Research article Special Issues

Dynamic analysis and optimal control of leptospirosis based on Caputo fractional derivative

  • Received: 24 September 2024 Revised: 27 October 2024 Accepted: 05 November 2024 Published: 11 November 2024
  • Caputo fractional derivative solves the fractional initial value problem in Riemann-Liouville (R-L) fractional calculus. The definition of a Caputo-type derivative is in the same form as the definition of an integral differential equation, including the restriction of the value of the integral derivative to the value of the unknown function at the endpoint t=a. Therefore, this paper introduced the Caputo fractional derivative (CFD) to establish the transmission model of leptospirosis. First, to ensure that the model had a particular significance, we proved the dynamic properties of the model, such as nonnegative, boundedness, and stability of the equilibrium point. Second, according to the existence mode and genetic characteristics of pathogenic bacteria of leptospirosis, and from the perspective of score optimal control, we put forward measures such as wearing protective clothing, hospitalization, and cleaning the environment to prevent and control the spread of the disease. According to the proposed control measures, a control model of leptospirosis was established, and a forward-backward scanning algorithm (FB algorithm) was introduced to optimize the control function. Three different disease control strategies were proposed. Finally, the numerical simulation of different fractional orders used the fde12 (based on Adams–Bashforth–Moulton scheme) solver. The three optimized strategies, A, B, and C, were compared and analyzed. The results showed that the optimized control strategy could shorten the transmission time of the disease by about 80 days. Therefore, the above methods contributed to the study of leptospirosis and the World Health Organization.

    Citation: Ling Zhang, Xuewen Tan, Jia Li, Fan Yang. Dynamic analysis and optimal control of leptospirosis based on Caputo fractional derivative[J]. Networks and Heterogeneous Media, 2024, 19(3): 1262-1285. doi: 10.3934/nhm.2024054

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  • Caputo fractional derivative solves the fractional initial value problem in Riemann-Liouville (R-L) fractional calculus. The definition of a Caputo-type derivative is in the same form as the definition of an integral differential equation, including the restriction of the value of the integral derivative to the value of the unknown function at the endpoint t=a. Therefore, this paper introduced the Caputo fractional derivative (CFD) to establish the transmission model of leptospirosis. First, to ensure that the model had a particular significance, we proved the dynamic properties of the model, such as nonnegative, boundedness, and stability of the equilibrium point. Second, according to the existence mode and genetic characteristics of pathogenic bacteria of leptospirosis, and from the perspective of score optimal control, we put forward measures such as wearing protective clothing, hospitalization, and cleaning the environment to prevent and control the spread of the disease. According to the proposed control measures, a control model of leptospirosis was established, and a forward-backward scanning algorithm (FB algorithm) was introduced to optimize the control function. Three different disease control strategies were proposed. Finally, the numerical simulation of different fractional orders used the fde12 (based on Adams–Bashforth–Moulton scheme) solver. The three optimized strategies, A, B, and C, were compared and analyzed. The results showed that the optimized control strategy could shorten the transmission time of the disease by about 80 days. Therefore, the above methods contributed to the study of leptospirosis and the World Health Organization.



    In this paper, we deal with a new variational problem suggested by applications to satellite image segmentation. The satellite images are an important source for extracting landscape boundaries and other vegetation structures, which can provide extremely useful insights for applications in environmental monitoring, agriculture, forestry, and other related fields (see, for instance, [1,2,3]). In particular, in agricultural crop field classification, one fundamental problem is to provide a disjunctive decomposition of a fixed domain ΩR2 onto a finite number of nonempty subsets Ω=Ω1Ω2ΩK such that each of these subsets could be associated with a crop that is grown in this area, with forest regions, water zones, and so on, and this correspondence must be established at a rather a high level of accuracy. Up-to-date and accurate crop maps (or crop field classification) are needed to update agricultural statistics, to provide agricultural crop yield prediction, and are often used in environmental modeling. Typically, such an association between a given region and some agricultural crop can be made through the detection and quantitative assessment of green vegetation, which is one of the major applications of remote sensing studies. The information obtained in this way is a source of knowledge used for environmental resources management. One of the ways to get such information is the determination of the so-called vegetation indices (see [4] for a review and development). Since over the years many vegetation indices have been proposed for determining the vigor and health of vegetation, the reliability of information about vegetation directly and strictly depends on the fidelity, preciseness, and smoothness of the corresponding vegetation indices within each particular crop field.

    The most commonly used vegetation index (Ⅵ) is the so-called slope-based infrared percentage vegetation index (IPVI),

    IPVI:=u2,du1,d+u2,d,0IPVI1, (1.1)

    where ui,d=ui,d(x1,x2), i=1,2, with (x1,x2)Ω, are functions of two variables representing the intensity of red (Red) and near-infrared (NIR) reflectance of some region Ω of R2, respectively.

    Thus, each pixel x=(x1,x2)Ω of the original image can be associated with the corresponding IPVI-feature. The problem, which is suggested by application to remote sensing satellite image processing, consists of computing a decomposition

    Ω=Ω1Ω2ΩKK (1.2)

    of the domain of the image F:ΩR2 such that

    (a) the IPVI-characteristic varies smoothly and/or slowly within each Ωj;

    (b) the IPVI-characteristic varies discontinuously and/or rapidly across most of the boundary K between different Ωj.

    The distinguished features of this statement that do not permit to reduce it to the standard settings of the segmentation problem (see, for instance, the Mumford-Shah energy-based model [5] or the models proposed by Alvarez [6], Guichard [7], Lions, Morel [8], Caselles [9], and others) are the following ones:

    ● Each region Ω1, Ω2, , ΩK should consist of pixels that can be reasonably grouped according to the IPVI-characteristic. Simultaneously, these regions should be easy to differentiate according to the chosen image feature;

    ● The respective interiors of image regions should have a more or less simple geometry without gaps. Boundaries of image regions should be smooth enough but also accurate with respect to the chosen image feature;

    ● The most restrictive obstacle in the construction of such decomposition is the fact that these subdomains should not overlap the borders between fields or contain any fragments of such borders, meaning that they cannot take in even small parts of different fields with arguably different crops.

    All of these make the abovementioned segmentation problem rather challenging. It is enough to observe that a precise consideration of this problem demonstrates that the quantitative interpretation of remote sensing information from vegetation is a complex task. Many studies have limited this interpretation by assuming that the extracting vegetation information is uniformly and smoothly distributed within the particular crop fields. However, this assumption is broken when trying to apply these types of vegetation indices on heterogeneous canopies such as plantations with a mixed combination of soil, weeds, and other crops, or plantation where the vegetation of interest has different IPVI-characteristic due to spatial variability. The main idea, we realize in the new setting of the variational problem, can be briefly described as follows. We propose to make use of the so-called f-decomposition instead of the standard Chan-Vese "active contours without edges" model [10]. The role of the function f:ΩR in such decomposition of Ω has to guarantee that the new objects {Ωj}Kj=1 after the f-decomposition will have homogeneous values of the target function f within each separate field (a similar point of view can be found in [4,11]). In particular, in the case of the agricultural applications, where Ω stands for a zone of interest, the IPVI-characteristic can be considered as the main feature of this area, i.e., in this case f(x)=IPVI(x) for all xΩ. Thus, the main idea that we push forward is to formulate the segmentation problem as a constrained minimization problem in a special anisotropic functional space, with the "effect of anisotropy" we associate with the structure topology of IPVI-distribution. As a result, the main benefit of such an approach can be briefly described as follows:

    (i) It prevents the appearance of subdomains containing zones of discontinuity of f or places where this function tends to change rapidly by utilizing the main characteristic of the given function f — the unit normal vector field θ:ΩR2 to the level sets of f. This characteristic has been used to construct the so-called anisotropic diffusion tensor Mf, which can be defined as a square parametrized matrix function Mf(x)=[Iη2θ(x)θ(x)]. This matrix plays a central role in the process of f-decomposition of domain Ω, and we associate with it special anisotropic perimeters of the obtained subdomains (segments).

    (ii) The second characteristic feature of our approach is the fact that we apply the Jeffreys divergence to replace the standard Euclidean distance in the fidelity term of the objective functional. It is well-known that compared with Euclidean distance, Jeffreys divergence leads to more accurate results in information measurement (for the details of this metric and its advantages, we refer to [12,13,14]).

