Optical astronomical imaging

Optical astronomical imaging is concerned with imaging astronomical objects by detecting the light that they emit using (usually ground-based) telescopes. The objects of interest such as stars, planets, and galaxies are located at a large distance from the observer and hence the field at the telescope is described by Fraunhofer diffraction. The field at the aperture (pupil) plane is the Fourier transform of the astronomical source brightness dis- tribution which is to be imaged (Fig. 1.7). The telescope optical system forms an image in the image plane. The resolution of the image is determined by the dimensions of the aper- ture in the pupil plane. The resolution is quoted asangular resolution, the angle subtended at the telescope between two objects that can just be resolved.

The complication in optical astronomy arises from the passage of the radiation through the Earth’s atmosphere (Fig. 1.7). The atmosphere is a layer gases and particles surrounding the earth, and temperature fluctuations, variations in wind speed and direction and the concentration of water vapour generate turbulence, which results in a position- and time- varying refractive index in the atmosphere. The effect of the refractive index variations is that the wavefront, and in particular its phase, is distorted while passing through the atmosphere. At low resolution, or at small spatial frequencies or for wavefronts close to the optical axis (i.e., small apertures), the radiation passes through small lateral regions of the atmosphere where the refractive index is almost constant and phase distortions are not significant. Good images are therefore formed at low resolution, corresponding to<1 arc-sec at a good observing site and this is the resolution limit imposed by atmospheric tur- bulence [7]. This resolution limit is in practice quite small for astronomical purposes. At

Figure 1.7: Astronomical imaging. Left: A diffraction limited image in the absence of atmosphere turbulence. Right: A speckle image due to atmospheric turbulence.

1.6 Diffraction imaging 23

higher resolution the radiation passes through atmospheric “patches” of different refrac- tive index which introduces phase variations at the pupil plane. This leads to highly dis- torted images in the image plane. To extract higher resolution images from a large aperture that are diffraction-limited, both image-plane and diffraction (aperture)-plane processing must be used. Many techniques have been developed to overcome the effects of turbu- lence such as speckle interferometry, blind deconvolution and adaptive optics. Adaptive optics uses a flexible mirror to detect and correct for the wavefront distortions in real time. Although effective, it requires very expensive equipment.

Astronomical speckle interferometry uses phase retrieval algorithms to solve the astro- nomical problem. Many short exposure images are captured within milliseconds (less than the fluctuation time of the atmosphere), which are referred to as speckle images. The atmospheric turbulence is effectively frozen (static) during each image acquisition. The images have a speckle appearance as a result of the atmosphere existing as patches with approximately constant refractive index. Them-th speckle image is given by

sm(x, y) =f(x, y) ⊙ hm(x, y) +nm(x, y) (1.76)

wheref(x, y)is the high-resolution astronomical image,hm(x, y)is the psf of the atmosphere above the aperture plane at the time the image was taken andnm(x, y)is the noise. The Fourier transform of each speckle image is then given by

Sm(u, v) =F(u, v)Hm(u, v) +Nm(u, v), (1.77) whereHm(u, v)is them-th OTF (Eq. (1.66)). Averaging many speckle images simply gives a long exposure (low resolution) image. However, if the spectral intensity of the speckle images is averaged, high resolution information is retained, i.e.,

|Sm(u, v)|2=|F(u, v)|2|Hm(u, v)|2+|Nm(u, v)|2, (1.78)

where

|Hm(u, v)|2 is referred to as the speckle transfer function. The speckle transfer function can be estimated by viewing an unresolvable star (point object) under similar con- ditions and using the same processing.|F(u, v)_|2can be then calculated from Eq. (1.78) us- ing deconvolution. The high resolution imagef(x, y)can then be recovered from|F(u, v)_| using phase retrieval.

Another method to reconstruct a high resolution image from the speckle images is the triple-correlation (also called bispectrum) method. The triple correlation and the bispec- trum, which are a Fourier transform pair, of an image are defined as

ˆ

f(x1,x2) = Z ∞

−∞

and

ˆ

F(u1,u2) =F(u1)F(u2)F∗(u1+u2), (1.80)

respectively. The speckle bispectrum can be calculated from the speckle images and (ig- noring the noise) is given by

D ˆ Sm(u1,u2) E = ˆF(u1,u2) D ˆ H(u1,u2) E , (1.81) whereDHˆ(u1,u2) E

is called the bispectral speckle transfer function and can be determined in the similar way to that for the speckle transfer function. Therefore Fˆ(u1,u2) can be

calculated. An important property of the bispectral speckle transfer function is that it is positive [35, 36], i.e., its phase is zero, so that the phase of the object bispectrum is equal to the phase of the image bispectrum, i.e.,φnDSˆm(u1,u2)

Eo

=φnFˆ(u1,u2) o

. Using this relationship and Eq. (1.80) gives relationships between the phases of F(u), φ_{F(u)_}, at two pointsumandunin Fourier space of the form

φ_{F(un)}=−φ{F(um)} −φ{F(u−m−n)}+φ n ˆ F(um,un) o . (1.82)

There are many such equations for different um andun and these can be solved for the phasesφ_{F(um)}, allowing the image to be reconstructed [35, 37, 38, 39].

Another approach to astronomical phase retrieval that uses multiple data is phase diver- sity (Gonsalves [40] and Paxman [41, 42]). This method requires two or images of the same object. The first image is the focused image taken at the focal plane, which is a low resolu- tion image due to the phase distortion. Additional images of the same object are captured but intentionally defocused. Possible setups for acquiring images from different planes is described by a number of authors [41]. Phase diversity can also be applied to correct other phase aberrations such as misalignment [41, 42]. The image at the focal plane can be expressed as

g(x) =f(x) _⊙ h(x), (1.83)

and the Fourier transform of the image is

G(u) =F(u)H(u), (1.84) whereh(x)andH(u)are the psf and OTF of the system. The OTF of the defocused image is given by

1.6 Diffraction imaging 25

where the subscriptddenotes the defocused plane and∆φ(u)is the known phase shift due to defocusing. The Fourier transform of the defocused image is then

Gd(u) =F(u)Hd(u). (1.86) The goal is then to reconstructf(x)fromg(x)andgd(x)withh(x)unknown. This can be approached by minimising the mean-square error

M SE=

Z ∞

−∞

|G(u)₋Hˇ(u) ˇF(u))_|2+_|Gd(u)−Hˇ(u) exp (i∆φ(u)) ˇF(u)|2du (1.87)

over the estimates indicated byˇ. Various methods have been used to solve the minimi- sation problem including maximum likelihood estimates for Gaussian and Poisson noise models [42].