Geodesic Information Flows: Spatially-Variant Graphs and Their Application to Segmentation and Fusion

Clinical annotations, such as voxel-wise binary or probabilistic tissue segmentations, structural parcellations, pathological regionsof- interest and anatomical landmarks are key to many clinical studies. However, due to the time consuming nature of manually generating these annotations, they tend to be scarce and limited to small subsets of data. This work explores a novel framework to propagate voxel-wise annotations between morphologically dissimilar images by diffusing and mapping the available examples through intermediate steps. A spatially-variant graph structure connecting morphologically similar subjects is introduced over a database of images, enabling the gradual diffusion of information to all the subjects, even in the presence of large-scale morphological variability. We illustrate the utility of the proposed framework on two example applications: brain parcellation using categorical labels and tissue segmentation using probabilistic features. The application of the proposed method to categorical label fusion showed highly statistically significant improvements when compared to state-of-the-art methodologies. Significant improvements were also observed when applying the proposed framework to probabilistic tissue segmentation of both synthetic and real data, mainly in the presence of large morphological variability

DeepSphere: Efficient spherical Convolutional Neural Network with HEALPix sampling for cosmological applications

Convolutional Neural Networks (CNNs) are a cornerstone of the Deep Learning toolbox and have led to many breakthroughs in Artificial Intelligence. These networks have mostly been developed for regular Euclidean domains such as those supporting images, audio, or video. Because of their success, CNN-based methods are becoming increasingly popular in Cosmology. Cosmological data often comes as spherical maps, which make the use of the traditional CNNs more complicated. The commonly used pixelization scheme for spherical maps is the Hierarchical Equal Area isoLatitude Pixelisation (HEALPix). We present a spherical CNN for analysis of full and partial HEALPix maps, which we call DeepSphere. The spherical CNN is constructed by representing the sphere as a graph. Graphs are versatile data structures that can act as a discrete representation of a continuous manifold. Using the graph-based representation, we define many of the standard CNN operations, such as convolution and pooling. With filters restricted to being radial, our convolutions are equivariant to rotation on the sphere, and DeepSphere can be made invariant or equivariant to rotation. This way, DeepSphere is a special case of a graph CNN, tailored to the HEALPix sampling of the sphere. This approach is computationally more efficient than using spherical harmonics to perform convolutions. We demonstrate the method on a classification problem of weak lensing mass maps from two cosmological models and compare the performance of the CNN with that of two baseline classifiers. The results show that the performance of DeepSphere is always superior or equal to both of these baselines. For high noise levels and for data covering only a smaller fraction of the sphere, DeepSphere achieves typically 10% better classification accuracy than those baselines. Finally, we show how learned filters can be visualized to introspect the neural network.Comment: arXiv admin note: text overlap with arXiv:astro-ph/0409513 by other author

Maximal entropy random walk in community finding

The aim of this paper is to check feasibility of using the maximal-entropy random walk in algorithms finding communities in complex networks. A number of such algorithms exploit an ordinary or a biased random walk for this purpose. Their key part is a (dis)similarity matrix, according to which nodes are grouped. This study encompasses the use of the stochastic matrix of a random walk, its mean first-passage time matrix, and a matrix of weighted paths count. We briefly indicate the connection between those quantities and propose substituting the maximal-entropy random walk for the previously chosen models. This unique random walk maximises the entropy of ensembles of paths of given length and endpoints, which results in equiprobability of those paths. We compare performance of the selected algorithms on LFR benchmark graphs. The results show that the change in performance depends very strongly on the particular algorithm, and can lead to slight improvements as well as significant deterioration.Comment: 7 pages, 4 figures, submitted to European Physical Journal Special Topics following the 4-th Conference on Statistical Physics: Modern Trends and Applications, July 3-6, 2012 Lviv, Ukrain

