3D weak lensing with spin wavelets on the ball

We construct the spin flaglet transform, a wavelet transform to analyze spin signals in three dimensions. Spin flaglets can probe signal content localized simultaneously in space and frequency and, moreover, are separable so that their angular and radial properties can be controlled independently. They are particularly suited to analyzing of cosmological observations such as the weak gravitational lensing of galaxies. Such observations have a unique 3D geometrical setting since they are natively made on the sky, have spin angular symmetries, and are extended in the radial direction by additional distance or redshift information. Flaglets are constructed in the harmonic space defined by the Fourier-Laguerre transform, previously defined for scalar functions and extended here to signals with spin symmetries. Thanks to various sampling theorems, both the Fourier-Laguerre and flaglet transforms are theoretically exact when applied to bandlimited signals. In other words, in numerical computations the only loss of information is due to the finite representation of floating point numbers. We develop a 3D framework relating the weak lensing power spectrum to covariances of flaglet coefficients. We suggest that the resulting novel flaglet weak lensing estimator offers a powerful alternative to common 2D and 3D approaches to accurately capture cosmological information. While standard weak lensing analyses focus on either real or harmonic space representations (i.e., correlation functions or Fourier-Bessel power spectra, respectively), a wavelet approach inherits the advantages of both techniques, where both complicated sky coverage and uncertainties associated with the physical modeling of small scales can be handled effectively. Our codes to compute the Fourier-Laguerre and flaglet transforms are made publicly available.Comment: 24 pages, 4 figures, version accepted for publication in PR

Hydrodynamic Flows on Curved Surfaces: Spectral Numerical Methods for Radial Manifold Shapes

We formulate hydrodynamic equations and spectrally accurate numerical methods for investigating the role of geometry in flows within two-dimensional fluid interfaces. To achieve numerical approximations having high precision and level of symmetry for radial manifold shapes, we develop spectral Galerkin methods based on hyperinterpolation with Lebedev quadratures for $L^2$-projection to spherical harmonics. We demonstrate our methods by investigating hydrodynamic responses as the surface geometry is varied. Relative to the case of a sphere, we find significant changes can occur in the observed hydrodynamic flow responses as exhibited by quantitative and topological transitions in the structure of the flow. We present numerical results based on the Rayleigh-Dissipation principle to gain further insights into these flow responses. We investigate the roles played by the geometry especially concerning the positive and negative Gaussian curvature of the interface. We provide general approaches for taking geometric effects into account for investigations of hydrodynamic phenomena within curved fluid interfaces.Comment: 14 figure

DeepSphere: Efficient spherical Convolutional Neural Network with HEALPix sampling for cosmological applications

Convolutional Neural Networks (CNNs) are a cornerstone of the Deep Learning toolbox and have led to many breakthroughs in Artificial Intelligence. These networks have mostly been developed for regular Euclidean domains such as those supporting images, audio, or video. Because of their success, CNN-based methods are becoming increasingly popular in Cosmology. Cosmological data often comes as spherical maps, which make the use of the traditional CNNs more complicated. The commonly used pixelization scheme for spherical maps is the Hierarchical Equal Area isoLatitude Pixelisation (HEALPix). We present a spherical CNN for analysis of full and partial HEALPix maps, which we call DeepSphere. The spherical CNN is constructed by representing the sphere as a graph. Graphs are versatile data structures that can act as a discrete representation of a continuous manifold. Using the graph-based representation, we define many of the standard CNN operations, such as convolution and pooling. With filters restricted to being radial, our convolutions are equivariant to rotation on the sphere, and DeepSphere can be made invariant or equivariant to rotation. This way, DeepSphere is a special case of a graph CNN, tailored to the HEALPix sampling of the sphere. This approach is computationally more efficient than using spherical harmonics to perform convolutions. We demonstrate the method on a classification problem of weak lensing mass maps from two cosmological models and compare the performance of the CNN with that of two baseline classifiers. The results show that the performance of DeepSphere is always superior or equal to both of these baselines. For high noise levels and for data covering only a smaller fraction of the sphere, DeepSphere achieves typically 10% better classification accuracy than those baselines. Finally, we show how learned filters can be visualized to introspect the neural network.Comment: arXiv admin note: text overlap with arXiv:astro-ph/0409513 by other author

Harmonic Exponential Families on Manifolds

In a range of fields including the geosciences, molecular biology, robotics and computer vision, one encounters problems that involve random variables on manifolds. Currently, there is a lack of flexible probabilistic models on manifolds that are fast and easy to train. We define an extremely flexible class of exponential family distributions on manifolds such as the torus, sphere, and rotation groups, and show that for these distributions the gradient of the log-likelihood can be computed efficiently using a non-commutative generalization of the Fast Fourier Transform (FFT). We discuss applications to Bayesian camera motion estimation (where harmonic exponential families serve as conjugate priors), and modelling of the spatial distribution of earthquakes on the surface of the earth. Our experimental results show that harmonic densities yield a significantly higher likelihood than the best competing method, while being orders of magnitude faster to train.Comment: fixed typ

