Exact reconstruction with directional wavelets on the sphere

A new formalism is derived for the analysis and exact reconstruction of band-limited signals on the sphere with directional wavelets. It represents an evolution of the wavelet formalism developed by Antoine & Vandergheynst (1999) and Wiaux et al. (2005). The translations of the wavelets at any point on the sphere and their proper rotations are still defined through the continuous three-dimensional rotations. The dilations of the wavelets are directly defined in harmonic space through a new kernel dilation, which is a modification of an existing harmonic dilation. A family of factorized steerable functions with compact harmonic support which are suitable for this kernel dilation is firstly identified. A scale discretized wavelet formalism is then derived, relying on this dilation. The discrete nature of the analysis scales allows the exact reconstruction of band-limited signals. A corresponding exact multi-resolution algorithm is finally described and an implementation is tested. The formalism is of interest notably for the denoising or the deconvolution of signals on the sphere with a sparse expansion in wavelets. In astrophysics, it finds a particular application for the identification of localized directional features in the cosmic microwave background (CMB) data, such as the imprint of topological defects, in particular cosmic strings, and for their reconstruction after separation from the other signal components.Comment: 22 pages, 2 figures. Version 2 matches version accepted for publication in MNRAS. Version 3 (identical to version 2) posted for code release announcement - "Steerable scale discretised wavelets on the sphere" - S2DW code available for download at http://www.mrao.cam.ac.uk/~jdm57/software.htm

Gaussian Processes with Monotonicity constraints for Big Data

Tämän työn tarkoitus on kehittää menetelmä monotonisuusrajoitettujen Gaussisten Prosessien käyttämiseksi suurille aineistoille. Variaatiolaskentaan perustuvaa menetelmää testataan usealla simuloidulla ja oikealla aineistolla. Uuden menetelmän prediktiivistä kykyä verrataan expectation propagation menetelmään, sekä Markov chain Monte Carlo menetelmiin. Työssä saatujen tulosten perusteella voidaan päätellä, että uusi menetelmä toimii ja sitä voidaan käyttää, kun aineistot kasvavat liian suuriksia laskennallisesti raskaille menetelmille.In this thesis, we combine recent advances in monotonicity constraints for Gaussian processes with Big Data inference of Gaussian Proceses. The new variational inference based method is developed and experimented on several simulated and real world data sets by comparing the predictive performance to Expectation Propagation and Markov chain Monte Carlo methods. The results indicate that the new method produces good results and can be used when the data sets get so large that the computationally demanding methods cannot be used

Fast joint detection-estimation of evoked brain activity in event-related fMRI using a variational approach

In standard clinical within-subject analyses of event-related fMRI data, two steps are usually performed separately: detection of brain activity and estimation of the hemodynamic response. Because these two steps are inherently linked, we adopt the so-called region-based Joint Detection-Estimation (JDE) framework that addresses this joint issue using a multivariate inference for detection and estimation. JDE is built by making use of a regional bilinear generative model of the BOLD response and constraining the parameter estimation by physiological priors using temporal and spatial information in a Markovian modeling. In contrast to previous works that use Markov Chain Monte Carlo (MCMC) techniques to approximate the resulting intractable posterior distribution, we recast the JDE into a missing data framework and derive a Variational Expectation-Maximization (VEM) algorithm for its inference. A variational approximation is used to approximate the Markovian model in the unsupervised spatially adaptive JDE inference, which allows fine automatic tuning of spatial regularisation parameters. It follows a new algorithm that exhibits interesting properties compared to the previously used MCMC-based approach. Experiments on artificial and real data show that VEM-JDE is robust to model mis-specification and provides computational gain while maintaining good performance in terms of activation detection and hemodynamic shape recovery

Space-time Structure of Initial Parton Production in Ultrarelativistic Heavy Ion Collisions

The space and time evolution of initial parton production in ultrarelativistic heavy ion collisions is investigated within the framework of perturbative QCD which includes both initial and final state radiations. Uncertainty principle is used to relate the life time of a radiating parton to its virtuality and momentum. The interaction time of each hard or semihard parton scattering is also taken into account. For central $Au+Au$ collisions at $\sqrt{s}=200$ GeV, most of the partons are found to be produced within 0.5 fm/c after the total overlap of the two colliding nuclei. The local momentum distribution is approximately isotropical at that time. The implication on how to treat correctly the the secondary scattering in an ultimate parton cascading model is also discussed.Comment: 19 pages in REVTEX with 12 figures in separate uuencoded postscript files, LBL-3415