Inhomogeneous Noise Research Articles

Non-Gaussianity and Minkowski functionals: forecasts for Planck

We study Minkowski Functionals as probes of primordial non-Gaussianity in the Cosmic Microwave Background, specifically for the estimate of the primordial `local' bi-spectrum parameter f_NL, with instrumental parameters which should be appropriate for the Planck experiment. We use a maximum likelihood approach, which we couple with various filtering methods and test thoroughly for convergence. We included the effect of inhomogeneous noise as well as astrophysical biases induced by point sources and by the contamination from the Galaxy. We find that, when Wiener filtered maps are used (rather than simply smoothed with Gaussian), the expected error on the measurement of f_NL should be as small as \Delta f_NL \simeq 10 when combining the 3 channels at 100, 143 and 217 GHz in the Planck extended mission setup. This result is fairly insensitive to the non homogeneous nature of the noise, at least for realistic hit-maps expected from Planck. We then estimate the bias induced on the measurement of f_NL by point sources in those 3 channels. With the appropriate masking of the bright sources, this bias can be reduced to a negligible level in the 100 and 143 GHz channels. It remains significant in the 217 GHz channel, but can be corrected for. The galactic foreground biases are quite important and present a complex dependence on sky coverage: making them negligible will depend strongly on the quality of the component separation methods.

Image Segmentation Using a Local GMM in a Variational Framework

In this paper, we propose a new variational framework to solve the Gaussian mixture model (GMM) based methods for image segmentation by employing the convex relaxation approach. After relaxing the indicator function in GMM, flexible spatial regularization can be adopted and efficient segmentation can be achieved. To demonstrate the superiority of the proposed framework, the global, local intensity information and the spatial smoothness are integrated into a new model, and it can work well on images with inhomogeneous intensity and noise. Compared to classical GMM, numerical experiments have demonstrated that our algorithm can achieve promising segmentation performance for images degraded by intensity inhomogeneity and noise.

A hybrid approach to cosmic microwave background lensing reconstruction from all-sky intensity maps

Based on realistic simulations, we propose an hybrid method to reconstruct the lensing potential power spectrum, directly on PLANCK-like CMB frequency maps. It implies using a large galactic mask and dealing with a strong inhomogeneous noise. For l < 100, we show that a full-sky inpainting method, already described in a previous work, still allows a minimal variance reconstruction, with a bias that must be accounted for by a Monte-Carlo method, but that does not couple to the deflection field. For l>100 we develop a method based on tiling the cut-sky with local 10x10 degrees overlapping tangent planes (referred to in the following as "patches"). It requires to solve various issues concerning their size/position, non-periodic boundaries and irregularly sampled data after the sphere-to-plane projection. We show how the leading noise term of the quadratic lensing estimator applied onto an apodized patch can still be taken directly from the data. To not loose spatial accuracy, we developed a tool that allows the fast determination of the complex Fourier series coefficients from a bi-dimensional irregularly sampled dataset, without performing an interpolation. We show that the multi-patch approach allows the lensing power spectrum reconstruction with a very small bias, thanks to avoiding the galactic mask and lowering the noise inhomogeneities, while still having almost a minimal variance. The data quality can be assessed at each stage and simple bi-dimensional spectra build, which allows the control of local systematic errors.

Non-linear composite control with applications to 3-DOF planar motor

A 3-DOF (degree of freedom) planar motion motor using multiple linear motors as actuators is widely implemented in ultra-precision positioning systems. It is a multi-input multi-output (MIMO) system and requires an accurate decoupled model for controller design. In practice, there are modelling errors between the nominal model and the actual model, such as the force ripple, inhomogeneous air-gap thickness, measurement noise and uncertain disturbances. Due to the modelling errors, this MIMO system cannot be decoupled and the desired tracking performance specifications will not be achieved. In order to improve the servo performance of the planar motor with uncertainties, a non-linear composite controller consisting of u1 and u2 terms is proposed. u1 is designed to compensate for the modelling errors; therefore the coupling effects between multi-degrees of freedom are reduced and the robustness is improved. In order to reduce the large servo error amplitudes, which are caused by the low-frequency disturbances and high-frequency noise, an amplitude-based variable-gain function is applied in u2; thus the control action benefits from this operation by suppressing the low-frequency contribution and no extra control effect, to avoid the high-frequency noise. The convergence and stability of the control law are guaranteed by Lyapunov stability theory. The trajectory tracking experimental results show that the proposed control method has superior tracking performance and positioning ability compared with the well-known inverse dynamics controller. This indicates that the non-linear composite controller is an attractive approach to improve the servo performance of a system with uncertainties.

