Self-similar prior and wavelet bases for hidden incompressible turbulent motion

This work is concerned with the ill-posed inverse problem of estimating turbulent flows from the observation of an image sequence. From a Bayesian perspective, a divergence-free isotropic fractional Brownian motion (fBm) is chosen as a prior model for instantaneous turbulent velocity fields. This self-similar prior characterizes accurately second-order statistics of velocity fields in incompressible isotropic turbulence. Nevertheless, the associated maximum a posteriori involves a fractional Laplacian operator which is delicate to implement in practice. To deal with this issue, we propose to decompose the divergent-free fBm on well-chosen wavelet bases. As a first alternative, we propose to design wavelets as whitening filters. We show that these filters are fractional Laplacian wavelets composed with the Leray projector. As a second alternative, we use a divergence-free wavelet basis, which takes implicitly into account the incompressibility constraint arising from physics. Although the latter decomposition involves correlated wavelet coefficients, we are able to handle this dependence in practice. Based on these two wavelet decompositions, we finally provide effective and efficient algorithms to approach the maximum a posteriori. An intensive numerical evaluation proves the relevance of the proposed wavelet-based self-similar priors.Comment: SIAM Journal on Imaging Sciences, 201

Inferring Latent States and Refining Force Estimates via Hierarchical Dirichlet Process Modeling in Single Particle Tracking Experiments

Optical microscopy provides rich spatio-temporal information characterizing in vivo molecular motion. However, effective forces and other parameters used to summarize molecular motion change over time in live cells due to latent state changes, e.g., changes induced by dynamic micro-environments, photobleaching, and other heterogeneity inherent in biological processes. This study focuses on techniques for analyzing Single Particle Tracking (SPT) data experiencing abrupt state changes. We demonstrate the approach on GFP tagged chromatids experiencing metaphase in yeast cells and probe the effective forces resulting from dynamic interactions that reflect the sum of a number of physical phenomena. State changes are induced by factors such as microtubule dynamics exerting force through the centromere, thermal polymer fluctuations, etc. Simulations are used to demonstrate the relevance of the approach in more general SPT data analyses. Refined force estimates are obtained by adopting and modifying a nonparametric Bayesian modeling technique, the Hierarchical Dirichlet Process Switching Linear Dynamical System (HDP-SLDS), for SPT applications. The HDP-SLDS method shows promise in systematically identifying dynamical regime changes induced by unobserved state changes when the number of underlying states is unknown in advance (a common problem in SPT applications). We expand on the relevance of the HDP-SLDS approach, review the relevant background of Hierarchical Dirichlet Processes, show how to map discrete time HDP-SLDS models to classic SPT models, and discuss limitations of the approach. In addition, we demonstrate new computational techniques for tuning hyperparameters and for checking the statistical consistency of model assumptions directly against individual experimental trajectories; the techniques circumvent the need for "ground-truth" and subjective information.Comment: 25 pages, 6 figures. Differs only typographically from PLoS One publication available freely as an open-access article at http://journals.plos.org/plosone/article?id=10.1371/journal.pone.013763

Cosmic Bulk Flow and the Local Motion from Cosmicflows-2

Full sky surveys of peculiar velocity are arguably the best way to map the large scale structure out to distances of a few times 100 Mpc/h. Using the largest and most accurate ever catalog of galaxy peculiar velocities "Cosmicflows-2", the large scale structure has been reconstructed by means of the Wiener filter and constrained realizations assuming as a Bayesian prior model the LCDM model with the WMAP inferred cosmological parameters. The present paper focuses on studying the bulk flow of the local flow field, defined as the mean velocity of top-hat spheres with radii ranging out to R=500 Mpc/h. The estimated large scale structures, in general, and the bulk flow, in particular, are determined by the tension between the observational data and the assumed prior model. A prerequisite for such an analysis is the requirement that the estimated bulk flow is consistent with the prior model. Such a consistency is found here. At R=50(150) Mpc/h the estimated bulk velocity is 250+/-21 (239+/-38) km/s. The corresponding cosmic variance at these radii is 126(60)km/s, which implies that these estimated bulk flows are dominated by the data and not by the assumed prior model. The estimated bulk velocity is dominated by the data out to R~200 Mpc/h, where the cosmic variance on the individual Supergalactic Cartesian components (of the r.m.s. values) exceeds the variance of the Constrained Realizations by at least a factor of 2. The supergalactic SGX and SGY components of the CMB dipole velocity are recovered by the Wiener filter velocity field down to a very few km/s. The SGZ component of the estimated velocity, the one that is most affected by the Zone of Avoidance, is off by 126 km/s (an almost 2 sigma discrepancy).Comment: 10 pages, accepted for MNRA