Appendix A: Tests on the PSF

We have made a series of tests that tends to strengthen our confidence in the 59 Cyg's image as a correct PSF for P Cyg. These tests are negative tests in the sense that if the PSF fails to succeed one, it should be rejected as a bad PSF.

Let us assume that 59 Cyg is a good unbiased representation of the true PSF h(r). Differences between the blurred reconstructed image $b(r) = h(r) \otimes x(r)$ and the observed data y(r) must then be dominated by statistical noise fluctuations, with no bias term. In this relation, x(r) is our best reconstructed image, as shown in Fig. 4, and h(r) the PSF 59 Cyg of Fig. 1 of the body of the paper. We assume that the noise comes from a photo-detection process, and that y(r) is a realization of the Poisson process of mean b(r).

A.1. Test 1: Poisson-Mandel transform

The first test we have performed is a basic one, not very sensitive to the exact value of h(r), but that must be verified in any case. Let us denote y and b the values taken by y(r) and b(r). The total probability theorem ([Papoulis 1984]) allows us to write the unconditional probability P(y) of y(r) as the sum of $P(y/b)\times P(b)$ for all bvalues. For the Poisson process, the conditional probability of yassuming b is $P(y/b) = \frac{b^y}{y!} \exp{(-b)}$ where y is an integer (the number of photons) and b a continuous value.

As a consequence, P(y) is the Poisson-Mandel transform ([Mandel 1959]; [Mehta 1970]) of P(b):

$$P(y) = \int_{0}^{\infty} P(b) \frac{b^y}{y!} \exp{(-b)} \hspace{0.2cm}{\rm d}b.$$ (A1)

We have verified that our data correctly obeys relation A1. We have taken for P(b) the histogram of b(r), applied the above transformation and compared the result P(y) with the direct histogram of the values of y(r). The comparison is shown in Fig. A1. The results are consistent with the data.

A.2. Test 2: Rotations of the true PSF

The second test we have performed was to check the correctness of the small departures from circular symmetry of 59 Cyg. We use as the PSF $h_{\theta}(r)$ an image of 59 Cyg rotated by an angle $\theta$. Then the deconvolution procedure is carried out as previously and leads to an image $x_{\theta}(r)$. The blurred image $b_{\theta }(r)$is obtained as the convolution of $x_{\theta}(r)$ and $h_{\theta}(r)$, and we finally compute the Euclidean distance between y(r) and $b_{\theta }(r)$. The results are shown in Fig. A2 in a polar plot for $\theta$ varying from 0 to 2 $\pi$. The original PSF gives the best result. If the deviation from circular symmetry was purely random, the goodness of the deconvolution would not be affected by this rotation. Secondary minima of the curve appear for apparent symmetries of the PSF.

Figure A1: Thick line: Poisson-Mandel transform P(y) of P(b) calculated using relation A1. Thin line: Histogram of the values of y(r). A good agreement between the two curves is observed

Figure A2: Thick line: Polar plot of the Euclidean distance between y(r) and $b_{\theta }(r)$ for $\theta$ varying between 0 and 2$\pi$. Thin line: Circle with a radius of the minimum of the Euclidean distance between y(r) and $b_{\theta }(r)$. The minimum of the Euclidean distance is obtained for $\theta$ = 0

Figure A3: Representation of the sign of the difference $y(r)- b_{\theta }(r)$ for no rotation of the PSF (top) and for a rotation of $\pi /2$ (bottom). This later image is not a particular case, and is characteristic of what can be obtained when the PSF is rotated

Representations of the difference between y(r) and $b_{\theta }(r)$ are shown in Fig. A3. For the sake of clarity we have represented the sign of the difference $y(r)- b_{\theta }(r)$. For $\theta = 0$, we get a speckle-like pattern, roughly uniform over the whole image. For other values of $\theta$, this difference shows large patterns that indicates regions over which $b_{\theta }(r)$ does not correctly match y(r). These trends clearly show that $h_{\theta}(r)$ for $\theta \neq 0$is a biased version of h(r).

As a conclusion, 59 Cyg passed the tests and could not be rejected as a bad PSF.

Acknowledgements

We wish to acknowledge A. Labeyrie for having encouraged the present work. P Cyg's observations were done using the BOA Adaptive Optics provided by ONERA to us. The manuscript benefited from discussions and critics of J.P. Véran, J. De Freitas Pacheco, G. Ricort and the GI2T team.