Unified Heat Kernel Regression for Diffusion, Kernel Smoothing and Wavelets on Manifolds and Its Application to Mandible Growth Modeling in CT Images

We present a novel kernel regression framework for smoothing scalar surface data using the Laplace-Beltrami eigenfunctions. Starting with the heat kernel constructed from the eigenfunctions, we formulate a new bivariate kernel regression framework as a weighted eigenfunction expansion with the heat kernel as the weights. The new kernel regression is mathematically equivalent to isotropic heat diffusion, kernel smoothing and recently popular diffusion wavelets. Unlike many previous partial differential equation based approaches involving diffusion, our approach represents the solution of diffusion analytically, reducing numerical inaccuracy and slow convergence. The numerical implementation is validated on a unit sphere using spherical harmonics. As an illustration, we have applied the method in characterizing the localized growth pattern of mandible surfaces obtained in CT images from subjects between ages 0 and 20 years by regressing the length of displacement vectors with respect to the template surface.Comment: Accepted in Medical Image Analysi

Structured Sparsity Models for Multiparty Speech Recovery from Reverberant Recordings

We tackle the multi-party speech recovery problem through modeling the acoustic of the reverberant chambers. Our approach exploits structured sparsity models to perform room modeling and speech recovery. We propose a scheme for characterizing the room acoustic from the unknown competing speech sources relying on localization of the early images of the speakers by sparse approximation of the spatial spectra of the virtual sources in a free-space model. The images are then clustered exploiting the low-rank structure of the spectro-temporal components belonging to each source. This enables us to identify the early support of the room impulse response function and its unique map to the room geometry. To further tackle the ambiguity of the reflection ratios, we propose a novel formulation of the reverberation model and estimate the absorption coefficients through a convex optimization exploiting joint sparsity model formulated upon spatio-spectral sparsity of concurrent speech representation. The acoustic parameters are then incorporated for separating individual speech signals through either structured sparse recovery or inverse filtering the acoustic channels. The experiments conducted on real data recordings demonstrate the effectiveness of the proposed approach for multi-party speech recovery and recognition.Comment: 31 page

Aprendizado de variedades para a síntese de áudio espacial

Orientadores: Luiz César Martini, Bruno Sanches MasieroTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: O objetivo do áudio espacial gerado com a técnica binaural é simular uma fonte sonora em localizações espaciais arbitrarias através das Funções de Transferência Relativas à Cabeça (HRTFs) ou também chamadas de Funções de Transferência Anatômicas. As HRTFs modelam a interação entre uma fonte sonora e a antropometria de uma pessoa (e.g., cabeça, torso e orelhas). Se filtrarmos uma fonte de áudio através de um par de HRTFs (uma para cada orelha), o som virtual resultante parece originar-se de uma localização espacial específica. Inspirados em nossos resultados bem sucedidos construindo uma aplicação prática de reconhecimento facial voltada para pessoas com deficiência visual que usa uma interface de usuário baseada em áudio espacial, neste trabalho aprofundamos nossa pesquisa para abordar vários aspectos científicos do áudio espacial. Neste contexto, esta tese analisa como incorporar conhecimentos prévios do áudio espacial usando uma nova representação não-linear das HRTFs baseada no aprendizado de variedades para enfrentar vários desafios de amplo interesse na comunidade do áudio espacial, como a personalização de HRTFs, a interpolação de HRTFs e a melhoria da localização de fontes sonoras. O uso do aprendizado de variedades para áudio espacial baseia-se no pressuposto de que os dados (i.e., as HRTFs) situam-se em uma variedade de baixa dimensão. Esta suposição também tem sido de grande interesse entre pesquisadores em neurociência computacional, que argumentam que as variedades são cruciais para entender as relações não lineares subjacentes à percepção no cérebro. Para todas as nossas contribuições usando o aprendizado de variedades, a construção de uma única variedade entre os sujeitos através de um grafo Inter-sujeito (Inter-subject graph, ISG) revelou-se como uma poderosa representação das HRTFs capaz de incorporar conhecimento prévio destas e capturar seus fatores subjacentes. Além disso, a vantagem de construir uma única variedade usando o nosso ISG e o uso de informações de outros indivíduos para melhorar o desempenho geral das técnicas aqui propostas. Os resultados mostram que nossas técnicas baseadas no ISG superam outros métodos lineares e não-lineares nos desafios de áudio espacial abordados por esta teseAbstract: The objective of binaurally rendered spatial audio is to simulate a sound source in arbitrary spatial locations through the Head-Related Transfer Functions (HRTFs). HRTFs model the direction-dependent influence of ears, head, and torso on the incident sound field. When an audio source is filtered through a pair of HRTFs (one for each ear), a listener is capable of perceiving a sound as though it were reproduced at a specific location in space. Inspired by our successful results building a practical face recognition application aimed at visually impaired people that uses a spatial audio user interface, in this work we have deepened our research to address several scientific aspects of spatial audio. In this context, this thesis explores the incorporation of spatial audio prior knowledge using a novel nonlinear HRTF representation based on manifold learning, which tackles three major challenges of broad interest among the spatial audio community: HRTF personalization, HRTF interpolation, and human sound localization improvement. Exploring manifold learning for spatial audio is based on the assumption that the data (i.e. the HRTFs) lies on a low-dimensional manifold. This assumption has also been of interest among researchers in computational neuroscience, who argue that manifolds are crucial for understanding the underlying nonlinear relationships of perception in the brain. For all of our contributions using manifold learning, the construction of a single manifold across subjects through an Inter-subject Graph (ISG) has proven to lead to a powerful HRTF representation capable of incorporating prior knowledge of HRTFs and capturing the underlying factors of spatial hearing. Moreover, the use of our ISG to construct a single manifold offers the advantage of employing information from other individuals to improve the overall performance of the techniques herein proposed. The results show that our ISG-based techniques outperform other linear and nonlinear methods in tackling the spatial audio challenges addressed by this thesisDoutoradoEngenharia de ComputaçãoDoutor em Engenharia Elétrica2014/14630-9FAPESPCAPE