    The paper is organized as follows. In Section 2, we give some preliminaries related to the space of functions of bounded variation and other notions. Section 3 is devoted to the description of some specification of the standard BV(Ω) space. In particular, we introduce the so-called anisotropic version for the total variation of L1(Ω)-functions. At the end of this section, we show that some of the results of Samson et al. [15] can be extended to the case of subsets with a finite anisotropic perimeter.

    The precise setting of the main constrained minimization problem and its previous analysis are given in Section 4. We show that the proposed minimization problem can be interpreted as a special case of the piecewise-constant Mumford-Shah segmentation problem and the Chan-Vese active contour model without edges. We study this problem in the space of L1(Ω)-functions with bounded anisotropic total variation, where the type of anisotropy is closely related to the structure of the image f which is involved in the segmentation procedure. It is worth emphasizing that the anisotropic perimeter of the segments with uniform distribution of IPVI-characteristic can drastically differ from the standard one because the natural edges of the original image f can affect it significantly. Despite the "natural" setting of the proposed segmentation problem, the existence of its minimizers seems to be an open issue nowadays. The main reason is that the objective functional is neither coercive nor lower semicontinuous with respect to the weak- topology of BV(Ω) space. This circumstance stimulated us to introduce a special family of unconstrained two-parametric problems to approximate the original one. We show that each of those approximated problems is well-posed and has a nonempty set of minimizers.

    Section 5 aims to derive optimality conditions for approximated problems and provide their formal substantiation. In Section 6, we study the asymptotic behavior of the approximated problems and their solutions. The main question is to find out whether the convergence of minima of approximated problems is to minima of the original segmentation problem as small parameters tend to zero. Our main result of this section asserts that: If the original problem has a nonempty set of minimizers, then some of them can be successfully attained by the solution of approximated problems. Otherwise, we can come to a solution of the relaxed version of the original segmentation problem. In Section 7, the implementation of the proposed optimization problem is illustrated, providing numerical experiences with satellite images.

    A detailed description of the algorithm (including the method of marching squares for the generation of closed contours in the two-dimensional case) and finite-difference scheme for the proposed approach with the results of numerical simulation using the real-life satellite images will be considered in the forthcoming paper.

    We denote by L2 the Lebesgue 2-dimensional measure in R2 and by H1 the 1-dimensional Hausdorff measure. Let Ω be a bounded open subset of R2 with a Lipschitz boundary. For any subset EΩ, we denote by |E| its 2-dimensional Lebesgue measure L2(E). For a subset EΩ, let ¯E denote its closure and E its boundary. We define the characteristic function χE of E by

    χE(x):={1, for  xE,0, otherwise.

    For a function u, we denote by u|E its restriction to the set EΩ, and by uE its trace on E. Let C0(Ω) be the infinitely differentiable functions with compact support in Ω. The k-dimensional Hausdorff measure is denoted by Hk, and is the restriction of the measure μ to the set E. For a Banach space X, its dual is X and ,X,X is the duality form on X×X. By and , we denote the weak and weak convergence in normed spaces.

    We remind here of the most common definitions of some functional spaces that we will use later on.

    Throughout the paper, we will often use the concept of weak and strong convergence in L1(Ω). Let {fn}nN be a bounded sequence of functions in L1(Ω). We recall that {fn}nN is called equi-integrable on Ω if for any δ>0 there is a τ=τ(δ), such that S|fn|dx<δ for every measurable subset SΩ of Lebesgue measure |S|<τ. Then, the following assertions are equivalent for L1(Ω)-bounded sequences (see, for instance, [16,17]):

    (i) a sequence {fk}kN is weakly convergent in L1(Ω);

    (ii) the sequence {fk}kN is equi-integrable.

    The following theorem holds.

    Theorem 1. [16] If a bounded sequence {fk}kNL1(Ω) is equi-integrable and fkf almost everywhere in Ω, then fkf strongly in L1(Ω).

    We set |E|=L2(E), the Lebesgue measure of a measurable set ER2. Let M(Ω;R2) be the space of all R2-valued Borel measures which is, according to the Riesz theory, the dual of the space C0(Ω;R2) of all continuous vector-valued functions φ() with compact support in Ω and equipped with the uniform norm.

    φ=(2i=1supxΩ|φi(x)|2)1/2.

    Note that M(Ω;R2) is isomorphic to the product space

    M2(Ω):=2i=1M(Ω)

    and that μ=(μ1,μ2)M(Ω;R2) μi[C0(Ω)], i=1,2.

    Given a vector-valued measure μ:B(Ω)R2, we use the notation |μ| for its total variation. We recall that

    |μ|(E)=sup{2i=1Ωφidμi : φ=[φ1φ2]C0(E;R2), φ1}, (2.1)

    for all measurable EΩ.

    The usual weak- topology on M(Ω;R2) is defined as the weakest topology on M(Ω;R2), for which the maps μ2i=1Ωφidμi are continuous for every φC0(Ω;R2).

    By BV(Ω), we denote the space of all functions uL1(Ω), for which their distributional derivatives are representable by finite Borel measures in Ω, i.e.,

    Ωuϕxidx=ΩϕDiu,ϕC0(Ω), i=1,2

    for some R2-valued measure Du=(D1u,D2u)M2(Ω). It can be shown that BV(Ω), endowed with the norm uBV(Ω)=uL1(Ω)+|Du|(Ω), is a Banach space, where in view of Eq (2.1), the total variation of Du in Ω can be defined as

    |Du|(Ω):=Ω|Du|=sup{Ωudivφdx : φC10(Ω;R2), |φ(x)|1 for xΩ}. (2.2)

    We recall that the product topology of the strong topology of L1(Ω) for u and of the weak- topology of measures for Du is called the weak- topology of BV(Ω), and it is denoted BV-. As a result, a sequence {fk}k=1 -converges to f in BV(Ω) if, and only if, the two following conditions hold (see [18, p.124]): fkf strongly in L1(Ω) and DfkDf weakly- in M(Ω;R2), i.e.,

    limkΩ(ϕ,Dfk)=Ω(ϕ,Df),ϕC0(Ω;R2),

    where, in fact, Dfk=(Dx1fk,Dx2fk)M(Ω;R2) and, therefore, the notation ΩϕDfk should be interpreted as follows:

    Ω(ϕ,Dfk):=Ωϕ1Dx1fk+Ωϕ2Dx2fk.

    Moreover, if {fk}k=1BV(Ω) converges strongly to some f in L1(Ω) and supkNΩ|Dfk|<+, then (see, for instance, [16] and [18])

    (i) fBV(Ω)  and  Ω|Df|lim infkΩ|Dfk|;(ii) fkf  in  BV(Ω). (2.3)

    A simple criterion for the BV- convergence can be stated as follows (see [18, p.125], [19, Theorem 1.19]):

    Proposition 2. A sequence {uk}kNBV(Ω) BV--converges to u if, and only if, {uk}kN is bounded in BV(Ω) and {uk}kN converges to u strongly in L1(Ω).

    The following embedding result for the BV-function is very useful with respect to the variational problem that we study in this paper.

    Proposition 3. [16, p.378] Let Ω be an open bounded Lipschitz subset of R2. Then, the embedding BV(Ω)L2(Ω) is continuous and the embeddings BV(Ω)Lp(Ω) are compact for all p such that 1p<2. Moreover, there exists a constant Cem>0, which depends only on Ω and p such that for all u in BV(Ω),

    (Ω|u|pdx)1/pCemuBV(Ω),p[1,2].

    We also make use of the following property concerning to approximation of BV-functions by smooth ones.

    Theorem 4. [20] Assume fBV(Ω). Then, there exist a sequence {fk}k=1BV(Ω)C(Ω) such that

    fkf in L1(Ω),|Dfk|(Ω)|Du|(Ω) as k.

    Let E be an L2-measurable subset of R2 with finite Lebesgue measure. Let χE be its characteristic function. Following R. Caccioppoli [21], we say that E is a set with a finite perimeter in Ω if χEBV(Ω). This means that the distributional gradient DχE is a vector-valued measure with finite total variation. The total variation |DχE|(Ω) is called the perimeter of E in Ω, i.e., P(E,Ω)=|DχE|(Ω) and, therefore,

    P(E,Ω)=sup{ΩχEdivφdx : φC10(Ω;R2), φL(Ω;R2)1}. (2.4)

    We also notice that if Et:={xΩ : f(x)>t} stands for the level set for given fBV(Ω) and tR, then (see [20, Theorem 5.5.1]) Et has a finite perimeter for a.e. tR.

