1,822 research outputs found
The Penalized Lebesgue Constant for Surface Spline Interpolation
Problems involving approximation from scattered data where data is arranged
quasi-uniformly have been treated by RBF methods for decades. Treating data
with spatially varying density has not been investigated with the same
intensity, and is far less well understood. In this article we consider the
stability of surface spline interpolation (a popular type of RBF interpolation)
for data with nonuniform arrangements. Using techniques similar to those
recently employed by Hangelbroek, Narcowich and Ward to demonstrate the
stability of interpolation from quasi-uniform data on manifolds, we show that
surface spline interpolation on R^d is stable, but in a stronger, local sense.
We also obtain pointwise estimates showing that the Lagrange function decays
very rapidly, and at a rate determined by the local spacing of datasites. These
results, in conjunction with a Lebesgue lemma, show that surface spline
interpolation enjoys the same rates of convergence as those of the local
approximation schemes recently developed by DeVore and Ron.Comment: 20 pages; corrected typos; to appear in Proc. Amer. Math. So
Image Reconstruction from Undersampled Confocal Microscopy Data using Multiresolution Based Maximum Entropy Regularization
We consider the problem of reconstructing 2D images from randomly
under-sampled confocal microscopy samples. The well known and widely celebrated
total variation regularization, which is the L1 norm of derivatives, turns out
to be unsuitable for this problem; it is unable to handle both noise and
under-sampling together. This issue is linked with the notion of phase
transition phenomenon observed in compressive sensing research, which is
essentially the break-down of total variation methods, when sampling density
gets lower than certain threshold. The severity of this breakdown is determined
by the so-called mutual incoherence between the derivative operators and
measurement operator. In our problem, the mutual incoherence is low, and hence
the total variation regularization gives serious artifacts in the presence of
noise even when the sampling density is not very low. There has been very few
attempts in developing regularization methods that perform better than total
variation regularization for this problem. We develop a multi-resolution based
regularization method that is adaptive to image structure. In our approach, the
desired reconstruction is formulated as a series of coarse-to-fine
multi-resolution reconstructions; for reconstruction at each level, the
regularization is constructed to be adaptive to the image structure, where the
information for adaption is obtained from the reconstruction obtained at
coarser resolution level. This adaptation is achieved by using maximum entropy
principle, where the required adaptive regularization is determined as the
maximizer of entropy subject to the information extracted from the coarse
reconstruction as constraints. We demonstrate the superiority of the proposed
regularization method over existing ones using several reconstruction examples
Review of the mathematical foundations of data fusion techniques in surface metrology
The recent proliferation of engineered surfaces, including freeform and structured surfaces, is challenging current metrology techniques. Measurement using multiple sensors has been proposed to achieve enhanced benefits, mainly in terms of spatial frequency bandwidth, which a single sensor cannot provide. When using data from different sensors, a process of data fusion is required and there is much active research in this area. In this paper, current data fusion methods and applications are reviewed, with a focus on the mathematical foundations of the subject. Common research questions in the fusion of surface metrology data are raised and potential fusion algorithms are discussed
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