NEW VARIANTS OF STOCKWELL TRANSFORM WITH APPLICATION IN IMAGE PROCESSING

Abstract

This thesis deals with many new variants of the Stockwell transform (S-transform) in signal and image processing for solving mathematical problems. We study mathematical formulas and theorems for signal processing applications, i.e., Parseval’s formula, the inversion formula, the reproduction property, the convolution theorem, and the cross-correlation theorem. Additionally, signal reconstruction harnesses the concept of reconstructing phase from the magnitude of a signal transform. The proposed work also demonstrates the application of signal reconstruction, denoising, and compression in signal analysis. The later part of the thesis dedicates a significant amount of effort to designing various new variants of Stockwell transforms (continuous anddiscrete forms), drawinginspiration fromsome existing integral transforms. Thethesispresentsseveralmathematicalformulasforthe proposed transforms. The thesis begins with a general introduction of the proposed work and a brief motivation. A brief inspection and literature review of the existing integral transforms and their relevant mathematical results are summarized in the f irst chapter. The next chapter deals with proposing a relationship between the phase and the log-magnitude of the S-transform. These relations can be utilized for reconstructing the phase from themagnitudeoftheS-transformofanysignalandcan also be used to overcome the computational cost of the S-transform. The work also includes a relationship between partial derivatives of the real and imaginary parts of the wavelet and the S-transform for a couple of window functions. The third chapter of the thesis focuses ondesigning another transform, say, the spectral graph fractional Stockwell transform (SGFrST), to identify the structure of signals on weighted graphs with the help of SGFrWT. We propose the transform by defining a distinct form of the Laplacian matrix. First, we propose the spectral graph Stockwell transform by modulation in the graph wavelet transform and the corresponding approximation results of Stockwell coefficients. Furthermore, we present the inversion form of the proposed transform with the window function’s admissibility condition. This study also derives the ’Inner product theorem’ for a fixed window function in the vertex domain. Next, we introduce a transform for non-stationary signals named the non-isotropic angular fractional Stockwell transform (NIAFrST). Additionally, we derived important mathematical properties such as linearity, anti-linearity, translation, scaling, parity, and the conjugation property NIAFrST.