Colour image segmentation by the vector-valued Allen-Cahn phase-field model: a multigrid solution

We propose a new method for the numerical solution of a PDE-driven model for colour image segmentation and give numerical examples of the results. The method combines the vector-valued Allen-Cahn phase field equation with initial data fitting terms. This method is known to be closely related to the Mumford-Shah problem and the level set segmentation by Chan and Vese. Our numerical solution is performed using a multigrid splitting of a finite element space, thereby producing an efficient and robust method for the segmentation of large images.Comment: 17 pages, 9 figure

A New Implementation of the Magnetohydrodynamics-Relaxation Method for Nonlinear Force-Free Field Extrapolation in the Solar Corona

Magnetic field in the solar corona is usually extrapolated from photospheric vector magnetogram using a nonlinear force-free field (NLFFF) model. NLFFF extrapolation needs a considerable effort to be devoted for its numerical realization. In this paper we present a new implementation of the magnetohydrodynamics (MHD)-relaxation method for NLFFF extrapolation. The magneto-frictional approach which is introduced for speeding the relaxation of the MHD system is novelly realized by the spacetime conservation-element and solution-element (CESE) scheme. A magnetic field splitting method is used to further improve the computational accuracy. The bottom boundary condition is prescribed by changing the transverse field incrementally to match the magnetogram, and all other artificial boundaries of the computational box are simply fixed. We examine the code by two types of NLFFF benchmark tests, the Low & Lou (1990) semi-analytic force-free solutions and a more realistic solar-like case constructed by van Ballegooijen et al. (2007). The results show that our implementation are successful and versatile for extrapolations of either the relatively simple cases or the rather complex cases which need significant rebuilding of the magnetic topology, e.g., a flux rope. We also compute a suite of metrics to quantitatively analyze the results and demonstrate that the performance of our code in extrapolation accuracy basically reaches the same level of the present best-performing code, e.g., that developed by Wiegelmann (2004).Comment: Accept by ApJ, 45 pages, 13 figure

A Variational Stereo Method for the Three-Dimensional Reconstruction of Ocean Waves

We develop a novel remote sensing technique for the observation of waves on the ocean surface. Our method infers the 3-D waveform and radiance of oceanic sea states via a variational stereo imagery formulation. In this setting, the shape and radiance of the wave surface are given by minimizers of a composite energy functional that combines a photometric matching term along with regularization terms involving the smoothness of the unknowns. The desired ocean surface shape and radiance are the solution of a system of coupled partial differential equations derived from the optimality conditions of the energy functional. The proposed method is naturally extended to study the spatiotemporal dynamics of ocean waves and applied to three sets of stereo video data. Statistical and spectral analysis are carried out. Our results provide evidence that the observed omnidirectional wavenumber spectrum S(k) decays as k-2.5 is in agreement with Zakharov's theory (1999). Furthermore, the 3-D spectrum of the reconstructed wave surface is exploited to estimate wave dispersion and currents

Afivo: a framework for quadtree/octree AMR with shared-memory parallelization and geometric multigrid methods

Afivo is a framework for simulations with adaptive mesh refinement (AMR) on quadtree (2D) and octree (3D) grids. The framework comes with a geometric multigrid solver, shared-memory (OpenMP) parallelism and it supports output in Silo and VTK file formats. Afivo can be used to efficiently simulate AMR problems with up to about $10^{8}$ unknowns on desktops, workstations or single compute nodes. For larger problems, existing distributed-memory frameworks are better suited. The framework has no built-in functionality for specific physics applications, so users have to implement their own numerical methods. The included multigrid solver can be used to efficiently solve elliptic partial differential equations such as Poisson's equation. Afivo's design was kept simple, which in combination with the shared-memory parallelism facilitates modification and experimentation with AMR algorithms. The framework was already used to perform 3D simulations of streamer discharges, which required tens of millions of cells

Fast Solvers for Cahn-Hilliard Inpainting

We consider the efficient solution of the modified Cahn-Hilliard equation for binary image inpainting using convexity splitting, which allows an unconditionally gradient stable time-discretization scheme. We look at a double-well as well as a double obstacle potential. For the latter we get a nonlinear system for which we apply a semi-smooth Newton method combined with a Moreau-Yosida regularization technique. At the heart of both methods lies the solution of large and sparse linear systems. We introduce and study block-triangular preconditioners using an efficient and easy to apply Schur complement approximation. Numerical results indicate that our preconditioners work very well for both problems and show that qualitatively better results can be obtained using the double obstacle potential

A discrete graph Laplacian for signal processing

In this thesis we exploit diffusion processes on graphs to effect two fundamental problems of image processing: denoising and segmentation. We treat these two low-level vision problems on the pixel-wise level under a unified framework: a graph embedding. Using this framework opens us up to the possibilities of exploiting recently introduced algorithms from the semi-supervised machine learning literature. We contribute two novel edge-preserving smoothing algorithms to the literature. Furthermore we apply these edge-preserving smoothing algorithms to some computational photography tasks. Many recent computational photography tasks require the decomposition of an image into a smooth base layer containing large scale intensity variations and a residual layer capturing fine details. Edge-preserving smoothing is the main computational mechanism in producing these multi-scale image representations. We, in effect, introduce a new approach to edge-preserving multi-scale image decompositions. Where as prior approaches such as the Bilateral filter and weighted-least squares methods require multiple parameters to tune the response of the filters our method only requires one. This parameter can be interpreted as a scale parameter. We demonstrate the utility of our approach by applying the method to computational photography tasks that utilise multi-scale image decompositions. With minimal modification to these edge-preserving smoothing algorithms we show that we can extend them to produce interactive image segmentation. As a result the operations of segmentation and denoising are conducted under a unified framework. Moreover we discuss how our method is related to region based active contours. We benchmark our proposed interactive segmentation algorithms against those based upon energy-minimisation, specifically graph-cut methods. We demonstrate that we achieve competitive performance

Automatically extracting cellular structures from images generated via electron microscopy

In this paper, we consider mathematical techniques for locating cellular structures in digital images generated via electron microscopy. We approach this problem in two steps: a pre-processing denoising stage and a segmentation stage. For image denoising, we will limit our discussion to Partial Differential Equation (PDE) based methods, primarily focusing on diffusion and total variation methods. Segmentation will also b

A free surface capturing discretization for the staggered grid finite difference scheme

International audienceThe coupling that exists between surface processes and deformation within both the shallowcrust and the deeper mantle-lithosphere has stimulated the development of computationalgeodynamic models that incorporate a free surface boundary condition. We introduce a treatmentof this boundary condition that is suitable for staggered grid, finite difference schemesemploying a structured Eulerian mesh. Our interface capturing treatment discretizes the freesurface boundary condition via an interface that conforms with the edges of control volumes(e.g. a ‘staircase’ representation) and requires only local stencil modifications to be performed.Comparisons with analytic solutions verify that the method is first-order accurate. Additionalintermodel comparisons are performed between known reference models to further validateour free surface approximation. Lastly, we demonstrate the applicability of a multigrid solverto our free surface methodology and demonstrate that the local stencil modifications do notstrongly influence the convergence of the iterative solver