G-Lets: Signal Processing Using Transformation Groups
This provides a new theoretical framework for signal processing tasks such as feature extraction and compression, though it appears incremental as it builds on group theory concepts.
The paper tackles the problem of signal analysis by introducing G-Lets, an algorithm that uses transformation groups and irreducible representations to generate orthogonal bases for signals, enabling multiresolution and multiscale analysis with applications like edge detection and denoising, demonstrated on an ECG signal and the Lena image.
We present an algorithm using transformation groups and their irreducible representations to generate an orthogonal basis for a signal in the vector space of the signal. It is shown that multiresolution analysis can be done with amplitudes using a transformation group. G-lets is thus not a single transform, but a group of linear transformations related by group theory. The algorithm also specifies that a multiresolution and multiscale analysis for each resolution is possible in terms of frequencies. Separation of low and high frequency components of each amplitude resolution is facilitated by G-lets. Using conjugacy classes of the transformation group, more than one set of basis may be generated, giving a different perspective of the signal through each basis. Applications for this algorithm include edge detection, feature extraction, denoising, face recognition, compression, and more. We analyze this algorithm using dihedral groups as an example. We demonstrate the results with an ECG signal and the standard `Lena' image.