387 research outputs found
Performance and Robustness Analysis of Co-Prime and Nested Sampling
Coprime and nested sampling are well known deterministic sampling techniques that operate at rates significantly lower than the Nyquist rate, and yet allow perfect reconstruction of the spectra of wide sense stationary signals. However, theoretical guarantees for these samplers assume ideal conditions such as synchronous sampling, and ability to perfectly compute statistical expectations. This thesis studies the performance of coprime and nested samplers in spatial and temporal domains, when these assumptions are violated.
In spatial domain, the robustness of these samplers is studied by considering arrays with perturbed sensor locations (with unknown perturbations). Simplified expressions for the Fisher Information matrix for perturbed coprime and nested arrays are derived, which explicitly highlight the role of co-array. It is shown that even in presence of perturbations, it is possible to resolve under appropriate conditions on the size of the grid. The assumption of small perturbations leads to a novel ``bi-affine" model in terms of source powers and perturbations. The redundancies in the co-array are then exploited to eliminate the nuisance perturbation variable, and reduce the bi-affine problem to a linear underdetermined (sparse) problem in source powers.
This thesis also studies the robustness of coprime sampling to finite number of samples and sampling jitter, by analyzing their effects on the quality of the estimated autocorrelation sequence. A variety of bounds on the error introduced by such non ideal sampling schemes are computed by considering a statistical model for the perturbation. They indicate that coprime sampling leads to stable estimation of the autocorrelation sequence, in presence of small perturbations. Under appropriate assumptions on the distribution of WSS signals, sharp bounds on the estimation error are established which indicate that the error decays exponentially with the number of samples. The theoretical claims are supported by extensive numerical experiments
Statistical Performance Analysis of Sparse Linear Arrays
Direction-of-arrival (DOA) estimation remains an important topic in array signal processing. With uniform linear arrays (ULAs), traditional subspace-based methods can resolve only up to M-1 sources using M sensors. On the other hand, by exploiting their so-called difference coarray model, sparse linear arrays, such as co-prime and nested arrays, can resolve up to O(M^2) sources using only O(M) sensors. Various new sparse linear array geometries were proposed and many direction-finding algorithms were developed based on sparse linear arrays. However, the statistical performance of such arrays has not been analytically conducted. In this dissertation, we (i) study the asymptotic performance of the MUtiple SIgnal Classification (MUSIC) algorithm utilizing sparse linear arrays, (ii) derive and analyze performance bounds for sparse linear arrays, and (iii) investigate the robustness of sparse linear arrays in the presence of array imperfections. Based on our analytical results, we also propose robust direction-finding algorithms for use when data are missing.
We begin by analyzing the performance of two commonly used coarray-based MUSIC direction estimators. Because the coarray model is used, classical derivations no longer apply. By using an alternative eigenvector perturbation analysis approach, we derive a closed-form expression of the asymptotic mean-squared error (MSE) of both estimators. Our expression is computationally efficient compared with the alternative of Monte Carlo simulations. Using this expression, we show that when the source number exceeds the sensor number, the MSE remains strictly positive as the signal-to-noise ratio (SNR) approaches infinity. This finding theoretically explains the unusual saturation behavior of coarray-based MUSIC estimators that had been observed in previous studies.
We next derive and analyze the Cramér-Rao bound (CRB) for general sparse linear arrays under the assumption that the sources are uncorrelated. We show that, unlike the classical stochastic CRB, our CRB is applicable even if there are more sources than the number of sensors. We also show that, in such a case, this CRB remains strictly positive definite as the SNR approaches infinity. This unusual behavior imposes a strict lower bound on the variance of unbiased DOA estimators in the underdetermined case. We establish the connection between our CRB and the classical stochastic CRB and show that they are asymptotically equal when the sources are uncorrelated and the SNR is sufficiently high. We investigate the behavior of our CRB for co-prime and nested arrays with a large number of sensors, characterizing the trade-off between the number of spatial samples and the number of temporal samples. Our analytical results on the CRB will benefit future research on optimal sparse array designs.
We further analyze the performance of sparse linear arrays by considering sensor location errors. We first introduce the deterministic error model. Based on this model, we derive a closed-form expression of the asymptotic MSE of a commonly used coarray-based MUSIC estimator, the spatial-smoothing based MUSIC (SS-MUSIC). We show that deterministic sensor location errors introduce a constant estimation bias that cannot be mitigated by only increasing the SNR. Our analytical expression also provides a sensitivity measure against sensor location errors for sparse linear arrays. We next extend our derivations to the stochastic error model and analyze the Gaussian case. We also derive the CRB for joint estimation of DOA parameters and deterministic sensor location errors. We show that this CRB is applicable even if there are more sources than the number of sensors.
