Research article Special Issues

Spatial general autoregressive model-based image interpolation accommodates arbitrary scale factors

  • Received: 30 July 2020 Accepted: 14 September 2020 Published: 28 September 2020
  • This paper proposed a novel image interpolation algorithm with an arbitrary upscaling factor based on the spatial general autoregressive model. First, to accommodate arbitrary scale factors, a non-integer mapping method was modulated into the spatial general autoregressive model, which was employed to model the piecewise stationary pattern with a higher description capacity than autoregressive models. A gradient angle guided extension method was utilized to extend the spatial general autoregressive model, and more pixels in the neighborhood were included to estimate the parameters of the spatial general autoregressive model. To realize the high-accuracy estimation of the model parameters, a regularization method via an elastic network was adopted to maintain the complexity of the object function in a reasonable state and address the overfitting problem. We also introduced an iterative curvature method to refine the interpolation result of those image blocks with large variances of gray intensities. Experiments on 25 images were conducted with integer and non-integer magnification factors to systematically verify the objective and subjective measures of the proposed method. The visual artifacts were effectively suppressed by the proposed method, and a flexible interpolation method for arbitrary scale factors was implemented.

    Citation: Yuntao Hu, Fei Hao, Chao Meng, Lili Sun, Dashuai Xu, Tianqi Zhang. Spatial general autoregressive model-based image interpolation accommodates arbitrary scale factors[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 6573-6600. doi: 10.3934/mbe.2020343

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  • This paper proposed a novel image interpolation algorithm with an arbitrary upscaling factor based on the spatial general autoregressive model. First, to accommodate arbitrary scale factors, a non-integer mapping method was modulated into the spatial general autoregressive model, which was employed to model the piecewise stationary pattern with a higher description capacity than autoregressive models. A gradient angle guided extension method was utilized to extend the spatial general autoregressive model, and more pixels in the neighborhood were included to estimate the parameters of the spatial general autoregressive model. To realize the high-accuracy estimation of the model parameters, a regularization method via an elastic network was adopted to maintain the complexity of the object function in a reasonable state and address the overfitting problem. We also introduced an iterative curvature method to refine the interpolation result of those image blocks with large variances of gray intensities. Experiments on 25 images were conducted with integer and non-integer magnification factors to systematically verify the objective and subjective measures of the proposed method. The visual artifacts were effectively suppressed by the proposed method, and a flexible interpolation method for arbitrary scale factors was implemented.


    We investigate a stability analysis of an explicit finite scheme to numerically solve the Allen-Cahn (AC) model with a high-order polynomial potential [1]:

    ϕ(x,t)t=Fα(ϕ(x,t))ϵ2+Δϕ(x,t),xΩ,t>0, (1.1)
    nϕ(x,t)=0,xΩ,t>0, (1.2)

    where ϕ(x,t) is the order parameter at two-dimensional space position x=(x,y) and time t, ϵ is positive, Fα(ϕ) is a double-well potential, and n is normal to the boundary Ω. We consider the following high-order polynomial free energy potential [1,2,3]:

    Fα(ϕ(x,t))=14α2((ϕ(x,t))2α1)2, (1.3)

    where α is a positive integer. Equation (1.1) can be rewritten as follows:

    ϕ(x,t)t=1αϵ2(ϕ(x,t))2α1((ϕ(x,t))2α1)+Δϕ(x,t). (1.4)

    We note that if α=1, then Eq (1.4) becomes the classical AC equation [4], which is a partial differential equation describing phase separation in materials, such as solidification or liquid-vapor transitions. It models the evolution of a scalar order parameter to minimize interfacial energy and has been used in volume repair [5], oil pollution dynamics [6], 3D volume reconstruction from slice data [7,8], shape transformation [9], data assimilation [10], image segmentation [11], structural topology optimization [12], mathematical physics for phase transition studies [13], and numerous applications related to interfacial evolution.

