Deconvolving signals of the stretched exponential form

A solution is presented for parametrically deconvolving an axisymmetric intrinsic field signal when it is expected to conform to the stretched exponential family of functions (SEF). Except for the Gaussian SEF, computable analytical models for the forward Abel transform of SEFs did not exist until recently. I will highlight a novel mathematical identity that has facilitated this calculation and show how to use the 2D models for the Abel transform to reconstruct the 3D signal to an accuracy of ∼10^-6 (or better) under noise-free conditions (possibly even with zero error). Several deconvolution techniques have tested their reconstructions using a noise-free projection of the Gaussian. Under similar conditions, our reconstruction produces errors ∼1950 times lower than 10 other techniques; we get significantly lower errors for other SEFs as well using fewer computing resources. Additionally, unlike other methods, our approach works on unequally spaced data and does not encounter the problem of increasing errors with radii in the outer parts as seen in all other methods. I will describe applications in the imaging of diverse astrophysical and biological systems where SEFs have been used and also highlight the possibility of using the projection of SEFs as basis functions in image deconvolution algorithms.