10,776 research outputs found
Shape-Driven Interpolation With Discontinuous Kernels: Error Analysis, Edge Extraction, and Applications in Magnetic Particle Imaging
Accurate interpolation and approximation techniques for functions with discontinuities are key tools in many applications, such as medical imaging. In this paper, we study a radial basis function type of 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 results of the conducted numerical experiments are in line with 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 in order to obtain edge-preserving image reconstructions in which ringing artifacts are reduced
Use of Anisotropic Radial Basis Functions
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ผ๋ฌธ(๋ฐ์ฌ) -- ์์ธ๋ํ๊ต๋ํ์ : ์์ฐ๊ณผํ๋ํ ํต๊ณํ๊ณผ, 2021.8. ์คํฌ์.Spatial inhomogeneity along the one-dimensional curve makes two-dimensional data non-stationary. Curvelet transform, first proposed by Candes and Donoho (1999), is one of the most well-known multiscale methods to represent the directional singularity, but it has a limitation in that the data needs to be observed on equally-spaced sites. On the other hand, radial basis function interpolation is widely used to approximate the underlying function from the scattered data. However, the isotropy of the radial basis functions lowers the efficiency of the directional representation. This thesis proposes a new multiscale method that uses anisotropic radial basis functions to efficiently represent the direction from the noisy scattered data in two-dimensional Euclidean space. Basis functions are orthogonalized across the scales so that each scale can represent a global or local directional structure separately. It is shown that the proposed method is remarkable for representing directional scattered data through numerical experiments. Convergence property and practical issues in implementation are discussed as well.2์ฐจ์ ๊ณต๊ฐ์์ ๊ด์ธก๋๋ ๋น์ ์ ์๋ฃ๋ ๊ทธ ๊ณต๊ฐ์ ๋น๋์ง์ฑ์ด 1์ฐจ์ ๊ณก์ ์ ๋ฐ๋ผ ๋ํ๋๋ค. ์ด๋ฌํ ๋ฐฉํฅ์ ํน์ด์ฑ์ ํํํ๊ธฐ ์ํ ๋ค์ค์ฒ๋ ๋ฐฉ๋ฒ๋ก ์ผ๋ก๋ Candes and Donoho (1999)๊ฐ ์ฒ์ ์ ์ํ ์ปค๋ธ๋ ๋ณํ์ด ๋๋ฆฌ ์๋ ค์ ธ ์์ง๋ง ์ด๋ ์๋ฃ๊ฐ ์ผ์ ํ ๊ฐ๊ฒฉ์ผ๋ก ๊ด์ธก๋์ด์ผ ํ๋ค๋ ์ ์ฝ์ด ์๋ค. ํํธ ์ฐ์ฌ๋ ์๋ฃ์ ๋ด์ฌ๋ ํจ์๋ฅผ ๊ทผ์ฌํ๊ธฐ ์ํด์๋ ๋ฐฉ์ฌ๊ธฐ์ ํจ์๋ฅผ ์ด์ฉํ ๋ด์ฝ๋ฒ์ด ํํ ์ด์ฉ๋์ง๋ง ๋ฑ๋ฐฉ์ฑ์ด ์๋ ๋ฐฉ์ฌ๊ธฐ์ ํจ์๋ก๋ ๋ฐฉํฅ์ฑ์ ํจ์จ์ ์ผ๋ก ํํํ ์ ์๋ค. ๋ณธ ํ์๋
ผ๋ฌธ์์๋ 2์ฐจ์ ์ ํด๋ฆฌ๋ ๊ณต๊ฐ์์ ์ก์๊ณผ ํจ๊ป ์ฐ์ฌ๋์ด ๊ด์ธก๋๋ ๋ฐฉํฅ์ฑ ์๋ฃ์ ํจ์จ์ ์ธ ํํ์ ์ํด ๋น๋ฑ๋ฐฉ์ฑ ๋ฐฉ์ฌ๊ธฐ์ ํจ์๋ฅผ ์ด์ฉํ ์๋ก์ด ๋ค์ค์ฒ๋ ๋ฐฉ๋ฒ๋ก ์ ์ ์ํ๋ค. ์ด๋ ๊ฐ ์ค์ผ์ผ์์ ์ ๋ฐ์ ์ธ ๋ฐฉํฅ์ฑ ๊ตฌ์กฐ์ ๊ตญ์์ ์ธ ๋ฐฉํฅ์ฑ ๊ตฌ์กฐ๋ฅผ ๋ถ๋ฆฌํ์ฌ ํํํ๊ธฐ ์ํด ๊ธฐ์ ํจ์์ ์ค์ผ์ผ ๊ฐ ์ง๊ตํ๊ฐ ์ด๋ฃจ์ด์ง๋ค. ์ ์๋ ๋ฐฉ๋ฒ์ด ์ฐ์ฌ๋ ๋ฐฉํฅ์ฑ ์๋ฃ๋ฅผ ํํํ๋ ๋ฐ ์์ด ์ฐ์ํจ์ ๋ณด์ด๊ธฐ ์ํด ๋ชจ์์คํ๊ณผ ์ค์ ์๋ฃ์ ๋ํ ์์น์คํ์ ํ ๊ฒฐ๊ณผ๋ฅผ ์ ์ํ์๋ค. ํํธ ์ ์๋ ๋ฐฉ๋ฒ์ ์๋ ด์ฑ๊ณผ ์ค์ ๊ตฌํ ๋ฐฉ๋ฒ์ ๊ดํ ์ฌ์๋ค๋ ๋ค๋ฃจ์๋ค.1 Introduction 1
2 Multiscale Analysis 4
2.1 Classical wavelet transform 5
2.1.1 Continuous wavelet transform 5
2.1.2 Multiresolution analysis 7
2.1.3 Discrete wavelet transform 10
2.1.4 Two-dimensional wavelet transform 13
2.2 Wavelets for equally-spaced directional data 14
2.2.1 Ridgelets 15
2.2.2 Curvelets 16
2.3 Wavelets for scattered data 19
2.3.1 Lifting scheme 21
2.3.2 Spherical wavelets 23
3 Radial Basis Function Approximation 26
3.1 Radial basis function interpolation 27
3.1.1 Radial basis functions and scattered data interpolation 27
3.1.2 Compactly supported radial basis functions 29
3.1.3 Error bounds 32
3.2 Multiscale representation with radial basis functions 35
3.2.1 Multiscale approximation 35
3.2.2 Error bounds 37
4 Multiscale Representation of Directional Scattered Data 41
4.1 Anisotropic radial basis function approximation 41
4.1.1 Representation of a single linear directional structure 42
4.1.2 Representation of complex directional structure 46
4.1.3 Multiscale representation of the directional structure 46
4.2 Directional wavelets for scattered data 47
4.2.1 Directional wavelets 48
4.2.2 Estimation of coefficients 49
4.2.3 Practical issues in implementation 50
5 Numerical Experiments 57
5.1 Simulation study 57
5.1.1 Scattered observation sites 60
5.1.2 Equally-spaced observation sites 69
5.2 Real data analysis 70
5.2.1 Temperature data in South Korea 70
6 Concluding Remarks 74
6.1 Summary of results 74
6.2 Future research 74
Abstract (in Korean) 82๋ฐ
