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Shape-Driven Interpolation with Discontinuous Kernels: Error Analysis, Edge Extraction and Applications in MPI
TL;DR: An RBF type method for scattered data interpolation that incorporates discontinuities via a variable scaling function and an application to interpolation in magnetic particle imaging shows that the presented method is very promising.
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Abstract: Accurate interpolation and approximation techniques for functions with discontinuities are key tools in many applications as, for instance, medical imaging. In this paper, we study an RBF type method for scattered data interpolation that incorporates discontinuities via a variable scaling function. For the construction of the discontinuous basis of kernel functions, information on the edges of the interpolated function is necessary. We characterize the native space spanned by these kernel functions and study error bounds in terms of the fill distance of the node set. To extract the location of the discontinuities, we use a segmentation method based on a classification algorithm from machine learning. The conducted numerical experiments confirm the theoretically derived convergence rates in case that the discontinuities are a priori known. Further, an application to interpolation in magnetic particle imaging shows that the presented method is very promising.
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Citations
Jumping with variably scaled discontinuous kernels (VSDKs)
TL;DR: The main result is the construction of discontinuous kernel-based basis functions which lead to a very flexible tool which sensibly or completely reduces the well-known Gibbs phenomenon in reconstructing functions with jumps.
On the reconstruction of discontinuous functions using multiquadric RBF-WENO local interpolation techniques
TL;DR: This paper proposes a true MQ-RBF–WENO method that does not revert to the classical polynomial WENO approximation near discontinuities, as opposed to what was proposed in Guo and Jung (2017).
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Fake Nodes approximation for Magnetic Particle Imaging
Stefano De Marchi,Wolfgang Erb,Elisa Francomano,Francesco Marchetti,Emma Perracchione,Davide Poggiali +5 more
- 16 Jun 2020
TL;DR: A method for scattered data interpolation, named mapped bases or Fake Nodes approach, which incorporates discontinuities via a suitable mapping function which naturally mitigates the Gibbs phenomenon, as numerical evidence for reconstructing MPI images confirms.
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Polynomial mapped bases: theory and applications
TL;DR: The basic theory and the most important applications of a novel technique that has shown to be suitable for scattered data interpolation, quadrature, bio-imaging reconstruction and more are collected.
2
Variably Scaled Kernels Improve Classification of Hormonally-Treated Patient-Derived Xenografts
Francesco Marchetti,Fabio De Martino,Marie Shamseddin,Stefano De Marchi,Cathrin Brisken +4 more
- 01 May 2020
TL;DR: The Mouse INtraDuctal (MIND) model is used, an innovative patient-derived xenograft model, to characterize global gene expression changes, which are triggered by stimulation of dihydrotestosterone (DHT) and progesterone (P4) in vivo.
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