Multiclass Data Segmentation using Diffuse Interface Methods on Graphs

We present two graph-based algorithms for multiclass segmentation of high-dimensional data. The algorithms use a diffuse interface model based on the Ginzburg-Landau functional, related to total variation compressed sensing and image processing. A multiclass extension is introduced using the Gibbs simplex, with the functional's double-well potential modified to handle the multiclass case. The first algorithm minimizes the functional using a convex splitting numerical scheme. The second algorithm is a uses a graph adaptation of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates between diffusion and thresholding. We demonstrate the performance of both algorithms experimentally on synthetic data, grayscale and color images, and several benchmark data sets such as MNIST, COIL and WebKB. We also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph Laplacian, and take advantage of the sparsity of the matrix. Experiments indicate that the results are competitive with or better than the current state-of-the-art multiclass segmentation algorithms.Comment: 14 page

Highly automatic quantification of myocardial oedema in patients with acute myocardial infarction using bright blood T2-weighted CMR

<p>Background: T2-weighted cardiovascular magnetic resonance (CMR) is clinically-useful for imaging the ischemic area-at-risk and amount of salvageable myocardium in patients with acute myocardial infarction (MI). However, to date, quantification of oedema is user-defined and potentially subjective.</p> <p>Methods: We describe a highly automatic framework for quantifying myocardial oedema from bright blood T2-weighted CMR in patients with acute MI. Our approach retains user input (i.e. clinical judgment) to confirm the presence of oedema on an image which is then subjected to an automatic analysis. The new method was tested on 25 consecutive acute MI patients who had a CMR within 48 hours of hospital admission. Left ventricular wall boundaries were delineated automatically by variational level set methods followed by automatic detection of myocardial oedema by fitting a Rayleigh-Gaussian mixture statistical model. These data were compared with results from manual segmentation of the left ventricular wall and oedema, the current standard approach.</p> <p>Results: The mean perpendicular distances between automatically detected left ventricular boundaries and corresponding manual delineated boundaries were in the range of 1-2 mm. Dice similarity coefficients for agreement (0=no agreement, 1=perfect agreement) between manual delineation and automatic segmentation of the left ventricular wall boundaries and oedema regions were 0.86 and 0.74, respectively.</p&gt

Nonconvex Nonsmooth Low-Rank Minimization via Iteratively Reweighted Nuclear Norm

The nuclear norm is widely used as a convex surrogate of the rank function in compressive sensing for low rank matrix recovery with its applications in image recovery and signal processing. However, solving the nuclear norm based relaxed convex problem usually leads to a suboptimal solution of the original rank minimization problem. In this paper, we propose to perform a family of nonconvex surrogates of $L_0$-norm on the singular values of a matrix to approximate the rank function. This leads to a nonconvex nonsmooth minimization problem. Then we propose to solve the problem by Iteratively Reweighted Nuclear Norm (IRNN) algorithm. IRNN iteratively solves a Weighted Singular Value Thresholding (WSVT) problem, which has a closed form solution due to the special properties of the nonconvex surrogate functions. We also extend IRNN to solve the nonconvex problem with two or more blocks of variables. In theory, we prove that IRNN decreases the objective function value monotonically, and any limit point is a stationary point. Extensive experiments on both synthesized data and real images demonstrate that IRNN enhances the low-rank matrix recovery compared with state-of-the-art convex algorithms

A graph-based mathematical morphology reader

This survey paper aims at providing a "literary" anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research