    The main goal of this section is to introduce some specification to the standard space of functions with bounded variation BV(Ω). This option is mainly motivated by the natural application in image segmentation problems. In view of this, we introduce the so-called anisotropic version for the total variation of the BV-functions. In principle, the notion of anisotropic total variation is not new in the literature (we refer to [22,23] for more details), and our representation for anisotropic total variation can be viewed as some specification of the rule in [22]. The main interest is in the application of this concept to the generalization of the well-known results of Samson et al. [15]. Namely, we focus on the proof of the following relation:

    limε0Ω|MD[χE]ε|=Ω|MDχE|,

    where [χE]ε stands for a smooth approximation of the characteristic function of a given set EΩ. The main results of this section are presented in the form of Lemma 5 and its corollary.

    Let Ω be an open bounded and connected subset of R2 with a Lipschitz boundary Ω. Let M:ΩR2×2 be a given matrix function such that

    M(x)=Mt(x),β1|ξ|2(ξ,M(x)ξ)β|ξ|2ξR2,M()C(Ω;R2×2), (3.1)

    for some constant β>1, i.e., M(x) is a positive-definite symmetric matrix for each xΩ.

    We say that uL1(Ω) is a function with bounded anisotropic variation if

    sup{Ωudiv(Mφ)dx : φC10(Ω;R2), |φ(x)|1 xΩ)}<+.

    It means that there exists a Radon measure Du such that

    Ωudiv(Mφ)dx=Ω(φ,MDu)φC10(Ω;R2).

    Moreover, for the total variation of the measure MDu, we have the following representation:

    |MDu|(Ω):=Ω|MDu|=sup{Ωudiv(Mφ)dx : φC10(Ω;R2), |φ(x)|1 xΩ)}. (3.2)

    Then, property (3.1) implies that

    β1(uL1(Ω)+|MDu|(Ω))uL1(Ω)+|Du|(Ω)uBVM(Ω)β1(uL1(Ω)+|MDu|(Ω)), uBV(Ω). (3.3)

    Hence, the expression can be viewed as an equivalent norm to standard one BVM(Ω) on the space BV(Ω). As a result, the main properties of BV-functions (see, for instance, [18,19,20]) can be reformulated with respect to the new norm. In particular, let M:ΩR2×2 be a given matrix function with property (3.1). Then:

    (j) If {uk}kNBV(Ω) is a bounded sequence, then there exist a subsequence {uki}iN and a function uBV(Ω) such that

    ukiustrongly in  L1(Ω),MDukiMDu weakly-in M(Ω;R2);

    (jj) If {uk}k=1BV(Ω) converges strongly to some u in L1(Ω) and satisfies supkNΩ|MDuk|<+, then

    uku  in  BV(Ω), uBV(Ω),  and  Ω|MDu|lim infkΩ|MDuk|; (3.4)

    (jjj) Let uBV(Ω) be an arbitrary function. Then, there exists a sequence {uk}kNC(Ω)BV(Ω) such that ukuL1(Ω)0 and Ω|MDuk|Ω|MDu| as k.

    By analogy with the standard notion, we can also define an anisotropic version of the perimeter, namely, we say that an L2-measurable subset UΩ has a finite M-perimeter if |MDχU|(Ω)<+, where χU() stands for the characteristic function of the set U. In this case, we write down

    Per(U;M;Ω):=Ω|MDχU|=sup{Udiv(Mφ)dx:φC10(Ω;R2), |φ(x)|1 xΩ}. (3.5)

    Moreover, for any uBV(Ω) and M()C(Ω;R2×2) with property (3.1), the following anisotropic Coarea formula

    Ω|MDu|=+Per({u>t};M;Ω)dt (3.6)

    holds true (see [22]).

    It is clear that, for a given level parameter lR, the area of the region {xΩ : φ(x)>l} can be defined as

    A{xΩ : φ(x)l}=Ωχ{φ(x)l}dx.

    Here, χ{φ(x)l}(x) stands for the characteristic function of the set {xΩ:φ(x)l}. To obtain some approximation of this area, we can substitute χE by its smooth approximation. With that in mind, for a given small positive parameter ε, we fix a positive symmetric mollifier ηCc(R2), i.e., η(x) is zero outside a compact set B1={xR2:|x|1},

    B1η(x)dx=1,η(x)0, and  ν(x)=μ(|x|) for some function  μ:R+R,

    and set

    [χE]ε=ηεχEwithηε(x)=ε2η(xε), (3.7)

    that is,

    [χE]ε=ε2R2η(xzε)χE(z)dz=R2η(w)χE(x+εw)dw.

    Then, using the standard properties of mollifiers, it can be shown that

    (i) [χE]εχE in L1(Ω) for any measurable subset E of R2 as 0;

    (ii) 0[χE]ε(x)1 for all xR2;

    (iii) If ER2 is bounded and gL1(R2), then R2[χE]εgdx=R2χE[g]εdx;

    (iv) If εΩ, then supp[χE]εΩε={xR2:dist(x,Ω)ε}.

    The following property will be utilized in our further analysis (see Section 6) and it can be considered as a natural generalization of the results of Samson et al. [15] (see also for comparison [19, Proposition 1.15]).

    Lemma 5. Let E be an open set such that ¯EΩ and E has a finite M-perimeter Per(E;M;Ω), where M:ΩR2×2 is a given matrix function with property (3.1). Let [χE]ε be the mollified characteristic function described above. Then,

    limε0Ω|MD[χE]ε|=Ω|MDχE|. (3.8)

    Proof. Taking into account the standard properties of mollifiers, we have {[χE]ε}ε>0BV(Ω) and [χE]εχE in L1(Ω) as ε0. Then, inequality (3.4) implies that

    Ω|MDχE|lim infε0Ω|MD[χE]ε|. (3.9)

    To establish a reverse inequality, we fix an arbitrary function φC10(Ω;R2) with |φ(x)|1. Then, there exists a vector-valued function ζC10(Ω;R2) such that ζ=Mφ. Therefore, by (iii)-property of mollifiers, we have

    Ω[χE]εdiv(Mφ)dx=ΩχE[div(Mφ)]εdx=ΩχEdiv[Mφ]εdx=ΩχEdiv[ζ]εdxsup{ΩεχEdiv[ζ]εdx:ζC10(Ω;R2), ζ=Mφ, |M1[ζ]ε(x)|1 xΩε}sup{ΩεχEdiv(Mφ)dx:φC10(Ωε;R2), |φ(x)|1 xΩε}=Ωε|MDχE|. (3.10)

    Taking then the supremum over all such φ, we arrive at the relation

    Ω|MD[χE]ε|Ωε|MεDχE|.

    Hence,

    lim supε0Ω|MD[χE]ε|limε0Ωε|MDχE|.

    Since E and Ω are open sets and ¯EΩ, it follows that

    Ω|DχE|=0andΩ|MDχE|=0.

    Therefore,

    limε0Ωε|MDχE|=¯Ω|MDχE|=Ω|MDχE|+Ω|MDχE|=Ω|MDχE|,limε0Ωε|DχE|=¯Ω|DχE|=Ω|DχE|+Ω|DχE|=Ω|DχE|<+.

    As a result, we obtain

    lim supε0Ω|MD[χE]ε|Ω|MDχE|.

    It remains to combine this inequality with Eq (3.9).

    Arguing similarly, we can generalize the Eq (3.8) as follows:

    Corollary 6. Let EΩ and the matrix M:ΩR2×2 be the same as in Lemma 5. Then,

    limε0Ω|MεD[χE]ε|=Ω|MDχE|, (3.11)

    where {Mε}ε>0C(Ω;R2×2) stands for any smooth approximation of the matrix M such that

    limε0MεMC(Ωε;R2×2)=0,Mε(x)=Mtε(x),β1|ξ|2(ξ,Mε(x)ξ)β|ξ|2ξR2, ε>0.

