=Paper=
{{Paper
|id=Vol-3101/Paper1
|storemode=property
|title=On variational problem with nonstandard growth conditions for the restoration of clouds corrupted satellite images
|pdfUrl=https://ceur-ws.org/Vol-3101/Paper1.pdf
|volume=Vol-3101
|authors=Pavel Khanenko,Peter I. Kogut,Mykola Uvarov
|dblpUrl=https://dblp.org/rec/conf/citrisk/KhanenkoKU21
}}
==On variational problem with nonstandard growth conditions for the restoration of clouds corrupted satellite images==
https://ceur-ws.org/Vol-3101/Paper1.pdf
On Variational Problem with Nonstandard
Growth Conditions for the Restoration
of Clouds Corrupted Satellite Images
Pavel Khanenko1,2, Peter I. Kogut3,4 and Mykola Uvarov1,5
Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany
1
EOS Data Analytics Ukraine, Desyatynny lane, 5, 01001 Kyiv, Ukraine
2
3
Department of Differential Equations, Oles Honchar Dnipro National University,
Gagarin av., 72, 49010 Dnipro, Ukraine
4
EOS Data Analytics Ukraine, Gagarin av., 103a, Dnipro, Ukraine
5
Department of Computational Physics, G. V. Kurdyumov Institute for Metal Physics, Kyiv, Ukraine
Abstract
Sensitivity to weather conditions, and specially to clouds, is a severe limiting factor to the use of opti-
cal remote sensing for Earth monitoring applications. Typically, the optical satellite images are often
corrupted because of poor weather conditions. As a rule, the measure of degradation of optical images
is such that one can not rely even on the brightness inside of the damaged regions. As a result, some
subdomains of such images become absolutely invisible. So, there is a risk of information loss in optical
remote sensing data. In view of this, we propose a new variational approach for exact restoration of
multispectral satellite optical images. We discuss the consistency of the proposed variational model,
give the scheme for its regularization, derive the corresponding optimality system, and discuss the algo-
rithm for the practical implementation of the reconstruction procedure. Experimental results are very
promising and they show a significant g ain over b aseline m ethods u sing t he reconstruction through
linear interpolation between data available at temporally-close time instants.
Keywords
Risk of cloud distortion of satellite images, Risk of information loss, Image restoration, Variational ap-
proach
1. Introduction
A very serious obstacle to utilization of optical remote sensing satellite images is a risk of cloud
and cloud shadow distortion issue (referred to as cloud contamination, hereafter). It has been
reported in [1] that over 50% of all the Moderate Resolution Imaging Spectroradiometer (MODIS)
instrument aboard the Terra and Aqua satellites are covered by clouds or cloud-contaminated
globally. Moreover, it is a typical situation when the measure of degradation of optical images
2ππ International Workshop on Computational & Information Technologies for Risk-Informed Systems, CITRiskβ2021, September
16-17, 2021, Kherson, Ukraine
pavel.khanenko@eosda.com (P. Khanenko); peter.kogut@eosda.com (P. I. Kogut); nikolay.uvarov@eosda.com (M. Uvarov)
{ https://eos.com (P. Khanenko); https://kogut.uaua.us (P. I. Kogut); https://eos.com (M. Uvarov)
0000-0003-1593-0510 (P. I. Kogut)
Β© 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
is such that we can not rely even on the brightness inside of the damaged regions. As a result,
there is a risk of information loss, some subdomains of such images become absolutely invisible.
So, there is a great deal of missing information in optical remote sensing data, and a huge
gap still exists between the satellite data we acquire and the data we require. Therefore, the
reconstruction of missing information in remote sensing data becomes an active research field.
Many solutions have been developed to remove the clouds from multispectral images. (for the
technical review, we refer to [2]). Formally, the present traditional algorithms can be primarily
classified into four categories: 1) spatial-based methods, without any other auxiliary information
source; 2) spectral-based methods, which extract the complementary information from other
spectra; 3) multitemporal-based methods, which extract the complementary information from
other data acquired at the same position and at different time periods; and 4) hybrid methods,
which extract the complementary information by a combination of the three previous approaches.
In parallel to the traditional approach, data-driven machine learning algorithms are actively
developing since 2014 [3]. However, the necessity of large datasets and volatility to errors in
input data limits its performance.
Our main effort in this research is to develop a new variational model for the exact restoration
of the damaged multi-band optical satellite images that will meet demands from the agro
application, that is, it must be applicable for large areas in different climate zones, and preserves
the crop fields borders within damaged regions. In some sense, this model combines the spacial-
based method with the multitemporal one. Therefore, in contrast to the standard variational-
based methods that are often optimased for a specific region or significantly blur textures, (see,
for instance, [4] and the references therein), we focus on the global texture reconstruction inside
damage regions. With that in mind we assume that the texture of a corrupted image can be
predicted through a number of past cloud-free images of the same region from the time series.
In order to describe the texture of background surface in the damage region, we follow the
paper [5], where the authors experimentally checked the hypothesis that the essential geometric
contents of a color image is contained in the level lines of the corresponding total spectral
energy of such image.
We also pay much attention to the faithfulness of the reconstruction problem in the framework
of the proposed model and supply this approach by the rigorous mathematical substantiation.
The experiments undertaken in this study confirmed the efficacy of the proposed method and
revealed that it can acquire plausible visual performance and satisfactory quantitative accuracy
for agro scenes with rather complicated texture of background surface.
2. Preliminaries
Let β¦ β R2 be a bounded image domain with a Lipschitz boundary πβ¦. With each particular
image βπ’ = [π’1 (π₯), π’2 (π₯), . . . , π’π (π₯)]π‘ : β¦ β Rπ , where each coordinate represents the
intensity of the corresponding spectral channel, we associate the panchromatic image π’ (the
so-called total spectral energy of βπ’)
π’(π₯) = πΌ1 π’1 (π₯) + . . . πΌπ π’π (π₯). (1)
Here, πΌ1 , . . . , πΌπ are some weight coefficients.
Let π· β β¦ be a Borel set with non empty interior and sufficiently regular boundary and
such that |β¦ β π·| > 0. We call π· the damage region. Let βπ’0 β πΏ2 (β¦ β π·; Rπ ) be an image of
interest which is assumed to be corrupted by clouds, and π· is the zone of missing information.
As it was mentioned before, we deal with the case where we have no information about
the original image βπ’0 inside π·. Instead of this, we assume that the texture of background
surface in the damage region π· can be predicted with some accuracy by a number of past
cloud-free images of the same region from the time series of satellite images. Unlike the well-
known βchronochrome methodβ [6] which essentially assumes that the background in π· is
stationary in wide sense, we admit that the image time series follows smooth variations over land
(background), the time-series data are strictly chronological, and display regular fluctuations.
Let {π’β π‘β1 , . . . , βπ’π‘βπ } be a given collection of past cloud-free images of the same region,
where we set βπ’π‘ = βπ’0 . We suppose that each cloud-free image of this time series should be well
co-registered with βπ’0 β πΏ2 (β¦ β π·; Rπ ) in β¦ β π· [7]. With each particular image βπ’π‘βπ in this
series, we associate its total spectral energy π’π‘βπ using the standard rule (1). So, each element
of the time series {π’π‘ , π’π‘β1 , . . . , π’π‘βπ } is well-defined in β¦.
Let π’* be a predicted total spectral energy of βπ’0 in the damage region π·. This prediction can
be done following the regularized regression model and the available information in the time
series {π’π‘ , π’π‘β1 , . . . , π’π‘βπ } (for the details we refer to [8]).
β«οΈ [οΈ π
π½1 βοΈ ]οΈ2
β(π€) = π’π‘β1 β π€πβ1 π’π‘βπ ππ₯
|π·| π· π=2
β«οΈ πβ1
π½1 [οΈ βοΈ ]οΈ2
+ π’π‘ β π€π π’π‘βπ ππ₯ + πβπ€β2Rπβ1 β inf, (2)
|β¦ β π·| Ξ©βπ· π=1
where π > 0 is the regularization parameter, π½1 > 0 and π½2 > 0 are the parameters that control
the importance of the prediction and estimation terms, respectively. Seeing that the prediction
and estimation errors in (2) are normalized by the volume of samples contributing to each term,
we can constrain the values of π½1 = π½ β [0, 1] and π½2 = 1 β π½ to control their relevance with a
single parameter π½.
