Use of Anisotropic Radial Basis Functions

학위논문(박사) -- 서울대학교대학원 : 자연과학대학 통계학과, 2021.8. 오희석.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.2차원 공간에서 관측되는 비정상 자료는 그 공간적 비동질성이 1차원 곡선을 따라 나타난다. 이러한 방향적 특이성을 표현하기 위한 다중척도 방법론으로는 Candes and Donoho (1999)가 처음 제시한 커브렛 변환이 널리 알려져 있지만 이는 자료가 일정한 간격으로 관측되어야 한다는 제약이 있다. 한편 산재된 자료에 내재된 함수를 근사하기 위해서는 방사기저함수를 이용한 내삽법이 흔히 이용되지만 등방성이 있는 방사기저함수로는 방향성을 효율적으로 표현할 수 없다. 본 학위논문에서는 2차원 유클리드 공간에서 잡음과 함께 산재되어 관측되는 방향성 자료의 효율적인 표현을 위해 비등방성 방사기저함수를 이용한 새로운 다중척도 방법론을 제안한다. 이때 각 스케일에서 전반적인 방향성 구조와 국소적인 방향성 구조를 분리하여 표현하기 위해 기저함수의 스케일 간 직교화가 이루어진다. 제안된 방법이 산재된 방향성 자료를 표현하는 데 있어 우수함을 보이기 위해 모의실험과 실제 자료에 대한 수치실험을 한 결과를 제시하였다. 한편 제안된 방법의 수렴성과 실제 구현 방법에 관한 사안들도 다루었다.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) 82박

Compact elliptical basis functions for surface reconstruction

In this technical report I present a method to reconstruct a surface representation from a a set of EBF's, and in addition present an efficient top--down method to build an EBF representation from a point cloud representation of a surface. I also discuss the advantages and disadvantages of this approach

Empowering materials processing and performance from data and AI

[No abstract available

Radial Basis Functions: Biomedical Applications and Parallelization

Radial basis function (RBF) is a real-valued function whose values depend only on the distances between an interpolation point and a set of user-specified points called centers. RBF interpolation is one of the primary methods to reconstruct functions from multi-dimensional scattered data. Its abilities to generalize arbitrary space dimensions and to provide spectral accuracy have made it particularly popular in different application areas, including but not limited to: finding numerical solutions of partial differential equations (PDEs), image processing, computer vision and graphics, deep learning and neural networks, etc. The present thesis discusses three applications of RBF interpolation in biomedical engineering areas: (1) Calcium dynamics modeling, in which we numerically solve a set of PDEs by using meshless numerical methods and RBF-based interpolation techniques; (2) Image restoration and transformation, where an image is restored from its triangular mesh representation or transformed under translation, rotation, and scaling, etc. from its original form; (3) Porous structure design, in which the RBF interpolation used to reconstruct a 3D volume containing porous structures from a set of regularly or randomly placed points inside a user-provided surface shape. All these three applications have been investigated and their effectiveness has been supported with numerous experimental results. In particular, we innovatively utilize anisotropic distance metrics to define the distance in RBF interpolation and apply them to the aforementioned second and third applications, which show significant improvement in preserving image features or capturing connected porous structures over the isotropic distance-based RBF method. Beside the algorithm designs and their applications in biomedical areas, we also explore several common parallelization techniques (including OpenMP and CUDA-based GPU programming) to accelerate the performance of the present algorithms. In particular, we analyze how parallel programming can help RBF interpolation to speed up the meshless PDE solver as well as image processing. While RBF has been widely used in various science and engineering fields, the current thesis is expected to trigger some more interest from computational scientists or students into this fast-growing area and specifically apply these techniques to biomedical problems such as the ones investigated in the present work

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In this paper we develop a discrete Hierarchical Basis (HB) to efficiently solve the Radial Basis Function (RBF) interpolation problem with variable polynomial order. The HB forms an orthogonal set and is adapted to the kernel seed function and the placement of the interpolation nodes. Moreover, this basis is orthogonal to a set of polynomials up to a given order defined on the interpolating nodes. We are thus able to decouple the RBF interpolation problem for any order of the polynomial interpolation and solve it in two steps: (1) The polynomial orthogonal RBF interpolation problem is efficiently solved in the transformed HB basis with a GMRES iteration and a diagonal, or block SSOR preconditioner. (2) The residual is then projected onto an orthonormal polynomial basis. We apply our approach on several test cases to study its effectiveness, including an application to the Best Linear Unbiased Estimator regression problem