Oversampling Errors in Multimodal Medical Imaging Are Due to the Gibbs Effect

Poggiali, Davide; Cecchin, Diego; Campi, Cristina; Marchi, Stefano De

doi:10.3390/math9121348

Cited by 8 publications

(10 citation statements)

References 42 publications

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“…It turns out that the FNA for image resampling it is designed to reduce the Gibbs effect when oversampling the functional image. This has been proved by a tight error analysis in [66]. • Links: https://en.wikipedia.org/wiki/Runge%27s_phenomenon#S-Runge_algorithm_ without_resampling • As mentioned above, S-Runge and S-Gibbs have been improved in [33] via the GRASPA approach.…”

Section: Examplesmentioning

confidence: 88%

Padua points and fake nodes for polynomial approximation: old, new and open problems

Marchi

2022

Constructive Mathematical Analysis

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Padua points, discovered in 2005 at the University of Padua, are the first set of points on the square [−1, 1] 2 that are explicitly known, unisolvent for total degree polynomial interpolation and with Lebesgue constant increasing like log 2 (n) of the degree. One of the key features of the Padua points is that they lie on a particular Lissajous curve. Other important properties of Padua points are(1) in two dimensions, Padua points are a WAM for interpolation and for extracting approximate Fekete points and discrete Leja sequences. (2) in three dimensions, Padua points can be used for constructing tensor product WAMs on different compacts. Unfortunately, their extension to higher dimensions is still the biggest open problem.The concept of mapped bases has been widely studied (cf. e.g.[35] and references therein), which turns out to be equivalent to map the interpolating nodes and then construct the approximant in the classical form without the need of resampling. The mapping technique is general, in the sense that works with any basis and can be applied to continuous, piecewise or discontinuous functions or even images. All the proposed methods show convergence to the interpolant provided that the function is resampled at the mapped nodes. In applications, this is often physically unfeasible. An effective method for interpolating via mapped bases in the multivariate setting, referred as Fake Nodes Approach (FNA), has been presented in [37]. In this paper, some interesting connection of the FNA with Padua points and "families of relatives nodes", that can be used as "fake nodes" for multivariate approximation, are presented and we conclude with some open problems.

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Section: Examplesmentioning

confidence: 88%

Padua points and fake nodes for polynomial approximation: old, new and open problems

Marchi

2022

Constructive Mathematical Analysis

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“…In image resampling, since the number of voxels n 3 can be extremely large, it is necessary to make some assumptions on the basis functions to compute the resampling image in a relatively short time. Accordingly with [10,11,6], the basis should satisfy the following characteristics. I) Separable: given a function b the basis B ijk is then the product of translates of b at the samples, that is…”

Section: Interpolation For Image Resamplingmentioning

confidence: 99%

“…The reader can find some examples of basis functions satisfying all these characteristics in [10,6]. For instance the so-called trilinear interpolation corresponds to choose the function b as…”

Section: Ii)mentioning

confidence: 99%

“…In particular, it is possible to speed up the computation of the interpolant, as it is shown by the following Lemma, whose proof is given in [6].…”

Section: Ii)mentioning

confidence: 99%

“…In a recent paper [6] we showed that the latter is preferable due to the Gibbs effect that occurs in oversampling. However this approach is a waste of the time and efforts that are spent in producing accurate segments at full resolution [7].…”

Section: Introductionmentioning

confidence: 98%

See 2 more Smart Citations

Reducing the Gibbs effect in multimodal medical imaging by the Fake Nodes Approach

Poggiali¹,

Cecchin²,

Marchi³

2022

Preprint

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It is a common practice in multimodal medical imaging to undersample the anatomically-derived segmentation images to measure the mean activity of a co-acquired functional image. This practice avoids the resamplingrelated Gibbs effect that would occur in oversampling the functional image. As sides effect, waste of time and efforts are produced since the anatomical segmentation at full resolution is performed in many hours of computations or manual work. In this work we explain the commonly-used resampling methods and give errors bound in the cases of continuous and discontinuous signals. Then we propose a Fake Nodes scheme for image resampling designed to reduce the Gibbs effect when oversampling the functional image. This new approach is compared to the traditional counterpart in two significant experiments, both showing that Fake Nodes resampling gives smaller errors.

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Undersampling and Oversampling Strategies for Convolutional Neural Networks Classifier

Abdani

Zulkifley

2022

Lecture Notes in Electrical Engineering

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Oversampling Errors in Multimodal Medical Imaging Are Due to the Gibbs Effect

Cited by 8 publications

References 42 publications

Padua points and fake nodes for polynomial approximation: old, new and open problems

Padua points and fake nodes for polynomial approximation: old, new and open problems

Reducing the Gibbs effect in multimodal medical imaging by the Fake Nodes Approach

Undersampling and Oversampling Strategies for Convolutional Neural Networks Classifier

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