Graeme Pope

Recovery of Sparsely Corrupted Signals

We investigate the recovery of signals exhibiting a sparse representation in a general (i.e., possibly redundant or incomplete) dictionary that are corrupted by additive noise admitting a sparse representation in another general dictionary. This setup covers a wide range of applications, such as image inpainting, super-resolution, signal separation, and recovery of signals that are impaired by, e.g., clipping, impulse noise, or narrowband interference. We present deterministic recovery guarantees based on a novel uncertainty relation for pairs of general dictionaries and we provide corresponding practicable recovery algorithms. The recovery guarantees we find depend on the signal and noise sparsity levels, on the coherence parameters of the involved dictionaries, and on the amount of prior knowledge about the signal and noise support sets.

Probabilistic Recovery Guarantees for Sparsely Corrupted Signals

We consider the recovery of sparse signals subject to sparse interference, as introduced by Studer , IEEE T-IT, 2012. We present novel probabilistic recovery guarantees for this framework, covering varying degrees of knowledge of the signal and interference support, which are relevant for a large number of practical applications. Our results assume that the sparsifying dictionaries are characterized by coherence parameters and we require randomness only in the signal and/or interference. The obtained recovery guarantees show that one can recover sparsely corrupted signals with overwhelming probability, even if the sparsity of both the signal and interference scale (near) linearly with the number of measurements.

Real-time principal component pursuit

Robust principal component analysis (RPCA) deals with the decomposition of a matrix into a low-rank matrix and a sparse matrix. Such a decomposition finds, for example, applications in video surveillance or face recognition. One effective way to solve RPCA problems is to use a convex optimization method known as principal component pursuit (PCP). The corresponding algorithms have, however, prohibitive computational complexity for certain applications that require real-time processing. In this paper we propose a variety of methods that significantly reduce the computational complexity. Furthermore, we perform a systematic analysis of the performance/complexity tradeoffs underlying PCP. For synthetic data, we show that our methods result in a speedup of more than 365 times compared to a reference C implementation at only a small loss in terms of recovery error. To demonstrate the effectiveness of our approach, we consider foreground/background separation for video surveillance, where our methods enable real-time processing of a 640×480 color video stream at 12 frames per second (fps) using a quad-core CPU.

Sparse signal recovery from sparsely corrupted measurements

We investigate the recovery of signals exhibiting a sparse representation in a general (i.e., possibly redundant or incomplete) dictionary that are corrupted by additive noise admitting a sparse representation in another general dictionary. This setup covers a wide range of applications, such as image inpainting, super-resolution, signal separation, and the recovery of signals that are corrupted by, e.g., clipping, impulse noise, or narrowband interference. We present deterministic recovery guarantees based on a recently developed uncertainty relation and provide corresponding recovery algorithms. The recovery guarantees we find depend on the signal and noise sparsity levels, on the coherence parameters of the involved dictionaries, and on the amount of prior knowledge on the support sets of signal and noise.

Learning phase-invariant dictionaries

In this paper, we present a novel algorithm to learn phase-invariant dictionaries, which can be used to efficiently approximate a variety of signals, such as audio signals or images. Our approach relies on finding a small number of generating atoms that can be used-along with their phase-shifts-to sparsely approximate a given signal. Our method is inspired by the K-SVD algorithm, but imposes an extra constraint that the dictionaries we learn are phase-invariant. We show that the learned dictionaries achieve competitive approximation performance compared to that of state-of-the-art methods for audio signals and images, while substantially reducing the storage requirements and computational complexity.

Sparse signal separation in redundant dictionaries

We formulate a unified framework for the separation of signals that are sparse in “morphologically” different redundant dictionaries. This formulation incorporates the so-called “analysis” and “synthesis” approaches as special cases and contains novel hybrid setups. We find corresponding coherence-based recovery guarantees for an ℓ₁-norm based separation algorithm. Our results recover those reported in Studer and Baraniuk, ACHA, submitted, for the synthesis setting, provide new recovery guarantees for the analysis setting, and form a basis for comparing performance in the analysis and synthesis settings. As an aside our findings complement the D-RIP recovery results reported in Candès et al., ACHA, 2011, for the “analysis” signal recovery problem minimize _x ||Ψx̃||₁ subject to ||y - Ax̃||₂ ≤ ϵ by delivering corresponding coherence-based recovery results.

Coherence-based recovery guarantees for generalized basis-pursuit de-quantizing

This paper deals with the recovery of signals that admit an approximately sparse representation in some known dictionary (possibly over-complete) and are corrupted by additive noise. In particular, we consider additive measurement noise with bounded ℓ_p-norm for p ≥ 2, and we minimize the ℓ_q quasi-norm (with q ∈ (0, 1]) of the signal vector. We develop coherence-based recovery guarantees for which stable recovery via generalized basis-pursuit de-quantizing (BPDQ_p,q) is possible. We finally show that depending on the measurement-noise model and the choice of the ℓ_p-norm used in the constraint, (BPDQ_p,q) significantly outperforms classical basis pursuit de-noising (BPDN).

Coherence-based probabilistic recovery guarantees for sparsely corrupted signals

In this paper, we present novel probabilistic recovery guarantees for sparse signals subject to sparse interference, covering varying degrees of knowledge of the signal and interference support. Our results assume that the sparsifying dictionaries are characterized by coherence parameters and we require randomness only in the signal and/or interference. The obtained recovery guarantees show that one can recover sparsely corrupted signals with overwhelming probability, even if the sparsity of both the signal and interference scale (near) linearly with the number of measurements.

Light curtain localization via compressive sensing

Light curtains are presence detection devices, typically employed to detect when an object enters (or passes through) a region to initiate emergency security procedures. However, most existing light curtain implementations are not designed to detect the precise location where the barrier was broken. In this paper, we present a hardware prototype implementation that is able to identify the locations where the light curtain was broken, even if multiple objects penetrate the barrier simultaneously. To this end, we deploy techniques from sparse signal recovery and compressive sensing to perform detection and localization with only a few infrared transmitters and sensors. The proposed prototype implementation is scalable to large physical sizes and can be tailored to suit a particular application in terms of the number and size of objects to detect, as well as the desired spatial resolution.

Sparse signal recovery in Hilbert spaces

This paper reports an effort to consolidate numerous coherence-based sparse signal recovery results available in the literature. We present a single theory that applies to general Hilbert spaces with the sparsity of a signal defined as the number of (possibly infinite-dimensional) subspaces participating in the signals representation. Our general results recover uncertainty relations and coherence-based recovery thresholds for sparse signals, block-sparse signals, multi-band signals, signals in shift-invariant spaces, and signals in finite unions of (possibly infinite-dimensional) subspaces. Moreover, we improve upon and generalize several of the existing results and, in many cases, we find shortened and simplified proofs.