Local Wavelet Transform on a Helix Space Curve Based on Frenet Frame

Abstract

Based on the Frenet frame, the approximate Local Wavelet Transform (LWT) is investigated on a class of generalized helix curves by their tangent projections at specific points. Initially, parametric equations are formulated for various types of helix curves, including circular helices and generalized helices. A particular helix curve on a unit sphere is selected as a representative instance of a generalized helix, and its tangent projection at a designated point is computed. Subsequently, the dilation and translation operators for a wavelet function are rigorously defined to facilitate the study of the LWT on this curve. A corresponding reconstruction formula is also derived. The Morlet wavelet is projected onto the space curve at the specified point t = 0 by the tangent projection method. To validate the proposed approach, three numerical examples are presented, demonstrating the application of the LWT and its inverse LWT (reconstruction formula) to analyze localized signals. The simulation results are illustrated through graphical representations.