Flexible rational approximation and its application for matrix functions

This paper proposes a unique optimization approach for estimating the minimax rational approximation and its application for evaluating matrix functions. Our method enables the extension to generalized rational approximations and has the flexibility of adding constraints. In particular, the latter allows us to control specific properties preferred in matrix function evaluation. For example, in the case of a normal matrix, we can guarantee a bound over the condition number of the matrix, which one needs to invert for evaluating the rational matrix function. We demonstrate the efficiency of our approach for several applications of matrix functions based on direct spectrum filtering

Asymptotics of wavelets and filters

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998.Includes bibliographical references (p. 126-131).by Jianhong (Jackie) Shen.Ph.D

Diffeomorphic Transformations for Time Series Analysis: An Efficient Approach to Nonlinear Warping

The proliferation and ubiquity of temporal data across many disciplines has sparked interest for similarity, classification and clustering methods specifically designed to handle time series data. A core issue when dealing with time series is determining their pairwise similarity, i.e., the degree to which a given time series resembles another. Traditional distance measures such as the Euclidean are not well-suited due to the time-dependent nature of the data. Elastic metrics such as dynamic time warping (DTW) offer a promising approach, but are limited by their computational complexity, non-differentiability and sensitivity to noise and outliers. This thesis proposes novel elastic alignment methods that use parametric \& diffeomorphic warping transformations as a means of overcoming the shortcomings of DTW-based metrics. The proposed method is differentiable \& invertible, well-suited for deep learning architectures, robust to noise and outliers, computationally efficient, and is expressive and flexible enough to capture complex patterns. Furthermore, a closed-form solution was developed for the gradient of these diffeomorphic transformations, which allows an efficient search in the parameter space, leading to better solutions at convergence. Leveraging the benefits of these closed-form diffeomorphic transformations, this thesis proposes a suite of advancements that include: (a) an enhanced temporal transformer network for time series alignment and averaging, (b) a deep-learning based time series classification model to simultaneously align and classify signals with high accuracy, (c) an incremental time series clustering algorithm that is warping-invariant, scalable and can operate under limited computational and time resources, and finally, (d) a normalizing flow model that enhances the flexibility of affine transformations in coupling and autoregressive layers.Comment: PhD Thesis, defended at the University of Navarra on July 17, 2023. 277 pages, 8 chapters, 1 appendi

Nonlinear Approximations in Filter Design and Wave Propagation

This thesis has two parts. In both parts we use nonlinear approximations to obtain accurate solutions to problems where traditional numerical approaches rapidly become computationally infeasible. The first part describes a systematic method for designing highly accurate and efficient infinite impulse response (IIR) and finite impulse response (FIR) filters given their specifications. In our approach, we first meet the specifications by constructing an IIR filter, without requiring the filter to be causal, and possibly with a large number of poles. We then construct, for any given accuracy, an optimal IIR version of such filter. Finally, also for any given accuracy, we convert the IIR filter to an efficient FIR filter cascade. In this FIR approximation, the non-causal part of the IIR filter only introduces an additional delay. Because our IIR construction does not have to enforce causality, the filters we design are more efficient than filters designed by existing methods. The second part describes a fast algorithm to propagate, for any desired accuracy, a time-harmonic electromagnetic field between two planes separated by free space. The analytic formulation of this problem (circa 1897) requires the evaluation of the Rayleigh-Sommerfeld integral. If the distance between the planes is small, this integral can be accurately evaluated in the Fourier domain; if the distance is large, it can be accurately approximated by asymptotic methods. The computational difficulties arise in the intermediate region where, in order to obtain an accurate solution, it is necessary to apply the oscillatory Rayleigh-Sommerfeld kernel as is. In our approach, we accurately approximate the kernel by a short sum of Gaussians with complex exponents and then efficiently apply the result to input data using the unequally spaced fast Fourier transform. The resulting algorithm has the same computational complexity as methods based on the Fresnel approximation. We demonstrate that while the Fresnel approximation may provide adequate accuracy near the optical axis, the accuracy deteriorates significantly away from the optical axis. In contrast, our method maintains controlled accuracy throughout the entire computational domain

Energy efficient enabling technologies for semantic video processing on mobile devices