As a result, setting π€0 = Argmin β(π€), the total spectral energy π’* in the damage region π·
can be estimated as follows π’* = π’ ΜοΈ in π·, where
πβ1
βοΈ
π’(π₯)
ΜοΈ = π€π0 π’π‘βπ (π₯), β π₯ β β¦.
π=1
In order to reconstruct the texture (or geometry) of βπ’0 in the damage region π·, we assume
that predicted total energy π’* is a function of bounded variation, i.e. π’* β π΅π (π·), and all
spectral channels of the damaged image should share the geometry of the panchromatic image
π’* β πΏ2 (π·) in π·. Hence, at most all points of almost all level sets of π’* βπ΅π (π·) we can define
a normal vector π(π₯), i.e., it formally satisfies (π, π’* ) = |βπ’* | and |π| β€ 1 a.e. in π·.
3. Problem Statement
In view of the risk of cloud distortion, the problem is to reconstruct the original multi-band
image βπ’0 in the damage region π· using the knowledge of its texture (geometry) on the subset
π· together with the exact information about this image in β¦ β π· (the undamaged region). We
say that a function βπ’ = [π’1 , π’2 , . . . , π’π ]π‘ : β¦ β Rπ is the result of restoration of a cloud
corrupted image βπ’0 : π· β Rπ if for given regularization parameters π>0, πΌ>1, and ππ >0,
π = 1, 2, each spectral component π’π is the solution of the following constrained minimization
problem with the nonstandard growth energy functional
β«οΈ β«οΈ
1 π
(π«π ) π½π (π£, π) := |βπ£(π₯)| π(π₯)
ππ₯ + |π£(π₯) β π’0,π (π₯)|πΌ ππ₯
Ξ© π(π₯) πΌ Ξ©βπ·
β«οΈ β β2 β«οΈ β (οΈ )οΈ βπΌ
β β₯
+ π1 βπΊ * (π£ β π’ ) ππ₯ + π π , βπ£ β ππ₯ ββ inf , (3)
β β β
β π 0,π β 2 β
Ξ©βπ· π· (π£,π)βΞ
where
β’ Ξ stands for the set of feasible solutions to the problem (3) which we define as follows
β π£ β π 1,π(Β·) (β¦), π β πΆ(β¦),
β§ β β«
βͺ
β¨ β βͺ
β¬
Ξ = (π£, π) β 1 β€ πΎ0 β€ π£(π₯) β€ πΎ1 a.a. in β¦,
β
βͺ β βͺ
β© β π(π₯) = F(π£(π₯)) in β¦. β
Here, π 1,π(Β·) (β¦) is the Sobolev-Orlicz space,
F(π£(π₯)) = 1 + π (|(βπΊπ * π£) (π₯)|) ,
and π:[0, β) β (0, β) is the edge-stopping function which we take it in the form of the
Cauchy law π(π‘) = 1+(π‘/π)
1
2 with an appropriate π > 0;
β’ π β πΏβ (π·, R2 ) is a given vector field such that
|π(π₯)| β€ 1 and (π(π₯), βπ’* (π₯))R2 = |βπ’* (π₯)| a.e. in π·;
β’ (πΊπ * π£) (π₯) determines the convolution of function π£ with the two-dimensional Gaussian
filter kernel πΊπ , where the parameter π>0 determines the spatial size of the image details
which are removed by this 2D filter;
By default we assume that the functions π’0,π and π’* are extended by zero outside of domains
β¦ β π· and π·, respectively.
The proposed model (3) provides a completely new approach to restoration of non-smooth
multi-spectral images βπ’0 with the gap in damage region. The main characteristic feature of this
model is that we involve into consideration the energy functional with the nonstandard growth
condition. The variable character of the exponent π(π₯) provides more flexibility in terms of
regularity for the recovered images. Since the first term in (3) is the regularization and the
second one is the so-called data fidelity, it is worth to emphasize the role of the rest terms in
(3). Taking into account the fact that the magnitude F(π£(π₯)) is close to one at those points,
where the spectral energy π£ is slowly varying, and this value is close to zero at the edges of π£,
it follows that the edge information in the non-damage zone for the reconstruction is derived
from the given image βπ’0 . So, in view of the estimate
β β
β 2 2 β
|(βπΊ * β |(βπΊ *
β«οΈ β«οΈ
2
β π π£)| π π’ 0,π )| β
|π(π₯) β F(π’0,π )| ππ₯ = π β
β (οΈ 2 )οΈ (οΈ )οΈ ββ ππ₯
Ξ©βπ· Ξ©βπ· β π + |(βπΊπ * π£)|2 π2 + |(βπΊπ * π’0,π )|2 β
β«οΈ
1 (οΈ )οΈβ β
β€ 2 |(βπΊπ * π£)| + |(βπΊπ * π’0,π )| β |(βπΊπ * π£)| β |(βπΊπ * π’0,π )| β ππ₯
β β
π Ξ©βπ·
3 (οΈ )οΈ 1
2βπΊπ βπΆ 1 (Ξ©βΞ©) πΎ1 |β¦| 2 β«οΈ β β2 2
β€ βπΊ * (π£ β π’ ) ππ₯ ,
β β
π 0,π
π2
β β
Ξ©βπ·
the third term is also fidelity term which forces the texture (or topological map) of minimizer π’
in domain β¦ β π· to stay close to the texture of a given spectral energy βπ’0,π .
As for the last term in (3), we notice that since π β πΏβ (π·, R2 ) is a vector field with indicated
properties, it follows that π(π₯) has the direction of the normal to the level lines of π’* . Therefore,
the counterclockwise rotation of angle π/2, denoted by πβ₯ , represents the tangent vector to the
level lines of π’* . In this case, if the spectral channels π’π share the geometry of the panchromatic
image π’* , we have (οΈ )οΈ
β₯
π , βπ’π 2 = 0, π = 1, . . . , π in π·.
R
Therefore, we impose them in the energy functional π½π in the form of the last term.
βΜοΈ
π’(π₯1 ,π₯2 )
In practice, at the discrete level, π can be defined by the relation π(π₯1 , π₯2 ) = |βΜοΈ
π’(π₯1 ,π₯2 )| when
π’(π₯1 , π₯2 ) ΜΈ= 0, and π(π₯1 , π₯2 ) = 0 when βΜοΈ
βΜοΈ π’(π₯1 , π₯2 ) = 0. However, a better choice would be
to compute π(π₯1 , π₯2 ) by first regularizing π’ ΜοΈ by the equation
(οΈ )οΈ
ππ£ βπ£
= div in (0, β) Γ β¦,
ππ‘ |βπ£|
coupled with the initial and Neumann boundary conditions
ππ£
π£(0, π₯1 , π₯2 ) = π’(π₯
ΜοΈ 1 , π₯2 ), for a.a. (π₯1 , π₯2 ) β β¦,
= 0 on πβ¦.
ππ
Then, for any π‘ β₯ 0, there is a vector field π(π‘) β πΏβ (β¦) with βπ(π‘)βπΏβ (Ξ©) β€ 1 such that (see
[9, 10] for the details)
(π(π‘), βπ£(π‘)) = |βπ£(π‘)| in β¦, (π(π‘), π) = 0 on πβ¦,
ππ£
and = div (π(π‘)) in the sense of distributions in (0, β) Γ β¦.
ππ‘
As a result, in order to characterize the texture of the cloud contaminated image βπ’0 in the
damage region π·, we may take π = π(π‘) for some small value of π‘ > 0. As was mentioned in
[9], following this way, we do not not distort the geometry of π’
ΜοΈ in an essential way.
So, the novelty of the model that we propose, is that the edge information for the multi-
spectral restoration in β¦ is accumulated in the variable exponent π(π₯) which we derive from
the time series and initial data.
4. Existence Result
In this section we show that constrained minimization problem (3) is consistent and admits at
least one solution (π’πππ
π , ππ ) β Ξ, where ππ (π₯) = F(π’π (π₯)) in β¦. We note that because of
πππ πππ πππ
the specific form of the energy functional π½π (π£, π) in (3), the standard approaches are no longer
applicable in its study, especially with respect to the existence of minimizers and their basic
properties. It makes the minimization problem (3) rather challenging.
We begin with some auxiliary results which will play a crucial role in the sequel.