    Let f:ΩR be a given function. Hereinafter, we associate the function f with a given gray scale image. We define a smoothed version of the original image using the convolution of f with a Gaussian kernel

    Gσ(x)=1(2πσ)2exp(|x|22σ2),σ>0, (4.1)

    i.e., fσ=(Gσf())(x):=ΩGσ(xy)f(y)dy. Here, σ>0 is a given small positive value.

    Since fσC(¯Ω), it follows that the boundaries of level sets {xΩ:fσ(x)λ}, for all feasible λ[0,Cf], can be described by smooth curves with finite length. So, at all points xΩ of each level sets of fσ, we can define a unit normal vector field θ(x) following the rule

    θ(x)={fσ(x)|fσ(x)|1,if  |fσ(x)|0,0,otherwise. (4.2)

    Then, we associate with the vector field θ:ΩR2 the following linear operator R:R2R2:

    Rξ:=ξη2(θ,ξ)θ=[Iη2θθ]ξ,ξR2, (4.3)

    where η(0,1) is a given threshold which should be sufficiently close to 1. Setting

    Mf(x)=[Iη2θ(x)θ(x)], (4.4)

    we see that

    Mf(x)=[Mf(x)]t,(1η2)|ξ|2(ξ,Mf(x)ξ)|ξ|2,Mf()C(Ω;R2×2), (4.5)

    i.e., Mf is a symmetric and positive-definite matrix on Ω and it satisfies property (3.1) with β=(1η2)1.

    Remark 7. Let's assume for a moment that ξ=v(x), where vW1,1(Ω) is a given function and xΩ is a Lebesgue point of f in which the original image f is not expected to change drastically in any direction, i.e., x is not close to a discontinuity of f or a zone where f tends to change rapidly. Then, Eq (4.4) implies that Mf can be represented at this point as a unit matrix. So, at this point, we have Mfvv.

    On the other hand, if we consider a point xΩ that is close to a discontinuity of f, then Mfv reduces to (1η2)v if the gradient v(x) at this point is colinear to θ(x), and to v(x) provided v(x) is orthogonal to θ(x). In view of this, the expression Mfv can be interpreted as the directional total variation of v along the vector field θ (see [24] for the details).

    Definition 8. We say that a gray scale image f:ΩR is feasible for the segmentation procedure using level sets if there exists a value γ>0 such that

    fL(Ω), f(x)γ>0 a.e.in Ω.

    We denote the set of all feasible images by Fγ.

    We are now in a position to state the main object of our interest in this paper. Let fFγ be a given image, and let φ:ΩR be a level set function such that φBV(Ω). We associate with this function a collection of m+2 distinct level values l0<l1<<lm<lm+1 such that l0φ(x)lm+1 almost everywhere in Ω.

    Then, the constrained optimization problem we are going to consider can be stated as follows:

    J(c,φ)=Ω(fc0)log(fc0)χ{φ(x)<l1}dx+m1j=1Ω(fcj)log(fcj)[χ{φ(x)>lj}χ{φ(x)>lj+1}]dx+Ω(fcm)log(fcm)χ{φ(x)>lm}dx+αmj=1Ω|MfDχ{φ(x)>lj}|infφΞ, (4.6)

    where α>0 is a weight coefficient and the set of feasible solutions is defined as follows:

    Ξ={(c,φ)Rm+1×BV(Ω)|l0φ(x)lm+1 a.e. in  Ω,c=(c0,c1,,cm), cj0, j=0,,m}. (4.7)

    It is worth noticing that the objective functional J is well-defined on the set Ξ. Indeed, in this case the assumption φBV(Ω) implies that the level sets Et={xΩ:φ(x)>t} have finite anisotropic perimeter Per({u>t};Mf;Ω) for L1 a.a. tR. Since

    Per({φ>lj};Mf;Ω)=Ω|MfDχ{φ(x)>lj}|<

    for each j=1,,m, it follows that J(c,φ)< for each (c,φ)Ξ.

    Minimization problems (4.6) and (4.7) can be interpreted as a special case of the piecewise-constant Mumford-Shah segmentation problem and the active contour model (see, for instance, [8,10,25,26]). We can indicate the following principle features of this statement:

    ● The problem is investigated in the space of L1(Ω)-functions with bounded anisotropic total variation and with additional pointwise constraints, where the matrix of anisotropy is closely related to the structure of the image f, which is involved in the segmentation procedure.

    As a result, the anisotropic perimeter of the region {xΩ:lj<φ(x)}, which is given by the term Ω|MfDχ{φ(x)>lj}|, can drastically differ from the standard one because the natural edges of the original image f can affect it significantly;

    ● To find a piecewise-constant approximation of the given image fFγ in the form

    u(x)=c0χ{φ(x)<l1}(x)+m1j=1cj[χ{φ(x)>lj}(x)χ{φ(x)>lj+1}(x)]+cmχ{φ(x)>lm}(x), (4.8)

    we utilize the Jeffreys divergence between two elements f,gL2(Ω) instead of the standard L2-norm of their difference fgL2(Ω). In spite of the fact that the trick of replacing the squared Euclidean norm with the Jeffreys distance does not alleviate the original problem from the point of view of its solvability and mathematical analysis, it makes the segmentation results more stable to the Poisson noise contaminated images (see [12,13,14] for the details);

    ● We consider the segmentation problems (4.6) and (4.7) as a constrained minimization problem in BV(Ω) space with the pointwise constraints l0φ(x)lm+1 on the set of feasible functions φ:ΩR.

    ● In the proposed statement of the segmentation problems (4.6) and (4.7), it is admitted that the set

    K=Ω[mj=0{xΩ:lj<φ(x)<lj+1}]

    may have a nonzero L2-measure.

    However, the existence of minimizers to the problems (4.6) and (4.7) seems to be an open issue nowadays because the standard application of the direct method of calculus of variation to this problem faces some unsolved challenges. To apply the direct method for proving the existence of minimizers, it is necessary to find a topology for which the functional (4.6) is lower semicontinuous while ensuring compactness of minimizing sequences. In view of the structure of the set of admissible solutions ΞRm+1×L1(Ω) (see Eq (4.7)), the natural topology, in this case, is the product of the norm topology in Rm+1 and the weak- topology in BV(Ω). However, the objective functional J:ΞR is not coercive and lower semicontinuous on Rm+1×L1(Ω) with respect to the above mentioned topology. Moreover, even if φkφ strongly in L1(Ω) as k, it does not imply the strong convergence in L1(Ω) of χ{φk(x)>lj} to χ{φ(x)>lj}. In particular, it is clear that the implication

    [limkφkφL1(Ω)=0][limkχ{φk(x)>lj}χ{φ(x)>lj}L1(Ω)=0]

    may hold true if only L2{xΩ:φ(x)=lj}=0.

    To overcome this difficulty, we make use of the following family of two-parametric approximated problems:

    Jε,τ(c,φ)=Ω(fc0)log(fc0)[χAτ0]εdx+m1j=1Ω(fcj)log(fcj)([χAτj]ε[χAτj+1]ε)dx+Ω(fcm)log(fcm)[χAτm]εdx+1ε[Ψ(l0φ)L1(Ω)+Ψ(φlm+1)L1(Ω)]+ε|MfDφ|(Ω)+αmj=1Ω2ε|MfD[χEτj]ε|infφΞε, (4.9)

    where τ and ε are small parameters, which vary within strictly decreasing sequences of positive numbers converging to 0. The functions [χEτj]ε()C(R), j=1,,m, are defined in Eq (3.7),

    Aτ0={xΩ:φ(x)l1τ},Aτj={xΩ:φ(x)lj+τ}, j=1,,m,Eτj={xΩ:φ(x)ljτ}, j=1,,m,} (4.10)
    Ω2ε={xR2:dist(x,Ω)2ε}, (4.11)
    Ψ(z)={zδ,if  z0,0,if  z<0,}with a given exponent δ(1,2), (4.12)

    and

    Ξε={(c,φ)Rm+1×BV(Ω)|c=(c0,c1,,cm), cj0, j=0,,m}. (4.13)

    Thus, a pair (c,φ) sounds as feasible to the problems (4.6) and (4.7) if (c,φ)Ξε, i.e.,

    cCad:={c=(c0,c1,,cm), cj0, j=0,,m},φGad:=BV(Ω).

    Before proceeding further, we make use of the following property.

