Summary

The storage requirements for the four representations are wildly different in many respects. For purposes of comparison, we use M=360 for the map size and N=4694 for the number of visibilities. The first half of Table 2 shows the storage space (in floating point numbers) as a function of N and M. The second half shows the corresponding actual numbers.

Table 2

Storage Requirements for Each Approach (All 9 Collimators)

Rectangular Direct HXT Style Polar

Visibilities: 2 N +1 2 N+1 N 2 N + 1

$$\sim 0.2N^2$$ Intermediate: 4 M N M² N 2MN

$$\sim 0.2N^2$$ Map pixels: M² M² M²

Visibilities: 37 KB 37 KB 18.7 KB 37 KB

Intermediate: 34 MB 54 MB 2433 MB 13 MB

Map pixels: 34 MB 0.5 MB 0.5 MB 0.5 MB

The number of equivalent floating-point operations for one transform from visibility space to map space is readily estimated. The table shows the ``brute force'' numbers for the direct and polar methods as well as the ``recurrence'' numbers described in Sections 3 and 5. The former uses the relative timing (in IDL 5.2) for Bessel functions, FFT and exponentials: beselj(d,100) : FFT(d,1) : exp(2*pi*i*d*x) : d*x = 450 : 54 : 49 : 1. When arrays of exponentials or arrays of Bessel functions are calculated using recurrence, both the exponential and Bessel function computation times can be reduced to about 12 multiplications.

Table 3

Computational Requirements for Each Approach (All 9 Collimators)

Rectangular Direct HXT Style Polar BRUTE FORCE:

Equiv. FP ops: 14*N² 50N M² 50N M² 500 N M

Equiv. FP ops: 3.e+08 3.e+10 3.e+10 8.e+08 RECURRENCE:

Equiv. FP ops: - ? * N M - 20 N M

Equiv. FP ops: - 9.e+09 - 4.e+07

These numbers should be considered accurate to about a factor of 2. Clearly the HXT-style representation is a major loser in computational speed, and with the use of recurrence, the polar representation is the winner.

Since the HXT-style representation scales as M², smaller maps could be made faster. A $60\times 60$ map, for example, would be made in only slightly more time than in the rectangular representation. But the storage requirements (67 MB) would still be larger than in the other methods.