Quantitative Evaluation of Activation State in Functional Brain Imaging

Neuronal activity can evoke the hemodynamic change that gives rise to the observed functional magnetic resonance imaging (fMRI) signal. These increases are also regulated by the resting blood volume fraction (V (0)) associated with regional vasculature. The activation locus detected by means of the change in the blood-oxygen-level-dependent (BOLD) signal intensity thereby may deviate from the actual active site due to varied vascular density in the cortex. Furthermore, conventional detection techniques evaluate the statistical significance of the hemodynamic observations. In this sense, the significance level relies not only upon the intensity of the BOLD signal change, but also upon the spatially inhomogeneous fMRI noise distribution that complicates the expression of the results. In this paper, we propose a quantitative strategy for the calibration of activation states to address these challenging problems. The quantitative assessment is based on the estimated neuronal efficacy parameter [Formula: see text] of the hemodynamic model in a voxel-by-voxel way. It is partly immune to the inhomogeneous fMRI noise by virtue of the strength of the optimization strategy. Moreover, it is easy to incorporate regional vascular information into the activation detection procedure. By combining MR angiography images, this approach can remove large vessel contamination in fMRI signals, and provide more accurate functional localization than classical statistical techniques for clinical applications. It is also helpful to investigate the nonlinear nature of the coupling between synaptic activity and the evoked BOLD response. The proposed method might be considered as a potentially useful complement to existing statistical approaches.

Optimal computation of 3-D similarity: Gauss–Newton vs. Gauss–Helmert

Because 3-D data are acquired using 3-D sensing such as stereo vision and laser range finders, they have inhomogeneous and anisotropic noise. This paper studies optimal computation of the similarity (rotation, translation, and scale change) of such 3-D data. We first describe two well known methods for this: the Gauss–Newton and the Gauss–Helmert methods, which are often regarded as different techniques. We then point out that they have similar mathematical structures and combine them to define a hybrid, which we call the modified Gauss–Helmert method. Doing stereo vision simulation, we demonstrate that the proposed method is superior to either of the two methods in convergence performance. Finally, we show an application to real GPS geodetic data and point out that the widely used homogeneous and isotropic noise model is insufficient. We also discuss some numerical issues about GPS data.

Power-law distributions and fluctuation-dissipation relation in the stochastic dynamics oftwo-variable Langevin equations

We show that the general two-variable Langevin equations with inhomogeneous noise andfriction can generate many different forms of power-law distributions. By solving thecorresponding stationary Fokker–Planck equation, we can obtain a condition under whichthese power-law distributions are accurately created in a system away from equilibrium.This condition is an energy-dependent relation between the diffusion coefficient and thefriction coefficient and thus it provides a fluctuation-dissipation relation for nonequilibriumsystems with power-law distributions. Further, we study the specific forms of theFokker–Planck equation that correctly lead to such power-law distributions, and thenpresent a possible generalization of the Klein–Kramers equation and the Smoluchowskiequation to a complex system, whose stationary-state solutions are exactly a Tsallisdistribution.

Algorithms for bispectra: forecasting, optimal analysis and simulation

We propose a factorizability ansatz for angular bispectra which permits fast algorithms for forecasting, analysis, and simulation, yet is general enough to encompass many interesting CMB bispectra. We describe a suite of general algorithms which apply to any bispectrum which can be represented in factorizable form. First, we present algorithms for Fisher matrix forecasts and the related problem of optimizing the factorizable representation, giving a Fisher forecast for Planck as an example. We show that the CMB can give independent constraints on the amplitude of primordial bispectra of both local and equilateral shape as well as those created by secondary anisotropies. We also show that the ISW-lensing bispectrum should be detected by Planck and could bias estimates of the local type of non-Gaussianity if not properly accounted for. Second, we implement a bispectrum estimator which is fully optimal in the presence of sky cuts and inhomogeneous noise, extends the generality of fast estimators which have been limited to a few specific forms of the bispectrum, and improves the running time of existing implementations by several orders of magnitude. Third, we give an algorithm for simulating random, weakly non-Gaussian maps with prescribed power spectrum and factorizable bispectrum.

