Wavelet-based estimation of a semiparametric generalized linear model of fMRI time-series

Characterization and correlation of signal drift in diffusion weighted MRI

Diffusion weighted MRI (DWMRI) and the myriad of analysis approaches (from tensors to spherical harmonics and brain tractography to body multi-compartment models) depend on accurate quantification of the apparent diffusion coefficient (ADC). Signal drift during imaging (e.g., due to b0 drift associated with heating) can cause systematic non-linearities that manifest as ADC changes if not corrected. Herein, we present a case study on two phantoms on one scanner. Different scan protocols exhibit different degrees of drift during similar scans and may be sensitive to the order of scans within an exam. Vos et al. recently reviewed the effects of signal drift in DWMRI acquisitions and proposed a temporal model for correction. We propose a novel spatial-temporal model to correct for higher order aspects of the signal drift and derive a statistically robust variant. We evaluate the Vos model and propose a method using two phantoms that mimic the ADC of the relevant brain tissue (0.36-2.2 × 10-3 mm²/s) on a single 3 T scanner. The phantoms are (1) a spherical isotropic sphere consisting of a single concentration of polyvinylpyrrolidone (PVP) and (2) an ice-water phantom with 13 vials of varying PVP concentrations. To characterize the impact of interspersed minimally weighted volumes ("b0's"), image volumes with b-value equal to 0.1 s/mm² are interspersed every 8, 16, 32, 48, and 96 diffusion weighted volumes in different trials. Signal drift is found to have spatially varying effects that are not accounted for with temporal-only models. The novel model captures drift more accurately (i.e., reduces the overall change per-voxel over the course of a scan) and results in more consistent ADC metrics.

Least Squares Time-Series Synchronization in Image Acquisition Systems

We consider an acquisition system constituted by an array of sensors scanning an image. Each sensor produces a sequence of readouts, called a time series. In this framework, we discuss the image estimation problem when the time series are affected by noise and by a time shift. In particular, we introduce an appropriate data model and consider the least squares (LS) estimate, showing that it has no closed form. However, the LS problem has a structure that can be exploited to simplify the solution. In particular, based on two known techniques, namely, separable nonlinear LS and alternating LS, we propose and analyze several practical estimation methods. As an additional contribution, we discuss the application of these methods to the data of the photodetector array camera and spectrometer, which is an infrared photometer onboard the Herschel satellite. In this context, we investigate the accuracy and the computational complexity of the methods, using both true and simulated data.

Wavelets and statistical analysis of functional magnetic resonance images of the human brain

Wavelets provide an orthonormal basis for multiresolution analysis and decorrelation or ‘whitening’ of nonstationary time series and spatial processes. Wavelets are particularly well suited to analysis of biological signals and images, such as human brain imaging data, which often have fractal or scale-invariant properties. We briefly define some key properties of the discrete wavelet transform (DWT) and review its applications to statistical analysis of functional magnetic resonance imaging (fMRI) data. We focus on time series resampling by ‘wavestrapping’ of wavelet coefficients, methods for efficient linear model estimation in the wavelet domain, and wavelet-based methods for multiple hypothesis testing, all of which are somewhat simplified by the decorrelating property of the DWT.

Bayesian Models for fMRI Data Analysis

Functional magnetic resonance imaging (fMRI), a noninvasive neuroimaging method that provides an indirect measure of neuronal activity by detecting blood flow changes, has experienced an explosive growth in the past years. Statistical methods play a crucial role in understanding and analyzing fMRI data. Bayesian approaches, in particular, have shown great promise in applications. A remarkable feature of fully Bayesian approaches is that they allow a flexible modeling of spatial and temporal correlations in the data. This paper provides a review of the most relevant models developed in recent years. We divide methods according to the objective of the analysis. We start from spatio-temporal models for fMRI data that detect task-related activation patterns. We then address the very important problem of estimating brain connectivity. We also touch upon methods that focus on making predictions of an individual's brain activity or a clinical or behavioral response. We conclude with a discussion of recent integrative models that aim at combining fMRI data with other imaging modalities, such as EEG/MEG and DTI data, measured on the same subjects. We also briefly discuss the emerging field of imaging genetics.

