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# (a) Original input image to be segmented; (b) K-means segmentation results; (c) Chan-Vese segmentation results; (d-f) Our model segmentation results after 1-3 iterations. respectively.

Source publication

In this paper we are addressing two main topics, as follows. First, a rigorous qualitative study is elaborated for a second-order parabolic problem, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction, as well as non-homogeneous Cauchy-Neumann boundary conditions. Under certain assumptions on the input data: f(t,x), w(t,x) an...

## Context in source publication

**Context 1**

... after one iteration, our segmentation is superior to K-means and Chan-Vese as the real galaxy boundaries are correctly identified in Figure 3d. Figure 4 (virus microscopy) brings together noise, blur and irregular boundaries. Again, after two iterations, the model successfully identifies all objects of interest and the results, starting with the first iteration, are better than the compared K-means method. ...

## Citations

... The secondary input feed is the result of mathematically segmenting the original input image. Our mathematical model, introduced in [16], is successfully removing most part of the noise in input images while still preserving a good approximation for the edges on the objects of interest. The model is based on the non-local form of reaction-diffusion equation: ...

... Following the results in [16] the following PDE scheme with non-homogeneous Cauchy-Neumann boundary conditions is proposed: ...

... The nonlinear anisotropic reaction-diffusion model (2) is well-posed, as it was proved in [16]. Consequently, it admits an unique classical solution I(t, x) ∈ W 1,2 p (Q), that represents the segmentation result applied on input image I(0, x) = I 0 (x). ...

In our current paper we are introducing a new method to enhance semantic image segmentation accuracy of a U-Net neural network model by integrating it with a mathematical model based on reaction-diffusion equations.
The methods currently used for semantic image segmentation, including U-Net neural networks, are processing images as blocks of pixels in which the boundaries, the colors and patterns are all mixed together as inputs to the transformations that take place inside the layers of the convolutional neural networks. In our method we are modifying the architecture of a U-Net network and introduce a new data input feed in parallel to the image feed that needs to be segmented. The new input feed is mathematically extracted from the input image and contains the edges (shape) information of the image to be processed. The new input feed it's used during the U-Net decoding phase in order to help shape more precisely the up-scaled output edges, thus leading to improved accuracy performance of the network.
Introducing the parallel feed shows an improvement of accuracy metrics up to 4% (if compared to the U-Net model) and has a limited impact on computational resources consumed at training, because we are only adding a small number of new parameters to be calculated.

... Particularly, the study of the solutions to the nonlocal problem (1) (or (2)-(4)), using the kernel Φ S1 (z) = 1 ε 3 Φ S1 z ε , can be found in [1], [2], [7], [13], [17], [33], [34], [37] and the numerical approximations in [4], [5], [11], [12], [13], [32]- [34] and [38]. The problem (1) (or (2)-(4)) is a nonlocal one due to the diffusion of the density S(t, x) that depends on all values of S through the convolution-like term (Φ S1 * S)(t, x) = Ω Φ S1 (x−y)S(t, y)dy (see [3] or [8]). ...

... As we can see, the local problem (9) (or (10)) is a particular case of the Allen-Cahn equation [21], considered in different modeling moving interface problems, such as the nucleation of solids, the vesicle membranes or the mixture of two incompressible fluids. For the existence, estimate, uniqueness and regularity of a solution to (9) (or (10)) in Sobolev spaces W 1,2 p (Q) (with various types of boundary conditions), we refer to [8], [9], [12]- [14], [22], [23] and [33]. Estimations by some different higher-order time-stepping schemes of problem (9) are given in [3], [6], [15], [20], [23]- [32] and [35]. ...

... The results of this paper may be applied in the quantitative study of different formulations of the system (1)-(5) in terms of mobility functions, boundary conditions (see [9], [19], [21]- [26], [33]) as well as in the analysis of distributed and/or boundary optimal control problems governed by such a second-order boundary value problems (see [36] and references therein). ...

In our current paper we are following the results obtained by Pavăl et al. in [36] and study a nonlocal form of the system they propose. First we are performing a qualitative analysis for the equivalent non-local second-order system of coupled PDEs, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction. As in [36] our PDEs system is also implementing a SEIRD (Susceptible, Exposed, Infected, Recovered, Deceased) epidemic model. In order to be able to compare with the before mentioned results we use the same hypothesis on the input data: \begin{document}$ S_0(x) $\end{document}, \begin{document}$ E_0(x) $\end{document}, \begin{document}$ I_0(x) $\end{document}, \begin{document}$ R_0(x) $\end{document}, \begin{document}$ D_0(x) $\end{document}, \begin{document}$ f(t,x) $\end{document} and \begin{document}$ w_{_i}(t,x), i = 1,2,3,4,5 $\end{document}, and we prove the well-posedness of a classical solution in \begin{document}$ C((0,T],C(\Omega)) $\end{document}, extending the types already proven by other authors.
Secondly we construct the implicit-explicit (IMEX) numerical approximation scheme which allows to compute the solution of the system of coupled PDEs. The results are then compared with the ones obtained by [36].

... The equation (1) is analogous to the local reaction-diffusion problem with inhomogeneous Neumann boundary conditions (see [1], [3], [5]- [8], [24]- [25]): ...

