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

Identification of the Beagle 2 lander on Mars

The 2003 Beagle 2 Mars lander has been identified in Isidis Planitia at 90.43° E, 11.53° N, close to the predicted target of 90.50° E, 11.53° N. Beagle 2 was an exobiology lander designed to look for isotopic and compositional signs of life on Mars, as part of the European Space Agency Mars Express (MEX) mission. The 2004 recalculation of the original landing ellipse from a 3-sigma major axis from 174 km to 57 km, and the acquisition of Mars Reconnaissance Orbiter High Resolution Imaging Science Experiment (HiRISE) imagery at 30 cm per pixel across the target region, led to the initial identification of the lander in 2014. Following this, more HiRISE images, giving a total of 15, including red and blue-green colours, were obtained over the area of interest and searched, which allowed sub-pixel imaging using super high-resolution techniques. The size (approx. 1.5 m), distinctive multilobed shape, high reflectivity relative to the local terrain, specular reflections, and location close to the centre of the planned landing ellipse led to the identification of the Beagle 2 lander. The shape of the imaged lander, although to some extent masked by the specular reflections in the various images, is consistent with deployment of the lander lid and then some or all solar panels. Failure to fully deploy the panels-which may have been caused by damage during landing-would have prohibited communication between the lander and MEX and commencement of science operations. This implies that the main part of the entry, descent and landing sequence, the ejection from MEX, atmospheric entry and parachute deployment, and landing worked as planned with perhaps only the final full panel deployment failing

Anisotropic Diffusion Partial Differential Equations in Multi-Channel Image Processing : Framework and Applications

We review recent methods based on diffusion PDE's (Partial Differential Equations) for the purpose of multi-channel image regularization. Such methods have the ability to smooth multi-channel images anisotropically and can preserve then image contours while removing noise or other undesired local artifacts. We point out the pros and cons of the existing equations, providing at each time a local geometric interpretation of the corresponding processes. We focus then on an alternate and generic tensor-driven formulation, able to regularize images while specifically taking the curvatures of local image structures into account. This particular diffusion PDE variant is actually well suited for the preservation of thin structures and gives regularization results where important image features can be particularly well preserved compared to its competitors. A direct link between this curvature-preserving equation and a continuous formulation of the Line Integral Convolution technique (Cabral and Leedom, 1993) is demonstrated. It allows the design of a very fast and stable numerical scheme which implements the multi-valued regularization method by successive integrations of the pixel values along curved integral lines. Besides, the proposed implementation, based on a fourth-order Runge Kutta numerical integration, can be applied with a subpixel accuracy and preserves then thin image structures much better than classical finite-differences discretizations, usually chosen to implement PDE-based diffusions. We finally illustrate the efficiency of this diffusion PDE's for multi-channel image regularization - in terms of speed and visual quality - with various applications and results on color images, including image denoising, inpainting and edge-preserving interpolation

A combined first and second order variational approach for image reconstruction

In this paper we study a variational problem in the space of functions of bounded Hessian. Our model constitutes a straightforward higher-order extension of the well known ROF functional (total variation minimisation) to which we add a non-smooth second order regulariser. It combines convex functions of the total variation and the total variation of the first derivatives. In what follows, we prove existence and uniqueness of minimisers of the combined model and present the numerical solution of the corresponding discretised problem by employing the split Bregman method. The paper is furnished with applications of our model to image denoising, deblurring as well as image inpainting. The obtained numerical results are compared with results obtained from total generalised variation (TGV), infimal convolution and Euler's elastica, three other state of the art higher-order models. The numerical discussion confirms that the proposed higher-order model competes with models of its kind in avoiding the creation of undesirable artifacts and blocky-like structures in the reconstructed images -- a known disadvantage of the ROF model -- while being simple and efficiently numerically solvable.Comment: 34 pages, 89 figure