An integrated framework for joint HRF and drift estimation and HbO/HbR signal improvement in fNIRS data

Nonparametric hemodynamic response function (HRF) estimation in functional near-infrared spectroscopy (fNIRS) data plays an important role when investigating the temporal dynamics of a brain region response during activations. Assuming the drift arising from both physical and physiological effects in fNIRS data is Lipschitz continuous; a novel algorithm for joint HRF and drift estimation is derived in this paper. The proposed algorithm estimates the HRF by applying a first-order differencing to the fNIRS time series samples in order to remove the drift effect. An estimate of the drift is then obtained using a wavelet thresholding technique applied to the residuals generated by removing the estimated induced activation response from the fNIRS time-series. It is shown that the proposed HRF estimator is √N consistent whereas the estimator of the drift is asymptotically optimal. The de-drifted fNIRS oxygenated (HbO) and deoxygenated (HbR) hemoglobin responses are then obtained by removing the corresponding estimated drifts from the fNIRS time-series. Its performance is assessed using both simulated and real fNIRS data sets. The application results reveal that the proposed joint HRF and drift estimation method is efficient both computationally and in terms of accuracy. In comparison to traditional model based methods used for HRF estimation, the proposed novel method avoids the selection of a model to remove the drift component. As a result, the proposed method finds an optimal estimate of the fNIRS drift and offers a model-free approach to de-drift the HbO/HbR responses.

A spatio-temporal nonparametric Bayesian variable selection model of fMRI data for clustering correlated time courses

In this paper we present a novel wavelet-based Bayesian nonparametric regression model for the analysis of functional magnetic resonance imaging (fMRI) data. Our goal is to provide a joint analytical framework that allows to detect regions of the brain which exhibit neuronal activity in response to a stimulus and, simultaneously, infer the association, or clustering, of spatially remote voxels that exhibit fMRI time series with similar characteristics. We start by modeling the data with a hemodynamic response function (HRF) with a voxel-dependent shape parameter. We detect regions of the brain activated in response to a given stimulus by using mixture priors with a spike at zero on the coefficients of the regression model. We account for the complex spatial correlation structure of the brain by using a Markov random field (MRF) prior on the parameters guiding the selection of the activated voxels, therefore capturing correlation among nearby voxels. In order to infer association of the voxel time courses, we assume correlated errors, in particular long memory, and exploit the whitening properties of discrete wavelet transforms. Furthermore, we achieve clustering of the voxels by imposing a Dirichlet process (DP) prior on the parameters of the long memory process. For inference, we use Markov Chain Monte Carlo (MCMC) sampling techniques that combine Metropolis-Hastings schemes employed in Bayesian variable selection with sampling algorithms for nonparametric DP models. We explore the performance of the proposed model on simulated data, with both block- and event-related design, and on real fMRI data.

A kernel machine-based fMRI physiological noise removal method

Functional magnetic resonance imaging (fMRI) technique with blood oxygenation level dependent (BOLD) contrast is a powerful tool for noninvasive mapping of brain function under task and resting states. The removal of cardiac- and respiration-induced physiological noise in fMRI data has been a significant challenge as fMRI studies seek to achieve higher spatial resolutions and characterize more subtle neuronal changes. The low temporal sampling rate of most multi-slice fMRI experiments often causes aliasing of physiological noise into the frequency range of BOLD activation signal. In addition, changes of heartbeat and respiration patterns also generate physiological fluctuations that have similar frequencies with BOLD activation. Most existing physiological noise-removal methods either place restrictive limitations on image acquisition or utilize filtering or regression based post-processing algorithms, which cannot distinguish the frequency-overlapping BOLD activation and the physiological noise. In this work, we address the challenge of physiological noise removal via the kernel machine technique, where a nonlinear kernel machine technique, kernel principal component analysis, is used with a specifically identified kernel function to differentiate BOLD signal from the physiological noise of the frequency. The proposed method was evaluated in human fMRI data acquired from multiple task-related and resting state fMRI experiments. A comparison study was also performed with an existing adaptive filtering method. The results indicate that the proposed method can effectively identify and reduce the physiological noise in fMRI data. The comparison study shows that the proposed method can provide comparable or better noise removal performance than the adaptive filtering approach.

