Wavefront Noise Reduction in a Shack-Hartmann Wavefront Sensor using Kalman filter Mikhail V. Konnik

James Welsh

School of Electrical Engineering and Computer Science, University of Newcastle, Callaghan 2308 Australia Email: [email protected]

School of Electrical Engineering and Computer Science, University of Newcastle, Callaghan 2308 Australia Email: [email protected]

Abstract—Accuracy of centroid coordinates is influenced by a noise in the intensity image that is received from a wavefront sensor. The noise on the intensity image introduces uncertainty to the centroid coordinates, and consequently to the reconstructed wavefront. The analytical model of wavefront noise caused by light noise is proposed. The model accounts an ADC resolution and centroid spots parameters that allow estimating the signaldependent noise for various quantisation levels. A Kalman-type filter was developed for noise reduction that uses the proposed model of wavefront noise. The comparison between the noised and filtered signals for various quantisation levels is provided.

I. I NTRODUCTION Wavefront sensors (WFS), which use CCD or CMOS photosensors, are the main sources of errors in adaptive optics systems. The photosensor inside the WFS converts the light into digital form, adding various types of noise (photon shot noise, photo response non-uniformity [PRNU], dark fixed pattern noise [FPN], etc.) to the intensity image. The noise on the intensity image causes the noise in centroid coordinates, which in turn introduces inaccuracy in the reconstructed wavefront, and therefore leads to overall performance degradation. Considerable efforts in both analytic and numerical models of the centroiding errors for the WFS have been made. The measurement error of Shack-Hartmann WFS was analysed in [1] and analytic results were obtained for centroid errors due to the photon shot noise and readout noise in CCD. The results in [2], [3] discuss the degradation factor of centroiding accuracy due to the pixels dimensions. The expression that evaluates the angular position error when a quadrant detector used in the Shack-Hartmann sensor was elaborated in [4]. The influence of photon noise and CCD truncation error on the detection accuracy was investigated in [5]. Other noise models were proposed [6], [7] that allow to explain different errors in centroiding. However, those models consider limited noise types and/or do not account the quantisation error. In this paper, we report preliminary results on modelling and reduction a photosensor-induced noise. The model for the light noise accounts an ADC resolution and signal-dependent noise is proposed. The Kalman-type filter has been developed for the noise reduction that uses the noise covariance estimation from the proposed noise model. The results can be used for prefiltering the wavefront slopes in small- and medium-aperture adaptive optics systems.

II. M ATHEMATICAL BACKGROUND The light spots produced by the lenslets of the ShackHartman WFS are converted to the intensity image I by the photosensor inside the WFS. The intensity image I is then used for the centroid coordinates (xk , yk ) calculation: P xi,j (Ii,j + N ) i,j P (1) xk + N1 = (Ii,j + N ) i,j

where N is the photosensor noise; N1 is the centroids noise due to the noise in the intensity image I, which is used for the calculation of turbulent (xck , yck ) and reference (xrk , yrk ) centroid coordinates. The wavefront angles can be calculated: βx ≈ (xrk − xck )

lpix , f

βy ≈ (yrk − yck )

lpix , f

(2)

where lpix is the size of the photosensor’s pixel, and f is the focal length of the lenslet. The wavefront slopes sx and sy are calculated as: sx = tan βx

sy = tan βy

(3)

Finally, the wavefront reconstruction can be performed using one of the geometries (for example, Fried [8] geometry): (φ2 + φ3 )/2 − (φ1 + φ4 )/2 h (φ + φ )/2 − (φ1 + φ2 )/2 4 3 , (4) sy1,1 = h where h = D/L; here D is the diameter of the aperture and the L is the number of lenslets. From the Eq. 4 one can find the wavefront phase φ using least-squares or FFT-based algorithms. sx1,1 =