    Lemma 9. Let fFγ be a given image, let c=(c0,c1,,cm) with cj>0, let {φk}k=1L1(Ω) be a strongly convergent sequence, and let φL1(Ω) be its limit. Then, for each j=1,,m1,

    limkΩ(fcj)log(fcj)([χ{φk(x)lj+τ}]ε[χ{φk(x)lj+1+τ}]ε)dx=Ω(fcj)log(fcj)([χ{φ(x)lj+τ}]ε[χ{φ(x)lj+1+τ}]ε)dx. (4.14)

    Proof. Since L1(Ω)φkφ as k, we may suppose that, up to a subsequence, φk(x)φ(x) almost everywhere in Ω. Then, taking into account the continuity of the function [χE]ε() and the fact that the passage in the inequality φk(x)lj+τ leads to φ(x)lj+τ for a.e xΩ), we see that

    [χ{φk(x)lj+τ}]ε[χ{φ(x)lj+τ}]ε a.e. in  Ω.

    Moreover, since |[χ{φk(x)lj+τ}]ε(x)|1 in Ω, it follows from the Lebesgue dominated theorem that

    [χ{φk(x)lj+τ}]ε()[χ{φ(x)lj+τ}]ε()strongly in  L1(Ω).

    Since fFγ and c=(c0,c1,,cm) with cj>0, it follows that the integrant (fcj)log(fcj) is dominated by some integrable function g in the sense that

    |(f(x)cj)log(f(x)cj)|g(x) a.e. in  Ω.

    In particular,

    0(f(x)cj)log(f(x)cj)([χ{φk(x)lj+τ}]ε[χ{φk(x)lj+1+τ}]ε)(f(x)cj)log(f(x)cj){|f(x)2c2j|cj,a.e. in  Ω{f(x)>cj},|logγcj||f(x)cj|,a.e. in  Ω{f(x)cj},}{1cjf2L(Ω)+cj,a.e. in  Ω{f(x)>cj},|logγcj|(fL(Ω)+cj),a.e. in  Ω{f(x)cj}.}

    As a result, Eq (4.14) is a direct consequence of the Lebesgue dominated theorem. Arguing similarly, we can establish the same assertion for the first and the third terms in Eq (4.9).

    To conclude this section, we give an existence result for the parametrized optimization problems (4.9)–(4.13).

    Theorem 10. Let fF(γ) be a given gray scale image, and let

    Mf=[Iη2θ(x)θ(x)],

    where the vector field θC(Ω;R2) is defined by the rule (4.2). Then, for each ε(0,1) and τ>0 small enough, the constrained minimization problems (4.9)–(4.13) admit at least one solution.

    Proof. Since the objective functional is bounded from below on ΞεRm+1×BV(Ω), it follows that there exists a minimizing sequence to problems (4.9)–(4.13), i.e.,

    inf(c,φ)ΞεJε,τ(c,φ)=limkJε,τ(ck,φk)C<+, (4.15)

    where C stands for a strictly positive constant that can be different from line to line. Without loss of generality, we can suppose that C=ζ+1.

    Taking into account the fact that

    φkLδ(Ω)=({φk<l0}|φk|δdx+{l0φk<lm+1}|φk|δdx+{φk>lm+1}|φk|δdx)1δ({φk<l0}|φk|δdx)1δ+({φk>lm+1}|φk|δdx)1δ+max{|l0|,|lm+1|}|Ω|1δ

    and

    {φk<l0}|φk|δdx2δ1Ψ(l0φk)L1(Ω)+2δ1|l0|δ|Ω|,{φk>lm+1}|φk|δdx2δ1Ψ(φklm+1)L1(Ω)+2δ1|lm+1|δ|Ω|,

    we see that

    φkδLδ(Ω)22δ2Ψ(l0φk)L1(Ω)+22δ2Ψ(φklm+1)L1(Ω)+22δ23(|l0|+|lm+1|)δ|Ω|ε22δ2Jε,τ(ck,φk)+22δ23(|l0|+|lm+1|)δ|Ω|. (4.16)

    Since

    φkδL1(Ω)φkδLδ(Ω)|Ω|δ1,

    and the function K:RR, given by K(z)=(fz)log(fz), is locally continuous and coercive, i.e.,

    limz+K(z)z=+,limz+0K(z)z=+, (4.17)

    it follows from Eqs (4.16) and (4.17) that there exists a constant C(ζ,K,m,|Ω|,li,δ)>0 such that

    mj=0c2j,k+φkδL1(Ω)+|Dφk|(Ω)C+ε22δ2Jε,τ(ck,φk)|Ω|δ1+1εJε,τ(ck,φk)C+ε22δ2(ζ+1)|Ω|δ1+1ε(ζ+1),kN.

    Hence, the sequence {(ck,φk)}kN is bounded in Rm+1×BV(Ω). Then, from the compactness property in BV-space and the fact that BV(Ω) is compactly embedded in Lδ(Ω), we can deduce the existence of a subsequence of {(ck,φk)}kN, that we denote in the same way, and a pair (c0,φ0)Rm+1×BV(Ω) such that

    ckc0  in Rm+1,φkφ0 strongly in Lδ(Ω), (4.18)
    φk(x)φ0(x)almost everywhere in  Ω, (4.19)
    DφkDφ0  weaklyin M(Ω;R2). (4.20)

    Without loss of generality, we can suppose that each of the functions {φk}kN and φ0 is extended by zero outside of Ω, and φk(x)φ0(x) for a.e. xΩ2ε. Then, in view of the standard properties of mollifiers, we have

    {[χ{φk(x)l}]ε}kNBV(Ω2ε)

    and

    [χ{φk(x)l}]εχ{φ0(x)l}  in L1(Ω2ε)ask.

    Then, property (3.4) implies that

    Ω2ε|MfDχ{φ0(x)l}|lim infkΩ2ε|MfD[χ{φk(x)l}]ε|, (4.21)
    limk|MfDφk|(Ω)by (4.18), (4.20)|MfDφ0|(Ω). (4.22)

    Besides, in view of the properties (4.18)–(4.20) and the fact that ΨC1loc(R), we have the pointwise convergence

    Ψ(l0φk)(x)Ψ(l0φ0)(x)a.e. in  Ω,Ψ(φklm+1)(x)Ψ(φ0lm+1)(x)a.e. in Ω.

    Since

    Ψ(l0φk)|l0φk|δandΨ(φklm+1)|φklm+1|δa.e. in  Ω,

    and the sequences {|l0φk|δ}kN and {|φklm+1|δ}kN are bounded in Lp(Ω) with p=2/δ>1 (by the continuous embedding BV(Ω)L2(Ω)), it follows from Vitali's lemma that

    Ψ(l0φk)Ψ(l0φ0) and Ψ(φklm+1)Ψ(φ0lm+1)  in Lr(Ω) 1r<p.

    Hence,

    limk[Ψ(l0φk)L1(Ω)+Ψ(φklm+1)L1(Ω)]=Ψ(l0φ0)L1(Ω)+Ψ(φ0lm+1)L1(Ω). (4.23)

    As a result, the lower semicontinuity property of the minimizing sequence

    inf(c,φ)ΞεJε,τ(c,φ)=limkJε,τ(ck,φk)lim infkJε,τ(ck,φk)Jε,τ(c0,φ0). (4.24)

    is a direct consequence of relations (4.21)–(4.23) and Lemma 9.

    It remains to notice that due to the pointwise convergence (4.18), we have

    c0=(c00,c01,,c0m)  with  c0j0  for all j=0,,m.

    Hence, the limit pair (c0,φ0) is a feasible solution, i.e., (c0,φ0)Ξε, and, therefore,

    Jε(c0,φ0)=inf(c,φ)ΞεJε(c,φ)C<+.

    Thus, (c0,φ0) is a minimizer to the problems (4.9)–(4.13).

    This section aims to derive some optimality conditions for the minimization problems (4.9)–(4.13). Let ε and τ be given small positive values. With that in mind, we study the differentiability properties of the objective functional Jε,τ(c,φ) in order to specify its local behavior in the immediate vicinity of its minimum point. The corresponding Euler-Lagrange system is presented in Theorem 11.