A wavelet-based Bayesian approach to regression models with long memory errors and its application to FMRI data

This article considers linear regression models with long memory errors. These models have been proven useful for application in many areas, such as medical imaging, signal processing, and econometrics. Wavelets, being self-similar, have a strong connection to long memory data. Here we employ discrete wavelet transforms as whitening filters to simplify the dense variance-covariance matrix of the data. We then adopt a Bayesian approach for the estimation of the model parameters. Our inferential procedure uses exact wavelet coefficients variances and leads to accurate estimates of the model parameters. We explore performances on simulated data and present an application to an fMRI data set. In the application we produce posterior probability maps (PPMs) that aid interpretation by identifying voxels that are likely activated with a given confidence.

Bayesian hierarchical multi-subject multiscale analysis of functional MRI data

We develop a methodology for Bayesian hierarchical multi-subject multiscale analysis of functional Magnetic Resonance Imaging (fMRI) data. We begin by modeling the brain images temporally with a standard general linear model. After that, we transform the resulting estimated standardized regression coefficient maps through a discrete wavelet transformation to obtain a sparse representation in the wavelet space. Subsequently, we assign to the wavelet coefficients a prior that is a mixture of a point mass at zero and a Gaussian white noise. In this mixture prior for the wavelet coefficients, the mixture probabilities are related to the pattern of brain activity across different resolutions. To incorporate this information, we assume that the mixture probabilities for wavelet coefficients at the same location and level are common across subjects. Furthermore, we assign for the mixture probabilities a prior that depends on a few hyperparameters. We develop an empirical Bayes methodology to estimate the hyperparameters and, as these hyperparameters are shared by all subjects, we obtain precise estimated values. Then we carry out inference in the wavelet space and obtain smoothed images of the regression coefficients by applying the inverse wavelet transform to the posterior means of the wavelet coefficients. An application to computer simulated synthetic data has shown that, when compared to single-subject analysis, our multi-subject methodology performs better in terms of mean squared error. Finally, we illustrate the utility and flexibility of our multi-subject methodology with an application to an event-related fMRI dataset generated by Postle (2005) through a multi-subject fMRI study of working memory related brain activation.

Concurrent bias correction in hemodynamic data assimilation

Low-frequency drift in fMRI datasets can be caused by various sources and are generally not of interest in a conventional task-based fMRI experiment. This feature complicates the assimilation approach that is always under specific assumption on statistics of system uncertainties. In this paper, we present a novel approach to the assimilation of nonlinear hemodynamic system with stochastic biased noise. By treating the drift variation as a random-walk process, the assimilation problem was translated into the identification of a nonlinear system in the presence of time-varying bias. We developed a bias aware unscented Kalman estimator to efficiently handle this problem. In this framework, the estimates of bias-free states and drift are separately carried out in two parallel filters, the optimal estimates of the system states then are corrected from bias-free states with drift estimates. The approach can simultaneously deal with the fMRI responses and drift in an assimilation cycle in an on-line fashion. It makes no assumptions of the structure and statistics of the drift, thereby is particularly suited for fMRI imaging where the formulation of real drift remains difficult to acquire. Experiments with synthetic data and real fMRI data are performed to demonstrate feasibility of our approach and to explore its potential advantages over classic polynomial approach. Moreover, we include the comparison of the variability of observables from the scanner and of normalized signal used in assimilation procedure in Appendix.

HRF estimation in fMRI data with an unknown drift matrix by iterative minimization of the Kullback-Leibler divergence

Hemodynamic response function (HRF) estimation in noisy functional magnetic resonance imaging (fMRI) plays an important role when investigating the temporal dynamic of a brain region response during activations. Nonparametric methods which allow more flexibility in the estimation by inferring the HRF at each time sample have provided improved performance in comparison to the parametric methods. In this paper, the mixed-effects model is used to derive a new algorithm for nonparametric maximum likelihood HRF estimation. In this model, the random effect is used to better account for the variability of the drift. Contrary to the usual approaches, the proposed algorithm has the benefit of considering an unknown and therefore flexible drift matrix. This allows the effective representation of a broader class of drift signals and therefore the reduction of the error in approximating the drift component. Estimates of the HRF and the hyperparameters are derived by iterative minimization of the Kullback-Leibler divergence between a model family of probability distributions defined using the mixed-effects model and a desired family of probability distributions constrained to be concentrated on the observed data. The performance of proposed method is demonstrated on simulated and real fMRI data, the latter originating from both event-related and block design fMRI experiments.