Lemma 4.1. Let {π£π }πβN be a sequence of measurable non-negative functions π£π : β¦ β
[πΎ0 , β) such that {π£π }πβN are uniformly bounded in πΏ1 (β¦) and π£π (π₯) β π£(π₯) almost every-
where in β¦ for some π£ β πΏ1 (β¦). Let {ππ = 1 + π (|(βπΊπ * π£π )|)}πβN be the corresponding
sequence of variable exponents. Then
ππ (Β·) β π(Β·) = 1 + π (|(βπΊπ * π£) (Β·)|) uniformly in β¦ as π β β,
πΌ := 1 + πΏ β€ ππ (π₯) β€ π½ := 2, β π₯ β β¦, β π β N, (4)
where
π2
πΏ= .
π2 + βπΊπ β2πΆ 1 (Ξ©βΞ©) supπβN βπ£π β2πΏ1 (Ξ©)
Proof. Since {π£π }πβN is the bounded sequence in πΏ1 (β¦), by smoothness of the Gaussian filter
kernel πΊπ , it follows that
β«οΈ
|(βπΊπ * π£π ) (π₯)| β€ |βπΊπ (π₯ β π¦)| π£π (π¦) ππ¦ β€ βπΊπ βπΆ 1 (Ξ©βΞ©) βπ£π βπΏ1 (Ξ©) ,
Ξ©
π2
ππ (π₯) = 1 +
π2 + (|(βπΊπ * π£π ) (π₯)|)2
π2
β₯1+ 2 , β π₯ β β¦.
π + βπΊπ β2πΆ 1 (Ξ©βΞ©) βπ£π β2πΏ1 (Ξ©)
Then πΏ1 -boundedness of {π£π }πβN guarantees the existence of a positive value πΏ β (0, 1) such
that estimate (4) holds true for all π β N.
Moreover, as follows from the estimate
β β
β 2 2 β
|(βπΊπ * π£π ) (π₯)| β |(βπΊπ * π£π ) (π¦)|
|ππ (π₯) β ππ (π¦)| β€ π2 ββ (οΈ
β β
)οΈ (οΈ )οΈ ββ
β π2 + |(βπΊπ * π£π ) (π₯)|2 π2 + |(βπΊπ * π£π ) (π¦)|2 β
2βπΊπ βπΆ 1 (Ξ©βΞ©) βπ£π βπΏ1 (Ξ©)
β€ ||(βπΊπ * π£π ) (π₯)| β |(βπΊπ * π£π ) (π¦)||
π2
2βπΊπ βπΆ 1 (Ξ©βΞ©) πΎ12 |β¦| β«οΈ
β€ |βπΊπ (π₯ β π§) β βπΊπ (π¦ β π§)| ππ§, β π₯, π¦ β β¦
π2 Ξ©
and smoothness of the function βπΊπ (Β·), there exists a positive constant πΆπΊ > 0 independent
of π such that
2βπΊπ βπΆ 1 (Ξ©βΞ©) πΎ12 |β¦|πΆπΊ
|ππ (π₯) β ππ (π¦)| β€ |π₯ β π¦|, β π₯, π¦ β β¦.
π2
Setting
2βπΊπ βπΆ 1 (Ξ©βΞ©) πΎ12 |β¦|πΆπΊ
πΆ := , (5)
π2
we see that
{οΈ β }οΈ
β |β(π₯) β β(π¦)| β€ πΆ|π₯ β π¦|, β π₯, π¦, β β¦,
{ππ (Β·)} β S = β β πΆ 0,1 (β¦) β
β
β 1 < πΌ β€ β(Β·) β€ π½ in β¦.
Since maxπ₯βΞ© |ππ (π₯)| β€ π½ and each element of the sequence {ππ }πβN has the same modulus of
continuity, it follows that this sequence is uniformly bounded and equi-continuous. Hence, by
ArzelΓ βAscoli Theorem the sequence {ππ }πβN is relatively compact with respect to the strong
topology of πΆ(β¦). Taking into account that the set S is closed with respect to the uniform
convergence and π£π (π₯) β π£(π₯) almost everywhere in β¦, we deduce: ππ (Β·) β π(Β·) uniformly in
β¦ as π β β, where π(π₯) = 1 + π (|(βπΊπ * π£) (π₯)|) in β¦. The proof is complete.
Following in many aspects the resent studies [11, 12], we give the following existence result.
Theorem 4.2. For each π = 1, . . . , π and given π>0, π1 >0, π2 >0, π β πΏβ (π·, R2 ), and
π’0,π β πΏ2 (β¦ β π·), the minimization problem (3) admits at least one solution (π’πππ πππ
π , ππ ) β Ξ.
Proof. Since Ξ ΜΈ= β
and 0 β€ π½π (π£, π) < +β for all (π£, π) β Ξ, it follows that there exists a
non-negative value π β₯ 0 such that π = inf π½π (π£, π). Let {(π£π , ππ )}πβN β Ξ be a minimizing
(π£,π)βΞ
sequence to the problem (3), i.e.
(π£π , ππ ) β Ξ, ππ (π₯) = 1 + π (|(βπΊπ * π£π ) (π₯)|) in β¦ β π β N, and lim π½π (π£π , ππ ) = π.
πββ
So, without lost of generality, we can suppose that π½π (π£π , ππ ) β€ π + 1 for all π β N. From this
and the initial assumptions, we deduce
β«οΈ β«οΈ
|π£π (π₯)|πΌ ππ₯ β€ πΎ1πΌ ππ₯ β€ πΎ1πΌ |β¦|, β π β N,
Ξ© Ξ©
β«οΈ β«οΈ
1
|βπ£π (π₯)|ππ (π₯)
ππ₯ β€ 2 |βπ£π (π₯)|ππ (π₯) ππ₯ < 2(π + 1), β π β N, (6)
Ξ© π
Ξ© π (π₯)
where
[οΈ ]οΈ
sup sup ππ (π₯) β€ 2.
πβN π₯βΞ©
Utilizing the fact that π£π (π₯) β€ πΎ1 for almost all π₯ β β¦, we infer the following estimate
βπ£π βπΏ1 (Ξ©) β€ πΎ1 |β¦|, β π β N.
Then arguing as in Lemma 4.1 it can be shown that ππ β πΆ 0,1 (β¦) and
πΌ := 1 + πΏ β€ ππ (π₯) β€ π½ := 2, β π₯ β β¦, β π β N, (7)
π2
with πΏ= . (8)
π2 + βπΊπ β2πΆ 1 (Ξ©βΞ©) πΎ12 |β¦|2
Taking this fact into account, we deduce from (6), (7), and (??) that
(οΈβ«οΈ )οΈ1/πΌ
πΌ πΌ
βπ£π βπ 1,πΌ (Ξ©) = [|π£π (π₯)| + |βπ£π (π₯)| ] ππ₯
Ξ©
(οΈβ«οΈ [οΈ ]οΈ )οΈ1/πΌ
1/πΌ ππ (π₯) ππ (π₯)
β€ (1 + |β¦|) |π£π (π₯)| + |βπ£π (π₯)| ππ₯ + 2
Ξ©
)οΈ]οΈ1/πΌ
β€ (1 + |β¦|) πΎ12 |β¦| + 2π + 4
[οΈ (οΈ
uniformly with respect to π β N. Therefore, there exists a subsequence of {π£π }πβN , still denoted
by the same index, and a function π’πππ
π β π 1,πΌ (β¦) such that
π£π β π’πππ
π strongly in πΏπ (β¦) for all π β [1, πΌ* ),
π£π β π’πππ
π weakly in π 1,πΌ (β¦) as π β β,
where, by Sobolev embedding theorem, πΌ* = 2βπΌ 2πΌ
= 2+2πΏ
1βπΏ > 2.
Moreover, passing to a subsequence if necessary, we have (see Proposition A.3 and Lemma 4.1):
π (π₯) a.e. in β¦.
π£π (π₯) β π’πππ (9)
π£π β π’πππ
π weakly in πΏ ππ (Β·)
(β¦),
βπ£π β βπ’πππ
π weakly in πΏππ (Β·) (β¦; R2 ),
ππ (Β·) β ππππ πππ
π (Β·) = 1 + π (|(βπΊπ * π’π ) (Β·)|) uniformly in β¦ as π β β,
πππ
where π’πππ
π β π 1,π (Β·) (β¦) with ππππ (π₯) = 1 + π (|(βπΊπ * π’πππ π ) (π₯)|) in β¦.
Since πΎ0 β€ π£π (π₯) β€ πΎ1 a.a. in β¦ for all π β N, it follows from (9) that the limit function π’πππ
π
is also subjected the same restriction. Thus, π’πππ
π is a feasible solution to minimization problem
(3).
