Image Estimation in the Presence of Irregular Sampling, Noise, and Pointing Jitter

We consider an acquisition system where a continuous image is reconstructed from a set of irregularly distributed noisy samples. Moreover, the system is affected by a random pointing jitter which makes the actual sampling positions different from the nominal ones. We develop a model for the system and derive the optimal, minimum variance unbiased (MVU) estimate. Unfortunately, the latter estimate is not practical to compute when the data size is large. Therefore, we develop a simplified, low resolution model and derive the corresponding MVU estimate, which has a drastically lower complexity. Moreover, we analyze the estimators' performance by using both theoretical analysis and simulations. Finally, we discuss the application to the data of the Photodetector Array Camera and Spectrometer (PACS) instrument, which is an infrared photometer on board the Herschel satellite.

Least Squares Image Estimation for Large Data in the Presence of Noise and Irregular Sampling

We consider an acquisition system where a continuous, band-limited image is reconstructed from a set of irregularly distributed, noisy samples. An optimal estimator can be obtained by exploiting Least Squares, but it is not practical to compute when the data size is large. A simpler, widely used estimate can be obtained by properly rounding off the pointing information, but it is suboptimal and is affected by a bias, which may be large and thus limits its applicability. To solve this problem, we develop a mathematical model for the acquisition system, which accounts for the pointing information round off. Based on the model, we derive a novel optimal estimate, which has a manageable computational complexity and is largely immune from the bias, making it a better option than the suboptimal one. Moreover, the model opens a new, fruitful point of view on the estimation performance analysis. Finally, we consider the application of the novel estimate to the data of the Photodetector Array Camera and Spectrometer instrument. In this paper, we discuss several implementation aspects and investigate the performance by using both true and simulated data.

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.

A semi-parametric nonlinear model for event-related fMRI

Nonlinearity in evoked hemodynamic responses often presents in event-related fMRI studies. Volterra series, a higher-order extension of linear convolution, has been used in the literature to construct a nonlinear characterization of hemodynamic responses. Estimation of the Volterra kernel coefficients in these models is usually challenging due to the large number of parameters. We propose a new semi-parametric model based on Volterra series for the hemodynamic responses that greatly reduces the number of parameters and enables "information borrowing" among subjects. This model assumes that in the same brain region and under the same stimulus, the hemodynamic responses across subjects share a common but unknown functional shape that can differ in magnitude, latency and degree of interaction. We develop a computationally-efficient strategy based on splines to estimate the model parameters, and a hypothesis test on nonlinearity. The proposed method is compared with several existing methods via extensive simulations, and is applied to a real event-related fMRI study.

A semi-parametric model of the hemodynamic response for multi-subject fMRI data

A semi-parametric model for estimating hemodynamic response function (HRF) from multi-subject fMRI data is introduced within the context of the General Linear Model. The new model assumes that the HRFs for a fixed brain voxel under a given stimulus share the same unknown functional form across subjects, but differ in height, time to peak, and width. A nonparametric spline-smoothing method is developed to evaluate this common functional form, based on which subject-specific characteristics of the HRFs can be estimated. This semi-parametric model explicitly characterizes the common properties shared across subjects and is flexible in describing various brain hemodynamic activities across different regions and stimuli. In addition, the temporal differentiability of the employed spline basis enables an easy-to-compute way of evaluating latency and width differences in hemodynamic activity. The proposed method is applied to data collected as part of an ongoing study of socially mediated emotion regulation. Comparison with several existing methods is conducted through simulations and real data analysis.

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.

A sparse and spatially constrained generative regression model for fMRI data analysis

In this study, we present an advanced Bayesian framework for the analysis of functional magnetic resonance imaging (fMRI) data that simultaneously employs both spatial and sparse properties. The basic building block of our method is the general linear regression model that constitutes a well-known probabilistic approach. By treating regression coefficients as random variables, we can apply an enhanced Gibbs distribution function that captures spatial constrains and at the same time allows sparse representation of fMRI time series. The proposed scheme is described as a maximum a posteriori approach, where the known expectation maximization algorithm is applied offering closed-form update equations for the model parameters. We have demonstrated that our method produces improved performance and functional activation detection capabilities in both simulated data and real applications.

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.