Techniques for approximating optimal linear estimators of multidimensional data

Multidimensional data are utilized in applications ranging from remote sensing to communications to astronomy to biomedical imaging. Due to a variety of factors, the fidelity of acquired data is often insufficient for interpretation or analysis of specific features. For example, in functional magnetic resonance imaging (fMRI), neural activity can be inferred from a four-dimensional dataset, but only in regions where the blood oxygenation level dependent (BOLD) contrast-to-noise (CNR) ratio is sufficiently large. The ability to accurately detect functional activation in areas of low CNR is important for both presurgical planning and neuroscience research.; Classic single-dimensional signal estimation techniques can typically be extended to multidimensional data. However, the reliance on a priori information or the sheer size of modern multidimensional datasets often make such methods impractical for use without some modification. This work explores techniques for constructing a blind approximation to the optimal linear estimator of a multidimensional signal and develops a general multidimensional estimation framework.; This framework is used to create estimators for four distinct applications. First, we create a blind estimator for hyperspectral and multispectral data that improves the average channel signal-to-noise ratio of a 0 dB observation by 16 dB. Second, we consider the problem of estimating a time-series of optical coherence tomography images and propose a blind estimator that improves visual image quality by reducing the speckle noise that is characteristic of coherent imaging. Next, a blind estimator for fMRI data is constructed that significantly improves the ability to detect low CNR functional activation in small regions of activation without a compromise to the false detection rate. Finally, the concepts developed for the multidimensional estimation framework are used to illustrate how regularized reconstruction of noisy projection data can be improved by exploiting the angular correlation of the true data. In the setting of a filtered back-projection (FBP) reconstruction scheme, this corresponds to performing the filtering step of the well known FBP method in a non-Radon domain. Doing so greatly improves the reconstruction quality of highly noisy projection data.