Group Seminar: Inverse Problems and Mathematical Imaging
Thursday, June 13, 2024, 09:00
Zoom Meeting
Modern Approaches to Wavelet Approximation
Abstract: In this talk, we will discuss modern methodologies in approximation theory using wavelets based on my
previous works. Wavelet frames are versatile and practical structures that provide the perfect recon-
struction property. Tight wavelet frames in L2(Rn) are particularly advantageous in signal processing, as
their exact dual is essentially the same as the given frame. We will begin by introducing a new method
for constructing tight wavelet frames, specifically addressing the challenges faced by existing methods,
with illustrative examples constructed using this new method. Conversely, an exact dual of a non-tight
wavelet frame may not always exhibit a wavelet structure. Instead of finding an exact dual with a wavelet
structure for a given frame, one can utilize the concept of approximate duals. This generalized concept of
exact duals enables wavelet series expansions in L2(Rn). We will demonstrate a wavelet series expansion
in Hardy spaces using approximate duals and present sufficient conditions for mother wavelets to ensure
the resulting systems are approximate duals. In recent years, convolutional neural networks have shown
robust performance on various vision tasks, such as image classification. The scattering transform is a
simple mathematical model for a convolutional neural network that involves cascading convolutions with
predefined wavelets followed by the modulus operator. This operator is translation invariant and has
Lipschitz continuity with respect to diffeomorphisms. We will introduce some other deformations beyond
the original work for univariate signals and propose an experimental method for classifying these signals
using the scattering transform and the modulus of the Fourier transform.
Zoom (Thu, Jun 13, 2024, 09:00 AM (CET))
oeaw-ac-at.zoom.us/j/66804179454
Meeting-ID: 668 0417 9454
Password: cbk5v9