Research article Special Issues

Restored texture segmentation using Markov random fields

  • Received: 16 November 2022 Revised: 27 February 2023 Accepted: 07 March 2023 Published: 29 March 2023
  • Texture segmentation plays a crucial role in the domain of image analysis and its recognition. Noise is inextricably linked to images, just like it is with every signal received by sensing, which has an impact on how well the segmentation process performs in general. Recent literature reveals that the research community has started recognizing the domain of noisy texture segmentation for its work towards solutions for the automated quality inspection of objects, decision support for biomedical images, facial expressions identification, retrieving image data from a huge dataset and many others. Motivated by the latest work on noisy textures, during our work being presented here, Brodatz and Prague texture images are contaminated with Gaussian and salt-n-pepper noise. A three-phase approach is developed for the segmentation of textures contaminated by noise. In the first phase, these contaminated images are restored using techniques with excellent performance as per the recent literature. In the remaining two phases, segmentation of the restored textures is carried out by a novel technique developed using Markov Random Fields (MRF) and objective customization of the Median Filter based on segmentation performance metrics. When the proposed approach is evaluated on Brodatz textures, an improvement of up to 16% segmentation accuracy for salt-n-pepper noise with 70% noise density and 15.1% accuracy for Gaussian noise (with a variance of 50) has been made in comparison with the benchmark approaches. On Prague textures, accuracy is improved by 4.08% for Gaussian noise (with variance 10) and by 2.47% for salt-n-pepper noise with 20% noise density. The approach in the present study can be applied to a diversified class of image analysis applications spanning a wide spectrum such as satellite images, medical images, industrial inspection, geo-informatics, etc.

    Citation: Sanjaykumar Kinge, B. Sheela Rani, Mukul Sutaone. Restored texture segmentation using Markov random fields[J]. Mathematical Biosciences and Engineering, 2023, 20(6): 10063-10089. doi: 10.3934/mbe.2023442

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  • Texture segmentation plays a crucial role in the domain of image analysis and its recognition. Noise is inextricably linked to images, just like it is with every signal received by sensing, which has an impact on how well the segmentation process performs in general. Recent literature reveals that the research community has started recognizing the domain of noisy texture segmentation for its work towards solutions for the automated quality inspection of objects, decision support for biomedical images, facial expressions identification, retrieving image data from a huge dataset and many others. Motivated by the latest work on noisy textures, during our work being presented here, Brodatz and Prague texture images are contaminated with Gaussian and salt-n-pepper noise. A three-phase approach is developed for the segmentation of textures contaminated by noise. In the first phase, these contaminated images are restored using techniques with excellent performance as per the recent literature. In the remaining two phases, segmentation of the restored textures is carried out by a novel technique developed using Markov Random Fields (MRF) and objective customization of the Median Filter based on segmentation performance metrics. When the proposed approach is evaluated on Brodatz textures, an improvement of up to 16% segmentation accuracy for salt-n-pepper noise with 70% noise density and 15.1% accuracy for Gaussian noise (with a variance of 50) has been made in comparison with the benchmark approaches. On Prague textures, accuracy is improved by 4.08% for Gaussian noise (with variance 10) and by 2.47% for salt-n-pepper noise with 20% noise density. The approach in the present study can be applied to a diversified class of image analysis applications spanning a wide spectrum such as satellite images, medical images, industrial inspection, geo-informatics, etc.



    Domestic pigs originated from the Eurasian wild boar (Sus scrofa), which first appeared about 9000 years ago [1]. They are essential for the transmission of swine influenza. Human beings raise domestic pigs, and then slaughter them for pork [2]. Domestic pigs grow in the food and environment provided by human beings, while human beings get the necessary nutrients by eating them. Consequently, in the breeding process, the swine flu virus is transmitted to human beings through domestic pig-human contact [2,3]. According to this process, a new breed-slaughter model with swine influenza transmission can be proposed as a model (1.1).

    {S1(x,t)t=D12S1(x,t)x2+(B12N2(x,t)ω0N1(x,t))N1(x,t)s0S1(x,t)                 β11I1(x,t)S1(x,t)+γ1I1(x,t),xR,t>0,I1(x,t)t=D12I1(x,t)x2+β11I1(x,t)S1(x,t)(s0+γ1)I1(x,t),xR,t>0,S2(x,t)t=D22S2(x,t)x2+(b2r2N2(x,t)K2+B21N1(x,t))N2(x,t)d2S2(x,t)                  2j=1β2jIj(x,t)S2(x,t)+γ2I2(x,t),xR,t>0,I2(x,t)t=D22I2(x,t)x2+2j=1β2jIj(x,t)S2(x,t)[e2+γ2+d2]I2(x,t),xR,t>0,Ni(x,t)=Si(x,t)+Ii(x,t),i=1,2,xR,t>0. (1.1)

    Domestic pig population N1(x,t) and human population N2(x,t) are assumed to be divided into 2 epidemiological compartments: susceptibles (Si(x,t)) and infectives (Ii(x,t)) at time t and location x, i=1,2. Susceptibles can become infected by means of intra-species or inter-species transmission and then recover as new susceptibles. The notation B12 represents the human breeding parameter for the population growth of domestic pigs, while B21 represents the nutrients from eating domestic pigs to increase the birth rate of human beings. The notation s0 represents the slaughter rate of domestic pigs. It's noteworthy that domestic pigs cannot survive independently without human beings, but human beings can still survive well without the supply of pork [2]. Restrictions on the development of human population mainly come from intra-species competition.

    For humans, the notation r2=b2d2 is the intrinsic growth rate of humans, where b2 and d2 represents the natural natality rate and mortality rate, respectively. K2 is the environmental carrying capacity of human population without domestic pig supply. e2 is the additional mortality rate of humans caused by swine flu. For domestic pigs, ω0 represents the intraspecific competition. During the spread of swine flu, the parameters βij represent the per capita incidence rate from species j to species i, where i,j=1,2. γi denote the recovery rate for domestic animals and humans, i=1,2. D1 and D2 are the diffusion coefficients for domestic animals and humans. It is noteworthy that all parameters mentioned above is positive.

    The main purpose of this paper is to propose a new breed-slaughter model with swine influenza transmission, and study the dynamic behavior of it. And then, we focus on the invasion process of infected domestic animals into the habitat of humans. Under certain conditions as in Theorem 2, we construct a propagating terrace linking human habitat to animal-human coexistent habitat, then to swine flu natural foci, which is divided by spreading speeds. Firstly, we calculate the equilibrium points of the model without spatial heterogeneity as a model (1.2) and analyze the existence of them by the persistence theory. Secondly, we discuss their stability by the basic reproduction number. Thirdly, we use these equilibrium points to construct a propagating terrace linking them by spreading speeds.