Unsupervised dealiasing and denoising of color-Doppler data

Color Doppler imaging (CDI) is the premiere modality to analyze blood flow in clinical practice. In the prospect of producing new CDI-based tools, we developed a fast unsupervised denoiser and dealiaser (DeAN) algorithm for color Doppler raw data. The proposed technique uses robust and automated image post-processing techniques that make the DeAN clinically compliant. The DeAN includes three consecutive advanced and hands-off numerical tools: (1) statistical region merging segmentation, (2) recursive dealiasing process, and (3) regularized robust smoothing. The performance of the DeAN was evaluated using Monte-Carlo simulations on mock Doppler data corrupted by aliasing and inhomogeneous noise. Fifty aliased Doppler images of the left ventricle acquired with a clinical ultrasound scanner were also analyzed. The analytical study demonstrated that color Doppler data can be reconstructed with high accuracy despite the presence of strong corruption. The normalized RMS error on the numerical data was less than 8% even with signal-to-noise ratio as low as 10dB. The algorithm also allowed us to recover highly reliable Doppler flows in clinical data. The DeAN is fast, accurate and not observer-dependent. Preliminary results showed that it is also directly applicable to 3-D data. This will offer the possibility of developing new tools to better decipher the blood flow dynamics in cardiovascular diseases.

Libfbi: a C++ implementation for fast box intersection and application to sparse mass spectrometry data

Algorithms for sparse data require fast search and subset selection capabilities for the determination of point neighborhoods. A natural data representation for such cases are space partitioning data structures. However, the associated range queries assume noise-free observations and cannot take into account observation-specific uncertainty estimates that are present in e.g. modern mass spectrometry data. In order to accommodate the inhomogeneous noise characteristics of sparse real-world datasets, point queries need to be reformulated in terms of box intersection queries, where box sizes correspond to uncertainty regions for each observation. This contribution introduces libfbi, a standard C++, header-only template implementation for fast box intersection in an arbitrary number of dimensions, with arbitrary data types in each dimension. The implementation is applied to a data aggregation task on state-of-the-art liquid chromatography/mass spectrometry data, where it shows excellent run time properties. The library is available under an MIT license and can be downloaded from http://software.steenlab.org/libfbi. marc.kirchner@childrens.harvard.edu Supplementary data are available at Bioinformatics online.

New optimized estimators for the primordial trispectrum

Cosmic microwave background studies of non-Gaussianity involving higher-order multispectra can distinguish between early universe theories that predict nearly identical power spectra. However, the recovery of higher-order multispectra is difficult from realistic data due to their complex response to inhomogeneous noise and partial sky coverage, which are often difficult to model analytically. A traditional alternative is to use one-point cumulants of various orders, which collapse the information present in a multispectrum to one number. The disadvantage of such a radical compression of the data is a loss of information as to the source of the statistical behaviour. A recent study by Munshi & Heavens (2009) has shown how to define the skew spectrum (the power spectra of a certain cubic field, related to the bispectrum) in an optimal way and how to estimate it from realistic data. The skew spectrum retains some of the information from the full configuration-dependence of the bispectrum, and can contain all the information on non-Gaussianity. In the present study, we extend the results of the skew spectrum to the case of two degenerate power-spectra related to the trispectrum. We also explore the relationship of these power-spectra and cumulant correlators previously used to study non-Gaussianity in projected galaxy surveys or weak lensing surveys. We construct nearly optimal estimators for quick tests and generalise them to estimators which can handle realistic data with all their complexity in a completely optimal manner. We show how these higher-order statistics and the related power spectra are related to the Taylor expansion coefficients of the potential in inflation models, and demonstrate how the trispectrum can constrain both the quadratic and cubic terms.

