Family of scaling chirp functions, diffraction, and holography

It is observed that diffraction is a convolution operation with a chirp kernel whose argument is scaled. Family of functions obtained from a prototype by shifting and argument scaling form the essential ground for wavelet framework. Therefore, a connection between diffraction and wavelet transform is developed. However, wavelet transform is essentially prescribed for time-frequency and/or multiresolution analysis which is irrelevant in our case. Instead, the proposed framework is useful in various location-depth type of analysis in imaging. The linear transform when the analyzing functions are the chirps is called the scaling chirp transform. The scaled chirp functions do not satisfy the commonly used admissibility condition for wavelets. However, it is formally shown that these neither band nor time limited signals can be used as wavelet functions and the inversion is still possible. Diffraction and in-line holography are revisited within the scaling chirp transform context. It is formally proven that a volume in-line hologram gives perfect reconstruction. The developed framework for wave propagation based phenomena has the potential of advancing both signal processing and optical applications