Let us show that (π’πππ
π , ππ ) is a minimizer of this problem. With that in mind we note that
πππ
in view of the obvious inequality
|π£π (π₯) β π’0,π (π₯)|πΌ β€ 2πΌβ1 (|π£π (π₯)|πΌ + |π’0,π (π₯)|πΌ )
and the fact that π’0,π β πΏ2 (β¦ β π·), we have: the sequence {π£π (π₯) β π’0,π (π₯)}πβN is bounded in
πΏπΌ (β¦ β π·) and converges weakly in πΏπΌ (β¦ β π·) to π’πππ π β π’0,π . Hence, by Proposition A.3 (see
πππ (Β·)
(32)), π’π β π’0,π β πΏ
πππ π (β¦ β π·) and
β«οΈ β«οΈ
lim inf πΌ
|π£π (π₯) β π’0,π (π₯)| ππ₯ β₯ |π’πππ πΌ
π (π₯) β π’0,π (π₯)| ππ₯. (10)
πββ Ξ©βπ· Ξ©βπ·
As for the rest terms in (3), in view of the strong convergence π£π β π’ππππ in πΏπ (β¦) with π > 2,
we have
β«οΈ β β2 β«οΈ (οΈβ«οΈ )οΈ2
πππ β πππ
ββπΊπ * (π£π β π’π )β ππ₯ β€ |βπΊπ (π₯ β π¦)| |π£π (π¦) β π’π (π¦)| ππ¦ ππ₯
β
Ξ© Ξ© Ξ©
β«οΈ (οΈβ«οΈ )οΈ2β 2
π π
β€ |βπΊπ (π₯ β π¦)| πβ1 ππ¦ ππ₯ βπ£π β π’πππ 2
π βπΏπ (Ξ©)
Ξ© Ξ©
3β 2
π βπΏπ (Ξ©) β 0 as π β β.
β€ βπΊπ βπΆ 1 (Ξ©βΞ©) |β¦| π βπ£π β π’πππ
2
Hence,
β«οΈ β β2 β«οΈ β β2
lim inf ββπΊπ * (π£π β π’0,π )β ππ₯ = πππ
ββπΊπ * (π’π β π’0,π )β ππ₯, (11)
β β β β
πββ Ξ©βπ· Ξ©βπ·
β«οΈ β (οΈ )οΈ βπΌ β«οΈ β (οΈ )οΈ βπΌ
β β₯ β β₯
lim inf β π , βπ£π β ππ₯ β₯ β π , βπ’πππ β ππ₯ ππ₯, (12)
β β
π
πββ π· π·
As a result, utilizing relations (10), (11), (12), and the lower semicontinuity property (32), we
finally obtain
π = inf π½π (π£, π) = lim π½π (π£π , ππ ) = lim inf π½π (π£π , ππ ) β₯ π½π (π’πππ πππ
π , ππ ).
(π£,π)βΞ πββ πββ
Thus, (π’πππ
π , ππ ) is a minimizer to the problem (3), whereas its uniqueness remains as an
πππ
open question.
5. On Relaxation of the Restoration Problem
It is clear that because of the nonstandard energy functional and its non-convexity, constrained
minimization problem (3) is not trivial in its practical implementation. The main difficulty in its
study comes from the state constraints
1 β€ πΎ0 β€ π£(π₯) β€ πΎ1 a.a. in β¦, π(π₯) = 1 + π (|(βπΊπ * π’) (π₯)|)
that we impose on the set of feasible solutions Ξ. This motivates us to pass to some relaxation.
In view of this, we propose the following iteration procedure which is based on the concept of
relaxation of extremal problems and their variational convergence [13, 14, 15, 16]. At the first
step we set up
1 + π (|(βπΊπ * π’0,π ) (π₯)|) , if π₯ββ¦ β π·,
{οΈ }οΈ
π0 (π₯) =
1 + π (|(βπΊπ * π’* ) (π₯)|) , if π₯βπ·,
π’0 = Argmin π½π (π£, π0 (Β·)).
π£ββ¬π0 (Β·)
Then, for each π β₯ 1, we set
(οΈβ(οΈ )οΈ β)οΈ
ππ (π₯) = 1 + π β βπΊπ * π’ πβ1
(π₯)β , β π₯ β β¦, π’π = Argmin π½π (π£, ππ (Β·)). (13)
β β
π£ββ¬ππ (Β·)
Here, β¬π(Β·) = π£ β π 1,π(Β·) (β¦) : 1β€πΎ0 β€ π£(π₯) β€ πΎ1 a.a. in β¦ .
{οΈ }οΈ
Before proceeding further, we set
{οΈ β }οΈ
β |β(π₯) β β(π¦)| β€ πΆ|π₯ β π¦|, β π₯, π¦ β β¦,
S = β β πΆ(β¦) β β
πΌ := 1 + πΏ β€ β(π₯) β€ π½ := 2, β π₯ β β¦,
where πΆ > 0 and πΏ > 0 are defined by (5) and (8), respectively.
Arguing as in the proof of Theorem 1 and using the convexity arguments, it can be shown
0,π(Β·) 0,π(Β·)
that, for each π(Β·) β S, there exists a unique element π’π β β¬π(Β·) such that π’π =
Argminπ£ββ¬π(Β·) π½π (π£, π(Β·)). Moreover, it can be shown that, for given π = 1, . . . , π , π > 0,
π1 >0, π2 > 0, π’* β π 1,πΌ (π·), and βπ’0 β πΏ2 (β¦βπ·, Rπ ), the sequence π’π β π 1,ππ (Β·) (β¦) πβN
{οΈ }οΈ
is compact with respect to the weak topology of π 1,πΌ (β¦), whereas the exponents {ππ }πβN are
compact with respect to the strong topology of πΆ(β¦).
We say that a pair (ΜοΈ ΜοΈ is a weak solution to the original problem (3) if
π’π , π)
π’
ΜοΈπ = Argmin π½π (π£, π(Β·)),
ΜοΈ ΜοΈπ β β¬π(Β·)
π’ ΜοΈ , π(π₯)
ΜοΈ = 1 + π (|(βπΊπ * π’
ΜοΈπ ) (π₯)|) , β π₯ β β¦.
π£ββ¬π(Β·)
ΜοΈ
Our main result can be stated as follows:
Theorem 5.2. Let π>0, π1 >0, π2 >0, π’* βπ΅π (π·), and βπ’0 βπΏ2 (β¦βπ·, π
{οΈ πR ) }οΈbe given data.
Then, for each π β {1, . . . , π }, the sequence of approximated solutions (π’ , ππ ) πβN possesses
the asymptotic properties:
π’π (π₯) β π’(π₯)
ΜοΈ a.e. in β¦,
π’π β π’
ΜοΈ in πΏπΌ (β¦), and βπ’π β βΜοΈ
π’ in πΏπΌ (β¦; R2 ),
ππ β πΜοΈ = F(ΜοΈ
π’(π₯)) strongly in πΆ(β¦) as π β β,
where (ΜοΈ ΜοΈ is a weak solution to the original problem (3), that is,
π’, π)
ΜοΈ β β¬π(Β·)
π’ ΜοΈ , π’
ΜοΈ = Argmin π½π (π£, π(Β·)),
ΜοΈ
π£ββ¬π(Β·)
ΜοΈ
and, in addition, the following variational property holds true
[οΈ ]οΈ
lim π½π (π’π , ππ (Β·)) = lim inf π½π (π£, ππ (Β·)) = inf π½π (π£, π(Β·))
ΜοΈ = π½π (ΜοΈ
π’, π(Β·)).
ΜοΈ (14)
πββ πββ π£ββ¬ππ (Β·) π£ββ¬π(Β·)
ΜοΈ
Proof. Letβs assume the converse β namely, there is a function π’β β β¬π(Β·)
ΜοΈ such that
π½π (π’β , π(Β·))
ΜοΈ = inf π½π (π£, π(Β·))
ΜοΈ < π½π (ΜοΈ
π’, π(Β·)).
ΜοΈ (15)
π£ββ¬π(Β·)
ΜοΈ
Using the procedure of the direct smoothing, we set
(οΈ )οΈ
π₯ β π§ ΜοΈβ
β«οΈ
1
π’π (π₯) = 2 πΎ π’ (π§) ππ§,
π R2 π(π)
where π > 0 is a small parameter, πΎ is a positive compactly supported smooth function with
properties β«οΈ
πΎ β πΆ0β (R2 ), πΎ(π₯) ππ₯ = 1, and πΎ(π₯) = πΎ(βπ₯),
R2
and π’
ΜοΈβ is zero extension of π’β outside of β¦.