III. P ROBLEM F ORMULATION AND N UMERICAL S IMULATIONS D ESCRIPTION The noise in centroid coordinates affects the accuracy of the wavefront reconstruction, leading to incomplete compensation of the atmospheric turbulence. For instance, in the worst case (e.g., photon shot noise + PRNU 5% + dark noise and 10 bit ADC), noised centroid coordinates cause considerable errors in the wavefront reconstruction: the normalised root mean squared error can be up to 0.76 or, equivalently, 76% from the noiseless measurements in this example. Therefore, fast

pre-filtering methods can be considered to increase accuracy of wavefront reconstruction. In modern high-end CCD photosensors for the wavefront sensors, the PRNU is usually less than 1-2%. However, those are very expensive custom-made devices. The adaptive optics systems for medium and small-aperture telescopes usually use lower-grade photosensors, where dark noise and PRNU are higher. The dark noise can be reduced using a threshold [9] or threshold centre of gravity (TCoG) centroiding algorithm [10]. The noise in centroid coordinates has two properties [11] that make it difficult to suppress: 1) the probability density function appears to be Gaussian-like but non-symmetrical, and 2) the noise is signal-dependent. Such noise properties can limit the efficiency of centroiding algorithms that assume the noise to be Gaussian. As one of the solutions for the light noise in the wavefront signals, the Kalman filter can be used. There are many techniques that can increase both speed (e.g., steady-state solution [12]) and numerical precision (e.g., U-D factorisation [13]) of the Kalman filter. Other solutions include sophisticated centroiding algorithms (iterative [10], correlation-based [14]), which can be computationally-intensive. A. Proposed solution In this subsection, we discuss the applicability of the Kalman filter to the problem of noise reduction in wavefront reconstruction. Using the developed model of light noise, the Kalman filter can reduce both PRNU-induced noise and photon shot noise in the wavefront slopes. The model of the process is assumed to be a random walk1 : x[k + 1] = x[k] + q[k] y[k] = x[k] + r[k]

(5) (6)

where: q[k] ∼ N (0, Q[k]),

r[k] ∼ N (0, R[k])

(7)

The covariances of the process and measurement noise were modelled: Q[k] ∼ 10−9 lpix 2 2 R[k] ∼ [σc.photon.shot + σc.prnu ]· f

(8) (9)

where lpix is the size of the pixel in the photosensor, and f is the focal length of the lenslet. We developed a variant of the Kalman filter for the noise reduction, which consists of the following steps: 1) compute the Kalman gain: Kk = Pk− (Pk− + Rk )−1 2) update the estimate x ˆk with the measurement zk : x ˆk = αKalman · x ˆ− ˆ− k + (1 − αKalman )zk + Kk (zk − x k) 1 The random walk model was chosen deliberately: we wanted to simulate the performance of Kalman-type filtering in case of dynamics mismodelling. This allows evaluating the performance of the Kalman-type filtering in common case of simplified dynamic assumptions. As it will be shown in Section IV, even in such case the Kalman-type filter provides good performance.

3) compute error covariance Pk for the updated estimate: Pk = (I − Kk )Pk− 4) project ahead the process estimation x ˆ− ˆk and its k+1 = x − covariance Pk+1 = Pk + Qk , where: •

• • • •

zk is the measurement of the wavefront angles (βx and βy ); x ˆk is the estimation of the wavefront angles at time tk ; x ˆ− k is the prior estimation of the wavefront angles; Qk and Rk are the covariances estimated from the Eq. 8; the parameter αKalman is used for the attenuation of smoothing (larger values of αKalman introduce more aggressive smoothing).

The Kalman filter is initialised with the prior estimation of − x ˆ− 0 = β0 and its covariance P0 = R0 . B. Description of Numerical Experiments The numerical simulations were performed to develop the wavefront noise model. One turbulence layer with von K´arm´an power spectrum density was numerically simulated. Then the matrix of turbulence layer was used as an input for the model of the Shack-Hartmann WFS. Centroids were calculated using the WCoG algorithm: first, centroid coordinates were calculated for the case when the photosensor adds no noise. Next, the coordinates for the same wavefront were calculated for the three cases, when the photosensors adds: 1) photon shot noise only, 2) PRNU noise only, and 3) both photon shot noise and PRNU. To facilitate the development of the model of the noise in centroid coordinates, we fixed the positions of the light spots. This means that the wavefront remains the same and does not evolve with time. Hence the noise introduced to the wavefront angles βx and βy (and consequently to wavefront slopes) are due to the noise in the intensity image I. The signal from four different groups of lenslets is chosen to show how the noise in centroid coordinates varies with the signal. The signal level in those four groups was made equal to 75%, 40%, 20%, and 5% of the photosensor saturation level. Next, the variant of the Kalman filter (see Subsection III-A) was used for reduction of noise that is caused by the photosensor’s noise. The noise covariance for the Kalman filter (see Eq. 8) was estimated using the proposed wavefront noise model (discussed in Subsections IV-A and IV-B). The simulated light spots were sensed by the numerical model of a CMOS photosensor [15] with 5µm pixel size, 60% quantum efficiency, 50% fill factor, and 20000e− fullwell. The ADC resolution was varied as 16, 12 and 10 bit. The photosensor in the WFS was simulated with considerable noise: photon shot noise and the PRNU factor 5% of the signal value (the worst-case scenario is considered). The model of Shack-Hartmann WFS has 20 × 20 lenslets; each lenslet contains 32 × 32 pixels, the pixel size is lpix = 5µm, and the focal distance of the lenslet is f = 10 mm.