Cooperative Position and Orientation Estimation with Multi-Mode Antennas

Robotic multi-agent systems are envisioned for planetary exploration and terrestrial applications. Autonomous operation of robots requires estimations of their positions and orientations, which are obtained from the direction-of-arrival (DoA) and the time-of-arrival (ToA) of radio signals exchanged among the agents. In this thesis, we estimate the signal DoA and ToA using a multi-mode antenna (MMA). An MMA is a single antenna element, where multiple orthogonal current modes are excited by different antenna ports. We provide a first study on the use of MMAs for cooperative position and orientation estimation, specifically exploring their DoA estimation capabilities. Assuming the agents of a cooperative network are equipped with MMAs, lower bounds on the achievable position and orientation accuracy are derived. We realize a gap between the theoretical lower bounds and real-world performance of a cooperative radio localization system, which is caused by imperfect antenna and transceiver calibration. Consequentially, we theoretically analyze in-situ antenna calibration, introduce an algorithm for the calibration of arbitrary multiport antennas and show its effectiveness by simulation. To also improve calibration during operation, we propose cooperative simultaneous localization and calibration (SLAC). We show that cooperative SLAC is able to estimate antenna responses and ranging biases of the agents together with their positions and orientations, leading to considerably better position and orientation accuracy. Finally, we validate the results from theory and simulation by experiments with robotic rovers equipped with software-defined radios (SDRs). In conclusion, we show that DoA estimation with an MMA is feasible, and accuracy can be improved by in-situ calibration and SLAC

Sound Source Localization and Modeling: Spherical Harmonics Domain Approaches

Sound source localization has been an important research topic in the acoustic signal processing community because of its wide use in many acoustic applications, including speech separation, speech enhancement, sound event detection, automatic speech recognition, automated camera steering, and virtual reality. In the recent decade, there is a growing interest in the research of sound source localization using higher-order microphone arrays, which are capable of recording and analyzing the soundfield over a target spatial area. This thesis studies a novel source feature called the relative harmonic coefficient, that easily estimated from the higher-order microphone measurements. This source feature has direct applications for sound source localization due to its sole dependence on the source position. This thesis proposes two novel sound source localization algorithms using the relative harmonic coefficients: (i) a low-complexity single source localization approach that localizes the source' elevation and azimuth separately. This approach is also appliable to acoustic enhancement for the higher-order microphone array recordings; (ii) a semi-supervised multi-source localization algorithm in a noisy and reverberant environment. Although this approach uses a learning schema, it still has a strong potential to be implemented in practice because only a limited number of labeled measurements are required. However, this algorithm has an inherent limitation as it requires the availability of single-source components. Thus, it is unusable in scenarios where the original recordings have limited single-source components (e.g., multiple sources simultaneously active). To address this issue, we develop a novel MUSIC framework based approach that directly uses simultaneous multi-source recordings. This developed MUSIC approach uses robust measurements of relative sound pressure from the higher-order microphone and is shown to be more suitable in noisy environments than the traditional MUSIC method. While the proposed approaches address the source localization problems, in practice, the broader problem of source localization has some more common challenges, which have received less attention. One such challenge is the common assumption of the sound sources being omnidirectional, which is hardly the case with a typical commercial loudspeaker. Therefore, in this thesis, we analyze the broader problem of analyzing directional characteristics of the commercial loudspeakers by deriving equivalent theoretical acoustic models. Several acoustic models are investigated, including plane waves decomposition, point source decomposition, and mixed source decomposition. We finally conduct extensive experimental examinations to see which acoustic model has more similar characteristics with commercial loudspeakers