Polar Linear Canonical Wavelet Transform: Theory and Its Application

The article delves into the development of the Polar Linear Canonical Wavelet Transform (PLCWT), a innovative mathematical tool designed to overcome the limitations of traditional transforms in analyzing complex, high-dimensional signals. By integrating the strengths of the Polar Wavelet Transform and the Linear Canonical Transform, the PLCWT offers superior directional analysis and the ability to characterize two-dimensional signals on the time-LCT domain plane. The text thoroughly investigates the basic properties of the PLCWT, including linearity, scaling, and inversion formulas, and derives crucial convolution and correlation theorems. A notable application of the PLCWT is demonstrated in medical image edge detection, where it successfully identifies subtle blood vessels in retinal images that were overlooked by conventional methods such as the Canny algorithm and the Polar Wavelet Transform. The article also discusses the potential of the PLCWT in other fields, such as ECG signal analysis and audio signal processing, highlighting its versatility and effectiveness in handling non-stationary signals. Through detailed numerical experiments and comparisons, the article underscores the PLCWT's advantages in preserving local directional features and edges, making it a promising tool for advanced signal and image processing tasks.