... The study of the solutions of the non-v by the term (E 1 * v)(t, x) = Ω E 1 (x − y)v(t, y)dy. The second integral of (1) involves the given flux of individuals that enter or leave the set Ω by the sign of f (t, x), (t, x) ∈ (0, T ] × ∂Ω (the in-homogeneous Neumann boundary conditions) and the nonlinear term v(t, x) − v 3 (t, x) is the reaction term (see [5], [6], [7], [8], [10]- [13], [17]- [18], [21], [24] and [25]). Thus, the density v(t, x) verifies the equation (1) without any internal or external sources. ...

... (see [1], [3], [4], [24], [25]). For the numerical tests we set h = 0.01, I = J = 300 (i.e. ...

In this paper we study the existence and uniqueness of the solution in \begin{document}$ C((0, T], L^\infty(\Omega)) $\end{document} of a new nonlocal and nonlinear second-order anisotropic reaction-diffusion problem with in-homogeneous Neumann boundary conditions, generalizing other problems in the literature. Then, by using the finite difference method, we propose an explicit in time numerical scheme to approximate the unique solution of our problem. We also present some numerical simulations that come to show the performance of our theoretical model.

... For more general assumptions (with various types of boundary conditions), equation (6) has been numerically investigated in e.g., [18], [19] or [23]. For the existence, estimate, uniqueness and regularity of a solution in Sobolev spaces W 1,2 p (Q) we refer to [12], [13] and [19] . ...

... For more general assumptions (with various types of boundary conditions), equation (6) has been numerically investigated in e.g., [18], [19] or [23]. For the existence, estimate, uniqueness and regularity of a solution in Sobolev spaces W 1,2 p (Q) we refer to [12], [13] and [19] . ...

... 3. Existence and uniqueness of the solution to the problem (6). In this section we adapt some techniques from [12], [13] and [19] (and references therein) to our problem with the non-constant mobility in a particular form, M = M (t, x, u(t, x)). Let us write the problem (6) in an equivalent form ...

Here we investigate some reaction-diffusion models, with local or nonlocal diffusion and nonlinear reaction term. The boundary conditions are in both cases of Robin type. We study the existence and uniqueness of the solutions of our problems in appropriate spaces. Using finite difference method, we derive explicit numerical schemes in order to approximate the solutions of the studied models. Numerical experiments that illustrate the effectiveness of the theoretical results are provided. Some conclusions are given as well as new directions in order to extend the results and methods presented in this paper.

... A similar interpretation can be made for equations (2)-(4) (for more details, see [4], [10], [11], [14], [15], [23] and [34]). ...

... Using an implicit-explicit (IMEX) scheme, which treats the nonlinearity by partial lagging (see [34]), and making use of (24), the left term in (25) is approximated ...

... We use the Leray-Schauder principle to prove the existence and uniqueness results and the L p theory to obtain regularity properties of the solution. Moreover, the a priori estimates are done in L p (Q), implying a better estimates of S(t, x), E(t, x), I(t, x), R(t, x), D(t, x), (t, x) ∈ Q (see [4,6,10,11,14,15], [20]- [23], [27,31,34]). ...

The paper is concerned with two main topics, as follows. In the first instance, a serious qualitative analysis is performed for a second-order system of coupled PDEs, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction, as well as in-homogeneous Neumann boundary conditions. The PDEs system is implementing a SEIRD (Susceptible, Exposed, Infected, Recovered, Deceased) epidemic model. Under certain hypothesis on the input data: \begin{document}$ S_0(x) $\end{document}, \begin{document}$ E_0(x) $\end{document}, \begin{document}$ I_0(x) $\end{document} \begin{document}$ R_0(x) $\end{document}, \begin{document}$ D_0(x) $\end{document}, \begin{document}$ f(t,x) $\end{document} and \begin{document}$ w_{_i}(t,x), i = 1,2,3,4,5 $\end{document}, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space \begin{document}$ W^{1,2}_p(Q) $\end{document}, extending the types already proven by other authors. The nonlinear second-order anisotropic reaction-diffusion model considered here is then particularized to monitor the spread of an epidemic infection.
The second topic refers to the construction of the IMEX numerical approximation schemes in order to compute the solution of the system of coupled PDEs. We also show several simulation examples highlighting the present model's capabilities.

... G(t, x, z 1 , z 2 ) ≤ b 4 (1 + |z 1 | 2(r −1) + |z 2 | 2(r −1) ), ∀(t, x) ∈ Σ, z 1 , z 2 ∈ IR, for a constant b 4 > 0 and r ≥ 1 such that (see (26) ...

... In this regard, as applications of problem (1)-(3), we indicate the moving interface problems, e.g. phase separation and transition (see [2]- [6], [8]- [11], [15]- [26], [28], [29]), anisotropy effects (see [2], [9], [10], [13], [18], [26]), image denoising and segmentation (see [1], [11], [26] and references therein) etc. In addition, the general hypotheses formulated on g k , k = 1, 2, also allows to take in the dynamic boundary conditions a nonlinearity with a larger growth exponent r ≤ (n + 2)/(n + 2 − 2p) if n + 2 > 2p (see (5)), for each unknown functions u and ϕ. ...

... In this regard, as applications of problem (1)-(3), we indicate the moving interface problems, e.g. phase separation and transition (see [2]- [6], [8]- [11], [15]- [26], [28], [29]), anisotropy effects (see [2], [9], [10], [13], [18], [26]), image denoising and segmentation (see [1], [11], [26] and references therein) etc. In addition, the general hypotheses formulated on g k , k = 1, 2, also allows to take in the dynamic boundary conditions a nonlinearity with a larger growth exponent r ≤ (n + 2)/(n + 2 − 2p) if n + 2 > 2p (see (5)), for each unknown functions u and ϕ. ...