Maxima of moving sums in a Poisson random field

In this paper we examine the extremal tail probabilities of moving sums in a marked Poisson random field. These sums are computed by adding up the weighted occurrences of events lying within a scanning set of fixed shape and size. We also provide an alternative representation of the constants of the asymptotic formulae in terms of the occupation measure of the conditional local random field at zero, and extend these representations to the constants of asymptotic tail probabilities of Gaussian random fields.

Maxima of moving sums in a Poisson random field

In this paper we examine the extremal tail probabilities of moving sums in a marked Poisson random field. These sums are computed by adding up the weighted occurrences of events lying within a scanning set of fixed shape and size. We also provide an alternative representation of the constants of the asymptotic formulae in terms of the occupation measure of the conditional local random field at zero, and extend these representations to the constants of asymptotic tail probabilities of Gaussian random fields.

Statistical analysis of brain tissue images in the wavelet domain: wavelet-based morphometry

Wavelet-based methods have been developed for statistical analysis of functional MRI and PET data, where the wavelet transformation is employed as a tool for efficient signal representation. A number of studies using these approaches have reported better estimation capabilities, in terms of increased sensitivity and specificity, than the standard statistical analyses in the spatial domain. In line with these previous studies, the present report proposes a statistical analysis in the wavelet domain for the estimation of inter-group differences from structural MRI data. The procedure, called wavelet-based morphometry (WBM), was implemented under a voxel-based morphometry (VBM) style analysis. It was evaluated by comparing the gray-matter images of a group of 32 healthy subjects whose images were artificially altered to induce thinning of the cortex, with a different group of 32 healthy subjects whose images were unaltered. In order to quantify the performance of the reconstruction from a practical perspective, the same comparison was also conducted with standard VBM using SPM's Gaussian random fields and FSL's cluster-based statistics, family-wise error corrected, for datasets spatially-normalized via two different registration methods (i.e., SyN and FNIRT). The effect of using different amounts of smoothing, Battle-Lemarié filters and resolution levels in the wavelet transform was also investigated. Results support the proposed approach as a different and promising methodology to assess the structural morphometric differences between different populations of subjects.

Modeling brain activation in fMRI using group MRF

Noise confounds present serious complications to functional magnetic resonance imaging (fMRI) analysis. The amount of discernible signals within a single dataset of a subject is often inadequate to obtain satisfactory intra-subject activation detection. To remedy this limitation, we propose a novel group Markov random field (GMRF) that extends each subject's neighborhood system to other subjects to enable information coalescing. A distinct advantage of GMRF over standard fMRI group analysis is that no stringent one-to-one voxel correspondence is required. Instead, intra- and inter-subject neighboring voxels are jointly regularized to encourage spatially proximal voxels to be assigned similar labels across subjects. Our proposed group-extended graph structure thus provides an effective means for handling inter-subject variability. Also, adopting a group-wise approach by integrating group information into intra-subject activation, as opposed to estimating a single average group map, permits inter-subject differences to be characterized and studied. GMRF can be elegantly implemented as a single MRF, thus enabling all subjects' activation maps to be simultaneously and collaboratively segmented with global optimality guaranteed in the case of binary labeling. We validate our technique on synthetic and real fMRI data and demonstrate GMRF's superior performance over standard fMRI analysis.

TWO-STAGE EMPIRICAL LIKELIHOOD FOR LONGITUDINAL NEUROIMAGING DATA

Longitudinal imaging studies are essential to understanding the neural development of neuropsychiatric disorders, substance use disorders, and the normal brain. The main objective of this paper is to develop a two-stage adjusted exponentially tilted empirical likelihood (TAETEL) for the spatial analysis of neuroimaging data from longitudinal studies. The TAETEL method allows us to efficiently analyze longitudinal data without correctly modeling temporal correlation and to classify different time-dependent covariate types. To account for spatial dependence, the TAETEL method developed here specifically combines all the data in the neighborhood of each voxel (or pixel) on a 3 dimensional (3D) volume (or 2D surface) with appropriate weights to calculate adaptive parameter estimates and adaptive test statistics. Simulation studies are used to examine the finite sample performance of the adjusted exponential tilted likelihood ratio statistic and TAETEL. We demonstrate the application of our statistical methods to the detection of the difference in the morphological changes of the hippocampus across time between schizophrenia patients and healthy subjects in a longitudinal schizophrenia study.

Variability of fMRI-response patterns at different spatial observation scales

Functional organization units of the cerebral cortex exist over a wide range of spatial scales, from local circuits to entire cortical areas. In the last decades, scale-space representations of neuroimaging data suited to probe the multi-scale nature of cortical functional organization have been introduced and methodologically elaborated. For this purpose, responses are statistically detected over a range of spatial scales using a family of Gaussian filters, with small filters being related to fine and large filters-to coarse spatial scales. The goal of the present study was to investigate the degree of variability of fMRI-response patterns over a broad range of observation scales. To this aim, the same fMRI data set obtained from 18 subjects during a visuomotor task was analyzed with a range of filters from 4- to 16-mm full width at half-maximum (FWHM). We found substantial observation-scale-related variability. For example, using filter widths of 6- to 8-mm FWHM, in the group-level results, significant responses in the right secondary visual but not in the primary visual cortex were detected. However, when larger filters were used, the responses in the right primary visual cortex reached significance. Often, responses in probabilistically defined areas were significant when both small and large filters, but not intermediate filter widths were applied. This suggests that brain responses can be organized in local clusters of multiple distinct activation foci. Our findings illustrate the potential of multi-scale fMRI analysis to reveal novel features in the spatial organization of human brain responses.

