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Department of Mathematics (5)

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## Scholarly Works (8 results)

The graph Laplacian is widely used in the graph signal processing field. When attempting to design graph wavelet transforms, people have been using its eigenvalues and eigenvectors in place of the frequencies and complex exponentials that are the backbone of the Fourier theory on Euclidean domains. However, this viewpoint could be misleading since the Laplacian eigenvalues cannot be interpreted as the frequencies of the eigenvectors on a general graph. Instead, we introduce and review several "natural" metrics of graph Laplacian eigenvectors, and propose a new way to naturally organize the eigenvectors by incorporating these metrics into a "dual" graph. We then introduce a set of novel multiscale basis transforms for graph signals fully utilizing this dual graph, rather than simply using the eigenvalue ordering. These basis dictionaries can be seen as generalizations of the classical Shannon/Meyer wavelet packet dictionary to arbitrary graphs, and do not rely on the frequency interpretation of Laplacian eigenvalues. We describe the algorithms (involving vector rotations, or orthogonalizations, or lapped orthogonal projections) to efficiently approximate and compress signals through the best-basis algorithm, and demonstrate the strengths of these basis dictionaries for graph signals on sunflower graphs and road traffic networks. Lastly, we propose a way to modify the spectral filters in the spectral graph wavelet transform by utilizing the structure of the dual graph instead of using the eigenvalue-dependent smooth functions. By doing so, we generate a redundant wavelet frame, propose a way to reduce its redundancy, and discuss its potential for applications.

The Scattering Transform (ST) is a formalization of some potential properties that have made convolutional neural networks effective at a wide variety of image and signal processing problems.Classifying raw side angle sonar (SAS) data provides an interesting test case for the scattering transform, since in addition to being a worthwhile problem in its own right, it is possible to model explicitly and understand how changes in the parameters of the model effect the resulting signal. In this dissertation we both apply the scattering transform to real and synthetic sonar classification problems, attempt to deepen our understanding of the scattering transform, and then apply that to understanding the sonar classifiers.

We use several methods to interpret the ST coefficients; the principal one is creating signals which maximize the output of a particular coefficient, balanced against the norm of the signal, which we call a pseudo-inversion.This turns out to be a difficult optimization problem, which we solve using differential evolution. We also use the gradient, as this can provide local information about the coefficient maximization, and theoretical properties of wavelets with vanishing moments. As the number of vanishing moments corresponds to the order of the wavelet as a pseudo-differential operator, this allows us to frame the scattering transform as mixing various orders of derivatives.

To try to understand the role of nonlinearity, in 2D we construct the shearlet scattering (or shattering) transform.We extend the sparsity guarantees of the shearlet transform to the shattering transform, and to do so we need to place some constraints on the possible nonlinearities. These constraints end up explaining the variation in classification results on the MNIST and FashionMNIST datasets.

We create synthetic sonar signals by varying the target object's shape and internal wave speed, which corresponds to the material composition.We examine some simple geometric properties, such as the relation between the signal delays and the wave speed, as well as a characterization of both rotation and translation of the target object in terms of how they modify the signal. When we compute the pseudo-inverse of the classification of objects with varying wave speed, the signal delay appears as an important discrimination feature. On the other hand, discriminating the shape of the object with a fixed speed is a subtler problem, where the shape with higher variation in curvature has more variation in the behavior of the tail, relying on higher second layer frequency when using Morlet wavelets.