Near-optimal compressed sensing guarantees for anisotropic and isotropic total variation minimization

Needell, Deanna; Ward, Rachel

doi:10.5642/cmcfacpub/318

Cited by 5 publications

(2 citation statements)

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“…In particular, the reconstruction is exact if the gradient of the image is precisely sparse, i.e., the image is piecewise constant. This is even true for high dimensional signals, see [36].…”

Section: 1mentioning

confidence: 95%

Inertial primal-dual algorithms for structured convex optimization

Chan¹,

Ma²,

Yang³

2014

Preprint

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The primal-dual algorithm recently proposed by Chambolle & Pock (abbreviated as CPA) for structured convex optimization is very efficient and popular. It was shown by Chambolle & Pock in [16] and also by Shefi & Teboulle in [49] that CPA and variants are closely related to preconditioned versions of the popular alternating direction method of multipliers (abbreviated as ADM). In this paper, we further clarify this connection and show that CPAs generate exactly the same sequence of points with the so-called linearized ADM (abbreviated as LADM) applied to either the primal problem or its Lagrangian dual, depending on different updating orders of the primal and the dual variables in CPAs, as long as the initial points for the LADM are properly chosen. The dependence on initial points for LADM can be relaxed by focusing on cyclically equivalent forms of the algorithms. Furthermore, by utilizing the fact that CPAs are applications of a general weighted proximal point method to the mixed variational inequality formulation of the KKT system, where the weighting matrix is positive definite under a parameter condition, we are able to propose and analyze inertial variants of CPAs. Under certain conditions, global pointconvergence, nonasymptotic O(1/k) and asymptotic o(1/k) convergence rate of the proposed inertial CPAs can be guaranteed, where k denotes the iteration index. Finally, we demonstrate the profits gained by introducing the inertial extrapolation step via experimental results on compressive image reconstruction based on total variation minimization.

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Section: 1mentioning

confidence: 95%

Inertial primal-dual algorithms for structured convex optimization

Chan¹,

Ma²,

Yang³

2014

Preprint

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“…The recovery guarantees we derive apply to both anisotropic and isotropic total variation semi norms, and we use the notation x T V to refer to either choice of seminorm. For brevity of presentation, we provide details only for the isotropic total variation seminorm, but refer the reader to [37] for the analysis of the anisotropic variant. The total variation seminorm is a regularizer of choice in many image processing applications.…”

Section: Imaging With Csmentioning

confidence: 99%

Near-Optimal Compressed Sensing Guarantees for Total Variation Minimization

Needell

Ward

2013

IEEE Trans. on Image Process.

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107

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Consider the problem of reconstructing a multidimensional signal from an underdetermined set of measurements, as in the setting of compressed sensing. Without any additional assumptions, this problem is ill-posed. However, for signals such as natural images or movies, the minimal total variation estimate consistent with the measurements often produces a good approximation to the underlying signal, even if the number of measurements is far smaller than the ambient dimensionality. This paper extends recent reconstruction guarantees for twodimensional images x ∈ C N 2 to signals x ∈ C N d of arbitrary dimension d ≥ 2 and to isotropic total variation problems. To be precise, we show that a multidimensional signal x ∈ C N d can be reconstructed from O(sd log(N d )) linear measurements y = Ax using total variation minimization to within a factor of the best s-term approximation of its gradient. The reconstruction guarantees we provide are necessarily optimal up to polynomial factors in the spatial dimension d.

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