    Let (c0ε,τ,φ0ε,τ)Ξε be a local minimizer to problems (4.9)–(4.13). Then,

    c0ε,τ=(c0ε,τ,0,c0ε,τ,1,,c0ε,τ,m)

    with c0ε,τ,j0 for each j{0,,m} and φ0ε,τGad. In the objective functional Jε,τ, we distinguish three terms

    Jε,τ(c,φ)=Fε,τ(c0,,cm,φ)+Φε(φ)+jε,τ(φ)

    with

    Fε,τ(c0,,cm,φ)=Ω(fc0)log(fc0)[χAτ0]εdx+m1j=1Ω(fcj)log(fcj)([χAτj]ε[χAτj+1]ε)dx+Ω(fcm)log(fcm)[χAτm]εdx,Φε(φ)=1ε[Ψ(l0φ)L1(Ω)+Ψ(φlm+1)L1(Ω)],jε,τ(φ)=ε|MfDφ|(Ω)+αmj=1Ω2ε|MfD[χEτj]ε|.

    From the differentiability of (fci)log(fci) and [χAτj]ε, it is immediate that the functional Fε,τ is of the class C1. Hence, there exist linear continuous functionals

    DφFε,τ(c0ε,τ,φ0ε,τ):Rm+1×BV(Ω)RandDcjFε,τ(c0ε,τ,φ0ε,τ):Rm+1×BV(Ω)R,j=0,,m

    such that

    Fε,τ(c0ε,φ0ε,τ+λh)=Fε,τ(c0ε,τ,φ0ε,τ)+λDφFε,τ(c0ε,τ,φ0ε,τ)[h]+r(h,λ),Fε,τ(c0ε,τ+λμej,φ0ε,τ)=Fε(c0ε,τ,φ0ε,τ)+λDcjFε,τ(c0ε,τ,φ0ε,τ)[μej]+rj(μ,λ),j=0,,m,

    for any hBV(Ω) and μR, where |r(h,λ)|=o(|λ|) and |rj(μ,λ)|=o(|λ|) as λ0, and

    ej=(0,,1j-th slot,,0)tRm+1.

    Moreover, making use of the following representations

    χAτ0=H(l1τφ)andχAτj=H(φljτ), j=1,.m

    with H(z)={1, z00, z<0}, we have

    [χAτ0]ε=Hε(l1τφ)and[χAτj]ε=Hε(φljτ), j=1,.m,

    where Hε(z)=[ηεχE](z) stands for the smooth approximation of H(z) through the mollification.

    Then, direct calculations show that

    DφFε,τ(c,φ)[h]=Ω(fc0)log(fc0)Hε(l1τφ)hdx,+m1j=1Ω(fcj)log(fcj)[Hε(φljτ)Hε(φlj+1τ)]hdx,+Ω(fcm)log(fcm)Hε(φlmτ)hdx, (5.1)
    DcjFε,τ(c,φ)[μej]=μcjΩfΛj(φ)dxμΩlog(f)Λj(φ)dx+μcjΩΛj(φ)dx+μΩΛj(φ)dx,j=0,1,,m (5.2)

    with

    Λ0(φ)=Hε(l1φτ),Λm(φ)=Hε(φlmτ),Λj(φ)=Hε(φljτ)Hε(φlj+1τ),j=1,,m1.

    Since fFγ, it follows from Eq (5.2) that the unique solution of the system

    DcjFε,τ(c0ε,τ,φ0ε,τ)[μej]=0,j=0,,m (5.3)

    can be expressed as

    {c0ε,τ,0=c0(φ0ε,τ)=ΩfHε(l1φ0ε,ττ)dxΩHε(l1φ0ε,ττ)dx,c0ε,τ,j=cj(φ0ε,τ)=Ωf[Hε(φ0ε,τljτ)Hε(φ0ε,τlj+1τ)]dxΩ[Hε(φ0ε,τljτ)Hε(φ0ε,τlj+1τ)]dx,c0ε,τ,m=cm(φ0ε,τ)=ΩfHε(φ0ε,τlmτ)dxΩHε(φ0ε,τlmτ)dx (5.4)

    with c0ε,τ,j0 for all j=0,,m, i.e., c0ε,τCad.

    Arguing in a similar manner and taking into account that ΨC1loc(R), it can be shown that

    Φε(φ0ε,τ+λh)=Φε(φ0ε,τ)+λDφΦε(φ0ε,τ)[h]+r(h,λ),

    for any hBV(Ω), where |r(h,λ)|=o(|λ|) as λ0, and DφΦε(φ0ε,τ):BV(Ω)R is a linear continuous functional with the following representation:

    DφΦε(φ0ε,τ)[h]=1ε[ΩΨ(l0φ0ε,τ)hdx+ΩΨ(φ0ε,τlm+1)hdx]. (5.5)

    Here,

    Ψ(z)={δzδ1,if  z0,0,if  z<0.}

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

    Theorem 11. Given ε>0 and τ>0 small enough, fFγ, α>0, and a collection of m+2 distinct level values l0<l1<<lm<lm+1, let (c0ε,τ,φ0ε,τ)Ξε be a local minimizer to problems (4.9)–(4.13). Then, the pair (c0ε,τ,φ0ε,τ) satisfies the following Euler-Lagrange system:

    Ω(fc0ε,τ,0)log(fc0ε,τ,0)Hε(l1φ0ε,ττ)[φφ0ε,τ]dx+m1j=1Ω(fc0ε,τ,j)log(fc0ε,τ,j)[Hε(φ0ε,τljτ)Hε(φ0ε,τlj+1τ)][φφ0ε,τ]dx+Ω(fc0ε,τ,m)log(fc0ε,τ,m)Hε(φ0ε,τlmτ)[φφ0ε,τ]dx+1εΩ[Ψ(φ0ε,τlm+1)Ψ(l0φ0ε,τ)][φφ0ε,τ]dx+jε,τ(φ)jε,τ(φ0ε,τ)0,φGad, (5.6)

    where the constants c0ε,τ,0,c0ε,τ,1,,c0ε,τ,m are defined by the rule (5.4).

    Proof. Since (c0ε,τ,φ0ε,τ)Ξε is a local minimum point of Eqs (4.9)–(4.13), we have that

    Jε,τ(c0ε,τ+ρ(μ(cj)0ε,τ)ej,φ0ε,τ)Jε,τ(c0ε,τ,φ0ε,τ), (5.7)
    Jε,τ(c0ε,τ,φ0ε,τ+ρ(φφ0ε,τ))Jε,τ(c0ε,τ,φ0ε,τ) (5.8)

    for all ρ>0 small enough and any given μ0 and φGad.

    As a result, inequality (5.7) leads to the Eq (5.3), and, hence, to the representation (5.4), whereas Eq (5.8) together with the convexity of jε implies

    0Jε,τ(c0ε,τ,φ0ε,τ+ρ(φφ0ε,τ))Jε,τ(c0ε,τ,φ0ε,τ)ρ=Fε,τ(c0ε,τ,φ0ε,τ+ρ(φφ0ε,τ))Fε,τ(c0ε,τ,φ0ε,τ)ρ+jε,τ(φ0ε,τ+ρ(φφ0ε,τ))jε,τ(φ0ε,τ)ρ+Φε(φ0ε,τ+ρ(φφ0ε,τ))Φε(φ0ε,τ)ρFε,τ(c0ε,τ,φ0ε,τ+ρ(φφ0ε,τ))Fε,τ(c0ε,τ,φ0ε,τ)ρ+Φε(φ0ε,τ+ρ(φφ0ε,τ))Φε(φ0ε,τ)ρ+jε,τ(φ)jε,τ(φ0ε,τ).

    Now, passing to the limit as ρ0, we get

    0DφFε,τ(c0ε,τ,φ0ε,τ)[φφ0ε,τ]+DφΦε(φ0ε,τ)[φφ0ε,τ]+jε,τ(φ)jε,τ(φ0ε,τ).

    Finally, using the expression of the Gateaux derivatives DφFε,τ(c0ε,τ,φ0ε,τ) and DφΦε(φ0ε,τ) given by Eqs (5.1) and (5.5), respectively, we immediately arrive at the optimality system (5.6).