    {dS1(t)dt=(B12N2(t)ω0N1(t))N1(t)s0S1(t)β11I1(t)S1(t)+γ1I1(t),dI1(t)dt=β11I1(t)S1(t)(s0+γ1)I1(t),dS2(t)dt=(b2r2N2(t)K2+B21N1(t))N2(t)d2S2(t)2j=1β2jIj(t)S2(t)+γ2I2(t),dI2(t)dt=2j=1β2jIj(t)S2(t)[e2+γ2+d2]I2(t),Ni(t)=Si(t)+Ii(t),i=1,2. (1.2)

    In model (1.2), domestic pig population N1(t) and human population N2(t) are assumed to be divided into 2 epidemiological compartments: susceptibles (Si(t)) and infectives (Ii(t)) at time t, i=1,2. Other parameters are the same with model (1.1).

    At first, we focus on the breed-slaughter system without swine flu transmission and spatial heterogeneity.

    If I1(0)=I2(0)=0, N1(0)>0 and N2(0)>0, model (1.2) turns to a new breed-slaughter system without swine influenza transmission as model (2.1).

    {dN1(t)dt=(B12N2(t)ω0N1(t))N1(t)s0N1(t),dN2(t)dt=r2(1N2(t)K2)N2(t)+B21N1(t)N2(t),N1(0)>0,N2(0)>0, (2.1)

    Similar to the competition system in [4], breed-slaughter system also has abundant dynamic results. For the positive equilibrium point

    E=(N1,N2)=(r2(s0B12K2)B12B21K2ω0r2,K2(s0B21ω0r2)B12B21K2ω0r2)

    of model (2.1), we have three cases:

    (a). If B12B21<ω0r2K2 and s0<min{ω0r2B21,B12K2}, the positive equilibrium point E of model (2.1) is stable (Figure 2(a)).

    Figure 1.  Swine flu transmission route from pig to human.
    Figure 2.  Phase diagram of E.

    (b). If B12B21>ω0r2K2 and s0>max{ω0r2B21,B12K2}, the positive equilibrium point E of model (2.1) is unstable (Figure 2(b)).

    (c). Other than the condition as (a) or (b), the positive equilibrium point E of model (2.1) does not exist.

    In order to reflect the effect of interspecific interaction on swine influenza transmission during breeding process as model (1.2), we suppose that B12B21<ω0r2K2 and s0<min{ω0r2B21,B12K2} to guarantee the existence and the stability of the boundary equilibrium point

    E3=(r2(s0B12K2)B12B21K2ω0r2,0,K2(s0B21ω0r2)B12B21K2ω0r2,0)

    with I1(0)=I2(0)=0 in model (1.2).

    After calculation, we summarize that there are at most 6 equilibrium points in R4+ of the system (1.2): E0=(0,0,0,0), E1=(0,0,K2,0), E2=(0,0,¯S2,¯I2), E3=(N1,0,N2,0), E4=(N1,0,S2,I2), E5=(S1,I1,S2,I2), where ¯S2=e2+γ2+d2β22, ¯I2=β22K2(e2+γ2+d2)β22, N1=N1=r2(s0B12K2)B12B21K2ω0r2, N2=K2(s0B21ω0r2)B12B21K2ω0r2, S2=e2+γ2+d2β22, I2=β22K2(1+s0B21B12B21K2B12B21K2ω0r2)(e2+γ2+d2)β22. The exact expression of E5 is unknown. However, under certain conditions as in Theorem 2, we can obtain its existence by persistence theory [5,6,7].

    If there is no domestic pigs participation, namely N1(0)=S1(0)=I1(0)=0, The persistence and the stability of boundary equilibrium E2=(0,0,¯S2,¯I2) has been proved in [8]. Similarly, we define R0=β22K1b2+e2+γ2. And then, we can get the following lemma.

    Lemma 1. If N1(0)=S1(0)=I1(0)=0 and I2(0)>0, {0}×{0}×R2+ is a invariant set of system (1.2). The trivial equilibrium point E0 in model (1.2) is unstable, and we have following two cases:

    (a) If R01, the disease-free equilibrium point E1 of model (1.2) is stable;

    (b) If R0>1, model (1.2) has a unique equilibrium point E2 in the interior of {0}×{0}×R2+, which is stable, and E1 is unstable.

    Furthermore, we consider the transmission process of human influenza with domestic pigs participating, but not infected from them. Namely I1(0)=0, I2(0)>0 and Ni(0)>0, i=1,2. The persistence and the stability of boundary equilibrium E4=(N1,0,S2,I2) is similar to Lemma 1. Taking E3 as the original point by coordinate translation, we can get the following lemma by the persistence theory [5,9,10], when B12B21<ω0r2K2 and s0<min{ω0r2B21,B12K2}.

    According to the definition of basic reproduction number in a single population as [5,11,12], we define R1=β1N1s0+γ1 as the basic reproduction number of swine flu transmission in demotic pig population and R2=β22N2e2+γ2+d2 as the basic reproduction number of swine flu transmission in human population.

    Lemma 2. If N1(0)=S1(0)>0, I1(0)=0 and I2(0)>0, R+×{0}×R2+ is a invariant set of system (1.2). The trivial equilibrium point E0 and the boundary equilibrium point E1, E2 in model (1.2) are unstable when B12B21<ω0r2K2 and s0<min{ω0r2B21,B12K2}, and we have following two cases:

    (a) If R21, the disease-free equilibrium point E3 of model (1.2) is stable;

    (b) If R2>1, model (1.2) has a unique equilibrium point E4 in the interior of R+×{0}×R2+, which is stable, and E3 is unstable.

    Next we focus on the discussion about the existence and the stability of the positive equilibrium point E5=(S1,I1,S2,I2). At first, we define Rs=max{R1,R2}. Then, we get the theorem as the following.

    Theorem 1. If Ni(0)>0 and Ii(0)>0, i=1,2, R4+ is a invariant set of system (1.2). The trivial equilibrium point E0 and the boundary equilibrium point E1, E2 in model (1.2) are unstable when B12B21<ω0r2K2 and s0<min{ω0r2B21,B12K2}, and we have following three cases:

    (a) If Rs1, the disease-free equilibrium point E3 of model (1.2) is stable;

    (b) If Rs>1, R1<R2 and R11, model (1.2) has a unique equilibrium point E4 except for E0, E1, E2 and E3, which is stable, and E3 is unstable;

    (c) If Rs>1 and R1R2 (or R2>R1>1) model (1.2) has a unique equilibrium point E5 in the interior of R4+, which is stable, and E3 and E4 are unstable.

