The continuous wavelet transform in n -dimensions

Abstract

Daubechies obtained the n-dimensional inversion formula for the continuous wavelet transform of spherically symmetric wavelets in L2(n) with convergence interpreted in the L2-norm. From the wavelet u L2(n), Daubechies generated a doubly indexed family of wavelets ua,b by restricting the dilation parameter a to be a real number greater than zero and the translation parameter b belonging to n. We show that a can be chosen to be in n with none of the components aj vanishing. Further, we prove that if f and ua,b are continuous in n, then the convergence besides being in L2(n) is also pointwise in n. We advance our theory further to the case when f and u both belong to L2(n) then convergence of the wavelet inversion formula is pointwise at all points of continuity of f. This result significantly enhances the applicability of the wavelet inversion formula to the image processing. © 2016 World Scientific Publishing Company.