Since π’β β π 1,ΜοΈπ(Β·) (β¦) and π(π₯)
ΜοΈ β₯ πΌ = 1 + πΏ in β¦), it follows that π’β β π 1,πΌ (β¦). Then
π’π β πΆ0β (R2 ) for each π > 0,
π’π β π’β in πΏπΌ (β¦), βπ’π β βπ’β in πΏπΌ (β¦; R2 ) (16)
by the classical properties of smoothing operators (see [17]). From this we deduce that
π’π (π₯) β π’β (π₯) a.e. in β¦. (17)
Moreover, taking into account the estimates
β«οΈ β«οΈ
π’π (π₯) = ΜοΈβ (π₯ β π(π)π¦) ππ¦ β€ πΎ1
πΎ (π¦) π’ πΎ (π¦) ππ¦ = πΎ1 ,
R 2 R 2
β«οΈ β«οΈ
π’π (π₯) β₯ β
πΎ (π¦) π’ (π₯ β π(π)π¦) ππ¦ β₯ πΎ0
ΜοΈ πΎ (π¦) ππ¦ β₯ πΎ0 ,
π¦βπ(π)β1 (π₯βΞ©) π¦βπ(π)β1 (π₯βΞ©)
we see that each element π’π is subjected to the pointwise constraints
πΎ0 β€ π’π (π₯) β€ πΎ1 a.a. in β¦, β π > 0.
Since, for each π > 0, π’π β π 1,ππ (Β·) (β¦) for all π β N, it follows that π’π β β¬ππ (Β·) , i.e.,
each
β¨ element of the sequence
β© {π’π }π>0 is a feasible solution to all approximating problems
inf π£ββ¬π (Β·) π½π (π£, ππ (Β·)) . Hence,
π
π½π (π’π , ππ (Β·)) β€ π½π (π’π , ππ (Β·)), β π > 0, β π = 0, 1, . . . (18)
Further we notice that
lim inf π½π (π’π , ππ (Β·)) β₯ π½π (ΜοΈ
π’, π(Β·))
ΜοΈ (19)
πββ
by Proposition A.3 and Fatouβs lemma, and
β«οΈ β«οΈ
1 π
lim π½π (π’π , ππ (Β·)) = lim |βπ’π (π₯)|ππ (π₯)
ππ₯ + |π’π (π₯) β π’0,π (π₯)|πΌ ππ₯
πββ πββ Ξ© ππ (π₯) πΌ Ξ©βπ·
β«οΈ β β2 β«οΈ β (οΈ )οΈ βπΌ
β β₯
+ π1 ββπΊπ * (π’π β π’0,π )β ππ₯ + π2 β π , βπ’π β ππ₯. (20)
β β β
Ξ©βπ· π·
Since
1 1
|βπ’π (π₯)|ππ (π₯) β |βπ’π (π₯)|π(π₯)
ΜοΈ
uniformly in β¦ as π β β,
ππ (π₯) π(π₯)
ΜοΈ
it follows from the Lebesgue dominated convergence theorem and (20) that
lim π½π (π’π , ππ (Β·)) = π½π (π’π , π(Β·)),
ΜοΈ βπ > 0. (21)
πββ
As a result, passing to the limit in (18) and utilizing properties (19)β(21), we obtain
β«οΈ β«οΈ
1 π
π½π (ΜοΈ
π’, π(Β·))
ΜοΈ β€ π½π (π’π , π(Β·))
ΜοΈ = |βπ’π (π₯)|π(π₯)
ΜοΈ
ππ₯ + |π’π (π₯) β π’0,π (π₯)|πΌ ππ₯
Ξ© π(π₯)
ΜοΈ πΌ Ξ©βπ·
β«οΈ β β2 β«οΈ β (οΈ )οΈ βπΌ
β β₯
+ π1 ββπΊπ * (π’π β π’0,π )β ππ₯ + π2 β π , βπ’π β ππ₯, (22)
β β β
Ξ©βπ· π·
for all π > 0. Taking into account the pointwise convergence (see (17) and property (16))
|βπ’π (π₯)|π(π₯)
ΜοΈ
β |βπ’β (π₯)|π(π₯)
ΜοΈ
,
|π’π (π₯) β π’0,π (π₯)|πΌ β |π’β (π₯) β π’0,π (π₯)|πΌ ,
β β2 β β2
β
βπΊ * (π’ β π’ ) β βπΊ * (π’ β π’0,π β ,
)
β β β β
β π π 0,π β β π
β (οΈ )οΈ βπΌ β (οΈ )οΈ βπΌ
β β₯ β β₯ β β
π , βπ’ β π , βπ’
β
β π β β β
as π β 0, and the fact that, for π small enough,
|βπ’π (π₯)|π(π₯)
ΜοΈ
β€ (1 + |βπ’β (Β·)|)π(Β·)
ΜοΈ
β πΏ1 (β¦) a.e. in β¦,
[οΈ ]οΈ
|π’π (Β·) β π’0,π (Β·)|πΌ β€ 2 (1 + |π’β (Β·)|)πΌ + 2 (1 + |π’0,π (Β·)|)2 β πΏ1 (β¦) a.e. in β¦ β π·,
β β2
βπΊ * (π’ β π’ )(π₯) β β€ βπΊπ β2πΆ 1 (Ξ©βΞ©) |β¦|2 πΎ02 = const, β π₯ β β¦,
β β
β π π 0,π
β (οΈ )οΈ βπΌ
β β₯ β π(Β·)
π , βπ’ π β β€ βπβπΏβ (π·,R2 ) (1 + |βπ’ (Β·)|) β πΏ1 (β¦) a.e. in β¦,
β ΜοΈ
β
we can pass to the limit in (22) as π β 0 by the Lebesgue dominated convergence theorem. This
yields
π½π (ΜοΈ
π’, π(Β·))
ΜοΈ β€ lim π½π (π’π , π(Β·))
ΜοΈ = π½π (π’β , π(Β·)).
ΜοΈ
πβ0
Combining this inequality with (22) and (15), we finally get
π½π (π’β , π(Β·))
ΜοΈ = inf π½π (π£, π(Β·))
ΜοΈ < π½π (π’* , π(Β·))
ΜοΈ β€ π½π (π’β , π(Β·)),
ΜοΈ
π£ββ¬π(Β·),π
ΜοΈ
that leads us into conflict with the initial assumption. Thus,
π½π (ΜοΈ
π’, π(Β·))
ΜοΈ = inf π½π (π£, π(Β·))
ΜοΈ (23)
π£ββ¬π(Β·)
ΜοΈ
and, therefore, (ΜοΈ ΜοΈ is a weak solution to the original problem (3). As for the variational
π’, π)
property (14), it is a direct consequence of (23) and (21).
6. Optimality Conditions
To characterize the solution π’0,π(Β·) β β¬π(Β·) of the approximating optimization problem
β¨ β©
inf π£ββ¬π(Β·) π½π (π£, π(Β·)) , we check that the functional πΉπ(Β·) is GΓ’teaux differentiable, that is,
π½π (π’0,π(Β·) + π‘π£, π(Β·)) β π½π (π’0,π(Β·) , π(Β·))
β«οΈ (οΈ )οΈ
lim = |βπ’0,π(Β·) (π₯)|π(π₯)β2 βπ’0,π(Β·) (π₯), βπ£(π₯) ππ₯
π‘β0 π‘ Ξ©
β«οΈ β βπΌβ2
β 0,π(Β·)
+π βπ’ (π₯) β π’0,π (π₯)β π’0,π(Β·) (π₯)π£(π₯) ππ₯
β
Ξ©βπ·
β«οΈ β«οΈ β (οΈ )οΈ βπΌβ1 (οΈ )οΈ
β β₯
+2π1 Ξ(π₯)π£(π₯) ππ₯ + πΌπ2 β π , βπ’0,π(Β·) β πβ₯ , βπ£ ππ₯, (24)
β
Ξ© Ξ©
for all π£ β π 1,π(Β·) (β¦), where
β«οΈ β«οΈ (οΈ )οΈ
Ξ(π₯) = (βπΊπ (π¦ β π§), βπΊπ (π¦ β π₯)) π’0,π(Β·) (π§) β π’0,π (π§) πΞ©βπ· (π¦) ππ§ ππ¦.
