Wavelet analysis on a generalized helix space curves and its examples

Abstract

AbstractIn this paper, wavelet transform on a generalized helix space curves are investigated, including of local continuous wavelet transform at some point and discrete wavelet transform on a class of helix curves. Firstly, a class of helix space curves are introduced and the parameter equations are given. Then the dilation operator and translation on the function $$\psi^{{(\xi_{{0}} )}}$$ ψ ( ξ 0 ) is properly defined by the local projection at some point from a space curve on the unit sphere onto its tangent line. The local continuous wavelet transform and its reconstruction formula are deduced at some point of a space curve on the unit sphere. On the other hand, According to the discretization of length-preserving projection, discrete wavelet transform is lifted onto a helix space curve, such as a circular helix curve. Based on length-preserving projection, the some properties are discussed, such as two-scale sequences of scaling function and wavelet, orthogonality, decomposition formula and so on. Finally, two examples are given for our discussion. One example is illustrating the application of local continuous wavelet transform at some point of a space curve. The result shows the signal at some point of a space curve can be reconstructed by local continuous wavelet method. The norm of the error is 0.3783 between original signal and reconstructed signal in this example. The other numerical example is given for decomposing and reconstructing with the signal on a circular helix curve. The result shows the signal on a helix space curve can be decomposed and reconstructed by the length-preserving projection. The norm of the error is 8.0741 × 10−11 between original signal and reconstructed signal. The figures are shown for the simulation results.