Resolution Analysis of Imaging with $\ell_1$ Optimization

Borcea, Liliana; Kocyigit, Ilker

doi:10.1137/15m1028881

Cited by 13 publications

(34 citation statements)

References 28 publications

(33 reference statements)

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“…In principle, the steps h and h 3 may be chosen arbitrarily small, to avoid discretization error due to sources being off the mesh. However, we know from [7] and the analysis below and the numerical simulations that we cannot expect reconstructions at scales that are finer than the resolution limits in homogeneous media. This motivates us to formulate the inversion using the assumption that the sources are further apart than 3 R in cross-range and 3 R 3 in range.…”

Section: 2mentioning

confidence: 98%

“…The results obtained in [7] for imaging with l 1 optimization in homogeneous media show that it is not possible to improve the c o /B range resolution, unless the sources are very far apart in range. We cannot expect to do better in random media, so we do not seek any super-resolution in range, in the narrowband regime.…”

Section: Coherent Interferometric Imagingmentioning

confidence: 99%

“…We also consider the case of clusters of nearby sources, and show that the l 1 solution is useful when the clusters are well separated. The analysis is built on our recent results in [7]. For example, if the sources are all in the same cross-range plane, and the minimum distance between any two of them is H min , we may take H = H min and H 3 = D 3 .…”

Section: Resolution Analysismentioning

confidence: 99%

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Imaging in Random Media with Convex Optimization

Borcea¹,

Kocyigit

2017

SIAM J. Imaging Sci.

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Abstract. We study an inverse problem for the wave equation where localized wave sources in random scattering media are to be determined from time resolved measurements of the waves at an array of receivers. The sources are far from the array, so the measurements are affected by cumulative scattering in the medium, but they are not further than a transport mean free path, which is the length scale characteristic of the onset of wave diffusion that prohibits coherent imaging. The inversion is based on the Coherent Interferometric (CINT) imaging method which mitigates the scattering effects by introducing an appropriate smoothing operation in the image formation. This smoothing stabilizes statistically the images, at the expense of their resolution. We complement the CINT method with a convex (l 1 ) optimization in order to improve the source localization and obtain quantitative estimates of the source intensities. We analyze the method in a regime where scattering can be modeled by large random wavefront distortions, and quantify the accuracy of the inversion in terms of the spatial separation of individual sources or clusters of sources. The theoretical predictions are demonstrated with numerical simulations.Key words. waves in random media, coherent interferometric imaging, l 1 optimization, mutual coherence.1. Introduction. Waves measured by a collection of nearby sensors, called an array of receivers, carry information about their source and the medium through which they travel. We consider a typical remote sensing regime with sources of small (point-like) support, and study the inverse problem of determining them from the array measurements.When the waves travel in a known and non-scattering (e.g. homogeneous) medium, the sources can be localized with reverse time migration [4,5] also known as backprojection [20]. This estimates the source locations as the peaks of the image formed by superposing the array recordings delayed by travel times from the receivers to the imaging points. The accuracy of the estimates depends on the array aperture, the distance of the sources from the array, and the temporal support of the signals emitted by the sources. It may be improved under certain conditions by using l 1 optimization, which seeks to invert the linear mapping from supposedly sparse vectors of the discretized source amplitude on some mesh, to the array measurements. The fast growing literature of imaging with l 1 optimization in homogeneous media includes compressed sensing studies such as [19,18], synthetic radar imaging studies like [1,8], array imaging studies like [14], and the resolution study [7].In this paper we assume that the waves travel in heterogeneous media with fluctuations of the wave speed caused by numerous inhomogeneities. The amplitude of the fluctuations is small, meaning that a single inhomogeneity is a weak scatterer. However, there are many inhomogeneities that interact with the waves on their way from the sources to the receivers, and their scattering effect accumulates. Because in appli...

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Section: 2mentioning

confidence: 98%

Section: Coherent Interferometric Imagingmentioning

confidence: 99%

Section: Resolution Analysismentioning

confidence: 99%

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Imaging in Random Media with Convex Optimization

Borcea¹,

Kocyigit

2017

SIAM J. Imaging Sci.

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“…The solution ρ δ can be separated into two parts: the coherent part supported in the vicinities S j of the true solution, j ∈ T , and the incoherent part, which is small for low noise, and that is supported away from these vicinities. Other stability results can be found in [8,9,17,30,19,3].…”

Section: Resultsmentioning

confidence: 84%

Imaging with highly incomplete and corrupted data

et al. 2020

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We consider the problem of imaging sparse scenes from a few noisy data using an 1 -minimization approach. This problem can be cast as a linear system of the form A ρ = b, where A is an N ×K measurement matrix. We assume that the dimension of the unknown sparse vector ρ ∈ C K is much larger than the dimension of the data vector b ∈ C N , i.e, K N . We provide a theoretical framework that allows us to examine under what conditions the 1 -minimization problem admits a solution that is close to the exact one in the presence of noise. Our analysis shows that 1 -minimization is not robust for imaging with noisy data when high resolution is required. To improve the performance of 1 -minimization we propose to solve instead the augmented linear system [A | C]ρ = b, where the N × Σ matrix C is a noise collector. It is constructed so as its column vectors provide a frame on which the noise of the data, a vector of dimension N , can be well approximated. Theoretically, the dimension Σ of the noise collector should be e N which would make its use not practical. However, our numerical results illustrate that robust results in the presence of noise can be obtained with a large enough number of columns Σ ≈ 10K.

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“…This is the main difficulty in synthetic aperture imaging when the phases are not measured. We explain in the next subsection how to synchronize the signals in the frequency domain, i.e., how to recover N individual phases from phase difference measurements of the form (2). Once all the signals are synchronized, the imaging problem is trivial as we show in Section 3.…”

Section: Multi-frequency Field Cross-correlationsmentioning

confidence: 99%

Synthetic Aperture Imaging With Intensity-Only Data

Moscoso

Novikov

Papanicolaou

et al. 2020

IEEE Trans. Comput. Imaging

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We consider imaging the reflectivity of scatterers from intensity-only data recorded by a single moving transducer that both emits and receives signals, forming a synthetic aperture. By exploiting frequency illumination diversity, we obtain multiple intensity measurements at each location, from which we determine field cross-correlations using an appropriate phase controlled illumination strategy and the inner product polarization identity. The field cross-correlations obtained this way do not, however, provide all the missing phase information because they are determined up to a phase that depends on the receiver's location. The main result of this paper is an algorithm with which we recover the field cross-correlations up to a single phase that is common to all the data measured over the synthetic aperture, so all the data are synchronized. Thus, we can image coherently with data over all frequencies and measurement locations as if full phase information was recorded.

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Resolution Analysis of Imaging with $\ell_1$ Optimization

Cited by 13 publications

References 28 publications

Imaging in Random Media with Convex Optimization

Imaging in Random Media with Convex Optimization

Imaging with highly incomplete and corrupted data

Synthetic Aperture Imaging With Intensity-Only Data

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