Application of Wavelet Transform for Image Denoising of Spatially and Time Variable Astronomical Imaging Systems

We report on our efforts to formulate algorithms for image signal processing with the spatially and time variant Point-Spread Function (PSF) and inhomogeneous noise of real imaging systems. In this paper we focus on application of the wavelet transform for denoising of the astronomical images with complicated conditions. They influence above all accuracy of the measurements and the new source detection ability. Our aim is to test the usefulness ofWavelet transform (as the standard image processing technique) for astronomical purposes.

Ct3d: tracking microglia motility in 3D using a novel cosegmentation approach

Motivation: Cell tracking is an important method to quantitatively analyze time-lapse microscopy data. While numerous methods and tools exist for tracking cells in 2D time-lapse images, only few and very application-specific tracking tools are available for 3D time-lapse images, which is of high relevance in immunoimaging, in particular for studying the motility of microglia in vivo.Results: We introduce a novel algorithm for tracking cells in 3D time-lapse microscopy data, based on computing cosegmentations between component trees representing individual time frames using the so-called tree-assignments. For the first time, our method allows to track microglia in three dimensional confocal time-lapse microscopy images. We also evaluate our method on synthetically generated data, demonstrating that our algorithm is robust even in the presence of different types of inhomogeneous background noise.Availability: Our algorithm is implemented in the ct3d package, which is available under http://www.picb.ac.cn/patterns/Software/ct3d; supplementary videos are available from http://www.picb.ac.cn/patterns/Supplements/ct3d.Contact: axel@picb.ac.cn; forestdu@ion.ac.cnSupplementary information: Supplementary data are available at Bioinformatics online.

On the optimality of the spherical Mexican hat wavelet estimator for the primordial non-Gaussianity

We study the spherical Mexican hat wavelet (SMHW) as a detector of primordial non-Gaussianity of the local type on the Cosmic Microwave Background (CMB) anisotropies. For this purpose we define third order statistics based on the wavelet coefficient maps and the original map. We find the dependence of these statistics in terms of the non-linear coupling parameter fnl and the bispectrum of this type of non-Gaussianity. We compare the analytical values for these statistics with the results obtained with non-Gaussian simulations for an ideal full-sky CMB experiment without noise. We study the power of this method to detect fnl, i. e. the variance of this parameter, and compare it with the variance obtained from the primary bispectrum for the same experiment. Finally we apply our wavelet based estimator on WMAP-like maps with incomplete sky and inhomogeneous noise and compare with the optimal bispectrum estimator. The results show that the wavelet cubic statistics are as efficient as the bispectrum as optimal detectors of this type of primordial non-Gaussianity.

General CMB and primordial trispectrum estimation

We present trispectrum estimation methods which can be applied to general non-separable primordial and CMB trispectra. We present a general optimal estimator for the connected part of the trispectrum, for which we derive a quadratic term to incorporate the effects of inhomogeneous noise and masking. We describe a general algorithm for creating simulated maps with given arbitrary (and independent) power spectra, bispectra and trispectra. We propose a universal definition of the trispectrum parameter $T_{NL}$, so that the integrated bispectrum on the observational domain can be consistently compared between theoretical models. We define a shape function for the primordial trispectrum, together with a shape correlator and a useful parametrisation for visualizing the trispectrum. We derive separable analytic CMB solutions in the large-angle limit for constant and local models. We present separable mode decompositions which can be used to describe any primordial or CMB bispectra on their respective wavenumber or multipole domains. By extracting coefficients of these separable basis functions from an observational map, we are able to present an efficient estimator for any given theoretical model with a nonseparable trispectrum. The estimator has two manifestations, comparing the theoretical and observed coefficients at either primordial or late times. These mode decomposition methods are numerically tractable with order $l^5$ operations for the CMB estimator and approximately order $l^6$ for the general primordial estimator (reducing to order $l^3$ in both cases for a special class of models). We also demonstrate how the trispectrum can be reconstructed from observational maps using these methods.