    Due to the fact that analytical solutions for the AC equation are often impractical or unattainable, numerical methods are necessary to obtain approximate solutions for the equation. Recently, many numerical methods for the AC equation have been proposed. Lee [14] introduced a linear convex splitting scheme and an energy-minimizing method on the mass-conserving space for solving the AC equation. Numerical simulations verified phase separation, dissipation of energy, and mass conservation. Zhang et al. [15] developed a class of up to third-order explicit structure-preserving schemes for solving two modified conservative AC equations. Furthermore, the authors developed and analyzed an unconditionally structural-preserving parametric single-step method of up to fourth-order accuracy for the conservative AC equation in [16]. Feng et al. [17] developed a maximum-principle preserving, unconditionally stable, and second-order FDM for the AC equation. Zhang et al. [18] developed the Runge-Kutta method for temporal integration and FDM for spatial discretization to propose high-order maximum principle-preserving schemes. Characteristics such as maximum principle preservation and energy dissipation were demonstrated through numerical experiments. Lee [19] presented a mathematical model and numerical method of area-minimizing surfaces using the AC equation. Deng and Weng [20] developed the barycentric interpolation collocation scheme based on the Crank-Nicolson method for the AC model. Park et al. [21] presented an unconditionally stable computational algorithm for the AC equation with logarithmic free energy. Zhang et al. [22] proposed and analyzed a class of up to eighth-order inequality-preserving two-step integrating factors using the Runge-Kutta method to solve the semilinear parabolic equation. Typically, fourth-order equations such as the Cahn-Hilliard (CH) type equations [23,24] require implicit numerical schemes for stability due to their severe time step restrictions with explicit schemes.

    However, the AC equation is a second-order equation, and its explicit numerical scheme requires only a moderate time step condition for stability. Furthermore, as demonstrated in a series of previous research papers [25], the fully explicit numerical schemes of the AC equation have performed well in terms of both efficiency and accuracy. The primary objective of this paper is to study the stability condition of an explicit FDM to solve the AC model with high-order polynomial potential.

    The contents of this article are as follows: In Section 2, a fully explicit computational method is described and stability analysis is conducted for the AC equation with the high-order polynomial free energy. In Section 3, numerical tests are performed to validate the derived theorem. In Section 4, we conclude.

    Now, we present a fully explicit numerical scheme of the AC equation with the high-order (higher than fourth) polynomial free energies and derive its temporal step restriction, which satisfies the maximum principle. Let Ω=(Lx,Ux)×(Ly,Uy) be the domain, and Ωh={(xi,yj)|xi=Lx+(i0.5)h,yj=Ly+(i0.5)h,1iNx,1jNy} be its discrete domain, where h is spatial step size and Nx and Ny are integers. Let ϕnij=ϕ(xi,yj,nΔt), where Δt is the time step size. We define the discrete maximum norm by ϕn=max1iNx,1jNy|ϕnij|. The AC model (1.1) is discretized by using an explicit method as follows:

    ϕn+1ijϕnijΔt=(ϕnij)2α1(1(ϕnij)2α)αϵ2+ϕni+1,j+ϕni1,j+ϕni,j+1+ϕni,j14ϕnijh2, (2.1)

    where we use a 5-point stencil for the discrete Laplacian operator. We can use the 9-stencil isotropic finite difference scheme [26] for the isotropic solutions. Here, we use the homogeneous Neumann boundary condition:

    ϕni0=ϕni1,ϕni,Ny+1=ϕniNy for i=1,,Nx and ϕn0j=ϕn1j,ϕnNx+1,j=ϕnNxj for j=1,,Ny.

    Now, we shall prove the following maximum principle theorem of the fully explicit scheme for the AC equation with high-order polynomial potentials.

    Theorem 1. Assume that the initial condition satisfies ϕ01. Then, the fully explicit numerical method (2.1) maintains the boundedness of the numerical solutions at any discrete time:

    ϕn+11,forn0,

    if the temporal step size satisfies

    Δtϵ2h22(h2+2ϵ2). (2.2)

    Proof. We have ϕ01 by the assumption. By mathematical induction, suppose that ϕn1. The explicit numerical method for the AC model (2.1) can be reformulated as follows:

    ϕn+1ij=ϕnij+Δt((ϕnij)2α1(1(ϕnij)2α)αϵ2+ϕni1,j+ϕni+1,j+ϕni,j1+ϕni,j+14ϕnijh2). (2.3)

    Let us first consider the following case:

    Case 1.

    (ϕnij)2α1(1(ϕnij)2α)αϵ2+ϕni1,j+ϕni+1,j+ϕni,j1+ϕni,j+14ϕnijh20. (2.4)

    In Eq (2.3), we have ϕn+1ij1. From Eq (2.3), we can derive the following inequality:

    ϕn+1ijϕnij+Δt((ϕnij)2α1(1(ϕnij)2α)αϵ2+44ϕnijh2). (2.5)

    We want to determine the condition for Δt such that

    ϕnij+Δt((ϕnij)2α1(1(ϕnij)2α)αϵ2+44ϕnijh2)1.

    Then, it can be reformulated as

    Δt(1ϕnij)((ϕnij)2α1(1+ϕnij++(ϕnij)2α1)αϵ2+4h2)1ϕnij. (2.6)

    If ϕnij=1, the time step is free from any constraints; and otherwise, i.e., ϕnij<1, by dividing both sides of (2.6) by (1ϕnij)>0, we have

    Δt((ϕnij)2α1(1+ϕnij++(ϕnij)2α1)αϵ2+4h2)1.