TABLE I

IV. R ESULTS OF N UMERICAL S IMULATIONS

M ODELLING OF

The numerical experiments allowed formulating the analytical model of the noise. Models for the photon shot noise (see Subsection IV-A) and photo response non-uniformity noise (see Subsection IV-B) were formulated. Each model was tested for the consistency with the estimated noise parameters. Next, the numerical experiments on noise reduction using the developed Kalman filter were performed. The noise covariance for the Kalman filter was estimated by the proposed analytical noise model. Finally, the simulations were performed for the case of both photon shot noise and PRNU noise. A. Modelling and Reducing the Photon Shot Noise Photon shot noise is inevitable due to an uncertainty that arises from random arrival of photons collected by the photosensor. Such a noise is described by the Poisson probability distribution. The effect of photon shot noise on the wavefront slopes is noticeable even for the high-end photosensors with 16 bit ADC. √ Photon shot noise depends on the signal [16] as σshot ∼ S, where S is the signal value. The dependency of the centroids coordinates xc and yc from the signal is more complicated (see Eq. 1). According to our simulations, the centroids noise can be described by the model: σc.photon.shot = α = 2(16−M) ,

1 p , ln δ · αξS

ξ = M/16,

(10) (11)

where: • M - the resolution of an ADC, bits; • δ - the full width on a half heigh a light spot; • S - the peak signal value of the light spot. The model of the wavefront noise due to the photon shot noise was compared with the numerical simulations when only shot noise was turned on. From the Table I one can see a good agreement of the model with simulations. The model of the wavefront noise due to the photon shot noise was used for the noise reduction in wavefront slopes. 2 The measurements noise covariance Rk = [σc.photon.shot ]·η was substituted into the Kalman filter. Here η is the constant for the conversion of centroid coordinates into the wavefront angles, η = lpix /f , where lpix = 5µm is the length of the pixel in the photosensor, and f = 10mm is the focus length of the lenslet. The simulation was run and 100 samples of the wavefront slopes were calculated out of noised centroid coordinates. Then the same data were filtered using the developed Kalman filter. The results are presented in Fig. 1, where the worst case scenario is shown (lenslet with only 5% light signal). The decrease of the standard deviation between noised and denoised slope data was used as a performance metric. The results show considerable reduction of shot noise influence in the wavefront slopes after filtration. Quantitatively, for the αKalman = 0.75 the decrease in the standard deviation of noise is about 2.5 times, while for more aggressive smoothing with the Kalman parameter αKalman = 0.95 the noise

WAVEFRONT NOISE DUE TO THE PHOTON SHOT NOISE

Centroid peak value on intensity image (in % of saturation)

Estimated wavefront noise std. (Gaussian, 95% conference bounds)

Noise std. according to model

75% 40% 20% 5%

16 bit case 0.0024 . . . 0.0026 0.0031 . . . 0.0034 0.0047 . . . 0.0052 0.0093 . . . 0.0102

0.0023 0.0031 0.0047 0.0094

75% 40% 20% 5%

12 bit case 0.0024 . . . 0.0026 0.0031 . . . 0.0034 0.0048 . . . 0.0052 0.0098 . . . 0.0107

0.0027 0.0036 0.0052 0.0096

75% 40% 20% 5%

10 bit case 0.0025 . . . 0.0027 0.0036 . . . 0.0040 0.0054 . . . 0.0059 0.0123 . . . 0.0134

0.0029 0.0039 0.0058 0.0117

decreases by 5-6 times (see Table II). Increasing the smoothing parameter αKalman to 0.95 provides better noise attenuation, but can degrade the accuracy of centroiding. Overall, the Kalman filter provides a reasonable noise reduction in the case of the photon shot noise only. TABLE II D ENOISING THE SLOPES DATA USING K ALMAN FILTER : PHOTON SHOT NOISE CASE