Monotonicity preserving approximation of multivariate scattered data
This paper describes a new method of monotone interpolation and smoothing of multivariate scattered data. It is based on the assumption that the function to be approximated is Lipschitz continuous. The method provides the optimal approximation in the worst case scenario and tight error bounds. Smoothing of noisy data subject to monotonicity constraints is converted into a quadratic programming problem. Estimation of the unknown Lipschitz constant from the data by sample splitting and cross-validation is described. Extension of the method for locally Lipschitz functions is presented.<br /
Error bound for radial basis interpolation in terms of a growth function
We suggest an improvement of Wu-Schaback local error bound for radial basis interpolation by using a polynomial growth function. The new bound is valid without any assumptions about the density of the interpolation centers. It can be useful for the localized methods of scattered data fitting and for the meshless discretization of partial differential equation
Local RBF approximation for scattered data fitting with bivariate splines
In this paper we continue our earlier research [4] aimed at developing effcient methods of local approximation suitable for the first stage of a spline based two-stage scattered data fitting algorithm. As an improvement to the pure polynomial local approximation method used in [5], a hybrid polynomial/radial basis scheme was considered in [4], where the local knot locations for the RBF terms were selected using a greedy knot insertion algorithm. In this paper standard radial local approximations based on interpolation or least squares are considered and a faster procedure is used for knot selection, signicantly reducing the computational cost of the method. Error analysis of the method and numerical results illustrating its performance are given
Extending the range of error estimates for radial approximation in Euclidean space and on spheres
We adapt Schaback's error doubling trick [R. Schaback. Improved error bounds
for scattered data interpolation by radial basis functions. Math. Comp.,
68(225):201--216, 1999.] to give error estimates for radial interpolation of
functions with smoothness lying (in some sense) between that of the usual
native space and the subspace with double the smoothness. We do this for both
bounded subsets of R^d and spheres. As a step on the way to our ultimate goal
we also show convergence of pseudoderivatives of the interpolation error.Comment: 10 page
Scattered Data Interpolation on Embedded Submanifolds with Restricted Positive Definite Kernels: Sobolev Error Estimates
In this paper we investigate the approximation properties of kernel
interpolants on manifolds. The kernels we consider will be obtained by the
restriction of positive definite kernels on , such as radial basis
functions (RBFs), to a smooth, compact embedded submanifold \M\subset \R^d.
For restricted kernels having finite smoothness, we provide a complete
characterization of the native space on \M. After this and some preliminary
setup, we present Sobolev-type error estimates for the interpolation problem.
Numerical results verifying the theory are also presented for a one-dimensional
curve embedded in and a two-dimensional torus
On Prediction Properties of Kriging: Uniform Error Bounds and Robustness
Kriging based on Gaussian random fields is widely used in reconstructing
unknown functions. The kriging method has pointwise predictive distributions
which are computationally simple. However, in many applications one would like
to predict for a range of untried points simultaneously. In this work we obtain
some error bounds for the (simple) kriging predictor under the uniform metric.
It works for a scattered set of input points in an arbitrary dimension, and
also covers the case where the covariance function of the Gaussian process is
misspecified. These results lead to a better understanding of the rate of
convergence of kriging under the Gaussian or the Mat\'ern correlation
functions, the relationship between space-filling designs and kriging models,
and the robustness of the Mat\'ern correlation functions
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