Signal Reconstruction From Modulo Observations

Shah, Viraj; Hegde, Chinmay

doi:10.1109/globalsip45357.2019.8969100

Cited by 18 publications

(7 citation statements)

References 39 publications

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“…The authors developed a generalized approximate message passing approach to reconstruct discrete signals. Further bounds in context of Gaussian matrices were studied in [44]. d) Unlimited sampling of continuous-time sparse signals in canonical and Fourier domain, was discussed in our works [45] and [46], respectively.…”

Section: Contemporary Literature (2018 Onwards)mentioning

confidence: 99%

On Unlimited Sampling and Reconstruction

Bhandari,

Krahmer,

Raskar

2019

Preprint

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Shannon's sampling theorem is one of the cornerstone topics that is well understood and explored, both mathematically and algorithmically. That said, practical realization of this theorem still suffers from a severe bottleneck due to the fundamental assumption that the samples can span an arbitrary range of amplitudes. In practice, the theorem is realized using so-called analogto-digital converters (ADCs) which clip or saturate whenever the signal amplitude exceeds the maximum recordable ADC voltage thus leading to a significant information loss.In this paper, we develop an alternative paradigm for sensing and recovery, called the Unlimited Sampling Framework. It is based on the observation that when a signal is mapped to an appropriate bounded interval via a modulo operation before entering the ADC, the saturation problem no longer exists, but one rather encounters a different type of information loss due to the modulo operation. Such an alternative setup can be implemented, for example, via so-called folding or self-reset ADCs, as they have been proposed in various contexts in the circuit design literature.The key task that we need to accomplish in order to cope with this new type of information loss is to recover a bandlimited signal from its modulo samples. In this paper we derive conditions when this is possible and present an empirically stable recovery algorithm with guaranteed perfect recovery. The sampling density required for recovery is independent of the maximum recordable ADC voltage and depends on the signal bandwidth only.Numerical experiments validate our approach and indeed show that it is possible to perfectly recover functions that take values that are orders of magnitude higher than the ADC's threshold.Applications of the unlimited sampling paradigm can be found in a number of fields such as signal processing, communication and imaging.

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Section: Contemporary Literature (2018 Onwards)mentioning

confidence: 99%

On Unlimited Sampling and Reconstruction

Bhandari,

Krahmer,

Raskar

2019

Preprint

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“…While the aforementioned results [4,5,6,28] are for the nonparametric setting and with f being univariate, the setting where f is a d-variate linear function was considered by Shah and Hegde [30]. Assuming f to be sparse, exact recovery guarantees were provided (for the noiseless setting) in the regime n ≪ d for an alternating minimization based algorithm.…”

Section: Related Workmentioning

confidence: 99%

Error analysis for denoising smooth modulo signals on a graph

Tyagi¹

2020

Preprint

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In many applications, we are given access to noisy modulo samples of a smooth function with the goal being to robustly unwrap the samples, i.e., to estimate the original samples of the function. In a recent work, Cucuringu and Tyagi [11] proposed denoising the modulo samples by first representing them on the unit complex circle and then solving a smoothness regularized least squares problem -the smoothness measured w.r.t the Laplacian of a suitable proximity graph G -on the product manifold of unit circles. This problem is a quadratically constrained quadratic program (QCQP) which is nonconvex, hence they proposed solving its sphere-relaxation leading to a trust region subproblem (TRS). In terms of theoretical guarantees, ℓ 2 error bounds were derived for (TRS). These bounds are however weak in general and do not really demonstrate the denoising performed by (TRS).In this work, we analyse the (TRS) as well as an unconstrained relaxation of (QCQP). For both these estimators we provide a refined analysis in the setting of Gaussian noise and derive noise regimes where they provably denoise the modulo observations w.r.t the ℓ 2 norm. The analysis is performed in a general setting where G is any connected graph.

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“…In this paper, we propose a recovery algorithm for exact reconstruction of sparse signals from modulo measurements of the form (1). We refer our algorithm as MoRAM, short for Modulo Recovery using Alternating Minimization.…”

Section: Our Contributionsmentioning

confidence: 99%

Sparse Signal Recovery from Modulo Observations

Shah

Hegde

2020

Preprint

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We consider the problem of reconstructing a signal from under-determined modulo observations (or measurements). This observation model is inspired by a (relatively) less well-known imaging mechanism called modulo imaging, which can be used to extend the dynamic range of imaging systems; variations of this model have also been studied under the category of phase unwrapping. Signal reconstruction in the under-determined regime with modulo observations is a challenging ill-posed problem, and existing reconstruction methods cannot be used directly. In this paper, we propose a novel approach to solving the inverse problem limited to two modulo periods, inspired by recent advances in algorithms for phase retrieval under sparsity constraints. We show that given a sufficient number of measurements, our algorithm perfectly recovers the underlying signal and provides improved performance over other existing algorithms. We also provide experiments validating our approach on both synthetic and real data to depict its superior performance.

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Signal Reconstruction From Modulo Observations

Cited by 18 publications

References 39 publications

On Unlimited Sampling and Reconstruction

On Unlimited Sampling and Reconstruction

Error analysis for denoising smooth modulo signals on a graph

Sparse Signal Recovery from Modulo Observations

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