Ξ© Ξ©
To this end, we note that
|βπ’0,π(Β·) (π₯) + π‘βπ£(π₯)|π(π₯) β |βπ’0,π(Β·) (π₯)|π(π₯)
π(π₯)π‘
(οΈ )οΈ
β |βπ’0,π(Β·) (π₯)|π(π₯)β2 βπ’0,π(Β·) (π₯), βπ£(π₯) as π‘ β 0
almost everywhere in β¦. Since, by convexity,
|π|π β |π|π β€ 2π |π|πβ1 + |π|πβ1 |π β π|,
(οΈ )οΈ
it follows that
β β
β |βπ’0,π(Β·) (π₯) + π‘βπ£(π₯)|π(π₯) β |βπ’0,π(Β·) (π₯)|π(π₯) β
β β
π(π₯)π‘
β β
β β
(οΈ )οΈ
β€ 2 |βπ’0,π(Β·) (π₯) + π‘βπ£(π₯)|π(π₯)β1 + |βπ’0,π(Β·) (π₯)|π(π₯)β1 |βπ£(π₯)|
(οΈ )οΈ
β€ const |βπ’0,π(Β·) (π₯)|π(π₯)β1 + |βπ£(π₯)|π(π₯)β1 |βπ£(π₯)|. (25)
Taking into account that
by (33)
(οΈβ«οΈ )οΈ 1β²
π½
0,π(Β·) π(π₯)β1 0,π(Β·) π(π₯)
βπ’ (π₯)| βπΏπβ² (Β·) (Ξ©) β€ βπ’ (π₯)| ππ₯ + 1
Ξ©
by (??) (οΈ )οΈ 1β²
βπ’0,π(Β·) |2πΏπ(Β·) (Ξ©) + 2
π½
β€ ,
by (33)
β«οΈ
|βπ’0,π(Β·) (π₯)|π(π₯)β1 |βπ£(π₯)| ππ₯ β€ 2βπ’0,π(Β·) (π₯)|π(π₯)β1 βπΏπβ² (Β·) (Ξ©) βπ£(π₯)|βπΏπ(Β·) (Ξ©) ,
Ξ©
by (??)
and π(π₯) ππ₯ β€ βπ£β2 + 1, we see that the right hand side of inequality (25) is
β«οΈ
Ξ© |π£(π₯)| πΏπ(Β·) (Ξ©)
an πΏ1 (β¦) function. Therefore,
|βπ’0,π(Β·) (π₯) + π‘βπ£(π₯)|π(π₯) β |βπ’0,π(Β·) (π₯)|π(π₯)
β«οΈ
ππ₯
Ξ© π(π₯)π‘
β«οΈ (οΈ )οΈ
β |βπ’0,π(Β·) (π₯)|π(π₯)β2 βπ’0,π(Β·) (π₯), βπ£(π₯) ππ₯ as π‘ β 0
Ξ©
by the Lebesgue dominated convergence theorem.
Utilizing the similar arguments to the rest terms in (3), we deduce that the representation
(24) for the GΓ’teaux differential of π½π (Β·, π(Β·)) at the point π’0,π(Β·) β β¬π(Β·) is valid.
Thus, in order to derive some optimality conditions for the minimizing element π’0,π(Β·) β
β¬π(Β·) to the problem inf π½π (π£, π(Β·)), we note that β¬π(Β·) is a nonempty convex subset of
π£ββ¬π(Β·)
π 1,π(Β·) (β¦) and the objective functional π½π (Β·, π(Β·)) : β¬π(Β·) β R is strictly convex. Hence, the
well known classical result (see [18, Theorem 1.1.3]) and representation (24) lead us to the
following conclusion.
Theorem 6.1. Let ππ (Β·) β S be an exponent given by the iterative rule (13). Then the unique
minimizer π’π β β¬ππ (Β·) to the approximating problem inf π£ββ¬π (Β·) π½π (π£, ππ (Β·)) is characterized by
π
β«οΈ (οΈβ )οΈ β«οΈ
β π βππ (π₯)β2 π
β (οΈ )οΈ
π
ββπ’ (π₯)β βπ’ (π₯), βπ£(π₯) β βπ’ (π₯) ππ₯ + 2π1 Ξ(π₯) π£(π₯) β π’π (π₯) ππ₯
Ξ© Ξ©
β«οΈ β βπΌβ2 (οΈ )οΈ
β π π π
+π π’ (π₯) β π’ (π₯) π’ (π₯) π£(π₯) β π’ (π₯) ππ₯
β
β 0,π β
Ξ©βπ·
β«οΈ β (οΈ )οΈ βπΌβ1 (οΈ )οΈ
β β₯
+ πΌπ2 β π , βπ’0,π(Β·) β πβ₯ , βπ£ β βπ’π ππ₯ β₯ 0, β π£ β β¬ππ (Β·) .
β
Ξ©
7. Numerical Experiments
In order to illustrate the proposed algorithm for the restoration of satellite multi-spectral images
we have provided some numerical experiments. As input data we have used a series of Sentinel-2
L2A images over the Dnipro Airport area, Ukraine (see Fig. 1, 2). This region represents a typical
agricultural area with medium sides fields of various shapes.
Figure 1: Given collection of past cloud-free images. Date of generation: (left) - 2019/06/15, (right) -
2019/07/01
As a final result, we obtain in Fig. 3. Comparing the restored image and the contaminated
one we could see that the texture of original image is well preserved. However, overall colors
of different fields are shifted due to colorization part of algorithm. This problem has to be
addressed in the following research.
Figure 2: The could contaminated image with date of generation 2019/07/17
8. Conclusion
We propose a novel model for the restoration of satellite multi-spectral images. This model
is based on the solutions of special variational problems with nonstandard growth objective
functional. Because of the risk of information loss in optical images (see [19] for the details),
we do not impose any information about such images inside the damage region, but instead we
assume that the texture of these images can be predicted through a number of past cloud-free
images of the same region from the time series. So, the characteristic feature of variational
problems, which we formulate for each spectral channel separately, is the structure of their
objective functionals. On the one hand, we involve into consideration the energy functionals
with the nonstandard growth π(π₯), where the variable exponent π(π₯) is unknown a priori and
it directly depends on the texture of an image that we are going to restore. On the other hand,
the texture of an image βπ’, we are going to restore, can have rather rich structure in the damage
region π·. In order to identify it, we push forward the following hypothesis: the geometry of
each spectral channels of a cloud corrupted image in the damage region is topologically close to
the geometry of the total spectral energy that can be predicted with some accuracy by a number
of past cloud-free images of the same region. As a result, we impose this requirement in each
objective functional in the form of a special fidelity term. In order to study the consistency of
the proposed collection of non-convex minimization problems, we develop a special technique
and supply this approach by the rigorous mathematical substantiation.
Figure 3: Result of the restoration of image in Fig 2 by the proposed method.
Appendix A. On Orlicz Spaces
Let π(Β·) be a measurable exponent function on β¦ such that 1 β€ πΌ β€ π(π₯) β€ π½ < β a.e. in β¦,
π(Β·)
where πΌ and π½ are given constants. Let πβ² (Β·) = π(Β·)β1 be the corresponding conjugate exponent.
It is clear that
π½ πΌ
1β€ β€ πβ² (π₯) β€ a.e. in β¦,
π½β1 β ββ 1
πΌ
β β
π½β² πΌβ²
where π½ β² and πΌβ² stand for the conjugates of constant exponents. Denote by πΏπ(Β·) (β¦) the set of all
measurable functions π (π₯) on β¦ such that Ξ© |π (π₯)|π(π₯) ππ₯ < β. Then πΏπ(Β·) (β¦) is a reflexive
β«οΈ
separable Banach space with respect to the Luxemburg norm (see [20] for the details)
βπ βπΏπ(Β·) (Ξ©) = inf π > 0 : ππ (πβ1 π ) β€ 1 , (26)
{οΈ }οΈ
where ππ (π ) := Ξ© |π (π₯)|π(π₯) ππ₯.
β«οΈ
β²
It is well-known that πΏπ(Β·) (β¦) is reflexive provided πΌ > 1, and its dual is πΏπ (Β·) (β¦), that is,
any continuous functional πΉ = πΉ (π ) on πΏπ(Β·) (β¦) has the form (see [21, Lemma 13.2])
β«οΈ
β²
πΉ (π ) = π π ππ₯, with π β πΏπ (Β·) (β¦).
