Fig. 1 (a) shows an input object, a, containing two pulses, each shaped by (350,700,350) counts in a background of 20 counts. This is convolved with the point spread function (PSF) p shown in (b). The observable data, d, is Poisson, driven by the rate vector g, the convolution of a with p. A typical realization is shown in (c).
Fig. 1 (d), (e) and (f) are estimates of a for an inverse filter, a Wiener filter, and the Richardson-Lucy (RL) algorithm. The noise control parameter in the Wiener filter was chosen to yield approximately the same bias (in the peaks) as the RL algorithm.
Fig. 2 shows the sample mean and RMSE (square root of the MSE) for 100 trials of the three estimation procedures. It also shows the CR bound calculations for the inverse and Wiener estimates. On the left are the sample means. We see that the inverse estimate is unbiased and that both the Wiener and RL estimates underestimate the peaks by about 20%. The latter two are biased estimates.
On the right are the sample RMSE's and two of the bounds. The bounds are smooth. We see that the bounds for the inverse and for the Wiener estimates closely match the sample RMSE's. This provides verification that the CR bounds are useful.
We present no bound for the RL algorithm because we do not know an analytic form for the bias. We note the general superiority of the RL estimate, which is known to be optimal for Poisson statistics. The RMSE is considerably smaller and the bias is about the same as the Wiener filter estimate.
Finally we address the (inappropriate) use of a in calculating the bound. Of course, a is unknown. We are compelled to use the estimate, â, of a in the calculation and this leads to errors. In Fig. 3 we show the effect. On the left is the bound for a known; on the right is the bound calculated for a single estimate, â. The latter has the general shape and amplitude of the former. We have not yet analyzed the differences. However, the bound on the right, which is actually calculable, seems to be useful.