    Proof. If Rs1, E4 and E5 do not exist. Similar to the results of Lemma 2 (a), the disease-free equilibrium point E3 of model (1.2) is stable.

    Then we consider the results of system (1.2) when Rs>1 and B12B21<ω0r2K2 and s0<min{ω0r2B21,B12K2}. At first, we define

    D={(S1,I1,S2,I2)|0IiSi+IiNi,i=1,2},
    D1={(S1,I1,S2,I2)|I1=0 or I2=0 ,0Si+IiNi,i=1,2},
    D2=DD1,˜D2={(S1,I1,S2,I2)|0<IiSi+IiNi,i=1,2}.

    D2 and ˜D2 are forward invariant.

    Let Ω consists of equilibria E0, E1, E2, E3 and E4. These equilibria cannot be chained to each other in D1. By analyzing the flow in neighborhood of each equilibrium, it is easy to see that Ω is isolated in D and D1 is a uniform strong repeller for ˜D2.

    If x(t)=(S1(t),I1(t),S2(t),I2(t)) stays close to E2, we have two cases: if I1(0)=I2(0)=0, then I1(t)=I2(t)=0; if I1(0)>0 or I2(0)>0, then I2(t)>0. Therefore, E2 is isolated in D. Similarly, we can prove that E0, E1 and E3 are isolated in D.

    For E4 and E5, we have two cases: (A). R1<R2 and R11; (B). R1R2 or R2>R1>1.

    (A). R1<R2 and R11

    If R1<R2 and R11, E5 do not exist. Similar to the results of Lemma 2 (b), the boundary equilibrium point E4 of model (1.2) is stable.

    (B). R1R2 or R2>R1>1

    If x(t)=(S1(t),I1(t),S2(t),I2(t)) stays close to E4, we have two cases: if I1(0)=0, then I1(t)=0; if I1(0)>0, then I1(t)>0. Since (S1(t),I1(t),S2(t),I2(t)) satisfying system (1.2) has no invariant subset other than E4 in its neighborhood. E4 is isolated in D and a uniform weak repeller for ˜D2. Therefore, we can prove that E0, E1, E2, E3 and E4 are isolated in D.

    Using Proposition 4.3 in [5], we can prove that D1 is a uniform weak repeller for ˜D2; and using Theorem 4.5 in [5], we can prove that D1 is a uniform strong repeller for ˜D2.

    Then we get that there exists an ϵ>0 such that

    liminftmin{I1(t),I2(t)}>ϵ,

    with Ni(0)>0 and Ii(0)>0, i=1,2.

    Therefore, if B12B21<ω0r2K2, s0<min{ω0r2B21,B12K2} and R1R2 (R2>R1>1), there exists at least one internal equilibrium of system (1.2) [9,10,13].

    Next, we use Theorem 2 in [11] to discuss the basic reproduction number of system (1.2).

    The Jacobian matrix of (I1,I2) is

    J=(β11S1(s0+γ1)0β21S2β22S2(e2+γ2+d2)),

    Let J=FV, F be the rate of appearance of new infections in compartment I, V be the rate of transfer of individuals out of compartment I. Then, we get

    F=(β11S10β21S2β22S2),
    V= diag(s0+γ1e2+γ2+d2).

    We call FV1 be the next generation matrix for the model (1.2) and set Rs=ρ(FV1|E3), where ρ(A) denotes the spectral radius of a matrix A.

    Then we get

    Rs=max{β11N1s0+γ1,β22N2e2+γ2+d2}.

    Finally, using Theorem 2 in [11], we can prove Theorem 1.

    The basic reproduction number is an important threshold value in the research of the epidemic mathematical model, which determines the disease to break out or not. However, it is not sufficient to discuss the breed-slaughter model with interspecific interaction. The main purpose of this paper is to investigate invasion process of infected domestic animal into human habitat. And we construct a propagating terrace linking human habitat E1 to animal-human coexistent habitat E3, then to swine flu natural foci E4 (or E5), which is divided by certain spreading speeds. The propagating terrace can describe the spatio-temporal continuous change of the transmission of swine flu.

    Based on the heterogeneity of the population structure and the temporal and spatial continuity of the mammal movement, the population's spatial factor is considered in the spread of swine flu. If the swine flu host populations are distributed differently in space, the diffusion term may change their local population structure, thus change the swine flu epidemic. In order to describe the population invasion process, we set the initial value is zero in the area x(,x0)(x0,). The area of (x0,x0) is the original habitat of N, and N will invade to the area of x(,x0)(x0,) at the spreading speed s [14].

    The definition of spreading speed of a single population is the positive value s satisfied with the conditions as follows,

    limt+{sup|x|>ctN(x,t)}=0,c>s

    and

    limt+inf{inf|x|<ctN(x,t)}>0,c<s,

    in the model [4.1]

    {N(x,t)t=D2N(x,t)x2+rN(x,t)(1N(x,t)K),xR,t>0,N(x,0)=N0>0,x[x0,x0],N(x,0)=0,x(,x0)(x0,). (4.1)

    The biological description of spreading speed s has been shown in the third figure of Figure 3. The value of s approximates the inverse of the slope of the color lines. It is easy to see that the co-effect of diffusion and reproduction leads to the population territory expansion, in which the local diffusion rate D guarantees the population spatial invasion to new areas and the reproduction rate r guarantees its development on occupied areas. The spreading speed of a single population in the model [4.1] is expressed by s:=2Dr by [14]. However, it is not enough to study the swine flu with more than one host species [15,16,17,18,19,20]. We redefine the spreading speeds at the human-animal interface, as shown below.

    s1:=2D1(B12K2s0),
    s2:=max{2D1(β11N1s0γ1),2D2(β22N2e2γ2d2)}.
    Figure 3.  Effect of r and D on the local diffusion of a single population.

    Due to the participation of two populations, some notations need to be redefined. The notations s and x0 are replaced by si, xi, with i=1,2, corresponding to the two swine flu host populations.

    Then we construct a propagating terrace linking human habitat E1 to animal-human coexistent habitat E3, then to swine flu natural foci E4 (or E5), which is divided by certain spreading speeds. The propagating terrace can describe the spatio-temporal continuous change of the transmission of swine flu, which can be show in Theorem 2.

