Cyclic Schemes for PDE-Based Image Analysis

Abstract

The simplest scheme for parabolic, diffusion-like partial differential equations (PDEs) in image analysis is given by the explicit finite difference discretisation. If a fixed time step size is used, this scheme is known to be inefficient due to a severe stability restriction. In this paper we show that a slight modication can boost the efficiency of the explicit scheme by several orders of magnitude. All one has to do in this novel Fast Explicit Diffusion (FED) scheme is to replace the originally fixed time step size by cycles of varying step sizes. We derive these cycles from a factorisation of symmetric smoothing filters, and we show that a box filter gives a favourable compromise between high efficiency and good smoothing properties. However, such ideas are not restricted to parabolic problems only. They can also be adapted for solving elliptic PDEs that arise e.g. as Euler-Lagrange equations in variational image analysis. Also in this context, we propose a modification of the simplest iterative solver, namely the Jacobi method. This time, we introduce cycles of varying relaxation parameters that can be linked to our FED time step sizes. As a result, we obtain highly efficient methods for the elliptic case, so-called Fast-Jacobi (FJ) schemes. Our novel FED and FJ schemes are applicable to a large range of PDEbased image analysis problems and can beat classical methods such as additive operator splittings and (semi-)implicit schemes. Applications include isotropic nonlinear diffusion filters with widely varying diffusivities as well as anisotropic diffusion methods for image filtering, inpainting and regularisation in computer vision. Moreover, they are equally suited for higher dimensional problems as well as higher order PDEs. Implementations are extremely simple, since the underlying explicit scheme or Jacobi method can be used as a black box solver. Last but not least, these cyclic schemes are perfectly suited for modern parallel architectures such as GPUs. We also provide a public domain code package that allows users to experiment with our cyclic methods.

Accelerated Variational PDEs for Efficient Solution of Regularized Inversion Problems

“…heavy-ball) in that the next increment ∆u n is expressed as a weighted combination of the gradient ∇E n and the previous increment ∆u n−1 . Recursion (28) is equivalent to the first order in time, explicit update (23) using forward differences while the recursion (29) is equivalent to the second order in time, explicit update (21) using central differences, and as such they must adhere to the same CFL conditions (24) and (22) derived earlier for these corresponding schemes.…”

Turning Diffusion-Based Image Colorization Into Efficient Color Compression

“…For our implementation, we use the finite difference framework of Weickert et al [42]. We consider the evolution of the parabolic PDEs from the previous section and solve the inpainting problem iteratively with fast explicit diffusion (FED) [43] combined with a coarse-to-fine initialization. We stop the iterative scheme as soon as the norm of the residual has decreased by a factor 10 −5 .…”

Reconstructing the microstructure of polyimide–silicalite mixed‐matrix membranes and their particle connectivity using FIB‐SEM tomography

“…This field of image filtering has developed from relatively simple schemes with a variable scalar diffusivity (Perona-Malik's filter; Perona & Malik, 1990) to sophisticated concepts based on a diffusion tensor, e.g. edge-enhancing diffusion and coherenceenhancing diffusion, and on advanced numerical algorithms for integration in space and time (Weickert, 1997;Weickert et al, 2015). Since enhancing the phase interface was essential for reliable separation of silicalite-1 particles, we preferred the edge-enhancing anisotropic diffusion in 3D space to the other methods.…”

“…For the rectangular cuboid image domain associated with a cubic grid, the boundary conditions were implemented by mirroring the external voxels of the image. The numerical integration of (2) in the time interval [0, τ ] involved a locally semianalytic scheme for space discretization and a fast explicit diffusion method for controlling time steps (Welk et al, 2008;Grewenig et al, 2010;Weickert et al, 2015).…”