    Hence, Δt is subject to the following restriction:

    Δtαϵ2h2h2(ϕnij)2α1(1+ϕnij++(ϕnij)2α1)+4αϵ2.

    Here, because |ϕnij|1, we derive the following condition:

    Δtϵ2h22(h2+2ϵ2).

    Hence, when Δt0.5ϵ2h2/(h2+2ϵ2), we have ϕn+11. Next, let us consider the following case:

    Case 2.

    (ϕnij)2α1(1(ϕnij)2α)αϵ2+ϕni1,j+ϕni+1,j+ϕni,j1+ϕni,j+14ϕnijh2<0. (2.7)

    From Eq (2.3), ϕn+1ij1 always holds true. From Eq (2.3), we get the following inequality:

    ϕnij+Δt((ϕnij)2α1(1(ϕnij)2α)αϵ24+4ϕnijh2)ϕn+1ij. (2.8)

    From (2.8), to satisfy that 1ϕn+1ij, we consider

    1ϕnij+Δt(1+ϕnij)((ϕnij)2α1(1(ϕnij)α)(1ϕnij++(1)α1(ϕnij)α1)αϵ24h2). (2.9)

    It can be reformulated as

    (1+ϕnij)Δt(1+ϕnij)((ϕnij)2α1(1(ϕnij)α)(1ϕnij++(1)α1(ϕnij)α1)αϵ24h2). (2.10)

    If ϕnij=1, the time step is free from any constraints; and otherwise, i.e., ϕnij>1, by dividing both sides of Eq (2.10) by (1+ϕnij)<0, we have

    Δt(4h2(ϕnij)2α1(1(ϕnij)α)(1ϕnij++(1)α1(ϕnij)α1)αϵ2)1. (2.11)

    If 1ϕnij1, since 0(1(ϕnij)α)(1ϕnij++(1)α1(ϕnij)α1)2α, we obtain the following inequality as

    Δt(4h2+2ϵ2)1. (2.12)

    Thus, we derive the following condition:

    Δth2ϵ22(h2+2ϵ2). (2.13)

    Therefore, Δt0.5h2ϵ2/(h2+2ϵ2) results in ϕn+11.

    We note that the case of α=1 was proved in [27]. Here, we generalized the stability of the numerical methods of the AC model with high-order polynomial potentials for all α1 by mathematical induction. Additionally, this time step constraint can be evaluated by using Lemma 3.7 in [28].

    Next, we verify that the time step size condition derived from Theorem 1 is optimal; namely, the bound is the maximum time step size that satisfies the maximum principle. Let

    Δtmax=ϵ2h22(h2+2ϵ2)

    be the least upper bound of the time steps that satisfy the maximum principle for the AC equation. To verify Δtmax is optimal, let us assume Δt>Δtmax. Then, we have Δt>Δtmax, which implies 0<Δtmax/Δt<1. Let us consider the following polynomial function on (0,1):

    f(ψ)=4αϵ2ψ+h2ψ2α(1+ψ+ψ2++ψ2α1)2α(h2+2ϵ2),

    which is a strictly monotonically increasing function because f(ψ)>0 on (0,1). Because f(0)=0, f(1)=1, and f(ψ) is continuous, there exists a unique 0<ψ<1 satisfying Δtmax/Δt=f(ψ). Using this obtained value of ψ, let us consider an initial condition on Ωh as follows:

    ϕ(xi,yj,0)={ψ,ifi=p,j=q,1,otherwise,

    where 1<p<Nx and 1<q<Ny. From Eq (2.3), we have

    ϕ1pq=ϕ0pq+Δt((ϕ0pq)2α1(1(ϕ0pq)2α)αϵ2+ϕ0p+1,q+ϕ0p1,q+ϕ0p,q+1+ϕ0p,q14ϕ0pqh2)=ψ+Δtmaxf(ψ)(ψ2α1(1ψ2α)αϵ2+44ψh2)=ψ+αϵ2h24αϵ2ψ+h2ψ2α(1+ψ++ψ2α1)(ψ2α1(1ψ2α)αϵ2+44ψh2)=ψ+1ψ1>1,

    which implies that the maximum principle is not satisfied for any time step sizes larger than the maximum time step size Δtmax. Therefore, we can conclude that Δtmax is the optimal bound.

    In this section, we perform several characteristic computational tests to validate the maximum principle property, motion by mean curvature, and the effect of parameter α on the dynamics of the equation.

