Gregory Ely

Novel Methods for Multilinear Data Completion and De-noising Based on Tensor-SVD

In this paper we propose novel methods for completion (from limited samples) and de-noising of multilinear (tensor) data and as an application consider 3-D and 4- D (color) video data completion and de-noising. We exploit the recently proposed tensor-Singular Value Decomposition (t-SVD)[11]. Based on t-SVD, the notion of multilinear rank and a related tensor nuclear norm was proposed in [11] to characterize informational and structural complexity of multilinear data. We first show that videos with linear camera motion can be represented more efficiently using t-SVD compared to the approaches based on vectorizing or flattening of the tensors. Since efficiency in representation implies efficiency in recovery, we outline a tensor nuclear norm penalized algorithm for video completion from missing entries. Application of the proposed algorithm for video recovery from missing entries is shown to yield a superior performance over existing methods. We also consider the problem of tensor robust Principal Component Analysis (PCA) for de-noising 3-D video data from sparse random corruptions. We show superior performance of our method compared to the matrix robust PCA adapted to this setting as proposed in [4].

The LSC glitch group: monitoring noise transients during the fifth LIGO science run

The LIGO Scientific Collaboration (LSC) glitch group is part of the LIGO detector characterization effort. It consists of data analysts and detector experts who, during and after science runs, collaborate for a better understanding of noise transients in the detectors. Goals of the glitch group during the fifth LIGO science run (S5) included (1) offline assessment of the detector data quality, with focus on noise transients, (2) veto recommendations for astrophysical analysis and (3) feedback to the commissioning team on anomalies seen in gravitational wave and auxiliary data channels. Other activities included the study of auto-correlation of triggers from burst searches, stationarity of the detector noise and veto studies. The group identified causes for several noise transients that triggered false alarms in the gravitational wave searches; the times of such transients were identified and vetoed from the data generating the LSC astrophysical results.

Exploiting structural complexity for robust and rapid hyperspectral imaging

This paper presents several strategies for spectral de-noising of hyperspectral images and hypercube reconstruction from a limited number of tomographic measurements. In particular we show that the non-noisy spectral data, when stacked across the spectral dimension, exhibits low-rank. On the other hand, under the same representation, the spectral noise exhibits a banded structure. Motivated by these features we show that the de-noised spectral data and the unknown spectral noise and the respective bands can be simultaneously estimated through the use of a low-rank and simultaneous sparse minimization operation without prior knowledge of the noisy bands. This result is novel for for hyperspectral imaging applications. In addition, we show that imaging for the Computed Tomography Imaging Systems (CTIS) can be improved under limited angle tomography by using low-rank penalization. For both of these cases we exploit the recent results in the theory of low-rank matrix completion using nuclear norm minimization.

Robust Hydraulic Fracture Monitoring (HFM) of multiple time overlapping events using a generalized discrete radon transform

In this work we propose a novel algorithm for multiple-event localization for Hydraulic Fracture Monitoring (HFM) through the exploitation of the sparsity of the observed seismic signal when represented in a basis consisting of space time propagators. We provide explicit construction of these propagators using a forward model for wave propagation which depends non-linearly on the problem parameters - the unknown source location and mechanism of fracture, time and extent of event, and the locations of the receivers. Under fairly general assumptions and an appropriate discretization of these parameters we first build an over-complete dictionary of generalized Radon propagators and assume that the data is well represented as a linear superposition of these propagators. Exploiting this structure we propose sparsity penalized algorithms and workflow for super-resolution extraction of time overlapping multiple seismic events from single well data.

Methods for large scale hydraulic fracture monitoring

In this paper we propose computationally efficient and robust methods for estimating the moment tensor and location of micro-seismic event(s) for large search volumes. Our contribution is two-fold. First, we propose a novel joint-complexity measure, namely the sum of nuclear norms which while imposing sparsity on the number of fractures (locations) over a large spatial volume, also captures the rank-1 nature of the induced wavefield pattern. This wavefield pattern is modeled as the outer-product of the source signature with the amplitude pattern across the receivers from a seismic source. A rank-1 factorization of the estimated wavefield pattern at each location can therefore be used to estimate the seismic moment tensor using the knowledge of the array geometry. In contrast to existing work this approach allows us to drop any other assumption on the source signature. Second, we exploit the recently proposed first-order incremental projection algorithms for a fast and efficient implementation of the resulting optimization problem and develop a hybrid stochastic & deterministic algorithm which results in significant computational savings.

Complexity penalized hydraulic fracture localization and moment tensor estimation under limited model information

In this paper we present a novel technique for micro-seismic localization using a group sparse penalization that is robust to the focal mechanism of the source and requires only a velocity model of the stratigraphy rather than a full Greens function model of the earths response. In this technique we construct a set of perfect delta detector responses, one for each detector in the array, to a seismic event at a given location and impose a group sparsity across the array. This scheme is independent of the moment tensor and exploits the time compactness of the incident seismic signal. Furthermore we present a method for improving the inversion of the moment tensor and Greens function when the geometry of seismic array is limited. In particular we demonstrate that both Tikhonov regularization and truncated SVD can improve the recovery of the moment tensor and be robust to noise. We evaluate our algorithm on synthetic data and present error bounds for both estimation of the moment tensor as well as localiza...

Exploiting Algebraic and Structural Complexity for Single Snapshot Computed Tomography Hyperspectral Imaging Systems

This paper presents strategies for spectral de- noising of hyperspectral images and 3-D data cube reconstruction from a limited number of tomographic measurements, arising in single snapshot imaging systems. For de-noising the main idea is to exploit the incoherency between the algebraic complexity measure, namely the low rank of the noise-free hyperspectral data cube, and the sparsity structure of the spectral noise. In particular, the non-noisy spectral data, when stacked across the spectral dimension, exhibits low-rank due to a small number of species. On the other hand, under the same representation, the spectral noise exhibits a banded structure. Motivated by this we show that the de-noised spectral data and the unknown spectral noise and the respective bands can be simultaneously estimated through the use of a low-rank and simultaneous sparse minimization operation without prior knowledge of the noisy bands. This result is novel for hyperspectral imaging applications and we compare our results with several existing methods for noisy band recovery. For recovery under limited tomographic projections we exploit both the low algebraic and structural complexity of the data cube via joint rank penalization plus Total Variation/wavelet domain sparsity, which is novel for single snapshot hyperspectral imaging systems. We combine these two approaches for simultaneous spectral de-noising and data cube recovery under limited measurements. We perform extensive simulations and our main result indicates that exploiting both low algebraic and structural complexity has a superior performance compared to exploiting only the structural complexity. To address the computational challenges associated with the resulting optimization problem we adapt several recent developments in the area of convex optimization, specifically employing splitting and proximal point based methods.