ADC resolution 16 bit 12 bit 10 bit

std(Snoised )/std(Skalman ) αKalman = 0.75 αKalman = 0.95 2.6 4.8 2.5 6.1 2.6 4.5

B. Modelling and Reducing the noise caused by Photo Response Non-Uniformity Photo response non-uniformity (PRNU) is the type of noise caused by the variation of pixel-to-pixel size, which occurs due to an inaccuracy in the fabrication process of the photosensor [17]. Typically the PRNU is about 1-2% [18], but can be up to 5% in low-grade sensors. The noise introduced by the PRNU is directly proportional [18] to the illumination as σP RN U ∼ S, where S is the signal of the intensity image I. According to simulations, the centroids noise due to the

Slope values calculated from centroid coordinates

TABLE III M ODELLING OF WAVEFRONT NOISE DUE TO THE P HOTO R ESPONSE N ON -U NIFORMITY

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12 bit case, PRNU factor γ = 0.05 (5% PRNU) 75% 0.0093 . . . 0.0102 0.0095 40% 0.0096 . . . 0.0105 0.0101 20% 0.0098 . . . 0.0107 0.0109 5% 0.0099 . . . 0.0108 0.0124 PRNU factor γ = 0.01 (1% PRNU) 75% 0.0019 . . . 0.0021 0.0019 40% 0.0019 . . . 0.0021 0.0020 20% 0.0021 . . . 0.0022 0.0022 5% 0.0026 . . . 0.0029 0.0025

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16 bit case, PRNU factor γ = 0.05 (5% PRNU) 75% 0.0094 . . . 0.0102 0.0095 40% 0.0096 . . . 0.0105 0.0108 20% 0.0095 . . . 0.0104 0.0112 5% 0.0098 . . . 0.0107 0.0128 PRNU factor γ = 0.01 (1% PRNU) 75% 0.0019 . . . 0.0020 0.0019 40% 0.0019 . . . 0.0021 0.0020 20% 0.0019 . . . 0.0022 0.0022 5% 0.0020 . . . 0.0023 0.0025

Slopes calculated from noised centroid coordinates Denoised slope values by Kalman filter, α=0.75 Denoised slope values by Kalman filter, α=0.95 0

Estimated wavefront noise std. (Gaussian, 95% conference bounds)

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Fig. 1. Noise reduction in wavefront slope calculated from the centroid coordinates that contain photon shot noise only: a) 16 bit quantisation, b) 12 bit quantisation, c) 10 bit quantisation.

10 bit case, PRNU factor γ = 0.05 (5% PRNU) 75% 0.0095 . . . 0.0104 0.0095 40% 0.0099 . . . 0.0108 0.0100 20% 0.0101 . . . 0.0110 0.0109 5% 0.0117 . . . 0.0128 0.0129 PRNU factor γ = 0.01 (1% PRNU) 75% 0.0020 . . . 0.0022 0.0019 40% 0.0021 . . . 0.0023 0.0020 20% 0.0024 . . . 0.0026 0.0022 5% 0.0031 . . . 0.0034 0.0026

PRNU can be described: σc.prnu =

γ , 0.25 · ln δ · ln(αS) α = 2(16−M) ,

(12) (13)

where: • • • •

M - the resolution of an ADC, bits; δ - full width on a half height of a light spot; S - the peak signal value of the light spot. γ - the PRNU factor (e.g., 0.05 for 5% PRNU).

It is noteworthy that the standard deviation of a noise due to the PRNU is directly proportional to PRNU factor γ and weakly depends on the signal level (can be well described

by a natural logarithm of the signal). The model Eq. 12 of the wavefront noise caused by the PRNU induced noise was compared with the results of simulations (see Table III) and it was found to be adequate. However, the model overestimates the noise induced by the PRNU in case of 16 bit quantisation and underestimates the noise in case of 10 bit. Such behaviour is a subject for further research. The formulated model was used for the noise reduction in the wavefront slopes signal calculated from the noised centroid 2 coordinates. The noise covariance Rk = [σc.prnu ] · η was substituted into the Kalman filter, similarly to the previous subsection (η = lpix /f , where lpix = 5µm is the length of the

Slope values calculated from centroid coordinates

pixel in the photosensor, and f = 10mm is the focus length of the lenslet). The simulation was run and 100 samples of the slopes calculated out of the noised centroid coordinates were obtained. The results of the numerical simulations for the lenslet with low light (worst-case scenario) and for PRNU factor of 5% (low-grade photosensor) are presented in Fig. 2.

