Superresolution without Separation

This paper provides a theoretical analysis of diffraction-limited superresolution, demonstrating that arbitrarily close point sources can be resolved in ideal situations. Precisely, we assume that the incoming signal is a linear combination of M shifted copies of a known waveform with unknown shifts and amplitudes, and one only observes a finite collection of evaluations of this signal. We characterize properties of the base waveform such that the exact translations and amplitudes can be recovered from 2M + 1 observations. This recovery is achieved by solving a a weighted version of basis pursuit over a continuous dictionary. Our methods combine classical polynomial interpolation techniques with contemporary tools from compressed sensing.Comment: 23 pages, 8 figure

Solving Inverse Problems with Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity

A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAP-EM algorithm. A dual mathematical interpretation of the proposed framework with structured sparse estimation is described, which shows that the resulting piecewise linear estimate stabilizes the estimation when compared to traditional sparse inverse problem techniques. This interpretation also suggests an effective dictionary motivated initialization for the MAP-EM algorithm. We demonstrate that in a number of image inverse problems, including inpainting, zooming, and deblurring, the same algorithm produces either equal, often significantly better, or very small margin worse results than the best published ones, at a lower computational cost.Comment: 30 page

Towards Faster MRI Acquisition

Extensive research on accelerating Magnetic Resonance Imaging (MRI) has been done on two fronts: (i) hardware acceleration, and (ii) image post processing. We present the results of our work on image post processing, where the input is a sparsely sampled volumetric images, and our deep-learning based models seek to output the original, densely sampled images. Specifically, we propose two different methods at accelerating MRI, in two different aspects of the MR acquisition process. First, We propose a marginal super-resolution (MSR) approach based on 2D convolutional neural networks (CNNs) for interpolating an anisotropic brain magnetic resonance scan along the highly under-sampled direction. Previous methods for slice interpolation only consider data from pairs of adjacent 2D slices. The possibility of fusing information from the direction orthogonal to the 2D slices remains unexplored. Our approach performs MSR in both sagittal and coronal directions, which provides an initial estimate for slice interpolation. The interpolated slices are then fused and refined in the axial direction for improved consistency. Since MSR consists of only 2D operations, it is more feasible in terms of GPU memory consumption and requires fewer training samples compared to 3D CNNs. Our experiments demonstrate that the proposed method outperforms traditional linear interpolation and baseline 2D/3D CNN-based approaches. We conclude by showcasing the method's practical utility in estimating brain volumes from under-sampled brain MR scans through semantic segmentation. Secondly, although undersampled MR image recovery has been widely studied for accelerated MR acquisition, it has been mostly studied under a single sequence scenario, despite the fact that multi-sequence MR scan is common in practice. We aim to optimize multi-sequence MR image recovery from undersampled k-space data under an overall time constraint while considering the difference in acquisition time for various sequences. We first formulate it as a constrained optimization problem and then show that finding the optimal sampling strategy for all sequences and the best recovery model at the same time is combinatorial and hence computationally prohibitive. To solve this problem, we propose a blind recovery model that simultaneously recovers multiple sequences, and an efficient approach to find the near-optimal combination of sampling strategy and recovery model. Our experiments demonstrate that the proposed method not only outperforms sequence-wise recovery, but also sheds light on how to optimally undersample the k-space for each sequence within an overall time budget