Graph Spectral Image Processing

Recent advent of graph signal processing (GSP) has spurred intensive studies of signals that live naturally on irregular data kernels described by graphs (e.g., social networks, wireless sensor networks). Though a digital image contains pixels that reside on a regularly sampled 2D grid, if one can design an appropriate underlying graph connecting pixels with weights that reflect the image structure, then one can interpret the image (or image patch) as a signal on a graph, and apply GSP tools for processing and analysis of the signal in graph spectral domain. In this article, we overview recent graph spectral techniques in GSP specifically for image / video processing. The topics covered include image compression, image restoration, image filtering and image segmentation

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

A Method for Geometry Optimization in a Simple Model of Two-Dimensional Heat Transfer

This investigation is motivated by the problem of optimal design of cooling elements in modern battery systems. We consider a simple model of two-dimensional steady-state heat conduction described by elliptic partial differential equations and involving a one-dimensional cooling element represented by a contour on which interface boundary conditions are specified. The problem consists in finding an optimal shape of the cooling element which will ensure that the solution in a given region is close (in the least squares sense) to some prescribed target distribution. We formulate this problem as PDE-constrained optimization and the locally optimal contour shapes are found using a gradient-based descent algorithm in which the Sobolev shape gradients are obtained using methods of the shape-differential calculus. The main novelty of this work is an accurate and efficient approach to the evaluation of the shape gradients based on a boundary-integral formulation which exploits certain analytical properties of the solution and does not require grids adapted to the contour. This approach is thoroughly validated and optimization results obtained in different test problems exhibit nontrivial shapes of the computed optimal contours.Comment: Accepted for publication in "SIAM Journal on Scientific Computing" (31 pages, 9 figures

Edge-enhancing Filters with Negative Weights

In [DOI:10.1109/ICMEW.2014.6890711], a graph-based denoising is performed by projecting the noisy image to a lower dimensional Krylov subspace of the graph Laplacian, constructed using nonnegative weights determined by distances between image data corresponding to image pixels. We~extend the construction of the graph Laplacian to the case, where some graph weights can be negative. Removing the positivity constraint provides a more accurate inference of a graph model behind the data, and thus can improve quality of filters for graph-based signal processing, e.g., denoising, compared to the standard construction, without affecting the costs.Comment: 5 pages; 6 figures. Accepted to IEEE GlobalSIP 2015 conferenc