    Theorem 2. For system (1.1), if B12B21<ω0r2K2, B21N1<r2 and s0<min{ω0r2B21,B12K2}, the initial conditions satisfy that 0<S1(x,0)<N1, x[x1,x1]; S1(x,0)=0, x(,x1)(x1,), for some x1>0; 0<I1(x,0)<N1, x[x2,x2]; I1(x,0)=0, x(,x2)(x2,), for some x2>0; S2(x,0)=K2, I2(x,0)=0, xR.

    We set

    s1:=2D1(B12K2s0),s2:=max{2D1(β11N1s0γ1),2D2(β22N2e2γ2d2)}.

    Suppose that s1>s2, x1>x2, then there are three cases about the invasion process as following:

    (a) Rs1,

    limt+sup|x|>ct{|S1(x,t)|+|I1(x,t)|+|S2(x,t)K2|+|I2(x,t)|}=0,  c>s1,
    limt+sup|x|<ct{|S1(x,t)N1|+|I1(x,t)|+|S2(x,t)N2|+|I2(x,t)|}=0,  c<s1.

    The system (1.1) forms a propagating terrace, linking E1 to E3.

    (b) If Rs>1, R1<R2 and R11,

    limt+sup|x|>ct{|S1(x,t)|+|I1(x,t)|+|S2(x,t)K2|+|I2(x,t)|}=0,  c>s1,
    limt+supc2t<|x|<c1t{|S1(x,t)N1|+|I1(x,t)|+|S2(x,t)N2|+|I2(x,t)|}=0,  s2<c2<c1<s1,
    limt+sup|x|<ct{|S1(x,t)N1|+|I1(x,t)|+|S2(x,t)S2|+|I2(x,t)I2|}=0,  c<s2.

    The system (1.1) forms a propagating terrace, linking E1 to E3, then to E4.

    (c) If Rs>1 and R1R2 (or R2>R1>1),

    limt+sup|x|>ct{|S1(x,t)|+|I1(x,t)|+|S2(x,t)K2|+|I2(x,t)|}=0,  c>s1,
    limt+supc2t<|x|<c1t{|S1(x,t)N1|+|I1(x,t)|+|S2(x,t)N2|+|I2(x,t)|}=0,  s2<c2<c1<s1,
    limt+sup|x|<ct{|S1(x,t)S1|+|I1(x,t)I1|+|S2(x,t)S2|+|I2(x,t)I2|}=0,  c<s2.

    The system (1.1) forms a propagating terrace, linking E1 to E3, then to E5.

    Proof. The epidemic of swine flu originates in the interaction between humans and domestic animals in the breeding process, so the breaking out of swine flu would lag behind this process. Therefore, we first confirm the propagating terrace linking E1 and E3.

    The breed-slaughter system without swine flu transmission can be transferred to model (4.2).

    {N1(x,t)t=D12N1(x,t)t2+(B12N2(x,t)ω0N1(x,t))N1(x,t)s0N1(x,t),N2(x,t)t=D22N2(x,t)t2+r2(1N2(x,t)K2)N2(t)+B21N1(x,t)N2(x,t). (4.2)

    Let (N1,N2) be a solution to system (4.2) with the initial condition 0<N1(x,0)<N1, x[x1,x1]; N1(x,0)=0, x(,x1)(x1,), for some x1>0; N2(x,0)=K2, xR.

    If B12B21<ω0r2K2 and s0<min{ω0r2B21,B12K2}, we claim that (N1(x,t),N2(x,t))Σ, xR,t[0,), where

    Σ:={(N1,N2)[0,N1]×[0,N2]:B12N2(x,t)ω0N1(x,t)s00,r2(1N2(x,t)K2)+B21N1(x,t)0}.

    By the strong maximum principle, N10 for t>0. Then we get

    N2(x,t)tD22N2(x,t)t2+r2(1N2(x,t)K2)N2(x,t).

    By a comparison, N2X, where X is the solution to

    {X(x,t)t=D22X(x,t)t2+r2(1X(x,t)K2)X(x,t),X(x,0)=N2(x,0). (4.3)

    Then we get the result

    limt+infN2(x,t)limt+infX(x,t)=K2.

    Set u:=N1 and v:=N2K2. Then N2(x,t)t can be rewritten as

    v(x,t)t=D22v(x,t)t2r2v(x,t)K2(v(x,t)+K2)+B21u(x,t)(v(x,t)+K2).

    Due to u=N1[0,N1] and v=N2K20 then

    v(x,t)tD22v(x,t)t2(r2B21N1)v(x,t)+B21K2u(x,t).

    By the strong maximum principle, if follows that vY in R×[0,), where Y is the solution to

    {Y(x,t)t=D22Y(x,t)t2(r2B21N1)Y(x,t)+B21K2u(x,t),X(x,0)=0,xR. (4.4)

    Then we have

    Y(x,t)=B21K2t0{e(r2B21N1)(ts)Re(xy)2/[4(ts)]u(y,s)dy}ds.

    Given ϵ>0, we choose δ>0 small enough such that 2D1(B12K2s0+B12δ)<s1+ϵ.

    For this δ, we claim that there is τ1 such that Y(x,t)<δ+Mu(x,t), xR, tτ, for some positive constant M. Then it follows that N1 satisfies

    N1(x,t)tD12N1(x,t)t2+(B12K1s0+B12δ(B12M+ω0)N1(x,t))N1(x,t).

    Therefore, according to the comparison principle and the definition of spreading speed [14,15,16,19,21], for any c(2D1(B12K2s0+B12δ),s1+ϵ), it follows that limt+sup|x|>ctN1(x,t)=0, and then limt+sup|x|>ctN2(x,t)=K2.

    Because of the arbitrariness of ϵ, we get

    limt+sup|x|>ct{|N1(x,t)|+|N2(x,t)K2|}=0,c>s1.

    Thus, if the swine flu does not break out, namely Rs1, for system (1.1),

    limt+sup|x|>ct{|S1(x,t)|+|I1(x,t)|+|S2(x,t)K2|+|I2(x,t)|}=0,  c>s1.

    Then we set U:=N1N1 and V:=N2N2. Similar to the proof before, we can get

    limt+sup|x|<ct{|N1(x,t)N1|+|N2(x,t)N2|}=0,  c<s1.

    If Rs1, for system (1.1),

    limt+sup|x|<ct{|S1(x,t)N1|+|I1(x,t)|+|S2(x,t)N2|+|I2(x,t)|}=0,  c<s1.