Statistical equilibrium of tetrahedra from maximum entropy principle

Discrete formulations of (quantum) gravity in four spacetime dimensions build space out of tetrahedra. We investigate a statistical mechanical system of tetrahedra from a many-body point of view based on non-local, combinatorial gluing constraints that are modelled as multi-particle interactions. We focus on Gibbs equilibrium states, constructed using Jaynes' principle of constrained maximisation of entropy, which has been shown recently to play an important role in characterising equilibrium in background independent systems. We apply this principle first to classical systems of many tetrahedra using different examples of geometrically motivated constraints. Then for a system of quantum tetrahedra, we show that the quantum statistical partition function of a Gibbs state with respect to some constraint operator can be reinterpreted as a partition function for a quantum field theory of tetrahedra, taking the form of a group field theory.Comment: v3 published version; v2 18 pages, 4 figures, improved text in sections IIIC & IVB, minor changes elsewher

Depth-based Hypergraph Complexity Traces from Directed Line Graphs

In this paper, we aim to characterize the structure of hypergraphs in terms of structural complexity measure. Measuring the complexity of a hypergraph in a straightforward way tends to be elusive since the hyperedges of a hypergraph may exhibit varying relational orders. We thus transform a hypergraph into a line graph which not only accurately reflects the multiple relationships exhibited by the hyperedges but is also easier to manipulate for complexity analysis. To locate the dominant substructure within a line graph, we identify a centroid vertex by computing the minimum variance of its shortest path lengths. A family of centroid expansion subgraphs of the line graph is then derived from the centroid vertex. We compute the depth-based complexity traces for the hypergraph by measuring either the directed or undirected entropies of its centroid expansion subgraphs. The resulting complexity traces provide a flexible framework that can be applied to both hypergraphs and graphs. We perform (hyper)graph classification in the principal component space of the complexity trace vectors. Experiments on (hyper)graph datasets abstracted from bioinformatics and computer vision data demonstrate the effectiveness and efficiency of the complexity traces.This work is supported by National Natural Science Foundation of China (Grant no. 61503422). This work is supported by the Open Projects Program of National Laboratory of Pattern Recognition. Francisco Escolano is supported by the project TIN2012-32839 of the Spanish Government. Edwin R. Hancock is supported by a Royal Society Wolfson Research Merit Award

A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces

On doubling metric measure spaces endowed with a strongly local regular Dirichlet form, we show some characterisations of pointwise upper bounds of the heat kernel in terms of global scale-invariant inequalities that correspond respectively to the Nash inequality and to a Gagliardo-Nirenberg type inequality when the volume growth is polynomial. This yields a new proof and a generalisation of the well-known equivalence between classical heat kernel upper bounds and relative Faber-Krahn inequalities or localized Sobolev or Nash inequalities. We are able to treat more general pointwise estimates, where the heat kernel rate of decay is not necessarily governed by the volume growth. A crucial role is played by the finite propagation speed property for the associated wave equation, and our main result holds for an abstract semigroup of operators satisfying the Davies-Gaffney estimates

3D geological models and their hydrogeological applications : supporting urban development : a case study in Glasgow-Clyde, UK

Urban planners and developers in some parts of the United Kingdom can now access geodata in an easy-to-retrieve and understandable format. 3D attributed geological framework models and associated GIS outputs, developed by the British Geological Survey (BGS), provide a predictive tool for planning site investigations for some of the UK's largest regeneration projects in the Thames and Clyde River catchments. Using the 3D models, planners can get a 3D preview of properties of the subsurface using virtual cross-section and borehole tools in visualisation software, allowing critical decisions to be made before any expensive site investigation takes place, and potentially saving time and money. 3D models can integrate artificial and superficial deposits and bedrock geology, and can be used for recognition of major resources (such as water, thermal and sand and gravel), for example in buried valleys, groundwater modelling and assessing impacts of underground mining. A preliminary groundwater recharge and flow model for a pilot area in Glasgow has been developed using the 3D geological models as a framework. This paper focuses on the River Clyde and the Glasgow conurbation, and the BGS's Clyde Urban Super-Project (CUSP) in particular, which supports major regeneration projects in and around the City of Glasgow in the West of Scotland