Quantitative analysis of hemodynamic and metabolic changes in subcortical vascular dementia using simultaneous near-infrared spectroscopy and fMRI measurements

Subcortical vascular dementia (SVD) is a form of vascular dementia from small vessel disease with white matter lesions and lacunes. We hypothesized that hemodynamic and metabolic changes in the cortex during a simple motor task may reflect the impaired neurovascular coupling in SVD. We used fMRI and near-infrared spectroscopy (NIRS) simultaneously, which together provided multiple hemodynamic responses as well as a robust estimation of the cerebral metabolic rate of oxygen (CMRO(2)). During the task periods, the oxy-hemoglobin, total-hemoglobin, blood oxygenation level-dependent (BOLD) response, cerebral blood flow (CBF), and CMRO(2) decreased statistically significantly in the primary motor and somatosensory cortices of SVD patients, whereas the oxygen extraction fraction increased when compared with controls. Notably, the flow-metabolism coupling ratio, n representing the ratio of oxygen supply to its utilization, showed a robust reduction in the SVD patient group (n(Control)=1.99 ± 0.23; n(SVD)=1.08 ± 0.24), which implies a loss of metabolic reserve. These results support the pathological small vessel compromise, including an increased vessel stiffness, impaired vascular reactivity, and impaired neurovascular coupling in SVD. In conclusion, simultaneous measurement by NIRS and fMRI can reveal various hemodynamic and metabolic changes and may be used for as an early detection or monitoring of SVD.

Quantification of CMRO(2) without hypercapnia using simultaneous near-infrared spectroscopy and fMRI measurements

Estimation of the cerebral metabolic rate of oxygen (CMRO(2)) and cerebral blood flow (CBF) is important to investigate the neurovascular coupling and physiological components in blood oxygenation level-dependent (BOLD) signals quantitatively. Although there are methods that can determine CMRO(2) changes using functional MRI (fMRI) or near-infrared spectroscopy (NIRS), current approaches require a separate hypercapnia calibration process and have the potential to incur bias in many assumed model parameters. In this paper, a novel method to estimate CMRO(2) without hypercapnia is described using simultaneous measurements of NIRS and fMRI. Specifically, an optimization framework is proposed that minimizes the differences between the two forms of the relative CMRO(2)-CBF coupling ratio from BOLD and NIRS biophysical models, from which hypercapnia calibration and model parameters are readily estimated. Based on the new methods, we found that group average CBF, CMRO(2), cerebral blood volume (CBV), and BOLD changes within activation of the primary motor cortex during a finger tapping task increased by 39.5 +/- 21.4%, 18.4 +/- 8.7%, 12.9 +/- 6.7%, and 0.5 +/- 0.2%, respectively. The group average estimated flow-metabolism coupling ratio was 2.38 +/- 0.65 and the hypercapnia parameter was 7.7 +/- 1.7%. These values are within the range of values reported from other literatures. Furthermore, the activation maps from CBF and CMRO(2) were well localized on the primary motor cortex, which is the main target region of the finger tapping task.

Detection of small bowel slow-wave frequencies from noninvasive biomagnetic measurements

We report a novel method for identifying the small intestine electrical activity slow-wave frequencies (SWFs) from noninvasive biomagnetic measurements. Superconducting quantum interference device magnetometer measurements are preprocessed to remove baseline drift and high-frequency noise. Subsequently, the underlying source signals are separated using the well-known second-order blind identification (SOBI) algorithm. A simple classification scheme identifies and assigns some of the SOBI components to a section of small bowel. SWFs were clearly identified in 10 out of 12 test subjects to within 0.09-0.25 cycles per minute. The method is sensitive at the 40.3 %-55.9 % level, while false positive rates were 0 %-8.6 %. This technique could potentially be used to help diagnose gastrointestinal ailments and obviate some exploratory surgeries.