    Next we consider the propagating terrace linking E3 to (E4 or E5). Let (S1,I1,S2,I2) be a solution to system (1.1) with the initial condition S1(x,0)=N1, S2(x,0)=N2, I2(x,0)=0, xR. I1(x,0)>0, x[x2,x2]; I1(x,0)=0, x(,x2)(x2,), for some x2>0.

    If B12B21<ω0r2K2 and s0<min{ω0r2B21,B12K2}, we claim that (S1(x,t)+I1(x,t),S2(x,t)+I2(x,t))Σ, xR,t[0,).

    For the spreading speed when Rs>1, comparison principle and strong maximum principle are no longer applicable due to the complexity of system (1.1). However, we can calculate the minimum wave speed from largest eigenvalue of its linearized system at E3 as [22] to link E3 and E4 (or E5).

    For the following eigenvalue problem

    1λAληλ=cηλ,

    where

    Aλ=diag(Diλ2)+J|E3.

    J is the jacobian matrix,

    J=(B12N22ω0N1s0β11I1β11S1+γ1B12N10β11I1β11S1(s0+γ1)00B12N2β21S2r2(12N2K2)+B21N1(β21I1+β22I2)β22S2+γ20β21S2β21I1+β22I2β22S2(e2+γ2+d2)).

    For λ0, the eigenvalues of the matrix

    Aλ=(D1λ2ω0N1β11N1+γ1B12N100D1λ2+β11N1(s0+γ1)00B12N2β21N2D2λ2r2N2K2β22N2+γ20β21N20D2λ2+β22N2(e2+γ2+d2)).

    are D1λ2+β11N1(s1+γ1), D2λ2+β22N2(e2+γ2+d2) and other two impossible results, which cannot define positive wave speed.

    Thus, the minimum wave speed can be defined as follows, which can be divided the propagating terrace, linking E3 to E4 (or E5).

    s2=max{infλ>0D1λ2+β11N1(s0+γ1)λ,infλ>0D2λ2+β22N2(e2+γ2+d2)λ}=max{2D1(β11N1s0γ1),2D2(β22N2e2γ2d2)}.

    If Rs>1, there are two cases: (A). R1<R2 and R11; (B). R1R2 or R2>R1>1.

    (A). If R1<R2 and R11, then E5 does not exist. s2=2D2(β22N2e2γ2d2), then we get

    limt+sup|x|>c2t+x2{|S1(x,t)N1|+|I1(x,t)|+|S2(x,t)N2|+|I2(x,t)|}=0,  c>s2,
    limt+sup|x|<ct+x2{|S1(x,t)N1|+|I1(x,t)|+|S2(x,t)S2|+|I2(x,t)I2|}=0,  c<s2.

    Combining the results before, linking E1 to E3, then

    limt+sup|x|>ct+x1{|S1(x,t)|+|I1(x,t)|+|S2(x,t)K2|+|I2(x,t)|}=0,  c>s1,
    limt+supc2t+x2<|x|<c1t+x1{|S1(x,t)N1|+|I1(x,t)|+|S2(x,t)N2|+|I2(x,t)|}=0,  s2<c2<c1<s1,
    limt+sup|x|<ct+x2{|S1(x,t)N1|+|I1(x,t)|+|S2(x,t)S2|+|I2(x,t)I2|}=0,  c<s2.

    The system (1.1) forms a propagating terrace, linking E1 to E3, then to E4.

    (B). If R1R2 or R2>R1>1, set s2=max{2D1(β11N1s0γ1),2D2(β22N2e2γ2d2)}. then we get

    limt+sup|x|>ct+x1{|S1(x,t)|+|I1(x,t)|+|S2(x,t)K2|+|I2(x,t)|}=0,  c>s1,
    limt+supc2t+x2<|x|<c1t+x1{|S1(x,t)N1|+|I1(x,t)|+|S2(x,t)N2|+|I2(x,t)|}=0,  s2<c2<c1<s1,
    limt+sup|x|<ct+x2{|S1(x,t)S1|+|I1(x,t)I1|+|S2(x,t)S2|+|I2(x,t)I2|}=0,  c<s2.

    The system (1.1) forms a propagating terrace, linking E1 to E3, then to E5.

    If s1>s2, x1>x2 and Rs>1, R1>R2>1, then in Figure 4, the blue area represents the original habitat area of humans at the population size of E1. After domesticating pigs, the red part will be shared with the two species at E3. While after swine flu transmitting between domestic pigs and humans, the internal red part will be shared again with two populations at E5 with swine flu transmission. It is a biological description of propagating terrace of humans with swine flu transmission, which is the local spacial variation of the population.

    Figure 4.  If Rs>1, R1>R2>1, the propagating terrace from E1 to E3, then to E5. (a): The simulation of N2; (b): Contour line of N2.

    If s1>s2,x1>x2 and Rs1, then in Figure 5, the blue area represents the original habitat area of humans at the population size of E1. After domesticating pigs, the red part will be shared with the two species at E3. Because Rs1, there is no swine flu transmission during the breed and slaughter process. Then the propagating terrace links unstable equilibrium E1 and stable equilibrium E2.

    Figure 5.  If Rs1, the propagating terrace from E1 to E3. (a): The simulation of N2; (b): Contour line of N2.

    We establish a new swine flu mathematical model to reflect the dynamic process of swine flu transmission with interspecific action between domestic pigs and humans, in which the roles of different species will no longer be at the same level. Domestic pigs cannot survive independently without human beings, but human beings can still survive well without the supply of pork. By our new swine flu model, we find that the human-animal interface has promoted the cross-species transmission of swine flu and resulted in the prevalence of flu in humans. In addition, the threshold values of population development and disease transmission are also discussed in order to provide a scientific basis for future health decision makers in swine flu prevention and control. We propose the zoonotic basic reproduction number Rs, which is more applicable to the study of swine flu transmission. Then, it is analyzed that the spreading speed of different species forming propagating terraces is influenced by the intrinsic growth rate r and diffusion rate D.

    In this paper, the equilibrium points of the model are calculated and we analyze the existence of the equilibrium points by the persistence theory. Then we discuss their stability by the basic reproduction number. In addition, after redefining the spreading speed, we divide the propagating terrace with two populations, which is an unprecedented task. We concern with the invasion process of infected domestic animals into the habitat of humans. Under certain conditions as in Theorem 2, we construct a propagating terrace linking human habitat to animal-human coexistent habitat, then to swine flu natural foci, which is divided by spreading speeds.

