. From this point of view, the model in Equations (
11)–(
13) can be interpreted as some generalization of the well-known ones (in particular, the Perona–Malik model [
9] or the models with the
-Laplacian operator that was proposed in [
4]). However, in general, it would be erroneously to assert that all the above-mentioned models can be obtained as a particular case of Equations (
11)–(
13).
It is worth emphasizing that because of the variable character of exponent
p in Equation (
2), we have a gap between the coercivity and monotonicity conditions. In light of this, the problem in Equations (
1)–(
4) can be specified as an optimal control problem for the quasi-linear parabolic equations with variable growth conditions, and it can be interpreted as a generalization of the parabolic version of the
-Laplacian equation
with a variable exponent that depends only on
t and
x. We can indicate extensive research which is devoted to Equation (
15). A rather complete insight to the theory of parabolic
-Laplacian equations can be found in [
10,
11,
12,
13,
14,
15].
However, to the best of our knowledge, the above-mentioned results for the solvability issues for evolutional equations of the type in Equation (
2) mainly concern the parabolic IBVPs with exponents depending on
only, whereas hardly any attention has been paid to the IBVP of the form in Equation (
2) with
and the exponent
given by the rule in Equation (
7). Moreover, in contrast to most of the existing results, in this paper, we do not predefine
and
a priori. Instead, we associate these characteristics with a particular solution for the IBVPs (Equations (
2)–(
5)). Therefore, the unknown solution
u can affect the rate of nonlinearity of
p and the tensor
D. It is also worth mentioning that in contrast to most existing publications (see, for instance, [
10,
16]), we do not assume that the dependency of
and
on
u is local. We show that all weak solutions to this problem live in the corresponding ‘personal’ functional spaces, and, in light of the special assumptions for the structure of
and
, the problem in Equations (
2)–(
5) can have the weak solutions that do not possess the standard variational properties of solutions to the parabolic equations. In particular, it is unknown whether a weak solution to the above problem satisfies the standard energy equality and is unique.
In spite of the fact that a number of different regularizations have been suggested in the literature for optimal control problems related to the degenerate elliptic equations (see, for instance, [
17,
18,
19]), the question about solvability of the proposed optimal control problem is still open, to the best of our knowledge.
In light of this, our primary goal is to study the solvability issues for the OCP (Equations (
1)–(
5)). In particular, a couple of characteristic features of the proposed problem can be emphasized here. The first one is that the tensor of anisotropy
and the exponent
depend not only on
but also on
. The second feature is that the optimal control problem is formulated with
as the control cost (together with additional pointwise control constraints). As a result, the optimal control may have directional sparsity (i.e., its support is constant in time, whereas the control
v can be identically zero on some parts of the domain
).
This paper is structured as follows. The main assumptions for the structure of the anisotropic diffusion tensor
and variable exponent
and some preliminaries are given in
Section 2. We also discuss in this section the basic auxiliary results concerning the Sobolev–Orlicz spaces with a variable exponent. In
Section 3, we focus mainly on the solvability issues for the IBVPs (Equations (
2)–(
5)). To this end, we follow the indirect approach using a special technique of passing to the limit in the sequences of variational problems. Precise statement of the optimal control problems for a quasi-linear elliptic equation with the sparse control is discussed in
Section 4. We also discuss in this section the main topological properties of feasible solutions to the given OCP, and as a consequence, we derive the conditions where the set of optimal solutions is nonempty. As for the optimality conditions, their substantiation, and the results of numerical simulations, these issues are the subject of a forthcoming paper. With that in mind, we will realize the principle of variational convergence of constrained minimization problems and utilize some key ideas from [
20,
21,
22,
23].
2. Main Assumptions and Preliminaries
Let be a bounded, open, simply connected set and its boundary be sufficiently smooth. For simplicity, we assume that the unit’s outward normal is well defined for a.e., . Let be a given value. We also set . For any measurable subset , we denote by its two-dimensional Lebesgue measure . Let be its closure and stand for the boundary of D. We also make use of the following notation: .
For two vectors and , the notation stands for the standard vector inner product in , where stands for the transpose operator. As for the norm , we take this as the Euclidean norm given by the rule .
2.1. Functional Spaces
Let Y be a real Banach space endowed with the norm , and let be its dual. With ⇀ and , we denote the weak and weak convergence in the spaces Y and , respectively. Let be the duality form on .