    The main question we are going to discuss in this section is to find out whether the convergence of minima of Eq (4.9) is to minima of Eq (4.6) as ε and τ tend to zero. To this end, we make use of the basic results of the variational convergence of minimization problems and Γ-convergence theory (see, for instance, [27,28,29]). In particular, in Lemmas 12 and 13, we show that the standard properties of Γ-limits hold true for the objective functional Jε,τ with respect to the weak- topology of BV(Ω) space and the pointwise convergence in Rm+1. Utilizing these characteristic features, we establish the main variational property of the proposed approximation procedure (see Theorem 14). Namely, we prove that any sequence of optimal pairs to the approximated problems (4.9)–(4.13) is compact in the weak--topology of Rm+1×BV(Ω) and each cluster point is a solution of the problem

    Jτ(c,φ)inf(c,φ)ΞφBV(Ω),

    where the cost functional is defined in Eq (6.6).

    We begin with the following noteworthy result.

    Lemma 12. Let τ>0 be a given value such that τ1. Let {(cε,φε)Ξε}ε0 be a sequence of feasible pairs to the approximated problems (4.9)–(4.13), satisfying the conditions

    supε>0Jε,τ(cε,φε)<+andsupε>0φεBV(Ω)<+. (6.1)

    Then, there exist a subsequence {(cεj,φεj)}j=1 with εj0 as j, and a pair (c,φ)Rm+1×BV(Ω) such that

    cεjc in Rm+1 asj, (6.2)
    φεjφ strongly in L1(Ω) as j, (6.3)
    MfDφεjMfDφ weaklyin M(Ω;R2) as j, (6.4)
    (c,φ)ΞandJτ(c,φ)lim infjJεj,τ(cεj,φεj), (6.5)

    where

    Jτ(c,φ)=Ω(fc0)log(fc0)χAτ0dx+m1j=1Ω(fcj)log(fcj)[χAτjχAτj+1]dx+Ω(fcm)log(fcm)χAτmdx+αmj=1Ω|MfDχEτj|. (6.6)

    Proof. In view of the initial assumptions (see Eq (6.1)2), the sequence {(cε,φε)}ε0 is compact with respect to the product of norm topology of Rm+1 and the weak- convergence in BV(Ω). So, there exists a subsequence {(cεj,φεj)}j=1 with εj0 as j and a pair (c,φ)Rm+1×BV(Ω) such that, in addition to Eqs (6.2) and (6.3), we have

    φεj(x)φ(x)  a.e. in  Ω,DφεjDφ weaklyin M(Ω;R2) as j. (6.7)

    Since the matrix Mf is positive-definite on Ω, property (6.7) implies the weak- convergence (6.4). Moreover, from Eqs (6.1) and (6.7), we deduce the existence of a constant C>0 independent of ε such that

    supjN|MfDφεj|(Ω)supjN|Dφεj|(Ω)C,Ψ(l0φεj)L1(Ω)+Ψ(φεjlm+1)L1(Ω)εjC,jN.

    Hence,

    limj[εj|MfDφεj|(Ω)]=0, (6.8)
    limjΨ(l0φεj)L1(Ω)=0,limjΨ(φεjlm+1)L1(Ω)=0. (6.9)

    It means that the limit function φ satisfies the pointwise constraints l0φ(x)lm+1 a.e. in Ω. Thus, we see that (c,φ)Ξ.

    It remains to establish the inequality (6.5)2. With that in mind, we make use of the pointwise convergence Eqs (6.2), (6.7), and properties (ⅰ) and (ⅱ) of the smoothed characteristic functions [χE]ε. Then, arguing as in the proof of Lemma 9, we deduce that

    [χ{φεj(x)lk+τ}]εjχ{φ(x)lk+τ} a.e. in  Ω,k=1,,m(f(x)cεj,k)log(f(x)cεj.k)(f(x)ck)log(f(x)ck) a.e. in  Ω,k=0,,m|(f(x)cεj,k)log(f(x)cεj,k)|g(x) a.e. in  Ω,

    with some integrable function g. Hence, by the Lebesgue dominated theorem, we have:

    Ω(fcεj,0)log(fcεj,0)[χ{φεj(x)l1τ}]εdx+m1k=1Ω(fcεj,k)log(fcεj,k)([χ{φεj(x)lk+τ}]ε[χ{φεj(x)lk+1+τ}]ε)dx+Ω(fcεj,m)log(fcεj,m)[χ{φεj(x)lm+τ}]εdxjΩ(fc0)log(fc0)χ{φ(x)l1τ}dx+m1j=1Ω(fcj)log(fcj)(χ{φ(x)lk+τ}χ{φ(x)lk+1+τ})dx+Ω(fcm)log(fcm)χ{φ(x)lm+τ}dx. (6.10)

    To end the proof, we have to show that

    lim infjmk=1Ω2ε|MfD[χ{φεj(x)lkτ}]εj|mk=1Ω|MfDχ{φ(x)lkτ}| (6.11)

    With that in mind, we notice that

    +by (6.1)>lim infjmk=1Ω2ε|MfD[χ{φεj(x)lkτ}]εj|lim infjmk=1Ω|MfD[χ{φεj(x)lkτ}]εj|. (6.12)

    Taking into account that (see Eq (6.1)2), all level sets of the functions {φεj}jN have a finite perimeters, and we see that χ{φεj(x)lkτ}BV(Ω) for each k=1,,m. Moreover, in view of Eq (6.7), we have

    χ{φεj(x)lkτ}jχ{φ(x)lkτ} a.e. in Ω and strongly in L1(Ω), (6.13)

    for each k{1,,m}.

    Hence,

    [χ{φεj(x)lkτ}]εjjχ{φ(x)lkτ} a.e. in Ω and strongly in  L1(Ω),lim infjmk=1Ω|MfD[χ{φεj(x)lkτ}]εj|mk=1Ω|MfDχ{φ(x)lkτ}|. (6.14)

    Together with Eq (6.12), we arrive at the announced inequality (6.11). Thus, the desired property (6.5) is a direct consequence of of relations (6.8)–(6.11).

    Lemma 13. For every feasible pair (c,φ)Ξ, there can be found a sequence {(ˆcε,ˆφε)}ε>0 satisfying the properties

    (ˆcε,ˆφε)Ξεfor ε>0smallenough, (6.15)
    (ˆcεc in Rm+1,ˆφεφ in BV(Ω) asε0, (6.16)
    Jτ(c,φ)=limε0Jε,τ(ˆcε,ˆφε). (6.17)

    Proof. Let (c,φ)Ξ be a given pair. In view of Theorem 4, we can suppose that φC(Ω).

    Let {cε}ε>0 be an arbitrary sequence in Rm+1 such that

    cεCadε>0andcεc  as  ε0. (6.18)

    Then, we define the sequence {(ˆcε,ˆφε)}ε>0 as follows:

    ˆcε=cεandˆφε=φ,ε>0.

    Since φC(Ω), it follows that ˆφε has a bouded anisotropic total variation and, therefore, {(ˆcε,ˆφε)Ξε)}ε>0 are the collection of feasible solutions for the corresponding two-parametric approximated problems (4.9). Then, due to the fact that l0φ(x)lm+1 in Ω, we have:

    Ψ(l0ˆφε)L1(Ω)+Ψ(ˆφεlm+1)L1(Ω)=0,ε>0, (6.19)
    limε0|MfDˆφε|(Ω)limε0|Dˆφε|(Ω)=|Dφ|(Ω)<. (6.20)

    Furthermore, arguing as in the proof of Lemma 6 and taking into account the property (ⅰ) of mollifiers, we see that

    Ω(fˆcε,0)log(fˆcε,0)[χ{φ(x)l1τ}]εdx+m1k=1Ω(fˆcε,k)log(fˆcε,k)([χ{φ(x)lk+τ}]ε[χ{φ(x)lk+1+τ}]ε)dx+Ω(fˆcε,m)log(fˆcε,m)[χ{φ(x)lm+τ}]εdxjΩ(fc0)log(fc0)χ{φ(x)l1τ}dx+m1j=1Ω(fcj)log(fcj)(χ{φ(x)lk+τ}χ{φ(x)lk+1+τ})dx+Ω(fcm)log(fcm)χ{φ(x)lm+τ}dx. (6.21)

    Thus, in view of Eqs (6.19)–(6.21), in order to deduce the Eq (6.17), it remains to show that

    limε0mj=1Ω2ε|MfD[χ{φ(x)ljτ}]ε|=mj=1Ω|MfDχ{φ(x)ljτ}|. (6.22)

    Observing that (see Lemma 5)

    limε0Ω2ε|MfD[χ{φ(x)ljτ}]ε|=¯Ω|MfDχ{φ(x)ljτ}|=Ω|MfDχ{φ(x)ljτ}|+Ω|MfDχ{φ(x)ljτ}|, (6.23)

    where the last term in Eq (6.23) is equal to zero because {xΩ:φ(x)ljτ} is a closed subset of Ω.

