Abstract

Usually a wavelet transform is based on dilated–translated wavelets. We propose a symplectic-transformed–translated wavelet family ${\psi }_{r,s}^{*}(z-\kappa )$ ($r,s$ are the symplectic transform parameters, ${\mid s\mid }^2-{\mid r\mid }^2=1$, κ is a translation parameter) generated from the mother wavelet ψ and the corresponding wavelet transformation $W_{\psi }f(r,s;\kappa )={\int }_{-\infty }^{\infty }({\mathrm{d}}^{2}z∕\pi )f\left(z\right){\psi }_{r,s}^{*}(z-\kappa )$. This new transform possesses well-behaved properties and is related to the optical Fresnel transform in quantum mechanical version.

© 2006 Optical Society of America

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