A Generalized Design of the Mexican Hat and other Even-order Hermitian Wavelets in a Gaussian Scale Space

A generalized methodology of constructing a Mexican hat wavelet family involving even order Gaussian derivatives has been devised in a Gaussian scale space only. The optimization has been carried out in Fourier domain and the kernels in Gaussian scale space domain are found to be exact replica of their derivative wavelet counterpart for low as well as high order. Wavelet properties of the lowest order (2), has been discussed and the results are shown to be better and different from the well-known LOG-DOG equivalence of Marr-Hildreth. Such filters, simple to implement in Gaussian scale space, are likely to be important in vision, analysis of seismic signals, cosmic microwave background (CMB) maps and possibly in the general cases of signals from Gaussian point sources.