Ridgelets and the representation of mutilated Sobolev functions

We show that ridgelets, a system introduced in [E. J. Candes, Appl. Comput. Harmon. Anal., 6(1999), pp. 197–218], are optimal to represent smooth multivariate functions that may exhibit linear singularities. For instance, let {u · x − b > 0} be an arbitrary hyperplane and consider the singular function f(x) = 1{u·x−b>0}g(x), where g is compactly supported with finite Sobolev L2 norm ||g||Hs, s > 0. The ridgelet coefficient sequence of such an object is as sparse as if f were without singularity, allowing optimal partial reconstructions. For instance, the n-term approximation obtained by keeping the terms corresponding to the n largest coefficients in the ridgelet series achieves a rate of approximation of order n−s/d; the presence of the singularity does not spoil the quality of the ridgelet approximation. This is unlike all systems currently in use, especially Fourier or wavelet representations

Recovering edges in ill-posed inverse problems: optimality of curvelet frames

We consider a model problem of recovering a function $f(x_1,x_2)$ from noisy Radon data. The function $f$ to be recovered is assumed smooth apart from a discontinuity along a $C^2$ curve, that is, an edge. We use the continuum white-noise model, with noise level $\varepsilon$. Traditional linear methods for solving such inverse problems behave poorly in the presence of edges. Qualitatively, the reconstructions are blurred near the edges; quantitatively, they give in our model mean squared errors (MSEs) that tend to zero with noise level $\varepsilon$ only as $O(\varepsilon^{1/2})$ as $\varepsilon\to 0$. A recent innovation--nonlinear shrinkage in the wavelet domain--visually improves edge sharpness and improves MSE convergence to $O(\varepsilon^{2/3})$. However, as we show here, this rate is not optimal. In fact, essentially optimal performance is obtained by deploying the recently-introduced tight frames of curvelets in this setting. Curvelets are smooth, highly anisotropic elements ideally suited for detecting and synthesizing curved edges. To deploy them in the Radon setting, we construct a curvelet-based biorthogonal decomposition of the Radon operator and build "curvelet shrinkage" estimators based on thresholding of the noisy curvelet coefficients. In effect, the estimator detects edges at certain locations and orientations in the Radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain. We prove that the curvelet shrinkage can be tuned so that the estimator will attain, within logarithmic factors, the MSE $O(\varepsilon^{4/5})$ as noise level $\varepsilon\to 0$. This rate of convergence holds uniformly over a class of functions which are $C^2$ except for discontinuities along $C^2$ curves, and (except for log terms) is the minimax rate for that class. Our approach is an instance of a general strategy which should apply in other inverse problems; we sketch a deconvolution example

Directional edge and texture representations for image processing

An efficient representation for natural images is of fundamental importance in image processing and analysis. The commonly used separable transforms such as wavelets axe not best suited for images due to their inability to exploit directional regularities such as edges and oriented textural patterns; while most of the recently proposed directional schemes cannot represent these two types of features in a unified transform. This thesis focuses on the development of directional representations for images which can capture both edges and textures in a multiresolution manner. The thesis first considers the problem of extracting linear features with the multiresolution Fourier transform (MFT). Based on a previous MFT-based linear feature model, the work extends the extraction method into the situation when the image is corrupted by noise. The problem is tackled by the combination of a "Signal+Noise" frequency model, a refinement stage and a robust classification scheme. As a result, the MFT is able to perform linear feature analysis on noisy images on which previous methods failed. A new set of transforms called the multiscale polar cosine transforms (MPCT) are also proposed in order to represent textures. The MPCT can be regarded as real-valued MFT with similar basis functions of oriented sinusoids. It is shown that the transform can represent textural patches more efficiently than the conventional Fourier basis. With a directional best cosine basis, the MPCT packet (MPCPT) is shown to be an efficient representation for edges and textures, despite its high computational burden. The problem of representing edges and textures in a fixed transform with less complexity is then considered. This is achieved by applying a Gaussian frequency filter, which matches the disperson of the magnitude spectrum, on the local MFT coefficients. This is particularly effective in denoising natural images, due to its ability to preserve both types of feature. Further improvements can be made by employing the information given by the linear feature extraction process in the filter's configuration. The denoising results compare favourably against other state-of-the-art directional representations

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

Biometric Applications Based on Multiresolution Analysis Tools

This dissertation is dedicated to the development of new algorithms for biometric applications based on multiresolution analysis tools. Biometric is a unique, measurable characteristic of a human being that can be used to automatically recognize an individual or verify an individual\u27s identity. Biometrics can measure physiological, behavioral, physical and chemical characteristics of an individual. Physiological characteristics are based on measurements derived from direct measurement of a part of human body, such as, face, fingerprint, iris, retina etc. We focussed our investigations to fingerprint and face recognition since these two biometric modalities are used in conjunction to obtain reliable identification by various border security and law enforcement agencies. We developed an efficient and robust human face recognition algorithm for potential law enforcement applications. A generic fingerprint compression algorithm based on state of the art multiresolution analysis tool to speed up data archiving and recognition was also proposed. Finally, we put forth a new fingerprint matching algorithm by generating an efficient set of fingerprint features to minimize false matches and improve identification accuracy. Face recognition algorithms were proposed based on curvelet transform using kernel based principal component analysis and bidirectional two-dimensional principal component analysis and numerous experiments were performed using popular human face databases. Significant improvements in recognition accuracy were achieved and the proposed methods drastically outperformed conventional face recognition systems that employed linear one-dimensional principal component analysis. Compression schemes based on wave atoms decomposition were proposed and major improvements in peak signal to noise ratio were obtained in comparison to Federal Bureau of Investigation\u27s wavelet scalar quantization scheme. Improved performance was more pronounced and distinct at higher compression ratios. Finally, a fingerprint matching algorithm based on wave atoms decomposition, bidirectional two dimensional principal component analysis and extreme learning machine was proposed and noteworthy improvements in accuracy were realized