For a given exponent
, the Lebesgue space
is defined by the standard rule
Here,
for
. The inner product of two functions
g and
f in
with
is given by
Let
be the locally convex space of all infinitely differentiable functions with compact support in
. We also define the Banach space
with
as the closure of
with respect to the norm
We denote by the dual space of . Let us remark that in this case, the embedding is continuous.
Given a real separable Banach space Y, we denote by the space of all continuous functions from into Y.
We recall that a function
is said to be Lebesgue-measurable if there exists a sequence of step functions
(i.e.,
for a finite number
of Borel subsets
and with
) such that this sequence converges to
u almost everywhere with respect to the Lebesgue measure in
. Then,
, for
, is the space of all measurable functions
such that
As for
, it is the space of measurable functions such that
This choice makes
a Banach space and guarantees that its dual can be identified with
, where
and
is the dual space to
X. In particular, for functions
, the continuous Minkowski inequality yields
and moreover
Hence, we have
. The full presentation of this topic can be found in [
24].
2.2. Variable Exponent
Let
be a given function. Let
be the exponent that can be associated with
using the rule in Equation (
7).
Since
, it follows from Equation (
7) and from the absolute continuity of the Lebesgue integral that
in
and
, even if
v is just an absolutely integrable function in
. Then, we observe that for each
,
if
contains homogeneous features or is smooth enough, and
in those places of
where some discontinuities are present in
. Thus, the sparse texture of the function
v can be characterized by the exponent
.
For our further analysis, we make use of the following result. (For comparison, we refer to Lemma 2.1 in [
25]).
Lemma 1. Let be a given sequence of measurable functions. We assume that all elements of this sequence are extended by zero outside of and Letbe the corresponding sequence of exponents. Then, there exists a constant depending on G, Ω, and such that Proof. Since the sequence
is uniformly bounded in
, and the Gaussian kernel
is smooth, it follows that
where
Then, the
-boundedness of
guarantees the existence of a value
such that
and
. Hence, the estimate in Equation (
17) holds true for all
.
Moreover, as immediately follows from the relations
with
, and from the smoothness of the function
, there exists a positive constant
such that
does not depend on
k and, for each
, the following estimate
holds true. Arguing in a similar manner, we see that
with
.
Taking into account the estimates in Equation (
21)–(
22), and by setting
we obtain
Thus, we see that
. Since each element of the sequence
has the same modulus of continuity, and
, it follows that this sequence is equicontinuous and uniformly bounded. Hence, under the Arzelà–Ascoli theorem, the sequence
is relatively compact with respect to the norm topology of
. Then, in light of the estimate in Equation (
24), the fact that the set
is closed with respect to the uniform convergence, and
by definition of the weak convergence in
, we finally deduce that
uniformly in
as
, where
in
. □
2.3. Anisotropic Diffusion Tensor
Let be the set of symmetric quadratic matrices , (). We endow with the Euclidian scalar product and with the corresponding Euclidian norm . We also introduce the spectral norm of the matrices . Note that the following relation holds true for all .
Let
be a given function. Wee suppose that
v is zero-extended outside of
. With
, we denote its convolution with a Gaussian kernel
(see Equations (
9) and (
10)).
By analogy with [
26,
27], we associate with the function
the structure tensor
, using for that the following representation:
where
is defined in Equation (
9) and
It is easy to check that
is the positively semi-definite matrix. Moreover, this matrix is uniformly bounded in
. To check, it is enough to notice that
for any
, where
.
Having this in mind, we define the following diffusion tensor
:
where
is a small positive value and
stands for the unit matrix. In fact,
can be interpreted as some relaxation of the anisotropic tensor
(we refer to [
1,
2] for comparison).
Then, it easily follows from Equations (
26), (
27), and (
28) that for any distribution
, the estimate
holds true with
For simplicity, we suppose that .
Following in many aspects the proof of Lemma 1, it is easy to establish the following result:
Lemma 2. Let be measurable functions with the properties in Equation (16). We assume that each of these functions is extended by zero outside of . Let be a collection of the associated diffusion tensors. Then, we havewhere 2.4. On Orlicz Spaces
Let
be a given distribution. Let
be the corresponding exponent defined by Equation (
7). Then, we have
where
and
are the constants given by Equation (
17) (see Lemma 1). Let
be the conjugate exponent. Then, we have
Let
be the set of measurable functions
such that their modular is finite; in other words, let
which is equipped with the Luxembourg norm
Here,
becomes a Banach space (see [
28,
29] for details). The space
is a sort of Musielak–Orlicz space. In fact, it can be denoted by a generalized Lebesgue space because its main properties are inherited from the classical Lebesgue spaces. In particular, the two-sides of the inequality in Equation (
33) implies that
is reflexive and separable, and the set
is dense in
. Moreover, under the condition in Equation (
33),
is also dense in
.