Wavelet minimum description length detrending for near-infrared spectroscopy

Near-infrared spectroscopy (NIRS) can be employed to investigate brain activities associated with regional changes of the oxy- and deoxyhemoglobin concentration by measuring the absorption of near-infrared light through the intact skull. NIRS is regarded as a promising neuroimaging modality thanks to its excellent temporal resolution and flexibility for routine monitoring. Recently, the general linear model (GLM), which is a standard method for functional MRI (fMRI) analysis, has been employed for quantitative analysis of NIRS data. However, the GLM often fails in NIRS when there exists an unknown global trend due to breathing, cardiac, vasomotion, or other experimental errors. We propose a wavelet minimum description length (Wavelet-MDL) detrending algorithm to overcome this problem. Specifically, the wavelet transform is applied to decompose NIRS measurements into global trends, hemodynamic signals, and uncorrelated noise components at distinct scales. The minimum description length (MDL) principle plays an important role in preventing over- or underfitting and facilitates optimal model order selection for the global trend estimate. Experimental results demonstrate that the new detrending algorithm outperforms the conventional approaches.

1/f noise in diffuse optical imaging and wavelet-based response estimation

In diffuse optical imaging (DOI) data analysis, the functional response is contaminated with physiological noise as in functional magnetic resonance imaging (fMRI). In this work we extend a previously proposed method for fMRI to estimate the parameters of a linear model of DOI time series. The regression is performed in the wavelet domain to infer drift coefficients at different scales and to estimate the strength of the hemodynamic response function (HRF). This multiresolution approach benefits from the whitening property of the discrete wavelet transform (DWT), which approximately decorrelates long-memory noise processes. We also show that a more accurate estimation is obtained by removing some regressors correlating with the protocol. Moreover, we observe that this improvement is related to a quantitative measure of 1/f noise. The performances of the method are first evaluated against a standard spline-cosine drift approach with simulated HRF and real background physiology. Lastly, the technique is applied to experimental event-related data acquired by near-infrared spectroscopy (NIRS).

Data-driven haemodynamic response function extraction using Fourier-wavelet regularised deconvolution

Background

We present a simple, data-driven method to extract haemodynamic response functions (HRF) from functional magnetic resonance imaging (fMRI) time series, based on the Fourier-wavelet regularised deconvolution (ForWaRD) technique. HRF data are required for many fMRI applications, such as defining region-specific HRFs, effciently representing a general HRF, or comparing subject-specific HRFs.

Results

ForWaRD is applied to fMRI time signals, after removing low-frequency trends by a wavelet-based method, and the output of ForWaRD is a time series of volumes, containing the HRF in each voxel. Compared to more complex methods, this extraction algorithm requires few assumptions (separability of signal and noise in the frequency and wavelet domains and the general linear model) and it is fast (HRF extraction from a single fMRI data set takes about the same time as spatial resampling). The extraction method is tested on simulated event-related activation signals, contaminated with noise from a time series of real MRI images. An application for HRF data is demonstrated in a simple event-related experiment: data are extracted from a region with significant effects of interest in a first time series. A continuous-time HRF is obtained by fitting a nonlinear function to the discrete HRF coeffcients, and is then used to analyse a later time series.

Conclusion

With the parameters used in this paper, the extraction method presented here is very robust to changes in signal properties. Comparison of analyses with fitted HRFs and with a canonical HRF shows that a subject-specific, regional HRF significantly improves detection power. Sensitivity and specificity increase not only in the region from which the HRFs are extracted, but also in other regions of interest.

Classification of fMRI time series in a low-dimensional subspace with a spatial prior

We propose a new method for detecting activation in functional magnetic resonance imaging (fMRI) data. We project the fMRI time series on a low-dimensional subspace spanned by wavelet packets in order to create projections that are as non-Gaussian as possible. Our approach achieves two goals: it reduces the dimensionality of the problem by explicitly constructing a sparse approximation to the dataset and it also creates meaningful clusters allowing the separation of the activated regions from the clutter formed by the background time series. We use a mixture of Gaussian densities to model the distribution of the wavelet packet coefficients. We expect activated areas that are connected, and impose a spatial prior in the form of a Markov random field. Our approach was validated with in vivo data and realistic synthetic data, where it outperformed a linear model equipped with the knowledge of the true hemodynamic response.

