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Extended object wavefront sensing based on the correlation spectrum phase

Per A. Knutsson, Mette Owner-Petersen, and Chris Dainty

Open Access Open Access

Abstract

In this paper we investigate the performance of a Fourier based algorithm for fast subpixel shift determination of two mutually shifted images subjected to noise. The algorithm will be used for Shack-Hartmann based adaptive optics correction of images of an extended object subjected to dynamical atmospheric fluctuations. The performance of the algorithm is investigated both analytically and by Monte Carlo simulations. Good agreement is achieved in relation to how the precision of the shift estimate depends on image parameters such as contrast, photon counts and readout noise, as well as the dependence on sampling format, zero-padding and field of view. Compared to the conventional method for extended object wavefront sensing, a reduction of the computational cost is gained at a marginal expense of precision.

©2005 Optical Society of America

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Figures (6)
Fig. 1.
Fig. 1. Example of images and correlation spectrum for the case of m = 8 and a negative Gaussian function. (a) Reference image f(p,q) with noise (original noise-free high resolution image below). (b) Shifted image g(p,q) with noise (original noise-free high resolution image below). (c) Logarithm of normalized amplitude of correlation spectrum log(|C(u,v)|/|C(0,0)|). (d) Phase of correlation spectrum (|ϕC (u,v) in radians.

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Fig. 2.
Fig. 2. Example of images and correlation spectrum for the case of m = 8 and a more irregular image. See previous figure for explanations.

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Fig. 3.
Fig. 3. Estimated shifts from simulations. Each point represents a noise realisation for a separation given by the colors in the legend. Upper row, (a)–(c), represent estimates for the Gaussian function in Fig. 1 and lower row, (d)–(f), represent estimates for the irregular image used in Fig. 2. Leftmost estimates, (a) and (d), are based on Eq. (8) with u 1 = v 1 = 1. Middle estimates, (b) and (e), are based on Eq. (4) when setting all C(u,v) to zero but C(1,-1), C(1,0), C(1,1) and C(0,1). Rightmost plots, (c) and (f), give the estimates that are based on parabolic interpolation of the cross correlation peak.

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Fig. 4.
Fig. 4. Effects of varying parameters on the standard error. Lines - analytical predictions from Eq. (13) using the parameters specified in Section 3.2. Stars - simulated estimation using Eq. (8) with u 1 = 1. (a) Standard error as function of contrast. (b) Standard error as function of signal. (c) Standard error as function of readout noise.

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Fig. 5.
Fig. 5. Lines - analytical predictions from Eq. (13) using the parameters specified in Section 3.3. Stars - simulated estimation using Eq. (8). (a) Standard error, in fraction of the field of view, as a function of sampling points. (b) Standard error, in pixels, as function of zero-padded format. (c) Standard error, in pixels, as function of size of the field of view (stretched object).

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Fig. 6.
Fig. 6. (a) Time divided by m 2 log2 m 2 for conventional method as a function of format. (b) Time divided by m 2 for correlation phase method as a function of format. (c) Fraction of the times for the two methods as a function of format.

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Equations (25)

