Efficient Adjoint Computation for Wavelet and Convolution Operators [Lecture Notes]

Folberth, James; Becker, Stephen

doi:10.1109/msp.2016.2594277

Cited by 10 publications

(3 citation statements)

References 6 publications

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“…Furthermore, we require the adjoints of W and G for the gradients. We compute W * with the technique from [32], which involves handling the padding of the boundary conditions manually. Regarding the blurring operator, we have G * = G.…”

Section: Sampling the Exact Posteriormentioning

confidence: 99%

Certified coordinate selection for high-dimensional Bayesian inversion with Laplace prior

Flock,

Dong,

Uribe

et al. 2023

Preprint

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We consider high-dimensional Bayesian inverse problems with arbitrary likelihood and product-form Laplace prior for which we provide a certified approximation of the posterior density in the Hellinger distance. The approximate posterior density differs from the prior density only in a small number of relevant coordinates that contribute the most to the update from the prior to the posterior. We propose and analyze a gradient-based diagnostic to identify these relevant coordinates. Although this diagnostic requires computing an expectation with respect to the posterior, we propose tractable methods for the classical case of a linear forward model with Gaussian likelihood. Our methods can be employed to estimate the diagnostic before solving the Bayesian inverse problem via, e.g., Markov chain Monte Carlo (MCMC) methods. After selecting the coordinates, the approximate posterior density can be efficiently inferred since most of its coordinates are only informed by the prior. Moreover, specialized MCMC methods, such as the pseudo-marginal MCMC algorithm, can be used to obtain less correlated samples when sampling the exact posterior density. We show the applicability of our method using a 1D signal deblurring problem and a high-dimensional 2D super-resolution problem.

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Section: Sampling the Exact Posteriormentioning

confidence: 99%

Certified coordinate selection for high-dimensional Bayesian inversion with Laplace prior

Flock,

Dong,

Uribe

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…This produces a fairly accurate approximation of the adjoint operator and in our implementation is available by using the parameter "adjoint" when calling the function idualtree4 [12]. Further improvement could be obtained by a more detailed consideration of the boundary conditions of the discrete convolution, as explained in [11], but we leave this to future work.…”

Section: Adjoint 4d Dt-cwtmentioning

confidence: 99%

4D Dual-Tree Complex Wavelets for Time-Dependent Data

Bubba,

Heikkilä,

Siltanen

2021

Preprint

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The dual-tree complex wavelet transform (DT-CWT) is extended to the 4D setting. Key properties of 4D DT-CWT, such as directional sensitivity and shift-invariance, are discussed and illustrated in a tomographic application. The inverse problem of reconstructing a dynamic three-dimensional target from X-ray projection measurements can be formulated as 4D space-time tomography. The results suggest that 4D DT-CWT offers simple implementations combined with useful theoretical properties for tomographic reconstruction.

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“…In particular, no matrix factorization is required and these algorithms can be implemented using forward-adjoint oracles (FAOs), yielding matrix-free optimization algorithms [31,32]. Many signal processing applications can readily make use of FAOs yielding a substantial decrease of the memory requirements.…”

Section: Introductionmentioning

confidence: 99%

Proximal Gradient Algorithms: Applications in Signal Processing

Antonello¹,

Stella²,

Patrinos³

et al. 2018

Preprint

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Advances in numerical optimization have supported breakthroughs in several areas of signal processing. This paper focuses on the recent enhanced variants of the proximal gradient numerical optimization algorithm, which combine quasi-Newton methods with forward-adjoint oracles to tackle large-scale problems and reduce the computational burden of many applications. These proximal gradient algorithms are here described in an easy-to-understand way, illustrating how they are able to address a wide variety of problems arising in signal processing.A new high-level modeling language is presented which is used to demonstrate the versatility of the presented algorithms in a series of signal processing application examples such as sparse deconvolution, total variation denoising, audio de-clipping and others.

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Efficient Adjoint Computation for Wavelet and Convolution Operators [Lecture Notes]

Cited by 10 publications

References 6 publications

Certified coordinate selection for high-dimensional Bayesian inversion with Laplace prior

Certified coordinate selection for high-dimensional Bayesian inversion with Laplace prior

4D Dual-Tree Complex Wavelets for Time-Dependent Data

Proximal Gradient Algorithms: Applications in Signal Processing

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