    This work is supported by the National Natural Science Foundation of China, 11771044.

    The authors declared that they have no conflicts of interest to this work.



    [1] H. Liu, G. Huo, Q. Li, X. Guan, M. L. Tseng, Multiscale lightweight 3D segmentation algorithm with attention mechanism: Brain tumor image segmentation, Expert Syst. Appl., 214 (2023), 119166. https://doi.org/10.1016/j.eswa.2022.119166 doi: 10.1016/j.eswa.2022.119166
    [2] X. Li, S. Chen, J. Wu, J. Li, T. Wang, J. Tang, et al., Satellite cloud image segmentation based on lightweight convolutional neural network, Plos One, 18 (2023), e0280408. https://doi.org/10.1371/journal.pone.0280408 doi: 10.1371/journal.pone.0280408
    [3] L. Guo, P. Shi, L. Chen, C. Chen, W. Ding, Pixel and region level information fusion in membership regularized fuzzy clustering for image segmentation, Inf. Fusion, 92 (2023), 479–497. https://doi.org/10.1016/j.inffus.2022.12.008 doi: 10.1016/j.inffus.2022.12.008
    [4] R. M. Haralick, K. Shanmugan, I. Dinstein, Textural features for image classification, IEEE Trans. Syst. Man Cybern., 3 (1973), 610–621. https://doi.org/10.1109/TSMC.1973.4309314 doi: 10.1109/TSMC.1973.4309314
    [5] J. Chaki, N. Dey, Texture Feature Extraction Techniques for Image Recognition, Springer, Singapore, (2020).
    [6] A. Distante, C. Distante, Handbook of Image Processing and Computer Vision: Volume 3: From Pattern to Object, Springer International Publishing, Basel, Switzerland, (2020).
    [7] C. C. Hung, E. Song, Y. Lan, Image Texture Analysis. Foundations, Models and Algorithms, Springer International Publishing, Basel, Switzerland, (2019).
    [8] Y. Chen, E. R. Dougherty, Gray-scale morphological granulometric texture classification, Opt. Eng., 33 (1994), 2713–2722. https://doi.org/10.1117/12.173552 doi: 10.1117/12.173552
    [9] B. B. Chaudhuri, N. Sarkar, Texture segmentation using fractal dimension, IEEE Trans. Pattern Anal. Mach. Intell., 17 (1995), 72–77. https://doi.org/10.1109/34.368149 doi: 10.1109/34.368149
    [10] J. M. Keller, S. Chen, R. M. Crownover, Texture description and segmentation through fractal geometry, Comput. Vis. Graph. Image Process., 45 (1989), 150–166. https://doi.org/10.1016/0734-189X(89)90130-8 doi: 10.1016/0734-189X(89)90130-8
    [11] R. Chellappa, S. Chatterjee, Classification of textures using Gaussian Markov random fields, IEEE Trans. Acoust. Speech Signal Process., 33 (1985), 959–963. https://doi.org/10.1109/TASSP.1985.1164641 doi: 10.1109/TASSP.1985.1164641
    [12] G. R.Cross, A. K. Jain, Markov random field texture models, IEEE Trans. Pattern Anal. Mach. Intell., (1983), 25–39. https://doi.org/10.1109/tpami.1983.4767341. doi: 10.1109/tpami.1983.4767341
    [13] L. Alparone, F. Argenti, G. Benelli, Fast calculation of co-occurrence matrix parameters for image segmentation, Electron. Lett., 26 (1990), 23–24.
    [14] R. Davarzani, S. Mozaffari, K. Yaghmaie, Scale-and rotation-invariant texture description with improved local binary pattern features, Signal Process., 111 (2015), 274–293. https://doi.org/10.1016/j.sigpro.2014.11.005. doi: 10.1016/j.sigpro.2014.11.005
    [15] C. C. Gotlieb, H. E. Kreyszig, Texture descriptors based on co-occurrence matrices, Comput. Vis. Graph. Image Process., 51 (1990), 70–86. https://doi.org/10.1016/S0734-189X(05)80063-5 doi: 10.1016/S0734-189X(05)80063-5
    [16] M. Topi, O. Timo, P. Matti, S. Maricor, Robust texture classification by subsets of local binary patterns, in Proceedings of the 15th International Conference on Pattern Recognition ICPR-2000, Barcelona, Spain, 3 (2000), 935–938. https://doi.org/10.1109/ICPR.2000.903698
    [17] V. Durgamahanthi, R. Rangaswami, C. Gomathy, A. C. J. Victor, Texture analysis using wavelet-based multiresolution autoregressive model: Application to brain cancer histopathology, J. Med. Imaging Health Inf., 7 (2017), 1188–1195.
    [18] L. D. Jacobson, H. Wechsler, Joint Spatial/Spatial frequency representations, Signal Process., 14 (1988), 37–68. https://doi.org/10.1016/0165-1684(88)90043-6 doi: 10.1016/0165-1684(88)90043-6
    [19] C. S. Lu, P. C. Chung, C. F. Chen, Unsupervised texture segmentation via wavelet transform, Pattern Recognit., 30 (1997), 729–742. https://doi.org/10.1016/S0031-3203(96)00116-1 doi: 10.1016/S0031-3203(96)00116-1
    [20] T. Randen, J. H. Husoy, Filtering for texture classification:A comparative study, IEEE Trans. Pattern Anal. Mach. Intell., 21 (1999), 291–310. https://doi.org/10.1109/34.761261 doi: 10.1109/34.761261
    [21] K. J. Laws, Textured image segmentation, in University of Southern California Los Angeles Image Processing INST, (1980).
    [22] D. A. Clausi, K-means Iterative Fisher (KIF) unsupervised clustering algorithm applied to image texture segmentation, Pattern Recognit., 35 (2002), 1959–1972. https://doi.org/10.1016/S0031-3203(01)00138-8 doi: 10.1016/S0031-3203(01)00138-8
    [23] A. K. Jain, F. Farrokhnia, Unsupervised texture segmentation using Gabor filters, Pattern Recognit., 24 (1991), 1167–1186. https://doi.org/10.1016/0031-3203(91)90143-S doi: 10.1016/0031-3203(91)90143-S
    [24] S. Kinge, B. S. Rani, Mukul Sutaone, A multi-class fisher linear discriminant approach for the improvement in the accuracy of complex texture discrimination, Helix, 9 (2019), 5108–5121. https://doi.org/10.29042/2019-5108-5121 doi: 10.29042/2019-5108-5121
