Learning Filters Filterbanks Wavelets and Multiscale Representations

Abstract

The problem of filter design is ubiquitous. Frequency selective filters are used in speech/audio processing, image analysis, convolutional neural networks for tasks such as denoising, deblurring/deconvolution, enhancement, compression, etc. While traditional filter design methods use a structured optimization formulation, the advent of deep learning techniques and associated tools and toolkits enables the learning of filters through data-driven optimization. In this thesis, we consider the filter design problem in a learning setting in both data-dependent and data-independent flavors. Data-dependent filters have properties governed by a downstream task, for instance, filters in a convolutional dictionary used for the task of image denoising. On the contrary, data-independent filters have constraints imposed on their frequency responses, such as lowpass, having diamond-shaped support, satisfying perfect reconstruction property, ability to generate wavelet functions, etc. The contributions of this thesis are four-fold: (i) the formulation of filter, filterbank, and wavelet design as regression problems, allowing them to be designed in a learning framework; (ii) the design of contourlet-based scattering networks for image classification; (iii) the design of a deep unfolded network using composite regularization techniques for solving inverse problems in image processing; and (iv) a multiscale dictionary learning algorithm that learns one or more multiscale generator kernels to parsimoniously explain certain neural recordings. We begin by developing learning approaches for designing filters having data-independent specifications, for instance, filters with a specified frequency response, including an ideal filter. The problem of designing such filters is formulated as a regression problem, using a training set comprising cosine signals with frequencies sampled uniformly at random. The filters are optimized using the mean-squared error loss, and generalization bounds are provided. We demonstrate the applicability o...