The effective choice of the smoothing norm in regularization

Cullum, Jane

doi:10.1090/s0025-5718-1979-0514816-1

Cited by 33 publications

(5 citation statements)

References 16 publications

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“…This result was expected since the regularisation in H2[a. b] takes into account the a priori information about the regularity of the solution. This fact confirms what is already known in other situations (see [28]).…”

supporting

confidence: 92%

Two-phase regularisation method in Hilbert spaces: description and application to EXAFS problems

et al. 1988

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The authors propose a two-phase regularisation method to obtain structural information in EXAFS spectroscopy. The method is based on a Tikhonov regularisation over a convenient Hilbert space, together with an inf-convolution which allows recovery of the 'peaks' of the radial distribution functions by using some information obtained with the regularisation procedure. They compare four different procedures using this technique through several numerical experiments carried out on two test models.

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supporting

confidence: 92%

Two-phase regularisation method in Hilbert spaces: description and application to EXAFS problems

et al. 1988

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“…For example, a different order of derivative or, indeed, a combination of derivatives may provide a more appropriate constraint on the recovered amplitudes. Ideally, the chosen function should be directed by knowledge of the system since it will alter the efficacy of the regularisation [27]. Here, the choice was primarily motivated by ease of computation.…”

Section: Smoothed Exponential Series Methods (Sesm)mentioning

confidence: 99%

Analysis of time-correlated single photon counting data: a comparative evaluation of deterministic and probabilistic approaches

Smith

McKenzie

Jones

et al. 2017

Methods Appl. Fluoresc.

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We review various methods for analysing time-resolved fluorescence data acquired using the time-correlated single photon counting method in an attempt to evaluate their benefits and limitations. We have applied these methods to both experimental and simulated data. The relative merits of using deterministic approaches, such as the commonly used iterative reconvolution method, and probabilistic approaches, such as the smoothed exponential series method, the maximum entropy method and recently proposed basis pursuit denoising (compressed sensing) method, are outlined. In particular, we show the value of using multiple methods to arrive at the most appropriate choice of model. We show that the use of probabilistic analysis methods can indicate whether a discrete component or distribution analysis provides the better representation of the data.

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“…The optimal smoothing functional for deconvolution operators on an infinite domain was determined by Aref'eva (1974) (see also Cullum 1979) who showed that it is completely determined by the decay rate of the Fourier coefficients of the unknown function. Unfortunately, in real applications this result cannot be used directly because, in addition to data being available only on a finite domain, it requires the function one is trying to recover.…”

Section: Numerical Regularizationmentioning

confidence: 99%

Digital deblurring of CMB maps: Performance and efficient implementation

Vio¹,

Nagy

Andreani

et al. 2003

A&A

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Abstract. Digital deblurring of images is an important problem that arises in multifrequency observations of the CosmicMicrowave Background (CMB) where, because of the width of the point spread functions (PSF), maps at different frequencies suffer a different loss of spatial resolution. Deblurring is useful for various reasons: first, it helps to restore high frequency components lost through the smoothing effect of the instrument's PSF; second, emissions at various frequencies observed with different resolutions can be better studied on a comparable resolution; third, some map-based component separation algorithms require maps with similar level of degradation. Because of computational efficiency, deblurring is usually done in the frequency domain. But this approach has some limitations as it requires spatial invariance of the PSF, stationarity of the noise, and is not flexible in the selection of more appropriate boundary conditions. Deblurring in real space is more flexible but usually not used because of its high computational cost. In this paper (the first in a series on the subject) we present new algorithms that allow the use of real space deblurring techniques even for very large images. In particular, we consider the use of Tikhonov deblurring of noisy maps with applications to PLANCK. We provide details for efficient implementations of the algorithms. Their performance is tested on Gaussian and non-Gaussian simulated CMB maps, and PSFs with both circular and elliptical symmetry. Matlab code is made available.

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The effective choice of the smoothing norm in regularization

Cited by 33 publications

References 16 publications

Two-phase regularisation method in Hilbert spaces: description and application to EXAFS problems

Two-phase regularisation method in Hilbert spaces: description and application to EXAFS problems

Analysis of time-correlated single photon counting data: a comparative evaluation of deterministic and probabilistic approaches

Digital deblurring of CMB maps: Performance and efficient implementation

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