Its dual can be identified with
, and therefore, each continuous functional
on
has the following representation (see Lemma 13.2 in [
15]):
Since the relation between the modular in Equation (
35) and the norm in Equation (
36) is not so direct as in the classical Lebesgue spaces, it can be proven from its definitions in Equations (
35) and (
36) that
The following consequence of Equation (
37) is very useful:
Moreover, if
, then
See, for instance, [
28,
29,
30] for more information.
In generalized Lebesgue spaces, there holds a version of Young’s inequality
with some positive constant
and arbitrary
.
The next assertion can be interpreted as an analogue of the Hölder inequality in variable Lebesgue spaces (we refer to [
28,
29] for the details).
Proposition 1. If and , then and As a biproduct of Equation (
42), we have that, for a bounded domain
and for
satisfying Equation (
33), the following imbedding
is continuous.
Let
, and let
be a given sequence of exponents. Assume that
With these sequences of exponents, we associate another sequence
. As a result, we see that each element
lives in the individual Orlicz space
. In fact, we deal with a sequence in the scale of spaces
. The sequence
is bounded if
Definition 1. Let be a bounded sequence. Then, we say that this sequence weakly convergences in the variable space to a function , where is the limit of in the norm topology of , if In order to proceed further, we recall some results concerning the lower semicontinuity property of the norm in the variable
space with respect to the weak convergence in
. (For the detailed proof, we refer to Lemma 3.1 in [
31]). For comparison, see also Lemma 13.3 in [
15] and Lemma 2.1 in [
25].
Proposition 2. Assume that a sequence of exponents satisfies the condition in Equation (
33),
as a.e.in , and is a sequence that is bounded and weakly convergent in to g. Then, , in variable , and We also recall the inequality which is well known in the theory of
p-Laplace equations. If
, then for all
, the following estimate holds true:
2.5. On a Weighted Sobolev Space with a Variable Exponent
Let
be a given distribution. Let
be a diffusion tensor associated with
z by the rule in Equation (
28). We define the weighted Sobolev space
as the set of functions
such that
We define the norm on the space
as follows:
where the second term in Equation (
49) is the norm of the function
in
. Since
it follows that
is a reflexive Banach space. Since
is the Lipschitz continuous exponent, it follows that the smooth and compactly supported functions are dense in the weighted Sobolev space
(see [
32]). Thus,
can be represented as the closure of the set
with respect to the norm
.
2.6. On Passage to the Limit in Fluxes
A standard situation in the study of many variational problems can be described as follows. Let, for a given
,
be a solution in the sense of distributions of the following parabolic equation of a monotone type:
with
. It is assumed that for all
, we have the following property:
as
pointwise a.e. with respect to the first two arguments. We also assume that the corresponding flow
converges weakly such that
The main question is to find out whether a flux converges to a flux (i.e., check whether the following equality
holds true). Since the weak convergence
and the nonlinearity of the function
with respect to
u do not guarantee that the limit passage
is valid, it makes this situation rather difficult and nontrivial. Therefore, the important problem is to show that
. The following result gives the answer to the above question (for the proof, we refer to a celebrated paper [
33]):
Theorem 1. Let us assume that the following assumptions are satisfied:
- (C1)
and are continuous in for a.e. and measurable with respect to for each (i.e., they are -valued Carathéodory functions);
- (C2)
, and for a.e. ;
- (C3)
and for all and for a.e. ;
- (C4)
in , , and are bounded in ;
- (C5)
in , ;
- (C6)
for all , and ;
- (C7)
.
Then, weakly in the Lebesgue space as .
The next results reveal some other properties of the weak convergence in .
