Our method has been implemented in C and in IDL . The C version uses the library FFTW ("Fastest Fourier Transform in the West," version ) to perform discrete Fourier transforms (DFT). This library, written by Matteo Frigo and Steven G. Johnson, is considered the quickest DFT library publicly available. The performance of our direct method is compared with that of the over-relaxation method, also implemented in C . The procedure used in the tests is summarized in the following points:
The results obtained in the tests are the following:
Regarding the second item, we note that the error is related to the finite sampling scale of the method; the error affects only the outermost pixel because of the proper choice of the truncation (see comment at the end of Sect. 3).
The measured execution times are plotted in Fig. 1 for different values of N. These are the averaged CPU execution times for a single reconstruction on a SUN Ultra 1 workstation. From this figure it is clear that the direct method is much faster than the over-relaxation method. Here we should recall that, because of some characteristics of the FFTW library, the execution time of the direct method can change significantly even for neighbouring values of N. In particular, the inversion is faster when can be factorized with small prime numbers, and is slower in other cases (see Fig. 1). For example, the execution time (on a SUN Ultra 1) changes from 2.942 to 0.232 seconds when N changes from 121 to 122. Finally, we observe that our implementation of the direct method is not optimal: in fact, with a different (non-trivial) use of FFT one might gain an additional factor of 4 on the execution time.
Besides the appealing aspects of simplicity inherent to the direct method described in this paper, we should note that gaining three orders of magnitude in CPU time will make it possible to undertake a few long-term projects of simulated observations (in particular, with the goal of a statistically sound investigation of the quality of mass reconstruction; but other objectives might be formulated, e.g. in the cosmological context) that would remain practically out of reach for other intrinsically slow reconstruction methods.
© European Southern Observatory (ESO) 1999
Online publication: July 16, 1999