Confidence regions for images observed under the Radon transform

Bissantz, Nicolai; Holzmann, Hajo; Proksch, Katharina

doi:10.1016/j.jmva.2014.03.005

Cited by 4 publications

(5 citation statements)

References 35 publications

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“…in the third line of formula (5) in the proof on page 2372. This again seems questionable, so we propose to use the same error estimation method as before, which gives the quadrature error O(n −1 a −d−1 n h −1 ) in the third line and, finally,…”

Section: Corrected Assumptions For Asymptotic Normalitymentioning

confidence: 99%

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A note on confidence intervals for deblurred images

Biel¹,

Szkutnik²

2020

Opuscula Math.

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We consider pointwise asymptotic confidence intervals for images that are blurred and observed in additive white noise. This amounts to solving a stochastic inverse problem with a convolution operator. Under suitably modified assumptions, we fill some apparent gaps in the proofs published in [N. Bissantz, M. Birke, Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators, J. Multivariate Anal. 100 (2009), 2364-2375]. In particular, this leads to modified bootstrap confidence intervals with much better finite-sample behaviour than the original ones, the validity of which is, in our opinion, questionable. Some simulation results that support our claims and illustrate the behaviour of the confidence intervals are also presented.

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Section: Corrected Assumptions For Asymptotic Normalitymentioning

confidence: 99%

“…5.1.3] and [6]. The latter paper was also the first step towards the construction of confidence bands in inverse problems and was followed in recent years by several similar works [2,4,5,10,11,15,18,19].…”

Section: Introductionmentioning

confidence: 97%

A note on confidence intervals for deblurred images

Biel¹,

Szkutnik²

2020

Opuscula Math.

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“…() use directly along with a regularized inverse of

scriptR

with a kernel smoother; see Cavalier () and Bissantz et al. () for other applications of kernel smoothing in inverting the Radon operator. The estimator they propose achieves the optimal rate of convergence in a Sobolev class of functions.…”

Section: General Mixturesmentioning

confidence: 99%

“…Beran et al (1996) work with characteristic functions and obtain a consistent and asymptotically normal estimator for f β . However, Hoderlein et al (2010) use (3.7) directly along with a regularized inverse of R with a kernel smoother; see Cavalier (2000) and Bissantz et al (2014) for other applications of kernel smoothing in inverting the Radon operator. The estimator they propose achieves the optimal rate of convergence in a Sobolev class of functions.…”

Section: Random Coefficients Modelsmentioning

confidence: 99%

Using Mixtures in Econometric Models: A Brief Review and Some New Results

Compiani

Kitamura

2016

The Econometrics Journal

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Summary This paper is concerned with applications of mixture models in econometrics. Focused attention is given to semiparametric and nonparametric models that incorporate mixture distributions, where important issues about model specifications arise. For example, there is a significant difference between a finite mixture and a continuous mixture in terms of model identifiability. Likewise, the dimension of the latent mixing variables is a critical issue, in particular when a continuous mixture is used. We present applications of mixture models to address various problems in econometrics, such as unobserved heterogeneity and multiple equilibria. New nonparametric identification results are developed for finite mixture models with testable exclusion restrictions without relying on an identification‐at‐infinity assumption on covariates. The results apply to mixtures with both continuous and discrete covariates, delivering point identification under weak conditions.

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“…the ground truth function, has been established both at a fixed point and in a global L 2 norm. The rate of convergence for the maximal deviation of an estimator from its mean has been obtained for similar kernel-type estimators in [6]. The accuracy of pointwise asymptotically efficient kernel estimator in terms of minimax risk of a probability density f from noise-free RT data sampled on a random grid has been derived in [7,8].…”

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confidence: 99%

Analysis of Reconstruction from Discrete Radon Transform Data in R^3 When the Function Has Jump Discontinuities

Katsevich¹

2019

SIAM J. Appl. Math.

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In this paper we study reconstruction of a function f from its discrete Radon transform data in R 3 when f has jump discontinuities. Consider a conventional parametrization of the Radon data in terms of the affine and angular variables. The step-size along the affine variable is , and the density of measured directions on the unit sphere is O( 2 ). Let f denote the result of reconstruction from the discrete data. Pick any generic point x 0 (i.e., satisfying some mild conditions), where f has a jump. Our first result is an explicit leading term behavior of f in an O( )-neighborhood of x 0 as → 0.A closely related question is why can we accurately reconstruct functions with discontinuities at all? This is a fundamental question, which has not been studied in the literature in dimensions three and higher. We prove that the discrete inversion formula "works", i.e. if x 0 ∈ S := singsupp(f ) is generic, then f (x 0 ) → f (x 0 ) as → 0. The proof of this result reveals a surprising connection with the theory of uniform distribution (u.d.). This is a new phenomenon that has not been known previously. We also present some numerical experiments, which confirm the validity of the developed theory.

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Confidence regions for images observed under the Radon transform

Cited by 4 publications

References 35 publications

A note on confidence intervals for deblurred images

A note on confidence intervals for deblurred images

Using Mixtures in Econometric Models: A Brief Review and Some New Results

Analysis of Reconstruction from Discrete Radon Transform Data in R^3 When the Function Has Jump Discontinuities

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