Lemma 3 ([
30])
. Let Ψ
be a set of functions such that each of them is convex with respect to . These functions are measurable with respect to and satisfy the estimateIf and F belong to the set Ψ,
andthen the following lower semicontinuity property is valid:provided that in . Lemma 4 ([
34])
. Let and be -valued Carathéodory functions satisfying properties (C1–C3) such thatand . Then, we have 3. Existence Theorem for the Weak Solutions of Parabolic Equations with a Variable Order of Nonlinearity
In this section, we focus on the solvability issues for the following problem:
Here,
and
are given distributions, we have
the exponent
is defined in Equation (
7), the matrix
is given by Equation (
28),
is the outward normal derivative, and
stands for the control with the following class of admissible controls
:
As follows from Equations (
57) and (
28) and Lemma 1, the mapping
is a Carathéodory function for each fixed
(i.e.,
is measurable with respect to
for each
), and this function is continuous with respect to the third argument
. Moreover, the following conditions (monotonicity, coerciveness, and boundedness) hold for a.e.
[
15]:
However, if we have
, then
provides an example of a non-monotone, strongly nonlinear, and noncoercive operator in divergence form. In contrast to [
35], where there existence of strong solutions to the similar class of the initial boundary value problems (IBVPs) was proven, we make use of the concept of weak solutions to the above problem. However, the issue of their uniqueness is, apparently, still open [
36] (Chapter III).
Definition 2. Let , , and be given distributions. We say that a function u is a weak solution to the IBVPs in Equations (
54)–(
56)
if ; in other words, we haveand the integral identityholds for any , where . To clarify the sense in which the initial value
is assumed for the weak solutions, we give the following assertion (for the proof, we refer to Proposition 2.2 in [
25]):
Proposition 3. Let , and be given distributions. Let be a weak solution to the problem in Equations (
54)–(
56)
in the sense of Definition 2. Then, for any , the scalar function belongs to , and . We next recall some known results that have recently been proven based on the Schauder fixed-point theorem and using the perturbation technique (see Theorem 3.2 in [
25]).
Theorem 2. Given , , and , the problem in Equations (54)–(56) admits at least one weak solution for which the following energy inequalityholds for all . Using Equation (
64), we can derive the following estimates:
Since the uniqueness issue for the weak solutions of the initial boundary value problem in Equations (
54)–(
56) seems to be an open question, we adopt the following concept:
Definition 3. We say that a weak solution to the problem in Equations (
54)–(
56)
for given distributions , , and is -attainable if there exists a sequence converging to zero as such thatwhereand, for each , is the weak solutions to the following perturbed problem: Remark 1. It is worth emphasizing that (see the recent results in [25]) can now be specified as follows. Given , , and , the initial-boundary value problem in Equations (
54)–(
56)
admits at least one -attainable weak solution for which the energy inequality in Equation (
64)
holds true for all . Moreover, as follows from the estimates in Equations (
65)–(
68)
, this solution is bounded in . 4. Setting of the Optimal Control Problem and Existence Result
As was pointed out in the previous sections, the operator
provides an example of a nonlinear operator in divergence form which is neither monotone nor coercive. In this case (see Theorem 2), a weak solution to the initial boundary value problem (Equations (
54)–(
56)) under some admissible control
may not be unique. Moreover, it is unknown whether all weak solutions to Equations (
54)–(
56) satisfy the energy inequality in Equation (
64), which plays a crucial role in the derivation of a priori estimates (Equations (
65)–(
68)).
Our prime interest in this section is to consider the following optimal control problem of the tracking type:
where
is the original noise-corrupted image,
is the pre-denoised image when applying a median filter to
f, and
and
a.e. in
are given distributions.
We say that
is a feasible pair to this problem if
Let
be the set of feasible solutions to the problem in Equation (
75). Then, Theorem 2 implies that
. Since the main topological properties of the set
are unknown, in general, we begin with the following observation:
Theorem 3. Given and , the set is sequentially closed with respect to the weak topology of .
Proof. Let
be a sequence such that
Since the set is convex and closed, it follows from Mazur’s theorem that is sequentially closed with respect to the weak topology of . Therefore, . Let us show that . We will accomplish this in several steps.
Step 1. Under the initial assumptions, for each
, the pair
satisfies the energy inequality in Equation (
64), and
is a
-attainable weak solution for Equations (
54)–(
56). Hence, in light of Definition 3, we may always suppose that there exists a sequence
such that
are the weak solutions (in the sense of distributions) to Equations (
71)–(
73) with
and
, and
Moreover, the fact that the energy equality
is valid for all
implies the boundedness of the sequence
in the space
. Hence, by combining this fact with Equations (
77) and (
76), we deduce that
Step 2. By utilizing the energy equality in Equation (
79) and arguing as we did in Equations (
65)–(
68), we can derive the following a priori estimates:
for all
, where
Our main intention in this step is to establish the following asymptotic property:
Indeed, for any vector-valued test function
, we have
Hence, the sequence
is bounded in
. As a result, we obtain
Step 3. Let us show that in this case, the flux weakly converges in to the flux as . To accomplish this, it is enough to show that all preconditions (C1–C7) of Theorem 1 are fulfilled.
