The goal of this work is to recover a sequence of sparse vectors, st; and a sequence of dense vectors, ℓt, that lie in a “slowly changing” low dimensional subspace, from time-varying undersampled linear projections of their sum. This type of problem typically occurs when the quantity being imaged can be split into a sum of two layers, one of which is sparse and the other is low-dimensional. A key application where this problem occurs is in undersampled functional magnetic resonance imaging (fMRI) to detect brain activation patterns in response to a stimulus. The brain image at time t can be modeled as being a sum of the active region image, st, (equal to the activation in the active region and zero everywhere else) and the background brain image, ℓt, which can be accurately modeled as lying in a slowly changing low dimensional subspace. We introduce a novel solution approach called matrix completion projected compressive sensing or MatComProCS. Significantly improved performance of MatComProCS over existing work is shown for the undersampled fMRI based brain active region detection problem.

In this work, we study the application of compressive sensing (CS) based approaches for blood oxygenation level dependent (BOLD) contrast functional MR imaging (fMRI). In particular, we show, via exhaustive experiments on actual MR scanner data for brain fMRI, that our recently proposed approach for recursive reconstruction of sparse signal sequences, modified-CS-residual, outperforms other existing CS based approaches. Modified-CS-residual exploits the fact that the sparsity pattern of brain fMRI sequences and their signal values change slowly over time. It provides a fast, yet accurate, reconstruction approach that is able to accurately track the changes of the active pixels, while using only about 30% measurements per frame. Significantly improved performance over existing work is shown in terms of practically relevant metrics such as active pixel time courses, activation maps and receiver operating characteristic (ROC) curves.

An attractive property of wavelet bases is their ability to sparsely represent piecewise polynomial signals. The sparsity of a wavelet-domain representation depends on several factors such as the mother wavelet, the number of decomposition levels, and the structure of the original signal. We consider the problem of selecting an overcomplete or dyadic wavelet basis that can sparsely represent a sparse piecewise polynomial signal. Most existing applications that apply wavelet-domain processing techniques to signals that are inherently sparse have not considered the sparsity of underlying signal when selecting a wavelet basis. By accounting for the initial sparseness of a signal, the maximum wavelet filter length and number of decomposition levels can be computed. Selecting a wavelet basis that satisfies these maximum values guarantees that the resulting wavelet-domain representation will be at least as sparse as the original signal. This criteria for wavelet basis selection is of use in applications having sparse source signals.

We present a novel scheme for blind suppression of noise from a sequence of optical coherence tomography (OCT) images, such as those collected on a real-time OCT imaging system. In contrast to virtually all existing approaches to OCT denoising, our technique is specifically aimed at collections of images and is able to exploit the correlations among those images. The proposed method approximates the optimal linear denoising operator for log-transformed data using a 2-D discrete wavelet transform (DWT) to decorrelate in space and the discrete Fourier transform (DFT), or an estimated transform, to decorrelate in time. Decorrelated coefficients are then denoised and converted back to the image domain to produce denoised OCT images. Real-time OCT data processed with this technique shows significant reduction in noise

We present motivation for performing the filtering step of FBP in a non-Radon domain. Specifically, we show that for penalized-likelihood regularization, with a shift-invariant penalty function, filtering noisy projection data in a domain for which the true projection data is sparse yields filtered data that is more faithful to the ideal filtered data than directly filtering the Radon-domain data. In contrast to simply penalizing across angles, the proposed method exploits correlation in the angle dimension. This allows for simple penalty matrices to be constructed, enables penalty coefficient to be calculated in a straightforward manner, and results in easily an computed, closed-form solution for the regularizing filters. Reconstructions employing this transform-domain filtering are superior to their Radon-domain filtered counterparts.

Optimal estimation of a two-dimensional (2-D) multichannel signal ideally decorrelates the data in both channel and space and weights the resulting coefficients according to their SNR. Many scenarios exist where the required second-order signal and noise statistics are not known in which the decorrelation is difficult or expensive to calculate. An asymptotically optimal estimation scheme proposed here uses a 2-D discrete wavelet transform to approximately decorrelate the signal in space and the discrete Fourier transform to decorrelate between channels. The coefficient weighting is replaced with a wavelet-domain thresholding operation to result in an efficient estimation scheme for both stationary and nonstationary signals. In contrast to optimal estimation, this new scheme does not require second-order signal statistics, making it well suited to many applications. In addition to providing vastly improved visual quality, the new estimator typically yields signal-to-noise ratio gains 12 dB or higher for hyperspectral imagery and functional magnetic resonance images.

Optimal detection of a known FSK-modulated binary signal in additive white Gaussian noise using a matched filter receiver requires knowledge of second-order noise statistics. The dependence on noise statistics causes the probability of detection to be sensitive to errors in the noise variance value. This makes the optimal scheme of limited use in situations where the noise statistics are unknown and cannot be estimated reliably. We propose an alternative approach, which has a more easily calculated test statistic than the optimal method and yields a constant false detection rate, regardless of noise statistics. While this approach has sub-optimal detection probability, it is significantly advantageous in applications where a primary interest is to limit false detections strictly. In addition, it allows one to determine easily the required detection length (in symbols) to achieve a desired performance level.

In this paper, we present a new wavelet-based estimator for 2-D multichannel signals. Estimation of a 2-D multichannel signal relies on the ability to decorrelate the signal in both space and channel. The new estimator we present uses a 2-D discrete wavelet transform to approximately decorrelate the signal in space, allowing for efficient estimation of both stationary and nonstationary signals. When channel correlation is unknown, but depends only on channel separation, the DFT may be utilized to decorrelate the channel efficiently. In contrast to the optimal estimation scheme, our new estimator does not require second-order signal statistics, making it well suited to many applications. In addition to providing vastly improved visual quality, the new estimator typically yields signal-to-noise ratio gains of over 12 dB.

In this paper, we present a novel DFT- and wavelet-based estimation scheme for hyperspectral imagery. Optimal hyperspectral image estimation relies on the ability to decorrelate the signal in both space and channel at the cost of requiring second-order signal statistics. This statistical requirement is removed by the proposed estimator, which approximately decorrelates the signal in space using a 2D discrete wavelet transform and in channel using a discrete Fourier transform. In addition to allowing extremely efficient estimation, the proposed estimator vastly improves visual quality and yields typical signal-to-noise ratio gains of over 14 dB.