fMRI data analysis with nonstationary noise models: a Bayesian approach

The assumption of noise stationarity in the functional magnetic resonance imaging (fMRI) data analysis may lead to the loss of crucial dynamic features of the data and thus result in inaccurate activation detection. In this paper, a Bayesian approach is proposed to analyze the fMRI data with two nonstationary noise models (the time-varying variance noise model and the fractional noise model). The covariance matrices of the time-varying variance noise and the fractional noise after wavelet transform are diagonal matrices. This property is investigated under the Bayesian framework. The Bayesian estimator not only gives an accurate estimate of the weights in general linear model, but also provides posterior probability of activation in a voxel and, hence, avoids the limitations (i.e., using only hypothesis testing) in the classical methods. The performance of the proposed Bayesian methods (under the assumption of different noise models) are compared with the ordinary least squares (OLS) and the weighted least squares (WLS) methods. Results from the simulation studies validate the superiority of the proposed approach to the OLS and WLS methods considering the complex noise structures in the fMRI data.

Activation detection in diffuse optical imaging by means of the general linear model

Due to its non-invasive nature and low cost, diffuse optical imaging (DOI) is becoming a commonly used technique to assess functional activation in the brain. When imaging with DOI, two major issues arise in the data analysis: (i) the separation of noise of physiological origin and the recovery of the functional response; (ii) the tomographic image reconstruction problem. This paper focuses on the first issue. Although the general linear model (GLM) has been extensively used in functional magnetic resonance imaging (fMRI), DOI has mostly relied on filtering and averaging of raw data to recover brain functional activation. This is mainly due to the high temporal resolution of DOI which implies a new design of the drift basis modelling physiology. In this paper, we provide (i) a filtering method based on cosine functions that is more adapted than standard averaging techniques for DOI specifically; (ii) a new mode-locking technique to recover small signals and locate them temporally with high precision (shift method). Results on real data show the capability of the shift method to retrieve HbR and HbO(2) peak locations.

Medical Image Compression using Wavelet Transform on Mobile Devices with ROI coding support

Medical applications have already been integrated into mobile devices (e.g. Tablet PC's and PDA's) and are being used by medical personnel in treatment centers, for retrieving and examining patient data and medical images. Network transmission and image data processing are key issues in such platforms, due to the significant image file sizes. Wavelet transform has been considered to be a highly efficient technique of image compression resulting in both lossless and lossy compressed images of great accuracy, enabling its use on medical images. This paper discusses a Picture Archiving and Communication Systems (PACS) application designed for viewing DICOM compliant medical images using Wavelet compression with ROI coding support, on mobile devices. In addition, it presents initial results from its pilot application and demonstrates its performance over heterogeneous radio network segments, like IEEE 802.11b, GPRS and DVB-H.

An efficient lossless compression algorithm for FMRI data volume

This paper presents an efficient lossless compression scheme designed for the fMRI data volume. An fMRI time series is composed of trend, noise and external stimulus response, which have different characteristics in wavelet domain. Reversible integer wavelet transform is applied to the fMRI time series. The smooth trend signals are extracted from the low frequency wavelet bands and they are coded by a lossy compression scheme. The lossy compression residual, noise, and other components are inversely transformed into time domain followed by adaptive clustering and then lossless coding. The experimental results show that the proposed method outperforms many existing schemes.

Hidden Markov multiple event sequence models: A paradigm for the spatio-temporal analysis of fMRI data

This paper presents a novel, completely unsupervised fMRI brain mapping method that addresses the three problems of hemodynamic response function (HRF) variability, hemodynamic event timing, and fMRI response non-linearity. Spatial and temporal information are directly taken into account into the core of the activation detection process. In practice, activation detection at voxel v is formulated in terms of temporal alignment between sequences of hemodynamic response onsets (HROs) detected in the fMRI signal at v and in the spatial neighborhood of v, and the input sequence of stimuli or stimulus onsets. Event-related and epoch paradigms are considered. The multiple event sequence alignment problem is solved within the probabilistic framework of hidden Markov multiple event sequence models (HMMESMs), a new class of hidden Markov models. Results obtained on real and synthetic data significantly outperform those obtained with the popular statistical parametric mapping (SPM2) method without requiring any prior definition of the expected activation patterns, the HMMESM mapping approach being completely unsupervised.

