Search

Random walks on networks are widely used to model stochastic processes such as search strategies, transportation problems or disease propagation. A prominent example of such process is the guiding of naive T cells by the lymph node conduits network. Here,we propose a general framework to find network heterogeneities, which we define as connectivity patterns that affect the random walk. We propose to characterize and measure these heterogeneities by detecting communities in a way that is interpretable in terms of random walk. Moreover, we use an approximation to accurately and efficiently compute these quantities on large networks. Finally, we propose an interactive data visualization platform to follow the dynamics of the random walks and their characteristics on our datasets, and a ready-to-use pipeline for other datasets upon download. By computing quantitative feature of random walk informed communities detected within the network, we show that the lymph node conduit network is spatially coherent, however, despite its quasi-regularity, contains some random walk related heterogeneities. To evaluate these characteristics, we applied the same workflow of diffusion based community detection and analysis on the LNCN and a series of generated toy networks.

Wavelet decompositions of integral operators have proven their efficiency in reducing computing times for many problems, ranging from the simulation of waves or fluids to the resolution of inverse problems in imaging. Unfortunately, computing the decomposition is itself a hard problem which is oftentimes out of reach for large scale problems. The objective of this work is to design fast decomposition algorithms based on another representation called product-convolution expansion. This decomposition can be evaluated efficiently assuming that a few impulse responses of the operator are available, but it is usually less efficient than the wavelet decomposition when incorporated in iterative methods. The proposed decomposition algorithms, run in quasi-linear time and we provide some numerical experiments to assess its performance for an imaging problem involving space varying blurs.

We design an image quality measure independent of contrast changes, which are defined as a set of transformations preserving an order between the level lines of an image. This problem can be expressed as an isotonic regression problem. Depending on the definition of a level line, the partial order between adjacent regions can be defined through chains, polytrees or directed acyclic graphs. We provide a few analytic properties of the minimizers and design original optimization procedures together with a full complexity analysis. The methods worst case complexities range from O(n) for chains, to O(nlog n) for polytrees and O(n2ϵ) for directed acyclic graphs, where n is the number of pixels and ϵ is a relative precision. The proposed algorithms have potential applications in change detection, stereo-vision, image registration, color image processing or image fusion. A C++ implementation with Matlab headers is available at https://github.com/pierre-weiss/contrast_invariant_snr.

We provide a new estimator of integral operators with smooth kernels, obtained from a set of scattered and noisy impulse responses. The proposed approach relies on the formalism of smoothing in reproducing kernel Hilbert spaces and on the choice of an appropriate regularization term that takes the smoothness of the operator into account. It is numerically tractable in very large dimensions. We study the estimator's robustness to noise and analyze its approximation properties with respect to the size and the geometry of the dataset. In addition, we show minimax optimality of the proposed estimator.

We design an image quality measure independent of local contrast changes, which constitute simple models of illumination changes. Given two images, the algorithm provides the image closest to the first one with the component tree of the second. This problem can be cast as a specific convex program called isotonic regression. We provide a few analytic properties of the solutions to this problem. We also design a tailored first order optimization procedure together with a full complexity analysis. The proposed method turns out to be practically more efficient and reliable than the best existing algorithms based on interior point methods. The algorithm has potential applications in change detection, color image processing or image fusion. A Matlab implementation is available at http://www.math.univ-toulouse.fr/_weiss/PageCodes.html.

Restoration of images degraded by spatially varying blurs is an issue of increasing importance in the context of photography, satellite or microscopy imaging. One of the main difficulty to solve this problem comes from the huge dimensions of the blur matrix. It prevents the use of naive approaches for performing matrix-vector multiplications. In this paper, we propose to approximate the blur operator by a matrix sparse in the wavelet domain. We justify this approach from a mathematical point of view and investigate the approximation quality numerically. We finish by showing that the sparsity pattern of the matrix can be pre-defined, which is central in tasks such as blind deconvolution., Comment: 6 pages