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

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

Phase-field simulations of viscous fingering in shear-thinning fluids

A phase-field model for the Hele-Shaw flow of non-Newtonian fluids is developed. It extends a previous model for Newtonian fluids to a wide range of shear-dependent fluids. The model is applied to perform simulations of viscous fingering in shear- thinning fluids, and it is found to be capable of describing the complete crossover from the Newtonian regime at low shear rate to the strongly shear-thinning regime at high shear rate. The width selection of a single steady-state finger is studied in detail for a 2-plateaux shear-thinning law (Carreau law) in both its weakly and strongly shear-thinning limits, and the results are related to previous analyses. In the strongly shear-thinning regime a rescaling is found for power-law (Ostwald-de-Waehle) fluids that allows for a direct comparison between simulations and experiments without any adjustable parameters, and good agreement is obtained

Automatic solar feature detection using image processing and pattern recognition techniques

The objective of the research in this dissertation is to develop a software system to automatically detect and characterize solar flares, filaments and Corona Mass Ejections (CMEs), the core of so-called solar activity. These tools will assist us to predict space weather caused by violent solar activity. Image processing and pattern recognition techniques are applied to this system. For automatic flare detection, the advanced pattern recognition techniques such as Multi-Layer Perceptron (MLP), Radial Basis Function (RBF), and Support Vector Machine (SVM) are used. By tracking the entire process of flares, the motion properties of two-ribbon flares are derived automatically. In the applications of the solar filament detection, the Stabilized Inverse Diffusion Equation (SIDE) is used to enhance and sharpen filaments; a new method for automatic threshold selection is proposed to extract filaments from background; an SVM classifier with nine input features is used to differentiate between sunspots and filaments. Once a filament is identified, morphological thinning, pruning, and adaptive edge linking methods are applied to determine filament properties. Furthermore, a filament matching method is proposed to detect filament disappearance. The automatic detection and characterization of flares and filaments have been successfully applied on Hα full-disk images that are continuously obtained at Big Bear Solar Observatory (BBSO). For automatically detecting and classifying CMEs, the image enhancement, segmentation, and pattern recognition techniques are applied to Large Angle Spectrometric Coronagraph (LASCO) C2 and C3 images. The processed LASCO and BBSO images are saved to file archive, and the physical properties of detected solar features such as intensity and speed are recorded in our database. Researchers are able to access the solar feature database and analyze the solar data efficiently and effectively. The detection and characterization system greatly improves the ability to monitor the evolution of solar events and has potential to be used to predict the space weather

Evaluating the Differences of Gridding Techniques for Digital Elevation Models Generation and Their Influence on the Modeling of Stony Debris Flows Routing: A Case Study From Rovina di Cancia Basin (North-Eastern Italian Alps)

Debris \ufb02ows are among the most hazardous phenomena in mountain areas. To cope with debris \ufb02ow hazard, it is common to delineate the risk-prone areas through routing models. The most important input to debris \ufb02ow routing models are the topographic data, usually in the form of Digital Elevation Models (DEMs). The quality of DEMs depends on the accuracy, density, and spatial distribution of the sampled points; on the characteristics of the surface; and on the applied gridding methodology. Therefore, the choice of the interpolation method affects the realistic representation of the channel and fan morphology, and thus potentially the debris \ufb02ow routing modeling outcomes. In this paper, we initially investigate the performance of common interpolation methods (i.e., linear triangulation, natural neighbor, nearest neighbor, Inverse Distance to a Power, ANUDEM, Radial Basis Functions, and ordinary kriging) in building DEMs with the complex topography of a debris \ufb02ow channel located in the Venetian Dolomites (North-eastern Italian Alps), by using small footprint full- waveform Light Detection And Ranging (LiDAR) data. The investigation is carried out through a combination of statistical analysis of vertical accuracy, algorithm robustness, and spatial clustering of vertical errors, and multi-criteria shape reliability assessment. After that, we examine the in\ufb02uence of the tested interpolation algorithms on the performance of a Geographic Information System (GIS)-based cell model for simulating stony debris \ufb02ows routing. In detail, we investigate both the correlation between the DEMs heights uncertainty resulting from the gridding procedure and that on the corresponding simulated erosion/deposition depths, both the effect of interpolation algorithms on simulated areas, erosion and deposition volumes, solid-liquid discharges, and channel morphology after the event. The comparison among the tested interpolation methods highlights that the ANUDEM and ordinary kriging algorithms are not suitable for building DEMs with complex topography. Conversely, the linear triangulation, the natural neighbor algorithm, and the thin-plate spline plus tension and completely regularized spline functions ensure the best trade-off among accuracy and shape reliability. Anyway, the evaluation of the effects of gridding techniques on debris \ufb02ow routing modeling reveals that the choice of the interpolation algorithm does not signi\ufb01cantly affect the model outcomes