Wavelet-based estimation of hemodynamic response function from fMRI data

We present a new algorithm to estimate hemodynamic response function (HRF) and drift components of fMRI data in wavelet domain. The HRF is modeled by both parametric and nonparametric models. The functional Magnetic resonance Image (fMRI) noise is modeled as a fractional brownian motion (fBm). The HRF parameters are estimated in wavelet domain by exploiting the property that wavelet transforms with a sufficient number of vanishing moments decorrelates a fBm process. Using this property, the noise covariance matrix in wavelet domain can be assumed to be diagonal whose entries are estimated using the sample variance estimator at each scale. We study the influence of the sampling rate of fMRI time series and shape assumption of HRF on the estimation performance. Results are presented by adding synthetic HRFs on simulated and null fMRI data. We also compare these methods with an existing method,(1) where correlated fMRI noise is modeled by a second order polynomial functions.

A wavelet-based statistical analysis of FMRI data: I. motivation and data distribution modeling

We propose a new method for statistical analysis of functional magnetic resonance imaging (fMRI) data. The discrete wavelet transformation is employed as a tool for efficient and robust signal representation. We use structural magnetic resonance imaging (MRI) and fMRI to empirically estimate the distribution of the wavelet coefficients of the data both across individuals and spatial locations. An anatomical subvolume probabilistic atlas is used to tessellate the structural and functional signals into smaller regions each of which is processed separately. A frequency-adaptive wavelet shrinkage scheme is employed to obtain essentially optimal estimations of the signals in the wavelet space. The empirical distributions of the signals on all the regions are computed in a compressed wavelet space. These are modeled by heavy-tail distributions because their histograms exhibit slower tail decay than the Gaussian. We discovered that the Cauchy, Bessel K Forms, and Pareto distributions provide the most accurate asymptotic models for the distribution of the wavelet coefficients of the data. Finally, we propose a new model for statistical analysis of functional MRI data using this atlas-based wavelet space representation. In the second part of our investigation, we will apply this technique to analyze a large fMRI dataset involving repeated presentation of sensory-motor response stimuli in young, elderly, and demented subjects.

Managing variability in the summary and comparison of gait data

Variability in quantitative gait data arises from many potential sources, including natural temporal dynamics of neuromotor control, pathologies of the neurological or musculoskeletal systems, the effects of aging, as well as variations in the external environment, assistive devices, instrumentation or data collection methodologies. In light of this variability, unidimensional, cycle-based gait variables such as stride period should be viewed as random variables and prototypical single-cycle kinematic or kinetic curves ought to be considered as random functions of time. Within this framework, we exemplify some practical solutions to a number of commonly encountered analytical challenges in dealing with gait variability. On the topic of univariate gait variables, robust estimation is proposed as a means of coping with contaminated gait data, and the summary of non-normally distributed gait data is demonstrated by way of empirical examples. On the summary of gait curves, we discuss methods to manage undesirable phase variation and non-robust spread estimates. To overcome the limitations of conventional comparisons among curve landmarks or parameters, we propose as a viable alternative, the combination of curve registration, robust estimation, and formal statistical testing of curves as coherent units. On the basis of these discussions, we provide heuristic guidelines for the summary of gait variables and the comparison of gait curves.

Unsupervised learning and mapping of active brain functional MRI signals based on hidden semi-Markov event sequence models