    [25] N. Senin, R. K. Leach, S. Pini, L. A. Blunt, Texture-based segmentation with Gabor filters, wavelet and pyramid decompositions for extracting individual surface features from areal surface topography maps, Meas. Sci. Technol., 26 (2015), 095405. https://doi.org/10.1088/0957-0233/26/9/095405 doi: 10.1088/0957-0233/26/9/095405
    [26] M. Tuceryan, A. K. Jain, Texture Segmentation using voronoi polygons, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 211–216. https://doi.org/10.1109/34.44407 doi: 10.1109/34.44407
    [27] G. Matheron, Random Sets and Integral Geometry, Wiley Publications, New York, (1975).
    [28] H. Deng, D. A. Clausi, Unsupervised image segmentation using a simple MRF model with a new implementation scheme, Pattern Recognit., 37 (2004), 2323–2335. https://doi.org/10.1016/j.patcog.2004.04.015 doi: 10.1016/j.patcog.2004.04.015
    [29] A. K. Qin, D. A. Clausi, Multivariate image segmentation using semantic region growing with adaptive edge penalty, IEEE Trans. Image Process., 19 (2010), 2157–2170. https://doi.org/10.1109/TIP.2010.2045708 doi: 10.1109/TIP.2010.2045708
    [30] Z. Kato, H. T. C. Pong, A Markov random field image segmentation model for color textured images, Image Vision Comput., 24 (2006), 1103–1114. https://doi.org/10.1016/j.imavis.2006.03.005 doi: 10.1016/j.imavis.2006.03.005
    [31] M. Kiechle, M. Storath, A. Weinmann, M. Kleinsteuber, Model-based learning of local image features for unsupervised texture segmentation, IEEE Trans. Image Process., 27 (2018), 1994–2007. https://doi.org/10.1109/TIP.2018.2792904 doi: 10.1109/TIP.2018.2792904
    [32] M. Pereyra, S. McLaughlin, Fast unsupervised Bayesian image segmentation with adaptive spatial regularisation, IEEE Trans. Image Process., 26 (2017), 2577–2587. https://doi.org/10.1109/TIP.2017.2675165 doi: 10.1109/TIP.2017.2675165
    [33] L. Gatys, A. S. Ecker, M. Bethge, Texture synthesis using convolutional neural networks, Adv. Neural Inf. Process. Syst., (2015), 262–270.
    [34] X. Snelgrove, High-resolution multi-scale neural texture synthesis, in SIGGRAPH Asia Technical Briefs, (2017), 1–4. https://doi.org/10.1145/3145749.3149449
    [35] Y. Zhou, Z. Zhu, X. Bai, D. Lischinski, D. Cohen-Or, H. Huang, Non-stationary texture synthesis by adversarial expansion, preprint, arXiv: 1805.04487.
    [36] V. Andrearczyk, P. F. Whelan, Using filter banks in convolutional neural networks for texture classification, Pattern Recognit. Lett., 84 (2016), 63–69. https://doi.org/10.1016/j.patrec.2016.08.016 doi: 10.1016/j.patrec.2016.08.016
    [37] X. Bu, Y. Wu, Z. Gao, Y. Jia, Deep convolutional network with locality and sparsity constraints for texture classification, Pattern Recognit., 91 (2019), 34–46. https://doi.org/10.1016/j.patcog.2019.02.003 doi: 10.1016/j.patcog.2019.02.003
    [38] M. Cimpoi, S. Maji, I. Kokkinos, A. Vedaldi, Deep filter banks for texture recognition, description, and segmentation, Int. J. Comput. Vis., 118 (2016), 65–94. https://doi.org/10.1007/s11263-015-0872-3 doi: 10.1007/s11263-015-0872-3
    [39] U. Dixit, A. Mishra, A. Shukla, R. Tiwari, Texture classification using convolutional neural network optimized with whale optimization algorithm, SN Appl. Sci., 1 (2019). https://doi.org/10.1007/s42452-019-0678-y doi: 10.1007/s42452-019-0678-y
    [40] T. Y. Lin, S. Maji, Visualizing and understanding deep texture representations, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, LasVegas, USA, (2016), 2791–2799.
    [41] L. Liu, J. Chen, G. Zhao, P. Fieguth, X. Chen, M. Pietikä inen, Texture classification in extreme scale variations using GANet, IEEE Trans. Image Process., 28 (2019), 3910–3922. https://doi.org/10.1109/TIP.2019.2903300 doi: 10.1109/TIP.2019.2903300
    [42] A. Shahriari, Parametric learning of texture filters by stacked fisher autoencoders, in Proceedings of the 2016 International Conference on Digital Image Computing: Technique and Applications (DICTA), Gold Coast, Australia, (2016), 1–8.
    [43] Y. Song, F. Zhang, Q. Li, H. Huang, L. J. O'Donnell, W. Cai, Locally-transferred fisher vectors for texture classification, in Proceedings of the IEEE International Conference on Computer Vision, Venice, Italy, (2017), 4912–4920.
    [44] H. Zhang, J. Xue, K. Dana, Deep ten: Texture encoding network, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Honolulu, HI, USA, (2017), 708–717.
    [45] V. Andrearczyk, P. F. Whelan, Texture segmentation with fully convolutional networks, preprint, arXiv: 1703.05230.
    [46] C. Karabag, J. Verhoeven, N. Miller, C. Reyes-Aldasoro, Texture segmentation: an objective comparison between five traditional algorithms and a deep-learning U-Net architecture, Appl. Sci., 9 (2019), 3900. https://doi.org/10.3390/app9183900 doi: 10.3390/app9183900
    [47] Y. Huang, F. Zhou, J. Gilles, Empirical curvelet based fully convolutional network for supervised texture image segmentation, Neurocomputing, 349 (2019), 31–43. https://doi.org/10.1016/j.neucom.2019.04.021 doi: 10.1016/j.neucom.2019.04.021
    [48] R. Yamada, H. Ide, N. Yudistira, T. Kurita, Texture segmentation using Siamese network and hierarchical region merging, in Proceedings of the 2018 24th International Conference on Pattern Recognition (ICPR), Beijing, China, (2018), 2735–2740. https://doi.org/10.1109/ICPR.2018.8545348