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std(Snoised )/std(Skalman ) αKalman = 0.75 αKalman = 0.95 2.2 3.8 2.4 7.1 3.1 6.8

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αKalman = 0.75 the decrease in the standard deviation of noise is about 2 − 3 times, and for the parameter αKalman = 0.95 the noise can be decreased by 4-7 times (see Table IV). The Kalman filter provides good performance in denoising the slopes calculated from the noised centroids, when the noise is induced by PRNU. 0

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TABLE IV D ENOISING THE SLOPES DATA USING K ALMAN FILTER : THE CASE OF PRNU INDUCED NOISE

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In the previous subsections, it was demonstrated that the Kalman filter with the formulated noise model could be used for the noise reduction in the wavefront signals. In practice, however, photon shot noise and the PRNU noise occurs simultaneously. Therefore, the numerical experiments were performed with both types of noise turned on. The worst case scenario of a large PRNU-induced noise (PRNU factor γ = 0.05 or 5%) was considered. The conditions of the numerical simulations are the same as in previous subsections. The only difference is the covariance of noise Rk for the 2 2 Kalman filter now is Rk = [σc.photon.shot + σc.prnu ] · η. Results of the numerical experiments for 16, 12 and 10 bit quantisation case for low-light lenslet are presented in Fig. 3. One can see that the Kalman filter with the formulated PRNU and photon shot noise model reduces the noise considerably. Quantitative results are provided in Table V. Similarly to the previous results, the Kalman filter reduces the noise in slopes data by 2 − 3 times for αKalman = 0.75 and 5 − 7 times for more aggressive smoothing parameter αKalman = 0.95. TABLE V D ENOISING THE SLOPES DATA USING THE K ALMAN FILTER : THE CASE OF BOTH PRNU INDUCED AND PHOTON SHOT NOISE

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Fig. 2. Noise reduction in the wavefront slopes calculated from the noised (PRNU induced noise only) centroid coordinates: a) 16 bit quantisation, b) 12 bit quantisation, c) 10 bit quantisation.

From the results in Fig. 2 one can conclude that the Kalman filter with the formulated PRNU noise model provides good denoising performance. The quantitative results are summarised in Table IV. Similar to the results in the case of photon shot noise only, the model of PRNU noise in the centroid coordinates can be considered as adequate. For the Kalman filter parameter

ADC resolution 16 bit 12 bit 10 bit

std(Snoised )/std(Skalman ) αKalman = 0.75 αKalman = 0.95 2.7 4.4 2.4 7.6 2.5 5.3

Summarising the results of the numerical experiments we conclude that the Kalman filter can be used to increase the accuracy of the wavefront reconstruction from the noised centroid coordinates.

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R EFERENCES

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variable signal level, and a width of light spots produced by the Shack-Hartmann WFS. A variant of Kalman filter with a variable smoothing was developed for the noise reduction. The noise covariance for the Kalman filter was calculated from the proposed model of wavefront noise. The results show considerable noise reduction in the wavefront slopes signal, especially for 10-12 bit ADC resolution, which if often used in the low-order adaptive optics. The proposed light noise model for the noise in centroid coordinates and the variant of the Kalman filter can be used for the pre-filtering of wavefront slopes before the wavefront reconstruction. Results can be used for small- and mediumaperture adaptive optics systems.

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Fig. 3. Noise reduction in the wavefront slopes calculated from the noised (PRNU and photon shot noise) centroid coordinates: a) 16 bit quantisation, b) 12 bit quantisation, c) 10 bit quantisation.

V. C ONCLUSION A solid-state photosensor inside the Shack-Hartmann wavefront sensor, which converts the intensity image of light spots into a digital form, inevitably adds various types of noise. Consequently, centroid coordinates will be inaccurate due to the noise in the intensity image, which in turn introduces errors into the reconstructed wavefront. Although dark noise can be usually removed by a threshold, the light noise remains difficult to reduce. This paper reports the preliminary results on modelling and reduction of the noise in wavefront slopes due to a photosensor noise. The proposed analytical model for light noise in centroid coordinates accounts an ADC resolution, a

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