| 1. |
- Baravdish, George, et al.
(författare)
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Extension of p-Laplace Operator for Image Denoising
- 2016
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Ingår i: 27th IFIP TC 7 Conference, CSMO 2015, Sophia Antipolis, France, June 29 - July 3, 2015, Revised Selected Papers. - Cham : Springer. - 9783319557946 - 9783319557953 ; , s. 107-116, s. 107-116
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Bokkapitel (refereegranskat)abstract
- In this work we introduce a novel operator $$\displaystyle \varDelta _(p,q)$$ as an extended family of operators that generalize the p-Laplace operator. The operator is derived with an emphasis on image processing applications, and particularly, with a focus on image denoising applications. We propose a non-linear transition function, coupling p and q, which yields a non-linear filtering scheme analogous to adaptive spatially dependent total variation and linear filtering. Well-posedness of the final parabolic PDE is established via pertubation theory and connection to classical results in functional analysis. Numerical results demonstrates the applicability of the novel operator $$\displaystyle \varDelta _(p,q)$$ .
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| 2. |
- Baravdish, George, 1964-, et al.
(författare)
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Generalizations of p-Laplace operator for image enhancement : Part 2
- 2020
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Ingår i: Communications on Pure and Applied Analysis. - Springfield, MO, United States : American Institute of Mathematical Sciences. - 1534-0392 .- 1553-5258. ; 19:7, s. 3477-3500
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Tidskriftsartikel (refereegranskat)abstract
- We have in a previous study introduced a novel elliptic operator Delta((p,q))u = vertical bar del u vertical bar(q) Delta(1)u + (p - 1) vertical bar del u vertical bar(p-2)Delta(infinity) u, p >= 1, q >= 0, as a generalization of the p-Laplace operator. In this paper, we establish the well-posedness of the parabolic equation u(t) =vertical bar del u vertical bar(1-q) Delta((1+q,q)), where q = q(vertical bar del u vertical bar) is continuous and has range in [0, 1], in the framework of viscosity solutions. We prove the consistency and convergence of the numerical scheme of finite differences of this parabolic equation. Numerical simulations shows the advantage of this operator applied to image enhancement.
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| 3. |
- Baravdish, George, et al.
(författare)
-
On Backward p(x)-Parabolic Equations for Image Enhancement
- 2015
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Ingår i: Numerical Functional Analysis and Optimization. - : Taylor & Francis. - 0163-0563 .- 1532-2467. ; 36:2, s. 147-168
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Tidskriftsartikel (refereegranskat)abstract
- In this study, we investigate the backward p(x)-parabolic equation as a new methodology to enhance images. We propose a novel iterative regularization procedure for the backward p(x)-parabolic equation based on the nonlinear Landweber method for inverse problems. The proposed scheme can also be extended to the family of iterative regularization methods involving the nonlinear Landweber method. We also investigate the connection between the variable exponent p(x) in the proposed energy functional and the diffusivity function in the corresponding Euler-Lagrange equation. It is well known that the forward problems converges to a constant solution destroying the image. The purpose of the approach of the backward problems is twofold. First, solving the backward problem by a sequence of forward problems we obtain a smooth image which is denoised. Second, by choosing the initial data properly we try to reduce the blurriness of the image. The numerical results for denoising appear to give improvement over standard methods as shown by preliminary results.
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