Windowed Radon Transforms, Analytic Signals and the Wave Equation Appeared in Wavelets: A Tutorial in Theory and Applications

The act of measuring a physical signal or field suggests a generalization of the wavelet transform that turns out to be a windowed version of the Radon transform. A reconstruction formula is derived which inverts this transform. A special choice of window yields the " Analytic–Signal transform " (AST), which gives a partially analytic extension of functions from R n to C n. For n =1, this reduces to Gabor's classical definition of " analytic signals. " The AST is applied to the wave equation, giving an expansion of solutions in terms of wavelets specifically adapted to that equation and parametrized by real space and imaginary time coordinates (the Euclidean region).