arXiv subjects

Mathews Jacob

Publications and source records attributed to Mathews Jacob.

At least 73 records · Page 4Linked to original sources

arXiv2018

Recovery of Noisy Points on Band-limited Surfaces: Kernel Methods Re-explained

We introduce a continuous domain framework for the recovery of points on a surface in high dimensional space, represented as the zero-level set of a bandlimited function. We show that the exponential maps of the points on the surface satisfy annihilation relations, implying that they lie in a finite dimensional subspace. The subspace properties are used to derive sampling conditions, which will guarantee the perfect recovery of the surface from finite number of points. We rely on nuclear norm minimization to exploit the low-rank structure of the maps to recover the points from noisy measurements. Since the direct estimation of the surface is computationally prohibitive in very high dimensions, we propose an iterative reweighted algorithm using the "kernel trick". The iterative algorithm reveals deep links to Laplacian based algorithms widely used in graph signal processing; the theory and the sampling conditions can serve as a basis for discrete-continuous domain processing of signals on a graph.

cs.CV↗

arXiv2018

Free-breathing cardiac MRI using bandlimited manifold modelling

We introduce a novel bandlimited manifold framework and an algorithm to recover freebreathing and ungated cardiac MR images from highly undersampled measurements. The image frames in the free breathing and ungated dataset are assumed to be points on a bandlimited manifold. We introduce a novel kernel low-rank algorithm to estimate the manifold structure (Laplacian) from a navigator-based acquisition scheme. The structure of the manifold is then used to recover the images from highly undersampled measurements. A computationally efficient algorithm, which relies on the bandlimited approximation of the Laplacian matrix, is used to recover the images. The proposed scheme is demonstrated on several patients with different breathing patterns and cardiac rates, without requiring the need for manually tuning the reconstruction parameters in each case. The proposed scheme enabled the recovery of free-breathing and ungated data, providing reconstructions that are qualitatively similar to breath-held scans performed on the same patients. This shows the potential of the technique as a clinical protocol for free-breathing cardiac scans.

cs.CV↗

arXiv2018

A General Algorithm For Compensation Of Trajectory Errors: Application To Radial Imaging

Purpose: To reconstruct artifact-free images from measured k-space data, when the actual k-space trajectory deviates from the nominal trajectory due to gradient imperfections. Methods: Trajectory errors arising from eddy currents and gradient delays introduce phase inconsistencies in several fast scanning MR pulse sequences, resulting in image artifacts. The proposed algorithm provides a novel framework to compensate for this phase distortion. The algorithm relies on the construction of a multi-block Hankel matrix, where each block is constructed from k-space segments with the same phase distortion. In the presence of spatially smooth phase distortions between the segments, the complete block-Hankel matrix is known to be highly low-rank. Since each k-space segment is only acquiring part of the k-space data, the reconstruction of the phase compensated image from their partially parallel measurements is posed as a structured low-rank matrix optimization problem, assuming the coil sensitivities to be known. Results: The proposed formulation is tested on radial acquisitions in several settings including partial Fourier and golden-angle acquisitions. The experiments demonstrate the ability of the algorithm to successfully remove the artifacts arising from the trajectory errors, without the need for trajectory or phase calibration. The quality of the reconstruction was comparable to corrections achieved using the Trajectory Auto-Corrected Image Reconstruction (TrACR) for radial acquisitions. Conclusion: The proposed method provides a general framework for the recovery of artifact-free images from radial trajectories without the need for trajectory calibration.

physics.med-ph↗

arXiv2018

Recovery of Point Clouds on Surfaces: Application to Image Reconstruction

We introduce a framework for the recovery of points on a smooth surface in high-dimensional space, with application to dynamic imaging. We assume the surface to be the zero-level set of a bandlimited function. We show that the exponential maps of the points on the surface satisfy annihilation relations, implying that they lie in a finite dimensional subspace. We rely on nuclear norm minimization of the maps to recover the points from noisy and undersampled measurements. Since this direct approach suffers from the curse of dimensionality, we introduce an iterative reweighted algorithm that uses the "kernel trick". The resulting algorithm has similarities to iterative algorithms used in graph signal processing (GSP); this framework can be seen as a continuous domain alternative to discrete GSP theory. The use of the algorithm in recovering free breathing and ungated cardiac data shows the potential of this framework in practical applications.