    Before conducting computational experiments, we investigate the accuracy of the numerical scheme. Consider the following initial condition:

    ϕ(x,y,0)=cos(2πx)cos(2πy) (3.1)

    in the computational domain Ω=(1,1)×(1,1). The parameters used to investigate spatial accuracy are ϵ=1/32, Δt=ϵ2(1/512)2/(2((1/512)2+2ϵ2)), and T=2000Δt. We use the relative error. Table 1 shows the spatial error and convergence rate. When h=1/32, the error is the relative error between using h=1/32 and h=1/16.

    Table 1.  The spatial error and convergence rate.
    h=1/32 h=1/64 h=1/128 h=1/256
    m=1 error 6.94631e-03 1.74150e-03 4.35750e-04 1.08960e-04
    rate 1.996 1.999 2.000
    m=2 error 5.26517e-03 1.31358e-03 3.28425e-04 8.21080e-05
    rate 2.003 2.000 2.000
    m=3 error 4.95124e-03 1.24355e-03 3.10964e-04 7.77460e-05
    rate 1.993 2.000 2.000

     | Show Table
    DownLoad: CSV

    The temporal accuracy is also investigated using the same initial condition (3.1). The other parameters are used as follows: Nx=Ny=210, ϵ=h, Δtmax=ϵ2h2/(2(h2+2ϵ2)), and T=100Δtmax. Table 2 provides the temporal error and convergence rate.

    Table 2.  The temporal error and convergence rate.
    Δt=Δtmax Δt=Δtmax/2 Δt=Δtmax/4 Δt=Δtmax/8
    m=1 error 3.03760e-08 1.70626e-08 8.95919e-09 4.58087e-09
    rate 0.832 0.929 0.968
    m=2 error 3.47939e-03 1.75910e-03 8.84523e-04 4.43520e-04
    rate 0.984 0.992 0.996
    m=3 error 1.57539e-03 7.93956e-04 3.98577e-04 1.99693e-04
    rate 0.989 0.994 0.997

     | Show Table
    DownLoad: CSV

    Consequently, it was found that the method has second-order accuracy in space and first-order accuracy in time.

    We consider the maximum principle of the fully explicit method with Δtmax. The random initial condition is given as follows:

    ϕ(x,y,0)=0.5rand(x,y),(x,y)Ω=(1,1)×(1,1),

    where rand(x,y) is a random value between 1 and 1. Here, the parameters used are α=1, h=0.02, ϵ=h, and a final time T=2000Δtmax. Figure 1 displays the temporal progressions of the highest and lowest values of ϕ, using time step sizes Δt=Δtmax and Δt=1.21Δtmax. When employing a time step greater than Δtmax, it is confirmed that the ϕ value exceeds the bounds of 1 and 1.

    Figure 1.  Temporal progressions of the highest and lowest values of ϕn using Δt=Δtmax and Δt=1.21Δtmax.

    Motion by mean curvature in the AC equation means the evolution of interfaces between different phases. It describes how the interface between two distinct phases evolves over time due to the minimization of the total interface energy. The motion is governed by the mean curvature of the interface, which tends to smooth out irregularities and minimize the interface area. This phenomenon plays a very important role in various physical processes, including phase separation, pattern formation, and material microstructure evolution. We consider the motion by mean curvature using the fully explicit scheme of the AC equation with a high-order polynomial potential. The initial condition on the computational domain Ω=(1,1)×(1,1) is defined as

    ϕ(x,y,0)=tanh(rx2+y22ϵ).

    We used the following parameters: Nx=Ny=128, α=3, r=0.5, ϵ=0.013, Δt=0.5h2ϵ2/(h2+2ϵ2), and Nt=3400. The analytic radius is defined as R(t)=r22t. Figure 2 shows the temporal evolution of the theoretical radius R(t) and numerical radius with the zero-level filled contours of the numerical solutions at t=560Δt, 1600Δt, and 3100Δt. From the computational experiment, it becomes evident that the numerical results demonstrate a strong correspondence with the theoretical radius, indicating a high level of agreement between the two results.

    Figure 2.  Temporal evolution of the radius R(t) and numerical radius with the zero-level filled contours of the numerical solutions at t=560Δt, 1600Δt, and 3100Δt.

    Next, we consider the effect of α on the dynamics of the phase-field of separated four squares as an initial condition on the computational domain ϕ=(0,2π)×(0,2π) with the following numerical parameters: Nx=Ny=256, h=2π/256, and Δt=2.0×105. The initial condition is given as follows:

    ϕ(x,y,0)={1,if (x,y)(15h,140h)×(15h,120h)(15h,78h)×(132h,237h)(152h,237h)×(15h,120h)(90h,237h)×(132h,237h),1,otherwise.

