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## Wavefront Sensing II

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**Wavefront Sensing II**Richard Lane Department of Electrical and Computer Engineering University of Canterbury**Contents**• Session 1 – Principles • Session 2 – Performances • Session 3 – Wavefront Reconstruction**Session 2 Performances**• Geometrical wavefront sensing take 2 • The inverse problem • The astronomical setting • The basic methods**Geometric wavefront sensing(or curvature sensing without**curvature) Plane 1 Image Plane Plane 2 Improve sensitivity (signal stronger) Improve the number of modes measurable (signal weaker)**W(x)**z x Geometric optics model • Slopes in the wave-front causes the intensity distribution to be stretched like a rubber sheet • Aim is to map the distorted • distribution back to uniform**Geometric wavefront sensingTake 2**Intensity Plane 1 Plane 1 Image Plane Intensity Plane 2 Plane 2 Intensity distribution gives the probability distribution For the photon arrival**Difference**gives a slope estimate Final slope estimate Recovering the phase Intensity Plane 1 Integrate to Form CDF Intensity Plane 2 Choose level Probability density functions Integrate slope to find the phase Defocus!**InverseProblem**Performance is determined by amount of photons entering the aperture and assumptions about the object and turbulence**Multiple layers**For wide angle imaging we need to know the height of the turbulence Layer 1 h1 h2 Layer 2 Aperture Plane**The fundamental problem:**How to optimally estimate the optical effects of turbulence from a minimal set of measurements**Limiting Factors**• Technological • CCD read noise • Design of wavefront sensor (Curvature, Shack-Hartmann, Phase Diversity) • Fundamental • Photon Noise • Loss of information in measurements • Quality of prior knowledge**In Its Raw Form the Inverse Problem Is Always Insoluble**• There are always an infinite number of ways to explain data. • The problem is to explain the data in the most reasonable way • Example Shack-Hartmann sensing for estimating turbulence**Example – fit a curve to known slopes**• Solution requires assumptions on the nature of the turbulence • Use a limited set of basis functions • Assume Kolmogorov turbulence or smoothness**Parameter estimation**• Essentially we need to find a set of unknown parameters which describe the object and/or turbulence • The parameters can be in terms of pixels or coefficients of basis functions • Solution should not be overly sensitive to our choice of parameters. • Ideally it should be on physical grounds**Bayesian estimation 101An important problem**• Estimate • And if you know that it models two people splitting the bill in a restaurant?**Possible phase functionsZernike basis**Δ Zernike Polynomials Low orders are smooth Pixel basis, highest frequency = 1/(2Δ)**Estimation using Zernike polynomials**Measurement Interaction Zernike Polynomial vector matrix Coefficents phase weighting Zernikie polynomial • ith column of Θ corresponds to the measurement that would occur if the phase was the ith Zernike polynomial**Extension to many modes**• Provided the set of basis functions is complete, the answer is independent of the choice • The best functions are approximately given by the eigenfunctions of the covariance matrix C • These approximate the low order Zernike polynomials, hence their use. • Conventional approach is to use a least squares solution and estimate only the first M Zernikes when M ≈N/2 (N is the number of measurements)**Ordinary least squares**• Minimise Weighted least squares • Not all measurements are equally noisy • hence minimise**Conventional Results**• As the number M increases the wavefront error decreases then increases as M approaches N. • Reason when M=N there is no error and there should be as higher order modes exist and will be affecting the measurements**Phase estimation from the centroid**• Tilt and coma both produce displacement of the centroid • According to Noll for Kolmogorov turbulence • Variance of the tilt • Variance of the coma Ideally you should estimate a small amount of coma**Bayesian viewpoint**• The problem in the previous slide is that we are not modelling the problem correctly • Assuming that the higher order modes are zero, is forcing errors on the lower order modes • Need to estimate the coefficients of all the modes as random variables**Example of Bayesian estimation for underdetermined equations**• Measurement z is a linear function of two unknowns x,y • The estimate (denoted by ^) is a linear • function of z • We want to minimise the expected error Statistical expectation**Minimisation