Closed form of the steered elongated Hermite-Gauss wavelets

We provide a closed form, both in the spatial and in the frequency domain, of a family of wavelets which arise from steering elongated Hermite-Gauss filters. These wavelets have interesting mathematical properties, as they form new dyadic families of eigenfunctions of the 2D Fourier transform, and generalize the well known Laguerre-Gauss harmonics. A special notation introduced here greatly simplifies our proof and unifies the cases of even and odd orders. Applying these wavelets to edge detection increases the performance of about 12.5% with respect to standard methods, in terms of the Pratt's figure of merit, both for noisy and noise-free input images.