To begin with, we notice that the conclusion, similar to Equation (
80), can also be made with respect to the sequence
. Then, Lemmas 1 and 2 imply that
Moreover, we deduce from Equations (
61) and (
83) that the sequence
is bounded in
. Hence, there exists an element
such that
We also make use of the following observation: the sequence
is uniformly bounded in
. Indeed, this inference is a direct consequence of the estimates in Equations (
86) and (
83), and the following one:
Utilizing this fact together with the properties in Equations (
89), (
90), (
59), (
80), and
and taking into account that
, we see that all preconditions of Theorem 1 hold true. Hence, in light of the property in Equation (
88), the assertion in Equation (
90) can be rewritten as follows:
Step 4. At this stage, we show that the limit pair
is related with the integral identity in Equation (
63). First, we notice that
is a weak solution (in the sense of distributions) of Equations (
71)–(
73) with
,
, and
. Hence,
satisfies the integral identity
Then, utilizing the properties in Equations (
93), (
80), and (
76) and passing to the limit in Equation (
94) as
, we immediately arrive at the announced identity in Equation (
63).
Step 5. In order to show that the limit pair
satisfies the energy inequality in Equation (
64), we have to realize the limit passage as
in the following relation (see [
25]):
This can be viewed as the energy equality for the weak solutions to the problem in Equations (
71)–(
73) with
,
, and
. With that in mind, we notice that the weak convergence in Equation (
80), under the Sobolev embedding theorem, implies the pointwise convergence
Then, in light of the estimate in Equation (
85), we have the strong convergence
in
for a.e.
(under Lebesgue’s dominated convergence theorem), and therefore
Moreover, taking into account that the
norm is less semi-continuous with respect to the weak convergence in Equation (
81), we see that
We also notice that due to the properties in Equations (
88), (
93), and (
80), we have
Since
(see Equation (
92)), it follows from Lemma 4 (see also Proposition 2) that
Therefore, in order to pass to the limit in Equation (
95), the asymptotic behavior of the term
as
remains to be found. We prove this in the next step using the well-known Aubin–Lions lemma.
Step 6. We recall that the Aubin–Lions lemma states the criteria for when a set of functions is relatively compact in
, where
,
, and
B is a Banach space. The standard formulation of the Aubin–Lions lemma states that if
U is a bounded set in
, and
is bounded in
,
, then
U is relatively compact in
under the conditions that
By setting
, we deduce from Equations (
82)–(
85) that
Since, under the Sobolev embedding theorem,
compactly, it follows from Lebesgue’s dominated convergence theorem that the following embeddings are compact as well:
Furthermore, we have in mind the fact that for each
, the functions
are the solutions in
for the variational problem
We derive from this the following estimate:
Utilizing this fact together with Equations (
99) and (
100), we deduce from the the Aubin–Lions lemma that the set
is relatively compact in
. Hence, we can complement properties with the following one:
strongly in
as
. Since
U is bounded in
, it leads to the conclusion that
Hence, the term
is the product of weakly and strongly convergent sequences in
. As a result, we have
Thus, in light of the obtained collection of properties (see Equations (
96)–(
98) and (
106)), the limit passage in Equation (
95) as
finally leads us to the energy inequality in Equation (
64).
Step 7. To end the proof, it remains to notice that due to the properties in Equation (
62), which were established in the previous steps, we have
and
. Moreover, it has been proven that in this case, the sequence
satisfies all requirements that were mentioned in Definition 3. Hence,
is a
-attainable weak solution to the problem in Equations (
54)–(
56). The proof is complete. □
Taking this result into account, let us show that the original optimal control problem (Equation (
74)) has a solution. In fact, this issue immediately follows from Theorem 3 and the facts that the set of feasible solutions
is bounded in
(see the estimates in Equations (
65)–(
68) and (
87)) and the objective functional
is less semicontinuous with respect to the weak topology of
. Thus, as a direct consequence, we can finalize this inference as follows:
Corollary 1. Let , , and , a.e. in Ω be given distributions, and let , , , and be some constants. Then, the optimal control problem (Equation (
74))
admits at least one solution .