Spatial smoothing in fMRI using prolate spheroidal wave functions

The acquisition of functional magnetic resonance imaging (fMRI) data in a finite subset of k-space produces ring-artifacts and 'side lobes' that distort the image. In this article, we explore the consequences of this problem for functional imaging studies, which can be considerable, and propose a solution. The truncation of k-space is mathematically equivalent to convolving the underlying "true" image with a sinc function whose width is inversely related to the amount of truncation. Spatial smoothing with a large enough kernel can eliminate these artifacts, but at a cost in image resolution. However, too little spatial smoothing leaves the ringing artifacts and side lobes caused by k-space truncation intact, leading to a potential decrease in signal-to-noise ratio and statistical power. Thus, to make use of the high-resolution afforded by MRI without introducing artifacts, new smoothing filters are needed that are optimized to correct k-space truncation-related artifacts. We develop a prolate spheroidal wave function (PSWF) filter designed to eliminate truncation artifacts and compare its performance to the standard Gaussian filter in simulations and analysis of fMRI data on a visual-motor task. The PSWF filter effectively corrected truncation artifacts and resulted in more sensitive detection of visual-motor activity in expected brain regions, demonstrating its efficacy.

Diffusion-based spatial priors for functional magnetic resonance images

We recently outlined a Bayesian scheme for analyzing fMRI data using diffusion-based spatial priors [Harrison, L.M., Penny, W., Ashburner, J., Trujillo-Barreto, N., Friston, K.J., 2007. Diffusion-based spatial priors for imaging. NeuroImage 38, 677–695]. The current paper continues this theme, applying it to a single-subject functional magnetic resonance imaging (fMRI) study of the auditory system. We show that spatial priors on functional activations, based on diffusion, can be formulated in terms of the eigenmodes of a graph Laplacian. This allows one to discard eigenmodes with small eigenvalues, to provide a computationally efficient scheme. Furthermore, this formulation shows that diffusion-based priors are a generalization of conventional Laplacian priors [Penny, W.D., Trujillo-Barreto, N.J., Friston, K.J., 2005. Bayesian fMRI time series analysis with spatial priors. NeuroImage 24, 350–362]. Finally, we show how diffusion-based priors are a special case of Gaussian process models that can be inverted using classical covariance component estimation techniques like restricted maximum likelihood [Patterson, H.D., Thompson, R., 1974. Maximum likelihood estimation of components of variance. Paper presented at: 8th International Biometrics Conference (Constanta, Romania)]. The convention in SPM is to smooth data with a fixed isotropic Gaussian kernel before inverting a mass-univariate statistical model. This entails the strong assumption that data are generated smoothly throughout the brain. However, there is no way to determine if this assumption is supported by the data, because data are smoothed before statistical modeling. In contrast, if a spatial prior is used, smoothness is estimated given non-smoothed data. Explicit spatial priors enable formal model comparison of different prior assumptions, e.g., that data are generated from a stationary (i.e., fixed throughout the brain) or non-stationary spatial process. Indeed, for the auditory data we provide strong evidence for a non-stationary process, which concurs with a qualitative comparison of predicted activations at the boundary of functionally selective regions.

ROC evaluation of statistical wavelet-based analysis of brain activation in [15O]-H2O PET scans

This paper presents and evaluates a wavelet-based statistical analysis of PET images for the detection of brain activation areas. Brain regions showing significant activations were obtained by performing Student's t tests in the wavelet domain, reconstructing the final image from only those wavelet coefficients that passed the statistical test at a given significance level, and discarding artifacts introduced during the reconstruction process. Using Receiver Operating Characteristic (ROC) curves, we have compared this statistical analysis in the wavelet domain to the conventional image-domain Statistical Parametric Mapping (SPM) method. For obtaining an accurate assessment of sensitivity and specificity, we have simulated realistic single subject [15O]-H2O PET studies with different hyperactivation levels of the thalamic region. The results obtained from an ROC analysis show that the wavelet approach outperforms conventional SPM in identifying brain activation patterns. Using the wavelet method, activation areas detected were closer in size and shape to the region actually activated in the reference image.

Integrated wavelet processing and spatial statistical testing of fMRI data

We introduce an integrated framework for detecting brain activity from fMRI data, which is based on a spatial discrete wavelet transform. Unlike the standard wavelet-based approach for fMRI analysis, we apply the suitable statistical test procedure in the spatial domain. For a desired significance level, this scheme has one remaining degree of freedom, characterizing the wavelet processing, which is optimized according to the principle of minimal approximation error. This allows us to determine the threshold values in a way that does not depend on data. While developing our framework, we make only conservative assumptions. Consequently, the detection of activation is based on strong evidence. We have implemented this framework as a toolbox (WSPM) for the SPM2 software, taking advantage of multiple options and functions of SPM such as the setup of the linear model and the use of the hemodynamic response function. We show by experimental results that our method is able to detect activation patterns; the results are comparable to those obtained by SPM even though statistical assumptions are more conservative.