of the error**• Key step, rewrite in terms of and • Solution is a function of the covariance of the unknown • parameters**Vector solution for the phase**• Express the phase as a sum of orthogonal basis functions • Observed measurements are a linear function of the coefficients • Reconstructor depends on the covariance of a**Simple example for tilt D/r0=4**• From Noll • From Primot et al**Bayesian estimate of the wavefront**Minimizes**Summary Bayesian method**• When the data is noisy you need to put more emphasis on the prior. • For example, if the data is very bad, don’t try and estimate a large number of modes • When done properly the result does not depend strongly on C being exact • Error predicted to be where**Operation of a Bayesian estimator**• Minimizes • When D becomes very large, the data is very noisy then more weight is placed on the prior data prior • Ultimately as D→∞, a→0 (for very noisy data no estimate is made)**Bayesian examination question**• You are on a game show. • You can select one of three doors • Behind one door is $10000, behind the others nothing • After you select a door, the compere then opens one of the other doors revealing nothing. • You are given the option to change your choice • Should you?**Estimating the performance limits when it is non-Gaussian**• The preceding analysis is fine when the measurement errors can be modelled as a Gaussian random variable • On many equations you need to perform an analysis to work out the error in the analysis • Cramer-Rao bounds**Cramer-Rao bound**• Linear unbiased estimators only • Essentially the quality of the parameter estimate is given by the curvature of the pdf • Doesn’t tell you how to achieve the bound**Simple example**• Find the performance limit estimating the mean of a one-dimensional Gaussian from 1 sample**Points to note**• Limit is a lower bound. Clearly for 1 sample from the pdf it cannot be attained • The variance decays as 1/N with more samples • For a Gaussian asymptotically the centroid of the distribution can be shown to approach the Cramer-Rao bound**Estimation of a laser guidestar location, Cramer-Rao bound**Small projection telescope Large AO corrected projection telescope Large uncorrected projection telescope Key points: In the presence of saturation a focused spot may not be optimal Need to know the pattern to reach the limit**Optimal estimation of a parameterwavefront tilt**• Important because the wavefront tilt is the dominant form of phase aberration • A small error in estimating the tilt can be larger than the full variance of a higher order aberration.**Issues**• Displacement of the centroid of an image is proportional to the average tilt (not the least mean square) of the phase distortion • Will discuss this issue later, but for the moment concentrate on estimating the mean square tilt.**How do you estimate the centre of a spot?**• The performance of the Shack-Hartmann sensor depends on how well the displacement of the spot is estimated. • The displacement is usually estimated using the centroid (center-of-mass) estimator. • This is the optimal estimator for the case where the spot is Gaussian distributed and the noise is Poisson.**Why Not Use the Centroid?**• In practice the spot intensity decays as • This means that photons can still occur at points quite distant from the centre. • Estimator is divergent unless restricted to a finite region in the image plane**Diffraction-limited spot**• For a square aperture, the distribution is:**Solutions (1)**• Use a quad cell detector and discard the photons away from the centre • The signal from the outer cells is discarded because it adds too much noise**Solutions 2**• Use an optimal estimator that weights the information appropriately • Consider two measurements of an unknown parameter an estimate of a parameter with different variances • A weighted sum is always a better estimator • A non linear estimator is better still**Maximum-likelihood estimation**• If photons are detected at x1, x2…, xN, the estimate is the value that maximizes the expression • The Cramer-Rao lower bound for the variance is • For a large number of photons, N, the variance approaches the Cramer-Rao lower bound.**Centroid location by model fitting**• Technique relies on finding a model of the object • Not sensitive to the size of window (unlike the centroid) • Centroid is a closed form solution for fitting a Gaussian of variable width**Tilt estimation in curvature sensing**• The image is displaced by the atmospheric tilt, how well you can estimate it is determined by the shape of the image formed.**Tilt estimation in the curvature actual propagated**wavefronts