Particle-Velocity-Field Difference Smoothing for Coherent Source Localization in Spatially Nonuniform Noise

This communication considers the problem of estimating 2-D directions of arrival (DOAs) of multiple coherent signals under spatially nonuniform noise (spatially inhomogeneous temporary white noise) using an array of vector hydrophones. A novel preprocessing method called particle-velocity-field difference smoothing (PVFDS) is proposed. The key idea underlying the PVFDS is to remove the spatially nonuniform noise by using the matrix difference of pairs of particle-velocity data correlation matrices, and to decorrelate the coherent signals by summing these difference correlation matrices. Unlike most of other existing preprocessing techniques, such as spatial smoothing and forward-backward averaging, the PVFDS processing does not decrease the array aperture. For arbitrary array geometries, the PVFDS can resolve up to four coherent signals, and for centro-symmetric arrays, forward-backward averaging can double this number to eight. Monte Carlo simulations illustrate that the PVFDS-based eigenstructure algorithms can offer better performance than the particle-velocity-field smoothing (PVFS)-based counterparts.

Lensing reconstruction from Planck sky maps: inhomogeneous noise

We discuss the effects of inhomogeneous sky-coverage on CMB lens reconstruction, focusing on application to the recently launched Planck satellite. We discuss the mean-field which is induced by noise inhomogeneities, as well as three approaches to lens reconstruction in this context: an optimal maximum-likelihood approach which is computationally expensive to evaluate, and two suboptimal approaches which are less intensive. The first of these is only sub-optimal at the five per-cent level for Planck, and the second prevents biasing due to uncertainties in the noise model.

A new approach to probing primordial non-Gaussianity

We address the dual challenge of estimating deviations from Gaussianity arising in models of the Early Universe, whilst retaining information necessary to assess whether a detection of non-Gaussianity is primordial. We do this by constructing a new statistic, the bispectrum-related power spectrum, which is constructed from a map of the Cosmic Microwave Background. The estimator is optimised for primordial non-Gaussianity detection, but can also be useful in distinguishing primordial non-Gaussianity from secondary non-Gaussianity, such as may arise from unsubtracted point sources, or residuals from component separation. Extending earlier studies we present unbiased non-Gaussianity estimators optimised for partial sky coverage and inhomogeneous noise associated with realistic scan strategies, but which retain the ability to assess foreground contamination.

Maximum likelihood algorithm for parametric component separation in cosmic microwave background experiments

We discuss an approach to the component separation of microwave, multifrequency sky maps as those typically produced from cosmic microwave background (CMB) anisotropy data sets. The algorithm is based on the two-step, parametric, likelihood-based technique recently elaborated on by Eriksen et al., where the foreground spectral parameters are estimated prior to the actual separation of the components. In contrast with the previous approaches, we accomplish the former task with help of an analytically derived likelihood function for the spectral parameters, which, we show, yields estimates equal to the maximum likelihood values of the full multidimensional data problem. We then use these estimates to perform the second step via the standard, generalized-least-squares-like procedure. We demonstrate that the proposed approach is equivalent to a direct maximization of the full data likelihood, which is recast in a computationally tractable form. We use the corresponding curvature matrices to characterize statistical properties of the recovered parameters. We incorporate in the formalism some of the essential features of the CMB data sets, such as inhomogeneous pixel domain noise, unknown map offsets as well as calibration errors and study their consequences for the separation. We find that the calibration is likely to have a dominant effect on the precision of the spectral parameter determination for a realistic CMB experiment. We apply the algorithm to simulated data and discuss the results. Our focus is on partial sky, total intensity and polarization, CMB experiments such as planned balloon-borne and ground-based efforts, however, the techniques presented here should be also applicable to the full-sky data as for instance, those produced by the Wilkinson Microwave Anisotropy Probe (WMAP) satellite and anticipated from the Planck mission.

Superconducting atom chips: advantages and challenges

Superconductors are considered in view of applications to atom chip devices. The main features of magnetic traps based on superconducting wires in the Meissner and mixed states are discussed. The former state may mainly be interesting for improved atom optics, while in the latter, cold atoms may provide a probe of superconductor phenomena. The properties of a magnetic side guide based on a single superconducting strip wire placed in an external magnetic field are calculated analytically and numerically. In the mixed state of type II superconductors, inhomogeneous trapped magnetic flux, relaxation processes and noise caused by vortex motion are posing specific challenges for atom trapping.