cs.CV↗

arXiv2018

Clustering of Data with Missing Entries

The analysis of large datasets is often complicated by the presence of missing entries, mainly because most of the current machine learning algorithms are designed to work with full data. The main focus of this work is to introduce a clustering algorithm, that will provide good clustering even in the presence of missing data. The proposed technique solves an $\ell_0$ fusion penalty based optimization problem to recover the clusters. We theoretically analyze the conditions needed for the successful recovery of the clusters. We also propose an algorithm to solve a relaxation of this problem using saturating non-convex fusion penalties. The method is demonstrated on simulated and real datasets, and is observed to perform well in the presence of large fractions of missing entries.

cs.LG↗

arXiv2017

Separation-Free Super-Resolution from Compressed Measurements is Possible: an Orthonormal Atomic Norm Minimization Approach

We consider the problem of recovering the superposition of $R$ distinct complex exponential functions from compressed non-uniform time-domain samples. Total Variation (TV) minimization or atomic norm minimization was proposed in the literature to recover the $R$ frequencies or the missing data. However, it is known that in order for TV minimization and atomic norm minimization to recover the missing data or the frequencies, the underlying $R$ frequencies are required to be well-separated, even when the measurements are noiseless. This paper shows that the Hankel matrix recovery approach can super-resolve the $R$ complex exponentials and their frequencies from compressed non-uniform measurements, regardless of how close their frequencies are to each other. We propose a new concept of orthonormal atomic norm minimization (OANM), and demonstrate that the success of Hankel matrix recovery in separation-free super-resolution comes from the fact that the nuclear norm of a Hankel matrix is an orthonormal atomic norm. More specifically, we show that, in traditional atomic norm minimization, the underlying parameter values $\textbf{must}$ be well separated to achieve successful signal recovery, if the atoms are changing continuously with respect to the continuously-valued parameter. In contrast, for the OANM, it is possible the OANM is successful even though the original atoms can be arbitrarily close. As a byproduct of this research, we provide one matrix-theoretic inequality of nuclear norm, and give its proof from the theory of compressed sensing.

cs.IT↗

arXiv2017

Clustering of Data with Missing Entries using Non-convex Fusion Penalties

The presence of missing entries in data often creates challenges for pattern recognition algorithms. Traditional algorithms for clustering data assume that all the feature values are known for every data point. We propose a method to cluster data in the presence of missing information. Unlike conventional clustering techniques where every feature is known for each point, our algorithm can handle cases where a few feature values are unknown for every point. For this more challenging problem, we provide theoretical guarantees for clustering using a $\ell_0$ fusion penalty based optimization problem. Furthermore, we propose an algorithm to solve a relaxation of this problem using saturating non-convex fusion penalties. It is observed that this algorithm produces solutions that degrade gradually with an increase in the fraction of missing feature values. We demonstrate the utility of the proposed method using a simulated dataset, the Wine dataset and also an under-sampled cardiac MRI dataset. It is shown that the proposed method is a promising clustering technique for datasets with large fractions of missing entries.

cs.CV↗

arXiv2017

Recovery of damped exponentials using structured low rank matrix completion

We introduce a structured low rank matrix completion algorithm to recover a series of images from their under-sampled measurements, where the signal along the parameter dimension at every pixel is described by a linear combination of exponentials. We exploit the exponential behavior of the signal at every pixel, along with the spatial smoothness of the exponential parameters to derive an annihilation relation in the Fourier domain. This relation translates to a low-rank property on a structured matrix constructed from the Fourier samples. We enforce the low rank property of the structured matrix as a regularization prior to recover the images. Since the direct use of current low rank matrix recovery schemes to this problem is associated with high computational complexity and memory demand, we adopt an iterative re-weighted least squares (IRLS) algorithm, which facilitates the exploitation of the convolutional structure of the matrix. Novel approximations involving two dimensional Fast Fourier Transforms (FFT) are introduced to drastically reduce the memory demand and computational complexity, which facilitates the extension of structured low rank methods to large scale three dimensional problems. We demonstrate our algorithm in the MR parameter mapping setting and show improvement over the state-of-the-art methods.

