Myung Cho

Precise semidefinite programming formulation of atomic norm minimization for recovering d-dimensional (D ≥ 2) off-the-grid frequencies

Recent research in off-the-grid compressed sensing (CS) has demonstrated that, under certain conditions, one can successfully recover a spectrally sparse signal from a few time-domain samples even though the dictionary is continuous. In particular, atomic norm minimization was proposed in [1] to recover 1-dimensional spectrally sparse signal. However, in spite of existing research efforts [2], it was still an open problem how to formulate an equivalent positive semidefinite program for atomic norm minimization in recovering signals with d-dimensional (d ≥ 2) off-the-grid frequencies. In this paper, we settle this problem by proposing equivalent semidefinite programming formulations of atomic norm minimization to recover signals with d-dimensional (d ≥ 2) off-the-grid frequencies.

Spectral Super-Resolution With Prior Knowledge

We address the problem of super-resolution frequency recovery using prior knowledge of the structure of a spectrally sparse, undersampled signal. In many applications of interest, some structure information about the signal spectrum is often known. The prior information might be simply knowing precisely some signal frequencies or the likelihood of a particular frequency component in the signal. We devise a general semidefinite program to recover these frequencies using theories of positive trigonometric polynomials. Our theoretical analysis shows that, given sufficient prior information, perfect signal reconstruction is possible using signal samples no more than thrice the number of signal frequencies. Numerical experiments demonstrate great performance enhancements using our method. We show that the nominal resolution necessary for the grid-free results can be improved if prior information is suitably employed.

Block Iterative Reweighted Algorithms for Super-Resolution of Spectrally Sparse Signals

We propose novel algorithms that enhance the performance of recovering unknown continuous-valued frequencies from undersampled signals. Our iterative reweighted frequency recovery algorithms employ the support knowledge gained from earlier steps of our algorithms as block prior information to enhance frequency recovery. Our methods improve the performance of the atomic norm minimization which is a useful heuristic in recovering continuous-valued frequency contents. Numerical results demonstrate that our block iterative reweighted methods provide both better recovery performance and faster speed than other known methods.

Off-the-grid spectral compressed sensing with prior information

Recent research in off-the-grid compressed sensing (CS) has demonstrated that, under certain conditions, one can successfully recover a spectrally sparse signal from a few time-domain samples even though the dictionary is continuous. In this paper, we extend off-the-grid CS to applications where some prior information about spectrally sparse signal is known. We specifically consider cases where a few contributing frequencies or poles, but not their amplitudes or phases, are known a priori. Our results show that equipping off-the-grid CS with the known-poles algorithm can increase the probability of recovering all the frequency components.

Super-resolution line spectrum estimation with block priors

We address the problem of super-resolution line spectrum estimation of an undersampled signal with block prior information. The component frequencies of the signal are assumed to take arbitrary continuous values in known frequency blocks. We formulate a general semidefinite program to recover these continuous-valued frequencies using positive trigonometric polynomials. The proposed semidefinite program achieves super-resolution frequency recovery by taking advantage of known structures of frequency blocks. Numerical experiments show great performance enhancements using our method.

Fast alternating projected gradient descent algorithms for recovering spectrally sparse signals

We propose fast algorithms that speed up or improve the performance of recovering spectrally sparse signals from un-derdetermined measurements. Our algorithms are based on a non-convex approach of using alternating projected gradient descent for structured matrix recovery. We apply this approach to two formulations of structured matrix recovery: Hankel and Toeplitz mosaic structured matrix, and Hankel structured matrix. Our methods provide better recovery performance, and faster signal recovery than existing algorithms, including atomic norm minimization.

Phaseless super-resolution in the continuous domain

Phaseless super-resolution refers to the problem of super-resolving a signal from only its low-frequency Fourier magnitude measurements. In this paper, we consider the phaseless super-resolution problem of recovering a sum of sparse Dirac delta functions which can be located anywhere in the continuous time-domain. For such signals in the continuous domain, we propose a novel Semidefinite Programming (SDP) based signal recovery method to achieve the phaseless super-resolution. This work extends the recent work of Jaganathan et al. [1], which considered phaseless super-resolution for discrete signals on the grid.

Computable performance guarantees for compressed sensing matrices

The null space condition for ℓ1 minimization in compressed sensing is a necessary and sufficient condition on the sensing matrices under which a sparse signal can be uniquely recovered from the observation data via ℓ1 minimization. However, verifying the null space condition is known to be computationally challenging. Most of the existing methods can provide only upper and lower bounds on the proportion parameter that characterizes the null space condition. In this paper, we propose new polynomial-time algorithms to establish upper bounds of the proportion parameter. We leverage on these techniques to find upper bounds and further develop a new procedure—tree search algorithm—that is able to precisely and quickly verify the null space condition. Numerical experiments show that the execution speed and accuracy of the results obtained from our methods far exceed those of the previous methods which rely on linear programming (LP) relaxation and semidefinite programming (SDP).

2D phaseless super-resolution

In phaseless super-resolution, we only have the magnitude information of continuously-parameterized signals in a transform domain, and try to recover the original signals from these magnitude measurements. Optical microscopy is one application where the phaseless super-resolution for 2D signals arise. In this paper, we propose algorithms for performing phaseless super-resolution for 2D or higher-dimensional signals, and investigate their performance guarantees.

Fast dose optimization for rotating shield brachytherapy

Purpose: To provide a fast computational method, based on the proximal graph solver (POGS) – A convex optimization solver using the alternating direction method of multipliers (ADMM), for calculating an optimal treatment plan in rotating shield brachytherapy (RSBT). RSBT treatment planning has more degrees of freedom than conventional high‐dose‐rate brachytherapy due to the addition of emission direction, and this necessitates a fast optimization technique to enable clinical usage. Methods: The multi‐helix RSBT (H‐RSBT) delivery technique was investigated for five representative cervical cancer patients. Treatment plans were generated for all patients using the POGS method and the commercially available solver IBM ILOG CPLEX. The rectum, bladder, sigmoid colon, high‐risk clinical target volume (HR‐CTV), and HR‐CTV boundary were the structures included in our optimization, which applied an asymmetric dose‐volume optimization with smoothness control. Dose calculation resolution was 1 × 1 × 3 mm3 for all cases. The H‐RSBT applicator had 6 helices, with 33.3 mm of translation along the applicator per helical rotation and 1.7 mm spacing between dwell positions, yielding 17.5° emission angle spacing per 5 mm along the applicator. Results: For each patient, HR‐CTV D90, HR‐CTV D100, rectum D2cc, sigmoid D2cc, and bladder D2cc matched within 1% for CPLEX and POGS methods. Also, similar EQD2 values between CPLEX and POGS methods were obtained. POGS was around 18 times faster than CPLEX. For all patients, total optimization times were 32.1–65.4 s for CPLEX and 2.1–3.9 s for POGS. Conclusions: POGS reduced treatment plan optimization time approximately 18 times for RSBT with similar HR‐CTV D90, organ at risk (OAR) D2cc values, and EQD2 values compared to CPLEX, which is significant progress toward clinical translation of RSBT.