Divergence-Free Wavelet Frames

Bostan, Emrah; Unser, Michaël; Ward, John Paul

doi:10.1109/lsp.2015.2388794

Cited by 6 publications

(7 citation statements)

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“…Intuitively, such orthogonality partially satisfies the independency requirement. Our numerical results indicate that the SE predictions are reliable for various matrix ensembles (e.g., IID Gaussian, partial orthogonal and some ill-conditioned 1 The name is from [25], although the discussions therein are irrelevant to this paper.…”

Section: Introductionmentioning

confidence: 91%

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Orthogonal AMP

Ma¹,

Li²

2017

IEEE Access

351

367

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Approximate message passing (AMP) is a low-cost iterative signal recovery algorithm for linear system models. When the system transform matrix has independent identically distributed (IID) Gaussian entries, the performance of AMP can be asymptotically characterized by a simple scalar recursion called state evolution (SE). However, SE may become unreliable for other matrix ensembles, especially for ill-conditioned ones. This imposes limits on the applications of AMP.In this paper, we propose an orthogonal AMP (OAMP) algorithm based on de-correlated linear estimation (LE) and divergence-free non-linear estimation (NLE). The Onsager term in standard AMP vanishes as a result of the divergence-free constraint on NLE. We develop an SE procedure for OAMP and show numerically that the SE for OAMP is accurate for general unitarily-invariant matrices, including IID Gaussian matrices and partial orthogonal matrices. We further derive optimized options for OAMP and show that the corresponding SE fixed point coincides with the optimal performance obtained via the replica method. Our numerical results demonstrate that OAMP can be advantageous over AMP, especially for ill-conditioned matrices.Index Terms-Compressed sensing, approximate message passing (AMP), replica method, state evolution, unitarily-invariant, IID Gaussian, partial orthogonal matrix.

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Section: Introductionmentioning

confidence: 91%

“…The name is from[25], although the discussions therein are irrelevant to this paper. arXiv:1602.06509v3 [cs.IT] 24 Jan 2017…”

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confidence: 99%

Orthogonal AMP

Ma¹,

Li²

2017

IEEE Access

351

367

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“…One can view such an as an infinite sum of compactly supported functions , centered at the lattice points . The function contains the local information of near , and the wavelet coefficients of satisfy the estimate (10) in analogy with (7).…”

Section: B Measure Of Localizationmentioning

confidence: 99%

Optimal Isotropic Wavelets for Localized Tight Frame Representations

Ward

Pad

Unser

2015

IEEE Signal Process. Lett.

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“…Our wavelets, in contrast, do not have compact support. Closest to the present construction is recent work by Bostan, Unser and Ward [4] who also proposed a tight, divergence free wavelet frame using window functions that are separable in polar (or spherical) coordinates. In the 1990s a similar construction was proposed by Suter [28], but to our knowledge without an implementation.…”

Section: Introductionmentioning

confidence: 98%

Divergence Free Polar Wavelets for the Analysis and Representation of Fluid Flows

Lessig

2019

J. Math. Fluid Mech.

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We present a Parseval tight wavelet frame for the representation and analysis of velocity vector fields of incompressible fluids. Our wavelets have closed form expressions in the frequency and spatial domains, are divergence free in the ideal, analytic sense, have a multi-resolution structure and fast transforms, and an intuitive correspondence to common flow phenomena. Our construction also allows for well defined directional selectivity, e.g. to model the behavior of divergence free vector fields in the vicinity of boundaries or to represent highly directional features like in a von Kármán vortex street. We demonstrate the practicality and efficiency of our construction by analyzing the representation of different divergence free vector fields in our wavelets.Mathematics Subject Classification (2010). Primary 42C40; Secondary 76B99.

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Divergence-Free Wavelet Frames

Cited by 6 publications

References 20 publications

Orthogonal AMP

Orthogonal AMP

Optimal Isotropic Wavelets for Localized Tight Frame Representations

Divergence Free Polar Wavelets for the Analysis and Representation of Fluid Flows

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