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Li Yu, Wei Ji (Dept. of Physics, Nat'l Univ. of Singapore)
Hankel transform (HT) is very useful in geophysics, astronomy, electromagnetics, optics and other branches of physics. Previously there were many discussions on algorithms of HT. In general, these algorithms were not exact discrete-HT (DHT), for the reason that the sampled points are not directly decided by Bessel function. Quasi-discrete HT (QDHT), originally developed by us [Opt.Lett.23, 409,1998], is not only a very efficient algorithm, but also a theoretical framework of DHT in which sample theorem and discrete Parseval's theorem are included. However, QDHT is not an exact DHT, because the product of two QDHT matrixes is not an unitary matrix. This leads to increasing errors in the iterative calculations like phase retrieval problem. Our purpose is to introduce an operation of improving the QDHT matrix numerically, whose process is descibed as below. Let M0 and M0-1 denote a QDHT matrix and its inverse matrix respectively. One such operation includes two steps: first, let a new matrix\ M1 equal (M0+M0-1)/2; second, get its inverse matrix M1-1. M1 or M1-1 is the improved matrix. This operation can be repeated again and again until the improved matrix meets the prescribed exactness. In fact, the product of two improved-QDHT matrixes is much more closed to unitary matrix than = that of QDHT matrixes. For example, the error of 4 dimensional QDHT matrix is 10-7. And the error of improved matrix is about of 10-15 after 5 times of improving operations. The inverse matrix is easily got by software like QTR mathematica.