cs.CV↗

arXiv2017

A Fast Algorithm for Convolutional Structured Low-Rank Matrix Recovery

Fourier domain structured low-rank matrix priors are emerging as powerful alternatives to traditional image recovery methods such as total variation and wavelet regularization. These priors specify that a convolutional structured matrix, i.e., Toeplitz, Hankel, or their multi-level generalizations, built from Fourier data of the image should be low-rank. The main challenge in applying these schemes to large-scale problems is the computational complexity and memory demand resulting from lifting the image data to a large scale matrix. We introduce a fast and memory efficient approach called the Generic Iterative Reweighted Annihilation Filter (GIRAF) algorithm that exploits the convolutional structure of the lifted matrix to work in the original un-lifted domain, thus considerably reducing the complexity. Our experiments on the recovery of images from undersampled Fourier measurements show that the resulting algorithm is considerably faster than previously proposed algorithms, and can accommodate much larger problem sizes than previously studied.

math.NA↗

arXiv2017

Novel Structured Low-rank algorithm to recover spatially smooth exponential image time series

We propose a structured low rank matrix completion algorithm to recover a time series of images consisting of linear combination of exponential parameters at every pixel, from under-sampled Fourier measurements. The spatial smoothness of these parameters is exploited along with the exponential structure of the time series at every pixel, to derive an annihilation relation in the $k-t$ domain. This annihilation relation translates into a structured low rank matrix formed from the $k-t$ samples. We demonstrate the algorithm in the parameter mapping setting and show significant improvement over state of the art methods.

cs.CV↗

arXiv2017

Convex recovery of continuous domain piecewise constant images from non-uniform Fourier samples

We consider the recovery of a continuous domain piecewise constant image from its non-uniform Fourier samples using a convex matrix completion algorithm. We assume the discontinuities/edges of the image are localized to the zero levelset of a bandlimited function. This assumption induces linear dependencies between the Fourier coefficients of the image, which results in a two-fold block Toeplitz matrix constructed from the Fourier coefficients being low-rank. The proposed algorithm reformulates the recovery of the unknown Fourier coefficients as a structured low-rank matrix completion problem, where the nuclear norm of the matrix is minimized subject to structure and data constraints. We show that exact recovery is possible with high probability when the edge set of the image satisfies an incoherency property. We also show that the incoherency property is dependent on the geometry of the edge set curve, implying higher sampling burden for smaller curves. This paper generalizes recent work on the super-resolution recovery of isolated Diracs or signals with finite rate of innovation to the recovery of piecewise constant images.

cs.IT↗

arXiv2016

Off-the-Grid Recovery of Piecewise Constant Images from Few Fourier Samples

We introduce a method to recover a continuous domain representation of a piecewise constant two-dimensional image from few low-pass Fourier samples. Assuming the edge set of the image is localized to the zero set of a trigonometric polynomial, we show the Fourier coefficients of the partial derivatives of the image satisfy a linear annihilation relation. We present necessary and sufficient conditions for unique recovery of the image from finite low-pass Fourier samples using the annihilation relation. We also propose a practical two-stage recovery algorithm which is robust to model-mismatch and noise. In the first stage we estimate a continuous domain representation of the edge set of the image. In the second stage we perform an extrapolation in Fourier domain by a least squares two-dimensional linear prediction, which recovers the exact Fourier coefficients of the underlying image. We demonstrate our algorithm on the super-resolution recovery of MRI phantoms and real MRI data from low-pass Fourier samples, which shows benefits over standard approaches for single-image super-resolution MRI.

cs.CV↗

arXiv2016

Structured low-rank recovery of piecewise constant signals with performance guarantees

We derive theoretical guarantees for the exact recovery of piecewise constant two-dimensional images from a minimal number of non-uniform Fourier samples using a convex matrix completion algorithm. We assume the discontinuities of the image are localized to the zero level-set of a bandlimited function, which induces certain linear dependencies in Fourier domain, such that a multifold Toeplitz matrix built from the Fourier data is known to be low-rank. The recovery algorithm arranges the known Fourier samples into the structured matrix then attempts recovery of the missing Fourier data by minimizing the nuclear norm subject to structure and data constraints. This work adapts results by Chen and Chi on the recovery of isolated Diracs via nuclear norm minimization of a similar multifold Hankel structure. We show that exact recovery is possible with high probability when the bandlimited function describing the edge set satisfies an incoherency property. Finally, we demonstrate the algorithm on the recovery of undersampled MRI data.