Semantic object-based processing will play an increasingly important role in future multimedia systems due to the ubiquity of digital multimedia capture/playback technologies and increasing storage capacity. Although the object based paradigm has many undeniable benefits, numerous technical challenges remain before the applications becomes pervasive, particularly on computational constrained mobile devices. A fundamental issue is the ill-posed problem of semantic object segmentation. Furthermore, on battery powered mobile computing devices, the additional algorithmic complexity of semantic object based processing compared to conventional video processing is highly undesirable both from a real-time operation and battery life perspective. This thesis attempts to tackle these issues by firstly constraining the solution space and focusing on the human face as a primary semantic concept of use to users of mobile devices. A novel face detection algorithm is proposed, which from the outset was designed to be amenable to be offloaded from the host microprocessor to dedicated hardware, thereby providing real-time performance and reducing power consumption. The algorithm uses an Artificial Neural Network (ANN), whose topology and weights are evolved via a genetic algorithm (GA). The computational burden of the ANN evaluation is offloaded to a dedicated hardware accelerator, which is capable of processing any evolved network topology. Efficient arithmetic circuitry, which leverages modified Booth recoding, column compressors and carry save adders, is adopted throughout the design. To tackle the increased computational costs associated with object tracking or object based shape encoding, a novel energy efficient binary motion estimation architecture is proposed. Energy is reduced in the proposed motion estimation architecture by minimising the redundant operations inherent in the binary data. Both architectures are shown to compare favourable with the relevant prior art

New Techniques for the Modeling, Processing and Visualization of Surfaces and Volumes

With the advent of powerful 3D acquisition technology, there is a growing demand for the modeling, processing, and visualization of surfaces and volumes. The proposed methods must be efficient and robust, and they must be able to extract the essential structure of the data and to easily and quickly convey the most significant information to a human observer. Independent of the specific nature of the data, the following fundamental problems can be identified: shape reconstruction from discrete samples, data analysis, and data compression. This thesis presents several novel solutions to these problems for surfaces (Part I) and volumes (Part II). For surfaces, we adopt the well-known triangle mesh representation and develop new algorithms for discrete curvature estimation,detection of feature lines, and line-art rendering (Chapter 3), for connectivity encoding (Chapter 4), and for topology preserving compression of 2D vector fields (Chapter 5). For volumes, that are often given as discrete samples, we base our approach for reconstruction and visualization on the use of new trivariate spline spaces on a certain tetrahedral partition. We study the properties of the new spline spaces (Chapter 7) and present efficient algorithms for reconstruction and visualization by iso-surface rendering for both, regularly (Chapter 8) and irregularly (Chapter 9) distributed data samples

Wavelets and Subband Coding

First published in 1995, Wavelets and Subband Coding offered a unified view of the exciting field of wavelets and their discrete-time cousins, filter banks, or subband coding. The book developed the theory in both continuous and discrete time, and presented important applications. During the past decade, it filled a useful need in explaining a new view of signal processing based on flexible time-frequency analysis and its applications. Since 2007, the authors now retain the copyright and allow open access to the book

Accurate, Fast and Controllable Image and Point Cloud Registration

Registration is the process of establishing spatial correspondences between two objects. Many downstream tasks, e.g, in image analysis, shape animation, can make use of these spatial correspondences. A variety of registration approaches have been developed over the last decades, but only recently registration approaches have been developed that make use of and can easily process the large data samples of the big data era. On the one hand, traditional optimization-based approaches are too slow and cannot take advantage of very large data sets. On the other hand, registration users expect more controllable and accurate solutions since most downstream tasks, e.g., facial animation and 3D reconstruction, increasingly rely on highly precise spatial correspondences. In recent years, deep network registration approaches have become popular as learning-based approaches are fast and can benefit from large-scale data during network training. However, how to make such deep-learning-based approached accurate and controllable is still a challenging problem that is far from being completely solved. This thesis explores fast, accurate and controllable solutions for image and point cloud registration. Specifically, for image registration, we first improve the accuracy of deep-learning-based approaches by introducing a general framework that consists of affine and non-parametric registration for both global and local deformation. We then design a more controllable image registration approach that image regions could be regularized differently according to their local attributes. For point cloud registration, existing works either are limited to small-scale problems, hardly handle complicated transformations or are slow to solve. We thus develop fast, accurate and controllable solutions for large-scale real-world registration problems via integrating optimal transport with deep geometric learning.Doctor of Philosoph