    As a result, we obtain

    limε0Ω2ε|MfD[χ{φ(x)ljτ}]ε|=Ω|MfDχ{φ(x)ljτ}|,j=1,,m.

    This concludes the proof.

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

    Theorem 14. Let τ1 be a given positive value. Let {(c0ε,τ,φ0ε,τ)Ξε}ε0 be a sequence of optimal pairs to the approximated minimization problems (4.9)–(4.13). Assume that the sequence {φ0ε,τ}ε>0 is bounded in BV(Ω), and the relaxed problem

    Jτ(c,φ)inf(c,φ)ΞφBV(Ω) (6.24)

    has a nonempty set of minimizers for the given value τ>0. Then, there exists a pair (cτ,φτ)Ξ such that, up to a subsequence,

    c0ε,τcτ in Rm+1 asε0, (6.25)
    φ0ε,τφτ strongly in L1(Ω), (6.26)
    MfDφ0ε,τMfDφτ weaklyin M(Ω;R2), (6.27)
    inf(c,φ)ΞJτ(c,φ)=Jτ(cτ,φτ)=limε0Jε,τ(c0ε,τ,φ0ε,τ)=limε0inf(c,φ)ΞεJε,τ(c,φ), (6.28)

    where the objective functional Jτ:ΞR is defined in Eq (6.6).

    Proof. First, we observe that a given sequence of minimizers for approximating problems (4.9)–(4.13) is compact with respect to the convergences (6.25)–(6.27). Indeed, for an arbitrary test function ˆφCc(R2) and arbitrary vector ˆcRm+1 with positive components, we have:

    ˆcCadandˆφBV(Ω),ε>0.

    Let's assume that, in addition, the function ˆφ satisfies the pointwise constraints l0ˆφ(x)lm+1 in Ω. Then, (ˆc,ˆφ)Ξε for each ε>0, and, therefore,

    Jε,τ(c0ε,τ,φ0ε,τ)=inf(c,φ)ΞεJε,τ(c,φ)Jε,τ(ˆc,ˆφ)supε>0Jε,τ(ˆc,ˆφ)C<+ε>0.

    Hence,

    supε>0Jε,τ(c0ε,τ,φ0ε,τ)<+andsupε>0φ0ε,τBV(Ω)<+. (6.29)

    Thus, for the sequence of minimizers {(c0ε,τ,φ0ε,τ)Ξε}ε>0, all preconditions of Lemma 12 are fulfilled. Therefore, there exist a subsequence {(c0εk,τ,φ0εk,τ)Ξεk}kN of the sequence {(c0ε,τ,φ0ε,τ)Ξε}ε0 and a pair (cτ,φτ)Ξ, such that (c0εk,τ,φ0εk,τ)(cτ,φτ) is the sense of convergences (6.25)–(6.27) and

    Jτ(cτ,φτ))lim infkJεk,τ(c0εk,τ,φ0εk,τ).

    From this, we deduce that

    lim infkinf(c,φ)ΞekJεk,τ(c,φ)=lim infkJεk,τ(c0εk,τ,φ0εk,τ)Jτ(cτ,φτ)inf(c,φ)ΞφBV(Ω)Jτ(c,φ)=Jτ(c0τ,φ0τ), (6.30)

    where (c0τ,φ0τ) is a minimizer for the relaxed problem (6.24).

    On the other hand, Lemma 13 implies the existence of a realizing sequence {(ˆcε,ˆφε)}ε>0 such that (ˆcε,ˆφε)(c0τ,φ0τ) as ε0 in the sense of relations (6.16), and

    Jτ(c0τ,φ0τ)=limε0Jε,τ(ˆcε,ˆφε).

    Utilizing this fact, we get

    inf(c,φ)ΞφBV(Ω)Jτ(c,φ)=Jτ(c0τ,φ0τ)=lim supε0Jε,τ(ˆcε,ˆφε)lim supε0inf(c,φ)ΞεJε,τ(c,φ)lim supkinf(c,φ)ΞεkJεk,τ(c,φ)=lim supkJεk,τ(c0εk,τ,φ0εk,τ). (6.31)

    From this and Eq (6.30), we deduce that

    lim infkJεk(c0εk,τ,φ0εk,τ)lim supkJεk,τ(c0εk,τ,φ0εk,τ).

    As a result, we have

    Jτ(c0τ,φ0τ)=Jτ(cτ,φτ)=inf(c,φ)ΞφBV(Ω)Jτ((c,φ))=limkinf(c,φ)ΞεkJεk,τ(c,φ). (6.32)

    Using these relations and the fact that the problem (6.24) is solvable, we may suppose that

    (cτ,φτ)=(c0τ,φ0τ).

    Since Eq (6.32) holds for all subsequences of {(c0ε,τ,φ0ε,τ)Ξε}ε0, which are convergent in the sense of relations (6.25)–(6.27), it follows that these limits coincide and, therefore, (c0τ,φ0τ) is the limit of the whole sequence {(c0ε,τ,φ0ε,τ)}ε>0. Then, using the same argument for the entire sequence of minimizers, we finally obtain

    lim infε0inf(c,φ)ΞεJε,τ(c,φ)=lim infε0Jε,τ(c0ε,τ,φ0ε,τ)Jτ(c0τ,φ0τ)inf(c,φ)ΞφBV(Ω)Jτ(c,φ)=limε0Jε,τ(c0ε,τ,φ0ε,τ)lim supε0inf(c,φ)ΞεJε,τ(c,φ)=lim supε0Jε,τ(c0ε,τ,φ0ε,τ),

    and this concludes the proof.

    To illustrate the implementation of the proposed optimization problem (4.6) to the domain decomposition that corresponds to the homogeneity zones of a given function f:ΩR, we provided numerical experiences with images that have been delivered by satellite Sentinel-2. As input data, we have used an image over the Dnipro area, Ukraine, with a resolution of 10m/pixel (see the left panel in Figure 1). This region represents a typical agricultural area with medium-sized fields of various shapes. As follows from the picture given in Figure 1 (see also the corresponding histogram in Figure 2), the observed data suffer from noise and blurs. So, at the first step, we have realized the denoising and debluring procedure (see the right panel in Figure 1) following the variational approach that has been recently proposed in [30]. As Figure 2 indicates, the histogram of the smoothed image has a strongly marked compactly localized spectrum that can be considered as a "good option" for its piecewise constant approximation. To conduct the numerical simulations of the segmentation procedure for the given area, we have set f(x)=u2(x) in Ω, where u2 stands for the intensity of the de-blurred image (see Figure 1) in the green spectral channel, and

    m=4, l0=5000, l1=0, l2=1000, l3=2000, l4=300, lm=5000.
    Figure 1.  Left panel: The original satellite image. Right panel: The same image after denoising.
    Figure 2.  Histogram of the original image (left) and the smoothed data (right).

    In accordance with the results of Section 5, we have to solve the system (5.6) and find its solution φ0e for the corresponding function f. Since, in practical implementations, it is reasonable to define the solution of the problem (5.6) using a "gradient descent" strategy, we started with some initial level-set function φ0C(Ω) and passed to the corresponding initial-boundary value problem for quasi-linear parabolic equations with Neumann boundary conditions. For numerical simulations, we set ε=0.01, τ=10, α=1, σ=3, η=0.95, and the initial level set function φ0C(Ω) was defined as follows:

    φ0(x)={+d(x,S),xinsideS,0,xS,d(x,S),xoutsideS,

    where S is a circle of radius 20 with a center at a central point of Ω, and d(x,S) denotes the Euclidean distance from the point xΩ to the circle S. We report the level sets in Figures 3 and 4.

    Figure 3.  The level sets {xΩ:φ>l} with l1=0 (left) and with l2=1000 (right).
    Figure 4.  The level sets {xΩ:φ>l} with l3=2000 (left) and with l4=3000 (right).

    All the authors conceived the idea, designed the methodology and the main proofs. All the authors contributed equally in the writing of the article.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    Peter Kogut thanks the support of University of Salerno.

    The authors declare there is no conflict of interest.



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