Application of the Hotelling and ideal observers to detection and localization of exoplanets

Abstract

The ideal linear discriminant or Hotelling observer is widely used for detection tasks and image-quality assessment in medical imaging, but it has had little application in other imaging fields. We apply it to detection of planets outside of our solar system with long-exposure images obtained from ground-based or space-based telescopes. The statistical limitations in this problem include Poisson noise arising mainly from the host star, electronic noise in the image detector, randomness or uncertainty in the point-spread function (PSF) of the telescope, and possibly a random background. PSF randomness is reduced but not eliminated by the use of adaptive optics. We concentrate here on the effects of Poisson and electronic noise, but we also show how to extend the calculation to include a random PSF. For the case where the PSF is known exactly, we compare the Hotelling observer to other observers commonly used for planet detection; comparison is based on receiver operating characteristic (ROC) and localization ROC (LROC) curves.

Fig. 1

Configuration of an adaptive optics telescope.

Fig. 2

Log-scale simulation of (a) image ḡ₀ of the star and (b) image ḡ₁ of the star and the planet. The intensity of the planet was set to a = 10⁶ e⁻ only in this figure and to make the planet location visible.

Fig. 3

(Color online) ROC curves for the simulated data.

Fig. 4

Images used for the simulated data case and unknown r_p: (a) locations of the planets for the test with the simulated image (Airy disk) and (b) test locations.

Fig. 5

(Color online) LROC curves for simulated data (A = 10⁷ e⁻, a = 20 e⁻, b_m = 100 e⁻, σ_m = 10 e⁻).

Fig. 6

Images used for the real data case and unknown r_p: (a) real image ḡ₀ of the star (plotted on logarithmic scale), (b) locations of the planets, (c) test locations.

Fig. 7

(Color online) LROC curves for real data (A = 3,066,000 e⁻, a = 1533 e⁻, b_m = 250,680 e⁻, σ_m = 700 e⁻).