Diagnosing observation error correlations for Doppler radar radial winds in the Met Office UKV model using observation-minus-background and observation-minus-analysis statistics

With the development of convection-permitting numerical weather prediction the efficient use of high-resolution observations in data assimilation is becoming increasingly important. The operational assimilation of these observations, such as Doppler radar radial winds (DRWs), is now common, though to avoid violating the assumption of uncorrelated observation errors the observation density is severely reduced. To improve the quantity of observations used and the impact that they have on the forecast requires the introduction of the full, potentially correlated, error statistics. In this work, observation error statistics are calculated for the DRWs that are assimilated into the Met Office high-resolution UK model using a diagnostic that makes use of statistical averages of observation-minus-background and observation-minus-analysis residuals. This is the first in-depth study using the diagnostic to estimate both horizontal and along-beam observation error statistics. The new results obtained show that the DRW error standard deviations are similar to those used operationally and increase as the observation height increases. Surprisingly the estimated observation error correlation length-scales are longer than the operational thinning distance. They are dependent both on the height of the observation and on the distance of the observation away from the radar. Further tests show that the long correlations cannot be attributed to the background error covariance matrix used in the assimilation, although they are, in part, a result of using superobservations and a simplified observation operator. The inclusion of correlated error statistics in the assimilation allows less thinning of the data and hence better use of the high-resolution observations

An energy-based model approach to rare event probability estimation

The estimation of rare event probabilities plays a pivotal role in diverse fields. Our aim is to determine the probability of a hazard or system failure occurring when a quantity of interest exceeds a critical value. In our approach, the distribution of the quantity of interest is represented by an energy density, characterized by a free energy function. To efficiently estimate the free energy, a bias potential is introduced. Using concepts from energy-based models (EBM), this bias potential is optimized such that the corresponding probability density function approximates a pre-defined distribution targeting the failure region of interest. Given the optimal bias potential, the free energy function and the rare event probability of interest can be determined. The approach is applicable not just in traditional rare event settings where the variable upon which the quantity of interest relies has a known distribution, but also in inversion settings where the variable follows a posterior distribution. By combining the EBM approach with a Stein discrepancy-based stopping criterion, we aim for a balanced accuracy-efficiency trade-off. Furthermore, we explore both parametric and non-parametric approaches for the bias potential, with the latter eliminating the need for choosing a particular parameterization, but depending strongly on the accuracy of the kernel density estimate used in the optimization process. Through three illustrative test cases encompassing both traditional and inversion settings, we show that the proposed EBM approach, when properly configured, (i) allows stable and efficient estimation of rare event probabilities and (ii) compares favorably against subset sampling approaches

Resilience of three-dimensional sinusoidal networks in liver tissue

Can three-dimensional, microvasculature networks still ensure blood supply if individual links fail? We address this question in the sinusoidal network, a plexus-like microvasculature network, which transports nutrient-rich blood to every hepatocyte in liver tissue, by building on recent advances in high-resolution imaging and digital reconstruction of adult mice liver tissue. We find that the topology of the three-dimensional sinusoidal network reflects its two design requirements of a space-filling network that connects all hepatocytes, while using shortest transport routes: sinusoidal networks are sub-graphs of the Delaunay graph of their set of branching points, and also contain the corresponding minimum spanning tree, both to good approximation. To overcome the spatial limitations of experimental samples and generate arbitrarily-sized networks, we developed a network generation algorithm that reproduces the statistical features of 0.3-mm-sized samples of sinusoidal networks, using multi-objective optimization for node degree and edge length distribution. Nematic order in these simulated networks implies anisotropic transport properties, characterized by an empirical linear relation between a nematic order parameter and the anisotropy of the permeability tensor. Under the assumption that all sinusoid tubes have a constant and equal flow resistance, we predict that the distribution of currents in the network is very inhomogeneous, with a small number of edges carrying a substantial part of the flow-a feature known for hierarchical networks, but unexpected for plexus-like networks. We quantify network resilience in terms of a permeability-at-risk, i.e., permeability as function of the fraction of removed edges. We find that sinusoidal networks are resilient to random removal of edges, but vulnerable to the removal of high-current edges. Our findings suggest the existence of a mechanism counteracting flow inhomogeneity to balance metabolic load on the liver