In this paper, a novel functional magnetic resonance imaging (fMRI) brain mapping method is presented within the statistical modeling framework of hidden semi-Markov event sequence models (HSMESMs). Neural activation detection is formulated at the voxel level in terms of time coupling between the sequence of hemodynamic response onsets (HROs) observed in the fMRI signal, and an HSMESM of the hidden sequence of task-induced neural activations. The sequence of HRO events is derived from a continuous wavelet transform (CWT) of the fMRI signal. The brain activation HSMESM is built from the timing information of the input stimulation protocol. The rich mathematical framework of HSMESMs makes these models an effective and versatile approach for fMRI data analysis. Solving for the HSMESM Evaluation and Learning problems enables the model to automatically detect neural activation embedded in a given set of fMRI signals, without requiring any template basis function or prior shape assumption for the fMRI response. Solving for the HSMESM Decoding problem allows to enrich brain mapping with activation lag mapping, activation mode visualizing, and hemodynamic response function analysis. Activation detection results obtained on synthetic and real epoch-related fMRI data demonstrate the superiority of the HSMESM mapping method with respect to a real application case of the statistical parametric mapping (SPM) approach. In addition, the HSMESM mapping method appears clearly insensitive to timing variations of the hemodynamic response, and exhibits low sensitivity to fluctuations of its shape.

Wavelets and functional magnetic resonance imaging of the human brain

The discrete wavelet transform (DWT) is widely used for multiresolution analysis and decorrelation or "whitening" of nonstationary time series and spatial processes. Wavelets are naturally appropriate for analysis of biological data, such as functional magnetic resonance images of the human brain, which often demonstrate scale invariant or fractal properties. We provide a brief formal introduction to key properties of the DWT and review the growing literature on its application to fMRI. We focus on three applications in particular: (i) wavelet coefficient resampling or "wavestrapping" of 1-D time series, 2- to 3-D spatial maps and 4-D spatiotemporal processes; (ii) wavelet-based estimators for signal and noise parameters of time series regression models assuming the errors are fractional Gaussian noise (fGn); and (iii) wavelet shrinkage in frequentist and Bayesian frameworks to support multiresolution hypothesis testing on spatially extended statistic maps. We conclude that the wavelet domain is a rich source of new concepts and techniques to enhance the power of statistical analysis of human fMRI data.

Wavelet-based multifractal analysis of fMRI time series

Functional magnetic resonance imaging (fMRI) time series are investigated with a multifractal method based on the Wavelet Modulus Maxima (WTMM) method to extract local singularity ("fractal") exponents. The spectrum of singularity exponents of each fMRI time series is quantified by spectral characteristics including its maximum and the corresponding dimension. We found that the range of Hölder exponents in voxels with activation is close to 1, whereas exponents are close to 0.5 in white matter voxels without activation. The maximum dimension decreases going from white matter to gray matter, and is lower still for activated time series. The full-width-at-half-maximum of the spectra is higher in activated areas. The proposed method becomes particularly effective when combining these spectral characteristics into a single parameter. Using these multifractal parameters, it is possible to identify activated areas in the human brain in both hybrid and in vivo fMRI data sets without knowledge of the stimulation paradigm applied.

Detection and detrending in fMRI data analysis

This article addresses the impact that colored noise, temporal filtering, and temporal detrending have on the fMRI analysis situation. Specifically, it is shown why the detection of event-related designs benefit more from pre-whitening than blocked designs in a colored noise structure. Both theoretical and empirical results are provided. Furthermore, a novel exploratory method for producing drift models that efficiently capture trends and drifts in the fMRI data is introduced. A comparison to currently employed detrending approaches is presented. It is shown that the novel exploratory model is able to remove a major part of the slowly varying drifts that are abundant in fMRI data. The value of such a model lies in its ability to remove drift components that otherwise would have contributed to a colored noise structure in the voxel time series.

Physiological basis and image processing in functional magnetic resonance imaging: neuronal and motor activity in brain

Functional magnetic resonance imaging (fMRI) is recently developing as imaging modality used for mapping hemodynamics of neuronal and motor event related tissue blood oxygen level dependence (BOLD) in terms of brain activation. Image processing is performed by segmentation and registration methods. Segmentation algorithms provide brain surface-based analysis, automated anatomical labeling of cortical fields in magnetic resonance data sets based on oxygen metabolic state. Registration algorithms provide geometric features using two or more imaging modalities to assure clinically useful neuronal and motor information of brain activation. This review article summarizes the physiological basis of fMRI signal, its origin, contrast enhancement, physical factors, anatomical labeling by segmentation, registration approaches with examples of visual and motor activity in brain. Latest developments are reviewed for clinical applications of fMRI along with other different neurophysiological and imaging modalities.