Ξ©
As for the infimum in (26), we have the following result.
Proposition A.1. The infimum in (26) is attained if ππ (π ) > 0. Moreover
if π* := βπ βπΏπ(Β·) (Ξ©) > 0, then ππ (πβ1
* π ) = 1.
Taking this result and condition 1 β€ πΌ β€ π(π₯) β€ π½ into account, we see that
β π (π₯) βπ(π₯)
β«οΈ β«οΈ β β β«οΈ
1 π(π₯) 1
|π (π₯)| ππ₯ β€ β β ππ₯ β€ πΌ |π (π₯)|π(π₯) ππ₯,
ππ½*
β π* β π* Ξ©
Ξ© Ξ©
β«οΈ β«οΈ
1 π(π₯) 1
π½
|π (π₯)| ππ₯ β€ 1 β€ πΌ
|π (π₯)|π(π₯) ππ₯.
π* Ξ© π * Ξ©
Hence, (see [20] for the details)
β«οΈ
πΌ
βπ βπΏπ(Β·) (Ξ©) β€ |π (π₯)|π(π₯) ππ₯ β€ βπ βπ½πΏπ(Β·) (Ξ©) , if βπ βπΏπ(Β·) (Ξ©) > 1,
Ξ©
β«οΈ
π½
βπ βπΏπ(Β·) (Ξ©) β€ |π (π₯)|π(π₯) ππ₯ β€ βπ βπΌπΏπ(Β·) (Ξ©) , if βπ βπΏπ(Β·) (Ξ©) < 1,
Ξ©
and, therefore,
β«οΈ
βπ βπΌπΏπ(Β·) (Ξ©) β 1 β€ |π (π₯)|π(π₯) ππ₯ β€ βπ βπ½πΏπ(Β·) (Ξ©) + 1, β π β πΏπ(Β·) (β¦),
Ξ©
β«οΈ
βπ βπΏπ(Β·) (Ξ©) = |π (π₯)|π(π₯) ππ₯, if βπ βπΏπ(Β·) (Ξ©) = 1.
Ξ©
The following estimates are well-known: if π β πΏπ(Β·) (β¦) then
βπ βπΏπΌ (Ξ©) β€ (1 + |β¦|)1/πΌ βπ βπΏπ(Β·) (Ξ©) ,
β² π½
βπ βπΏπ(Β·) (Ξ©) β€ (1 + |β¦|)1/π½ βπ βπΏπ½ (Ξ©) , π½β² = , β π β πΏπ½ (β¦).
π½β1
Let {ππ }πβN β πΆ 0,πΏ (β¦), with some πΏ β (0, 1], be a given sequence of exponents. Hereinafter
in this subsection we assume that
1 β€ πΌ β€ ππ (π₯) β€ π½ < β a.e. in β¦ for π = 1, 2, . . . , and ππ (Β·) β π(Β·) in πΆ(β¦) as π β β.
(27)
We associate with this sequence the following collection ππ β πΏ π (Β·) (β¦) πβN . The characteris-
{οΈ }οΈ
π
tic feature of this set of functions is that each element π
}οΈπ lives in the corresponding Orlicz space
π (Β·) (β¦). We say that the sequence ππ β πΏ π (Β·) (β¦) πβN is bounded if (see [22, Section 6.2])
{οΈ
πΏ π π
β«οΈ
lim sup |ππ (π₯)|ππ (π₯) ππ₯ < +β. (28)
πββ Ξ©
Definition A.2. A bounded sequence ππ β πΏππ (Β·) (β¦) πβN is weakly convergent in the
{οΈ }οΈ
variable Orlicz space πΏππ (Β·) (β¦) to a function π β πΏπ(Β·) (β¦), where π β πΆ 0,πΏ (β¦) is the limit of
{ππ }πβN β πΆ 0,πΏ (β¦) in the uniform topology of πΆ(β¦), if
β«οΈ β«οΈ
lim ππ π ππ₯ = π π ππ₯, β π β πΆ0β (R2 ).
πββ Ξ© Ξ©
For our further analysis, we need some auxiliary results (we refer to [21, Lemma 13.3] for
comparison). We begin with the lower semicontinuity property of the variable πΏππ (Β·) -norm
with respect to the weak convergence in πΏππ (Β·) {οΈ (β¦).
Proposition A.3. If a bounded sequence ππ β πΏππ (Β·) (β¦) πβN converges weakly in πΏπΌ (β¦)
}οΈ
to π for some πΌ > 1, then π β πΏπ(Β·) (β¦), ππ β π in variable πΏππ (Β·) (β¦), and
β«οΈ β«οΈ
lim inf |ππ (π₯)|ππ (π₯)
ππ₯ β₯ |π (π₯)|π(π₯) ππ₯. (29)
πββ Ξ© Ξ©
Proof. In view of the fact that
ββ«οΈ β«οΈ β
β
β |ππ (π₯)| π (π₯) π(π₯) π (π₯)
β
π
ππ₯ β |ππ (π₯)| π
ππ₯ββ
Ξ© ππ (π₯)
β
Ξ©
β«οΈ
1
β€ βππ β πβπΆ(Ξ©) |ππ (π₯)|ππ (π₯) ππ₯
Ξ© ππ (π₯)
βππ β πβπΆ(Ξ©) β«οΈ by (28)
β€ |ππ (π₯)|ππ (π₯) ππ₯ β 0 as π β β,
πΌ Ξ©
we see that β«οΈ β«οΈ
ππ (π₯) π(π₯)
lim inf |ππ (π₯)| ππ₯ = lim inf |ππ (π₯)|ππ (π₯) ππ₯.
πββ Ξ© πββ Ξ© ππ (π₯)
β²
Using the Young inequality ππ β€ |π|π /π + |π|π /πβ² , we have
β«οΈ β«οΈ β«οΈ
π(π₯) π(π₯) πβ²π (π₯)
|ππ (π₯)|ππ (π₯) ππ₯ β₯ π(π₯)ππ (π₯)π(π₯) ππ₯ β β² (π₯) |π(π₯)| ππ₯, (30)
π
Ξ© π (π₯) Ξ© π
Ξ© π
for πβ²π (π₯) = ππ (π₯)/(ππ (π₯) β 1) and any π β πΆ0β (R2 ).
Then passing to the limit in (30) as π β β and utilizing property (27) and the fact that
β«οΈ β«οΈ
β²
lim ππ (π₯)π(π₯) ππ₯ = π (π₯)π(π₯) ππ₯ for all π β πΏπΌ (β¦), (31)
πββ Ξ© Ξ©
we obtain
β«οΈ β«οΈ β«οΈ
ππ (π₯) π(π₯) β²
lim inf |ππ (π₯)| ππ₯ β₯ π(π₯)π (π₯)π(π₯) ππ₯ β β²
|π(π₯)|π (π₯) ππ₯.
πββ Ξ© Ξ© Ξ© π (π₯)
β²
Since the last inequality is valid for all π β πΆ0β (R2 ) and the set πΆ0β (R2 ) is dense in πΏπ (Β·) (β¦),
β²
it follows that this relation holds true for π β πΏπ (Β·) (β¦). So, taking π = |π (π₯)|π(π₯)β2 π (π₯), we
arrive at the announced inequality (29). As an consequence of (29) and estimate (??), we get:
π β πΏπ(Β·) (β¦).
To end of the proof, it remains to observe that relation (31) holds true for π β πΆ0β (R2 ) as
well. From this the weak convergence ππ β π in the variable Orlicz space πΏππ (Β·) (β¦) follows.
Remark A.4. Arguing in a similar manner and using, instead of (30), the estimate
β«οΈ β«οΈ β«οΈ
1 ππ (π₯) 1 πβ² (π₯)
lim inf |ππ (π₯)| ππ₯ β₯ π(π₯)π (π₯)π(π₯) ππ₯ β β² (π₯) |π(π₯)| ππ₯,
πββ π
Ξ© π (π₯) Ξ© π
Ξ© π
it is easy to show that the lower semicontinuity property (29) can be generalized as follows
β«οΈ β«οΈ
1 1
lim inf |ππ (π₯)|ππ (π₯)
ππ₯ β₯ |π (π₯)|π(π₯) ππ₯. (32)
πββ Ξ© ππ (π₯) Ξ© π(π₯)
We need the following result that leads to the analog of the HΓΆlder inequality in Lebesgue
spaces with variable exponents (for the details we refer to [20]).