    [49] S. Mikes, M. Haindl, Texture segmentation benchmark, IEEE Trans. Pattern Anal. Mach. Intell., 41 (2021), 1–16. https://doi.org/10.1109/TPAMI.2021.3075916 doi: 10.1109/TPAMI.2021.3075916
    [50] A. Awad, Denoising images corrupted with impulse, Gaussian, or a mixture of impulse and Gaussian noise, Eng. Sci. Technol. Int. J., 22 (2019), 746–753. https://doi.org/10.1016/j.jestch.2019.01.012 doi: 10.1016/j.jestch.2019.01.012
    [51] R. C. Gonzalez, R. E. Woods, Digital Image Processing, Third Edition Pearson Education, 2009.
    [52] S. Walid, B. Xi, A neighborhood regression approach for removing multiple types of noises, EURASIP J. Image Video Process., 2018 (2018). https://doi.org/10.1186/s13640-018-0259-9
    [53] M. Alkhatib, A. Hafiane, Robust adaptive median binary pattern for noisy texture classification and retrieval, IEEE Trans. Image Process., 28 (2019), 5407–5418. https://doi.org/10.1109/TIP.2019.2916742 doi: 10.1109/TIP.2019.2916742
    [54] S. Dash, M. R. Senapati, Noise robust Law's filters based on fuzzy filters for texture classification, Egypt. Inf. J., 21 (2020), 37–49. https://doi.org/10.1016/j.eij.2019.10.003 doi: 10.1016/j.eij.2019.10.003
    [55] C. Vacar, J. Giovannelli, Unsupervised joint deconvolution and segmentation method for textured images:a Bayesian approach and an advanced sampling algorithm, EURASIP J. Image Video Process., 2019 (2019), 1–17. https://doi.org/10.1186/s13634-018-0597-x doi: 10.1186/s13634-018-0597-x
    [56] H. A. Nugroho, E. L. Frannita, I. Ardiyanto, L. Choridah, Computer aided diagnosis for thyroid cancer system based on internal and external characteristics, J. King Saud Univ. Comput. Inf. Sci., 33 (2021), 329–339. https://doi.org/10.1016/j.jksuci.2019.01.007 doi: 10.1016/j.jksuci.2019.01.007
    [57] A. Bhargava, A. Bansal, Fruits and vegetables quality evaluation using computer vision: A review, J. King Saud Univ. Comput. Inf. Sci., 33 (2021), 243–257. https://doi.org/10.1016/j.jksuci.2018.06.002 doi: 10.1016/j.jksuci.2018.06.002
    [58] S. Hossain, S. Serikawa, Texture databases—A comprehensive survey, Pattern Recognit. Lett., 34 (2013), 2007–2022. https://doi.org/10.1016/j.patrec.2013.02.009 doi: 10.1016/j.patrec.2013.02.009
    [59] K. Dabov, A. Foi, V. Katkovnik, K. Egiazarian, Color image denoising via sparse 3D collaborative filtering with grouping constraint in luminance-chrominance space, in Proceeding of IEEE International Conference on Image Processing, (2007), 313–316. https://doi.org/10.1109/ICIP.2007.4378954
    [60] K. Dabov, A. Foi, V. Katkovnik, K. Egiazarian, Image denoising by sparse 3D transform-domain collaborative filtering, IEEE Trans. Image process., 16 (2007), 2080–2095. https://doi.org/10.1109/TIP.2007.901238 doi: 10.1109/TIP.2007.901238
    [61] L. Fan, Z. Fan, H. Fan, C. Zhang, Brief review of image denoising techniques, Visual Comput. Ind., Biomed., Art, 2 (2019), 1–12. https://doi.org/10.1186/s42492-019-0016-7 doi: 10.1186/s42492-019-0016-7
    [62] S. Kinge, B. S. Rani, Mukul sutaone, quantitative restoration of noisy colour texture segmentation benchmark images using state-of-the-art algorithm, in 2020 4th International Conference on Intelligent Computing and Control Systems (ICICCS), IEEE, (2020), 37–42. https://doi.org/10.1109/ICICCS48265.2020.9120927
    [63] R. R. Nair, E. David, S. Rajagopal, A robust anisotropic diffusion filter with low arithmetic complexity for images, EURASIP J. Image Video Process., 2019 (2019), 1–14. https://doi.org/10.1186/s13640-019-0444-5 doi: 10.1186/s13640-019-0444-5
    [64] D. Tourtounis, N. Mitianoudis, G. C. Sirakoulis, Salt-n-pepper noise filtering using cellular automata, preprint, arXiv: 1708.05019.
    [65] Z. Haliche, K. Hammouche, O. Losson, L. Macaire, Fuzzy color aura matrices for texture image segmentation, J. Imaging, 8 (2022), 1–20. https://doi.org/10.3390/jimaging8090244 doi: 10.3390/jimaging8090244
    [66] C. Bontozoglou, P. Xiao, Applications of capacitive imaging in human skin texture and hair analysis, Appl. Sci., 10 (2020), 256. https://doi.org/10.3390/app10010256 doi: 10.3390/app10010256
    [67] Y. Liu, K. Xu, J. Xu, An improved MB-LBP defect recognition approach for the surface of steel plates, Appl. Sci., 9 (2020), 4222. https://doi.org/10.3390/app9204222 doi: 10.3390/app9204222
    [68] Z. Jeelani, F. Qadir, Cellular automata-based approach for salt-and-pepper noise filtration, J. King Saud Univ. Comput. Inf. Sci., 2018 (2018). https://doi.org/10.1016/j.jksuci.2018.12.006
    [69] S. Z. Li, Markov Random Field Modeling in Image Analysis, 2nd edition, Springer-Verlag, New York, 2009.
    [70] Z. Wang, A. C. Bovik, H. R. Sheikh, E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600–612. https://doi.org/10.1109/TIP.2003.819861 doi: 10.1109/TIP.2003.819861
    [71] A. Hoover, G. J. Baptiste, X. Jiang, P. J. Flynn, H. Bunke, D. B. Goldgof, et al., An experimental comparison of range image segmentation algorithms, IEEE Trans. Pattern Anal. Mach. Intell., 18 (1996), 673–689. https://doi.org/10.1109/34.506791 doi: 10.1109/34.506791
    [72] Z. Hu, Z. Wu, Q. Zhang, Q. Fan, J. Xu, A spatially-constrained color-texture model for hierarchical VHR image segmentation, IEEE Geosci. Remote Sens. Lett., 10 (2013), 120–124. https://doi.org/10.1109/LGRS.2012.2194693 doi: 10.1109/LGRS.2012.2194693
    [73] S. Sangwine, R. Horne, The Colour Image Processing Handbook, Chapman and Hall, 1998.
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