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ε 2 = ∫∫ − ∞ ∞ ∣ f ( r ) − g ( r + r 0 ) ∣ 2 d r = ∫∫ − ∞ ∞ ∣ F ( f ) − G ( f ) exp ( i 2 π f· r 0 ) ∣ 2 d f ,
∂ ε 2 ∂ x 0 = ℜ [ 4 πi ∫∫ − ∞ ∞ f x F ( f ) G * ( f ) exp ( − i 2 π f · r 0 ) d f ] = 0
∂ ε 2 ∂ y 0 = ℜ [ 4 πi ∫∫ − ∞ ∞ f y F ( f ) G * ( f ) exp ( − i 2 π f · r 0 ) d f ] = 0
x ̂ 0 = 1 2 π b x a yy − b y a xy a xx a yy − a xy 2 y ̂ 0 = 1 2 π b y a xx − b x a xy a xx a yy − a xy 2 , where
a xx = ∫∫ − ∞ ∞ ∣ C ( f ) ∣ f x 2 d f a yy = ∫∫ − ∞ ∞ ∣ C ( f ) ∣ f y 2 d f a xy = ∫∫ − ∞ ∞ ∣ C ( f ) ∣ f x f y d f
b x = ∫∫ − ∞ ∞ ∣ C ( f ) ∣ φ C ( f ) f x d f b y = ∫∫ − ∞ ∞ ∣ C ( f ) ∣ φ C ( f ) f y d f .
x ̂ 0 = m Δ 2 π b p a qq − b q a pq a pp a qq − a pq 2 y ̂ 0 = m Δ π b q a pp − b p a pq a pp a qq − a pq 2 , where
a pp = ∑ u = − m 2 m 2 − 1 ∑ v = − m 2 m 2 − 1 ∣ C u v ∣ u 2 a qq = ∑ u = − m 2 m 2 − 1 ∑ v = − m 2 m 2 − 1 ∣ C u v ∣ v 2 a pq = ∑ u = − m 2 m 2 − 1 ∑ v = − m 2 m 2 − 1 ∣ C u v ∣ uv
b p = ∑ u = − m 2 m 2 − 1 ∑ v = − m 2 m 2 − 1 ∣ C u v ∣ φ C u v u b q = ∑ u = − m 2 m 2 − 1 ∑ v = − m 2 m 2 − 1 ∣ C u v ∣ φ C u v v .
φ C ( f ) = φ C 0 ( f ) + φ N ( f ) , where
φ N ( f ) ≈ ∣ N 1 ( f ) ∣ sin ( φ N 1 ( f ) − φ G 0 ( f ) ) + ∣ N 2 ( f ) ∣ sin ( φ F 0 ( f ) − φ N 2 ( f ) ) ∣ F 0 ( f ) ∣ .
〈 φ N 2 ( u , v ) 〉 = 〈 ∣ N u v ∣ 2 〉 ∣ F 0 u v ∣ 2 and 〈 φ N u v 〉 = 0 .
〈 δ x 2 〉 = ∑ u = − m 2 m 2 − 1 ∑ v = − m 2 m 2 − 1 w x 2 u v 〈 φ N 2 u v 〉 〈 δ y 2 〉 = ∑ u = − m 2 m 2 − 1 ∑ v = − m 2 m 2 − 1 w y 2 u v 〈 φ N 2 u v 〉
w x u v = m Δ 2 π ∣ C u v ∣ ( ua qq − va pq ) a pp a qq − a pq 2 w y u v = m Δ 2 π ∣ C u v ∣ ( va pp − ua pq ) a pp a qq − a pq 2 .
x ̂ 0 = m Δ 2 π u 1 φ C u 1 0 y ̂ 0 = m Δ 2 π v 1 φ C 0 v 1 .
〈 δ x 2 〉 = ( m Δ 2 π u 1 ) 2 〈 ∣ N u 1 0 ∣ 2 〉 ∣ F 0 u 1 0 ∣ 2 〈 δ y 2 〉 = ( m Δ 2 π v 1 ) 2 〈 ∣ N 0 v 1 ∣ 2 〉 ∣ F 0 0 v 1 ∣ 2 .
∣ F 0 ( u , v ) ∣ 2 = ∣ V ( u , v ) ∣ 2 N tot 2 ,
〈 ∣ N i u v ∣ 2 〉 = Δ 4 ∑ p = − m 2 m 2 − 1 ∑ q = − m 2 m 2 − 1 〈 n i 2 p q 〉 ,
〈 ∣ N i u v ∣ 2 〉 = ∑ p = − m 2 m 2 − 1 ∑ q = − m 2 m 2 − 1 ( Δ 2 f 0 p q + 〈 n R 2 p q 〉 ) = N tot + m 2 〈 n R 2 〉 .
〈 δ x 2 〉 = ( m Δ 2 π u 1 ) 2 N tot + m 2 〈 n R 2 〉 ∣ V u 1 0 ∣ 2 N tot 2 〈 δ y 2 〉 = ( m Δ 2 π v 1 ) 2 N tot + m 2 〈 n R 2 〉 ∣ V 0 v 1 ∣ 2 N tot 2 .
α jit ≃ 0.6 λ D 1 6 r 0 5 6 ,
α pix = Δ≃ λ 2 D .
f p q = f 0 p q + ε P + ε R ,
t cc = t 1 m 2 log 2 m 2 + t 2 m 2 + t OH 1 .
t φ = t 3 m 2 + t OH 2
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