Lastly, we develop robust DOA estimators for cases with missing data. By exploiting the difference coarray structure, we introduce three algorithms to construct an augmented covariance matrix with enhanced degrees of freedom. By applying MUSIC to this augmented covariance matrix, we are able to resolve more sources than sensors. Our method utilizes information from all snapshots and shows improved estimation performance over traditional DOA estimators
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Interplay of Theory and Numerics for Deterministic and Stochastic Homogenization
The workshop has brought together experts in the broad field of partial differential equations with highly heterogeneous coefficients. Analysts and computational and applied mathematicians have shared results and ideas on a topic of considerable interest both from the theoretical and applied viewpoints. A characteristic feature of the workshop has been to encourage discussions on the theoretical as well as numerical challenges in the field, both from the point of view of deterministic as well as stochastic modeling of the heterogeneities
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
Deep Structured Layers for Instance-Level Optimization in 2D and 3D Vision
The approach we present in this thesis is that of integrating optimization problems
as layers in deep neural networks. Optimization-based modeling provides an additional set of tools enabling the design of powerful neural networks for a wide
battery of computer vision tasks. This thesis shows formulations and experiments
for vision tasks ranging from image reconstruction to 3D reconstruction.
We first propose an unrolled optimization method with implicit regularization
properties for reconstructing images from noisy camera readings. The method resembles an unrolled majorization minimization framework with convolutional neural networks acting as regularizers. We report state-of-the-art performance in image
reconstruction on both noisy and noise-free evaluation setups across many datasets.
We further focus on the task of monocular 3D reconstruction of articulated objects using video self-supervision. The proposed method uses a structured layer for
accurate object deformation that controls a 3D surface by displacing a small number
of learnable handles. While relying on a small set of training data per category for
self-supervision, the method obtains state-of-the-art reconstruction accuracy with
diverse shapes and viewpoints for multiple articulated objects.
We finally address the shortcomings of the previous method that revolve
around regressing the camera pose using multiple hypotheses. We propose a method
that recovers a 3D shape from a 2D image by relying solely on 3D-2D correspondences regressed from a convolutional neural network. These correspondences are
used in conjunction with an optimization problem to estimate per sample the camera pose and deformation. We quantitatively show the effectiveness of the proposed
method on self-supervised 3D reconstruction on multiple categories without the need for multiple hypotheses
On the Manifold: Representing Geometry in C++ for State Estimation
Manipulating geometric objects is central to state estimation problems in robotics. Typical algorithms must optimize over non-Euclidean states, such as rigid transformations on the SE(3) manifold, and handle measurements expressed in multiple coordinate frames. Researchers typically rely on C++ libraries for geometric tasks. Commonly used libraries range from linear algebra software such as Eigen to robotics-targeted optimization frameworks such as GTSAM, which provides manifold operations and automatic differentiation of arbitrary expressions. This thesis examines how geometric operations in existing software can be improved, both in runtime performance and in the expression of geometric semantics, to support rapid and error-free development of robotics algorithms.
This thesis presents wave_geometry, a C++ manifold geometry library providing representations of objects in affine, Euclidean, and projective spaces, and the Lie groups SO(3) and SE(3). It encompasses the main contributions of this work: an expression template-based automatic differentiation system and compile-time checking of coordinate frame semantics. The library can evaluate Jacobians of geometric expressions in forward and reverse mode with little runtime overhead compared to hand-coded derivatives, and exceeds the performance of existing libraries. While high performance is achieved by taking advantage of compile-time knowledge, the library also provides dynamic expressions which can be composed at runtime.
Coordinate frame conversions are a common source of mistakes in calculations.
However, the validity of operations can automatically be checked by tracking the coordinate frames associated with each object. A system of rules for propagating coordinate frame semantics though geometric operations, including manifold operations, is developed. A template-based method for checking coordinate frame semantics at compile time, with no runtime overhead, is presented.
Finally, this thesis demonstrates an application to state estimation, presenting a framework for formulating nonlinear least squares optimization problems as factor graphs. The framework combines wave_geometry expressions with the widely used Ceres Solver software, and shows the utility of automatically differentiated geometric expressions
Effects of edge roughness on optical scattering from periodic microstructures
Planar photonic crystals and other microstructured surfaces have important applications in a number of emerging technologies. However, these structures can be difficult to fabricate in a consistent manner. Rapid, precise measurements of critical parameters are needed to control the fabrication process, but current measurement techniques tend to be slow and often require that the sample be modified in order to make the measurement. Optical scattering can provide a rapid, non-destructive, and precise method for measuring these structures, and optical scatterometry is a good candidate technique for measuring micro-structured surfaces for process control. However, variations in the profile, such as those caused by edge roughness, can make significant contributions to the uncertainty in scatterometry measurements. Because of the multi- dimensional nature of the problem, modeling these variations can be computationally expensive. This dissertation examines the effects of edge roughness on optical scatterometry signals. Rigorous numerical simulations show that the effects of edge roughness are sensitive to the correlation length and the frequency content of the roughness as well as its amplitude. However, these rigorous calculations are computationally expensive. A less computationally expensive model based on a generalized Bruggeman effective medium approximation is developed and shown to be effective for modeling the effects of short correlation length edge roughness on optical scatterometry signals
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