    Figures 3(a),(b) show the temporal evolution of the zero-level filled contour representing the numerical solution with α=1 and α=10, respectively. We note that when α=1, the proposed model reduces to the conventional AC equation with fourth-order polynomial free energy. In Figure 3(a), the separated four squares merge into one structure. Conversely, for α=10, the four squares shrink individually without merging, as shown in Figure 3(b). Here, ϵ=0.0512 is used for α=1, and ϵ=0.0085 is used for α=10.

    Figure 3.  Temporal evolutions of the zero-level filled contour represent the numerical solution with (a) α=1 and (b) α=10, where filled regions indicate positive values.

    In this paper, we presented a detailed stability analysis of the fully explicit finite difference scheme to solve the AC equation with a high-order polynomial free energy, which was recently proposed to preserve a more intricate structure of interfaces. To demonstrate the accuracy and optimal estimation of our stability limitations, we conducted a series of numerical experiments using the derived time step formulas that ensure the maximum principle. The obtained time step estimation, which guarantees stability, can be useful for applying the AC equation in various practical applications, such as data classification, image inpainting, and image segmentation. In data classification, for instance, the equation's stability ensures robust and accurate categorization of complex datasets. In the context of image inpainting, where missing or damaged portions of images are reconstructed, the stability provided by our estimation enhances the fidelity of the inpainting process. Similarly, in image segmentation tasks aimed at partitioning images into meaningful regions, the reliability of the AC equation enables precise description of boundaries. In conclusion, the stability contributions outlined in this paper not only advance the theoretical understanding of stability in finite difference schemes but also provide the methodology for practical implementations across a range of applications.

    The following MATLAB code is used for the numerical simulation as shown in Figure 2 and can be accessed from the corresponding author's webpage: https://mathematicians.korea.ac.kr/cfdkim/open-source-codes/.

    Nx = 128; xL = -1; xR = 1; h = (xR-xL)/Nx; yL = -1; yR = 1; Ny = round((yR-yL)/h);
    x = linspace(xL-0.5*h,xR+0.5*h,Nx+2); y = linspace(yL-0.5*h,yR+0.5*h,Ny+2);
    alpha = 3; ep = 0.013; ep2 = alpha*ep^2; dt = h^2*ep^2/(2*(h^2+2*ep^2)); Nt = 3400;
    r = 0.5; phi = tanh((r-sqrt(x'.^2+y.^2))/(sqrt(2)*ep));
    for it = 1:Nt
    phi(1,:) = phi(2,:); phi(end,:) = phi(end-1,:);
    phi(:,1) = phi(:,2); phi(:,end) = phi(:,end-1);
    phi(2:end-1,2:end-1) = phi(2:end-1,2:end-1) ...
          +dt*(phi(2:end-1,2:end-1).^(2*alpha-1).*(1-phi(2:end-1,2:end-1).^(2*alpha))/ep2 ...
          +(phi(3:end,2:end-1)+phi(1:end-2,2:end-1)+phi(2:end-1,3:end)+phi(2:end-1,1:end-2) ...
          -4*phi(2:end-1,2:end-1))/h^2);
    analytic(it) = sqrt(r^2-2*dt*it);
    figure(1); clf; hold on; box on; grid on
    [c d] = contourf(x,y,phi,[0 0],'facecolor',[0.8500 0.3250 0.0980]);
    axis image; axis([-0.5 0.5 -0.5 0.5]); drawnow;
    radius(it) = mean(sqrt(sum(c(:,2:end).^2)));
    end
    figure(2); clf; hold on; box on;
    plot([0:Nt]*dt, [r real(analytic)],'MarkerIndices',[1:200:3400])
    plot([0:Nt]*dt, [r radius],'o-','MarkerIndices',[100:200:3400]);
    axis([0 dt*Nt 0 0.5]); leg = legend('Analytic radius','Numerical radius');

    Jaeyong Choi: Formal Analysis, Validation, Investigation, Writing-original draft, Writing-review & editing; Seokjun Ham: Software; Formal Analysis, Writing-original draft, Writing-review & editing; Soobin Kwak: Data curation, Software, Visualization, Writing-original draft; YoungJin Hwang: Formal Analysis, validation, Writing-original draft, Writing-review & editing; Junseok kim: Conceptualization, Supervision, Validation, Methodology, Writing-original draft, Writing-review & editing. All authors have read and approved the final version of the manuscript for publication.

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

    This work was supported by BK21 FOUR.

    The authors are grateful to the reviewers for their constructive feedback on this article.

    The authors declare that there is no conflict of interest.



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