β²
Proposition A.6. If π β πΏπ(Β·) (β¦)π and π β πΏπ (Β·) (β¦)π , then (π, π) β πΏ1 (β¦) and
β«οΈ
(π, π) ππ₯ β€ 2βπ βπΏπ(Β·) (Ξ©)π βπβπΏπβ² (Β·) (Ξ©)π . (33)
Ξ©
Appendix B. Sobolev Spaces with Variable Exponent
We recall here the well-known facts concerning the Sobolev spaces with variable exponent.
Let π(Β·) be a measurable exponent function on β¦ such that 1 < πΌ β€ π(π₯) β€ π½ < β a.e. in β¦,
where πΌ and π½ are given constants. We associate with it the so-called Sobolev-Orlicz space
{οΈ β«οΈ [οΈ ]οΈ }οΈ
1,π(Β·) 1,1 π(π₯) π(π₯)
π (β¦) := π¦ β π (β¦) : |π¦(π₯)| + |βπ¦(π₯)| ππ₯ < +β
Ξ©
and equip it with the norm βπ¦βπ 1,π(Β·) (Ξ©) = βπ¦βπΏπ(Β·) (Ξ©) + ββπ¦βπΏπ(Β·) (Ξ©;R2 ) .
0
It is well-known that, in general, unlike classical Sobolev spaces, smooth functions are not
1,π(Β·)
necessarily dense in π = π0 (β¦). Hence, with the given variable exponent π = π(π₯)
(1 < πΌ β€ π β€ π½) it can be associated another Sobolev space,
π» = π» 1,π(Β·) (β¦) as the closure of the set πΆ β (β¦) in π 1,π(Β·) (β¦)-norm.
Since the identity π = π» is not always valid, it makes sense to say that an exponent π(π₯) is
regular if πΆ β (β¦) is dense in π 1,π(Β·) (β¦).
The following result reveals an important property ensuring the regularity of exponent π(π₯).
Proposition B.1. Assume that there exists πΏ β (0, 1] such that π β πΆ 0,πΏ (β¦). Then the set
β
πΆ (β¦) is dense in π 1,π(Β·) (β¦), and, therefore, π = π».
Proof. Let π β πΆ 0,πΏ (β¦) be a given exponent. Since
lim |π‘|πΏ log(|π‘|) = 0 with an arbitrary πΏ β (0, 1], (34)
π‘β0
it follows from the HΓΆlder continuity of π(Β·) that
[οΈ ]οΈ
πΏ |π₯ β π¦|πΏ
|π(π₯) β π(π¦)| β€ πΆ|π₯ β π¦| β€ sup β1
π(|π₯ β π¦|), β π₯, π¦ β β¦,
π₯,π¦βΞ© log(|π₯ β π¦| )
where π(π‘) = πΆ/ log(|π‘|β1 ), and πΆ > 0 is some positive constant.
Then property (34) implies that π(Β·) is a log-HΓΆlder continuous function. So, to deduce the
density of πΆ β (β¦) in π 1,π(Β·) (β¦) it is enough to refer to Theorem 13.10 in [21].
References
[1] M. D. King, S. Platnick, P. Menzel, S. A. Ackerman, P. A. Hubanks, Spatial and temporal
distribution of clouds observed by modis onboard the terra and aqua satellites, IEEE Trans,
Geosci. Remote Sens, 51, 2013), pp. 3826β3852. doi:10.1109/TGRS.2012.2227333
[2] H. Shen, X. Li, Q. Chen, C. Zeng, G. Yang, H. Li, L. Zhang, Missing information reconstruc-
tion of remote sensing data: A technical review, IEEE Geoscience and Remote Sensing
Magazine, 3, 2015, pp. 61β85. doi:10.1109/MGRS.2015.2441912
[3] L. Ma, Y. Liu, X. Zhang, Y. Ye, G. Yin, B. A. Johnson, Deep learning in remote sensing
applications: A meta-analysis and review, ISPRS Journal of Photogrammetry and Remote
Sensing, 152, 2019, pp. 166β177. doi:10.1016/j.isprsjprs.2019.04.015
[4] H. Shen, H. Li, Y. Qian, L. Zhang, Q. Yuan, An effective thin cloud removal procedure for
visible remote sensing images, ISPRS Journal of Photogrammetry and Remote Sensing,
96, 2014, pp. 224β235. doi:10.1016/j.isprsjprs.2014.06.011
[5] V. Caselles, B. Coll, J.-M. Morel, Geometry and color in natural images, Journal of
Mathematical Imaging and Vision, 16 , 2002, pp. 89β105. doi:10.1023/
A:1013943314097
[6] A. P. Schaum, A. Stocker, Long-interval chronochrome target detection, in: Proceedings
of the 1997 International Symposium on Spectral Sensing Research, 1998, pp. 1760β1770
[7] V. Hnatushenko, P. Kogut, M. Uvarow, On flexible co-registration of optical and sar satellite
images, in: Lecture Notes in Computational Intelligence and Decision Making, 2021, pp.
515β534. doi:10.1007/978-3-030-26474-1
[8] L. GΓ³mez-Chova, J. Amoros-Lopez, G. Mateo-GarcΓa, J. Munoz-MarΓ, G. Camps-Valls, Cloud
masking and removal in remote sensing image time series, Journal of Applied Remote
Sensing, 11, 2017, Id:015005. doi:10.1117/1.JRS.11.015005
[9] F. Andreu, C. Ballester, V. Caselles, J. M. Mazon, Minimizing total variation flow, Comptes
Rendus de lβAcadΓ©mie des Sciences. Series I β Mathematics, 331, 2000, pp. 867β872.
doi:10. 1016/S0764-4442(00)01729-8
[10] F. Andreu, V. Caselles, J.-M. Mazon, Parabolic Quasilinear Equations Minimizing Linear
Growth Functionals, Progress in Mathematics, Springer Basel AG, Basel, 2004
[11] P. Kogut, On optimal and quasi-optimal controls in coefficients for multi-dimensional
thermistor problem with mixed dirichlet-neumann boundary conditions, Control and
Cybernetics, 48, 2019, pp. 31β68. doi:10.1515/9783110668520-006
[12] C. DβApice, U. DeMaio, P. Kogut, An indirect approach to the existence of quasi-optimal
controls in coefficients for multi-dimensional thermistor problem, in: V. Sadovnichiy,
M. Zgurovsky (Eds.), Contemporary Approaches and Methods in Fundamental Mathemat-
ics and Mechanics, 2021, pp. 489β522. doi:10.1007/978-3-030-50302-4_24
[13] P. Kogut, On approximation of an optimal boundary control problem for linear elliptic
equation with unbounded coefficients., Discrete & Continuous Dynamical Systems, 34,
2014, pp. 2105β2133. doi:10.3934/dcds.2014.34.2105
[14] P. Kogut, Variational s-convergence of minimization problems. part i. definitions and
basic properties, Problemy Upravleniya i Informatiki (Avtomatika), 5, 1996, pp. 29β42.
doi:10. 1615/JAutomatInfScien.v30.i4-5.140
[15] P. Kogut, S-convergence of the conditional optimization problems and its variational
properties, Problemy Upravleniya i Informatiki (Avtomatika), 7 , 1997, pp. 64β79
[16] P. Kogut, G. Leugering, On π -homogenization of an optimal control problem with control
and state constraints, Zeitschrift fΓΌr Analysis und ihre Anwendungen, 20, 2001, pp. 395β
429. doi:10.4171/ZAA/1023
[17] L. C. Evans, R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press,
New York, 1992
[18] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer,
Berlin, 1971
[19] M. Sudmanns, D. Tiede, H. Augustin, S. Lang, Assessing global sentinel-2 coverage
dynamics and data availability for operational earth observation (eo) applications using
the eo-compass, International Journal of Digital Earth, 13, 2020, pp. 768β784.
doi:10.1080/ 17538947.2019.1572799
[20] D. L. Lars, P. Harjulehto, P. Hasto, M. Ruzicka, Lebesgue and Sobolev Spaces with Variable
Exponents, Springer, New York, 2011
[21] V. V. Zhikov, On variational problems and nonlinear elliptic equations with nonstandard
growth conditions, Journal of Mathematical Sciences, 173, 2011, pp. 463β570.
doi:10.1007/s10958-011-0260-7
[22] P. Kogut, G. Leugering, Optimal Control Problems for Partial Differential Equations on
Reticulated Domains, Springer, New York, Berlin, 2011