cs.IT↗

arXiv2016

Multi-shot multi-channel diffusion data recovery using structured low-rank matrix completion

Purpose: To introduce a novel method for the recovery of multi-shot diffusion weighted (MS-DW) images from echo-planar imaging (EPI) acquisitions. Methods: Current EPI-based MS-DW reconstruction methods rely on the explicit estimation of the motion- induced phase maps to recover the unaliased images. In the new formulation, the k-space data of the unaliased DWI is recovered using a structured low-rank matrix completion scheme, which does not require explicit estimation of the phase maps. The structured matrix is obtained as the lifting of the multi-shot data. The smooth phase-modulations between shots manifest as null-space vectors of this matrix, which implies that the structured matrix is low-rank. The missing entries of the structured matrix are filled in using a nuclear-norm minimization algorithm subject to the data-consistency. The formulation enables the natural introduction of smoothness regularization, thus enabling implicit motion-compensated recovery of fully-sampled as well as under-sampled MS-DW data. Results: Our experiments on in-vivo data show effective removal of the ghosting artifacts arising from intershot motion in MS-DW data using the proposed method. The performance is comparable and better in certain cases than conventional phase-based methods. Conclusion: The proposed method can achieve effective unaliasing of fully/under-sampled MS-DW images without using explicit phase estimates.

physics.med-ph↗

arXiv2015

Two step recovery of jointly sparse and low-rank matrices: theoretical guarantees

We introduce a two step algorithm with theoretical guarantees to recover a jointly sparse and low-rank matrix from undersampled measurements of its columns. The algorithm first estimates the row subspace of the matrix using a set of common measurements of the columns. In the second step, the subspace aware recovery of the matrix is solved using a simple least square algorithm. The results are verified in the context of recovering CINE data from undersampled measurements; we obtain good recovery when the sampling conditions are satisfied.

stat.ML↗

arXiv2015

Subspace based low rank and joint sparse matrix recovery

We consider the recovery of a low rank and jointly sparse matrix from under sampled measurements of its columns. This problem is highly relevant in the recovery of dynamic MRI data with high spatio-temporal resolution, where each column of the matrix corresponds to a frame in the image time series; the matrix is highly low-rank since the frames are highly correlated. Similarly the non-zero locations of the matrix in appropriate transform/frame domains (e.g. wavelet, gradient) are roughly the same in different frame. The superset of the support can be safely assumed to be jointly sparse. Unlike the classical multiple measurement vector (MMV) setup that measures all the snapshots using the same matrix, we consider each snapshot to be measured using a different measurement matrix. We show that this approach reduces the total number of measurements, especially when the rank of the matrix is much smaller than than its sparsity. Our experiments in the context of dynamic imaging shows that this approach is very useful in realizing free breathing cardiac MRI.

math.NA↗

arXiv2015

Super-resolution MRI Using Finite Rate of Innovation Curves

We propose a two-stage algorithm for the super-resolution of MR images from their low-frequency k-space samples. In the first stage we estimate a resolution-independent mask whose zeros represent the edges of the image. This builds off recent work extending the theory of sampling signals of finite rate of innovation (FRI) to two-dimensional curves. We enable its application to MRI by proposing extensions of the signal models allowed by FRI theory, and by developing a more robust and efficient means to determine the edge mask. In the second stage of the scheme, we recover the super-resolved MR image using the discretized edge mask as an image prior. We evaluate our scheme on simulated single-coil MR data obtained from analytical phantoms, and compare against total variation reconstructions. Our experiments show improved performance in both noiseless and noisy settings.

cs.CV↗

arXiv2015

Recovery of Piecewise Smooth Images from Few Fourier Samples

We introduce a Prony-like method to recover a continuous domain 2-D piecewise smooth image from few of its Fourier samples. Assuming the discontinuity set of the image is localized to the zero level-set of a trigonometric polynomial, we show the Fourier transform coefficients of partial derivatives of the signal satisfy an annihilation relation. We present necessary and sufficient conditions for unique recovery of piecewise constant images using the above annihilation relation. We pose the recovery of the Fourier coefficients of the signal from the measurements as a convex matrix completion algorithm, which relies on the lifting of the Fourier data to a structured low-rank matrix; this approach jointly estimates the signal and the annihilating filter. Finally, we demonstrate our algorithm on the recovery of MRI phantoms from few low-resolution Fourier samples.

cs.CV↗