The Sherman Morrison iteration method is developed to solve regularized least squares problems. Notions of pivoting and splitting are deliberated on to make the method more robust. The Sherman Morrison iteration method is shown to be effective when dealing with an extremely underdetermined least squares problem. The performance of the Sherman Morrison iteration is compared to classic direct methods, as well as iterative methods, in a number of experiments. Specific Matlab implementation of the Sherman Morrison iteration is discussed, with Matlab codes for the method available in the appendix. / Master of Science

We develop a theoretical analysis of generalization performances of regularized least-squares on reproducing kernel Hilbert spaces for supervised learning. We show that the concept of effective dimension of an integral operator plays a central role in the definition of a criterion for the choice of the regularization parameter as a function of the number of samples. In fact, a minimax analysis is performed which shows asymptotic optimality of the above-mentioned criterion.

Imageamento acústico é um problema computacionalmente caro e mal-condicionado, que envolve estimar distribuições de fontes com grandes arranjos de microfones. O método clássico para imageamento acústico utiliza beamforming, e produz a distribuição de fontes de interesse convoluída com a função de espalhamento do arranjo. Esta convolução borra a imagem ideal, significativamente diminuindo sua resolução. Convoluções podem ser evitadas com técnicas de ajuste de covariância, que produzem estimativas de alta resolução. Porém, estas têm sido evitadas devido ao seu alto custo computacional. Nesta tese, admitimos um arranjo bidimensional com geometria separável, e desenvolvemos transformadas rápidas para acelerar imagens acústicas em várias ordens de grandeza. Estas transformadas são genéricas, e podem ser aplicadas para acelerar beamforming, algoritmos de deconvolução e métodos de mínimos quadrados regularizados. Assim, obtemos imagens de alta resolução com algoritmos estado-da-arte, mantendo baixo custo computacional. Mostramos que arranjos separáveis produzem estimativas competitivas com as de geometrias espirais logaritmicas, mas com enormes vantagens computacionais. Finalmente, mostramos como estender este método para incorporar calibração, um modelo para propagação em campo próximo e superfícies focais arbitrárias, abrindo novas possibilidades para imagens acústicas. / Acoustic imaging is a computationally intensive and ill-conditioned inverse problem, which involves estimating high resolution source distributions with large microphone arrays. The classical method for acoustic imaging consists of beamforming, and produces the source distribution of interest convolved with the array point spread function. This convolution smears the image of interest, significantly reducing its effective resolution. Convolutions can be avoided with covariance fitting methods, which have been known to produce robust high-resolution estimates. However, these have been avoided due to prohibitive computational costs. In this thesis, we assume a 2D separable array geometry, and develop fast transforms to accelerate acoustic imaging by several orders of magnitude with respect to previous methods. These transforms are very generic, and can be applied to accelerate beamforming, deconvolution algorithms and regularized least-squares solvers. Thus, one can obtain high-resolution images with state-of-the-art algorithms, while maintaining low computational cost. We show that separable arrays deliver accuracy competitive with multi-arm spiral geometries, while producing huge computational benefits. Finally, we show how to extend this approach with array calibration, a near-field propagation model and arbitrary focal surfaces, opening new and exciting possibilities for acoustic imaging.

Imageamento acústico é um problema computacionalmente caro e mal-condicionado, que envolve estimar distribuições de fontes com grandes arranjos de microfones. O método clássico para imageamento acústico utiliza beamforming, e produz a distribuição de fontes de interesse convoluída com a função de espalhamento do arranjo. Esta convolução borra a imagem ideal, significativamente diminuindo sua resolução. Convoluções podem ser evitadas com técnicas de ajuste de covariância, que produzem estimativas de alta resolução. Porém, estas têm sido evitadas devido ao seu alto custo computacional. Nesta tese, admitimos um arranjo bidimensional com geometria separável, e desenvolvemos transformadas rápidas para acelerar imagens acústicas em várias ordens de grandeza. Estas transformadas são genéricas, e podem ser aplicadas para acelerar beamforming, algoritmos de deconvolução e métodos de mínimos quadrados regularizados. Assim, obtemos imagens de alta resolução com algoritmos estado-da-arte, mantendo baixo custo computacional. Mostramos que arranjos separáveis produzem estimativas competitivas com as de geometrias espirais logaritmicas, mas com enormes vantagens computacionais. Finalmente, mostramos como estender este método para incorporar calibração, um modelo para propagação em campo próximo e superfícies focais arbitrárias, abrindo novas possibilidades para imagens acústicas. / Acoustic imaging is a computationally intensive and ill-conditioned inverse problem, which involves estimating high resolution source distributions with large microphone arrays. The classical method for acoustic imaging consists of beamforming, and produces the source distribution of interest convolved with the array point spread function. This convolution smears the image of interest, significantly reducing its effective resolution. Convolutions can be avoided with covariance fitting methods, which have been known to produce robust high-resolution estimates. However, these have been avoided due to prohibitive computational costs. In this thesis, we assume a 2D separable array geometry, and develop fast transforms to accelerate acoustic imaging by several orders of magnitude with respect to previous methods. These transforms are very generic, and can be applied to accelerate beamforming, deconvolution algorithms and regularized least-squares solvers. Thus, one can obtain high-resolution images with state-of-the-art algorithms, while maintaining low computational cost. We show that separable arrays deliver accuracy competitive with multi-arm spiral geometries, while producing huge computational benefits. Finally, we show how to extend this approach with array calibration, a near-field propagation model and arbitrary focal surfaces, opening new and exciting possibilities for acoustic imaging.

Sparse approximation is a Bayesian inference program with a wide number of signal processing applications, such as Compressed Sensing recovery used in medical imaging. Previous sparse coding implementations relied on digital algorithms whose power consumption and performance scale poorly with problem size, rendering them unsuitable for portable applications, and a bottleneck in high speed applications. A novel analog architecture, implementing the Locally Competitive Algorithm (LCA), was designed and programmed onto a Field Programmable Analog Arrays (FPAAs), using floating gate transistors to set the analog parameters. A network of 6 coefficients was demonstrated to converge to similar values as a digital sparse approximation algorithm, but with better power and performance scaling. A rate encoded spiking algorithm was then developed, which was shown to converge to similar values as the LCA. A second novel architecture was designed and programmed on an FPAA implementing the spiking version of the LCA with integrate and fire neurons. A network of 18 neurons converged on similar values as a digital sparse approximation algorithm, with even better performance and power efficiency than the non-spiking network. Novel algorithms were created to increase floating gate programming speed by more than two orders of magnitude, and reduce programming error from device mismatch. A new FPAA chip was designed and tested which allowed for rapid interfacing and additional improvements in accuracy. Finally, a neuromorphic chip was designed, containing 400 integrate and fire neurons, and capable of converging on a sparse approximation solution in 10 microseconds, over 1000 times faster than the best digital solution.

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