Abstract
We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the $H^{-1}$-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation.
Highlights
In this paper we propose directional operator splitting methods for the numerical solution of fourth–order nonlinear partial differential equations of the type (1.1)ut = ∇ · (h(u)∇q) q ∈ ∂E(u) in Ω × (0, ∞), u(t = 0) = u0 in Ω, alternating directional splitting (ADI) schemes for a fourth-order nonlinear PDE where E is the total variation (TV) seminorm (1.2)E(u) := |Du|(Ω) = sup u∇ · p dx.p∈C0∞(Ω;R2), p ≤1 Ω and ∂Eu is the subdifferential of E in u
Explicit numerical schemes solving TV gradient flows turn out to show restrictive stability conditions related to the strength of the nonlinearity in the TV subgradient, cf. [13, 19]
In what follows we provide numerical discussion for applying the directional splitting schemes introduced in Section 4 for the numerical solution of the regularised TV-H−1 equation (4.6)
Summary
In this paper we propose directional operator splitting methods for the numerical solution of fourth–order nonlinear partial differential equations of the type (1.1). Ut = ∇ · (h(u)∇q) q ∈ ∂E(u) in Ω × (0, ∞), u(t = 0) = u0 in Ω, ADI schemes for a fourth-order nonlinear PDE where E is the total variation (TV) seminorm (see, for instance, [1]) (1.2). P∈C0∞(Ω;R2), p ≤1 Ω and ∂Eu is the subdifferential of E in u (see [1]). Ω ⊂ R2 is open and bounded with Lipschitz boundary, h : R → R, and u0 a sufficiently regular initial condition. In our considerations equation (1.1) is endowed with periodic boundary conditions. The elements q of the subdifferential ∂E have the property that, if q ∈ ∂E(u), (see [52, Proposition 4.1]): (1.3) q= −∇ · ∇u |∇u|.
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