Centroid Detection by Gaussian Pattern Matching In Adaptive Optics

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1 010 International Journal of Computer Appliations ( ) Centroid Detetion by Gaussian Pattern Mathing In Adaptive Optis Akondi Vyas Indian Institute of Astrophysis Indian Institute of Siene M B Roopashree Indian Institute of Astrophysis B R Prasad Indian Institute of Astrophysis ABSTRACT Shak Hartmann wavefront sensor is a two dimensional array of lenslets whih is used to detet the inoming phase distorted wavefront through loal tilt measurements made by reording the spot pattern near the foal plane. Wavefront reonstrution is performed in two stages - (a) image entroiding to alulate loal slopes, (b) formation of the wavefront shape from loal slope measurement. Centroiding auray ontributes to most of the wavefront reonstrution error in Shak Hartmann sensor based adaptive optis system with readout and bakground noise. It beomes even more diffiult in atmospheri adaptive optis ase, where sintillation effets may also our. In this paper we used a denoising tehnique based on thresholded Zernike reonstrutor to minimize the effets due to readout and bakground noise. At low signal to noise ratio, this denoising tehnique an be improved further by taking the advantage of the shape of the spot. Assuming a Gaussian pattern for individual spots, it is shown that the entroiding auray an be improved in the presene of strong sintillations and bakground. The information of the shape of the onjugated wavefront is then ommuniated to the orretor. Although different wavefront measuring devies exist like urvature sensor, shearing interferometer, ommon path interferometer, most generally used sensor is the Shak Hartmann Sensor (SHS) whih is made up of tiny lenses arranged in a two dimensional array. A wavefront inident on the SHS forms an array of spots near the foal plane of the lenses. Unlike an ideal plane wavefront normal to the SHS whih forms well foused equidistant spots, a distorted wavefront forms spots that are not equidistant as shown in Fig. 1. From the measurement of the shift in the spot positions from ideal ase, it is possible to alulate the loal slopes (over eah lenslet/subaperture) of the wavefront. Using the slope values, the approximate shape of the wavefront is estimated. The two steps involved in wavefront reonstrution are hene (a) entroiding of individual spots to alulate loal slopes, (b) formation of the wavefront shape from loal slope measurements. Categories and Subjet Desriptors I.4.9 [Image Proessing and Computer Vision]: Appliations General Terms Algorithms, Measurement, Performane, Experimentation Keywords Adaptive Optis, Shak Hartmann Sensor, Wavefront Reonstrution, Centroiding 1.INTRODUCTION Adaptive Optis (AO) is a well developed tehnology that helps in quality imaging of objets hidden behind turbulent media. It is widely used in astronomial imaging, retinal imaging and free spae ommuniation systems [1-3]. A simplest AO system omprises of a wavefront sensor that detets the shape of the inoming optial wavefront and a wavefront orretor that ompensates the effets of wavefront distortions by imposing a onjugate wavefront on the inident wavefront [4]. The wavefront sensor and orretor are onneted via a proessor where the required wavefront reonstrution omputations are performed. Figure 1 Priniple of Shak Hartmann Sensor Centroid detetion beomes a diffiult problem in the presene of noise [5]. Readout noise, bakground noise, photon noise and sintillation effets are dominant effets in atmospheri AO. Due to the presene of noise, mathematially simplest entroiding algorithm like Center of Gravity (CoG) method leads to large wavefront reonstrution errors [6]. To improve upon this tehnique, Weighted Center of Gravity (WCoG), Iteratively Weighted Center of Gravity (IWCoG), Intensity Weighted Centroiding (IWC), orrelation based entroiding, mathed 3

2 filtering based entroiding were developed [7-9]. In this paper, we studied the performane of CoG, WCoG, IWCoG and IWC tehniques. The entroid, (x, y ) an be omputed in the ase of CoG using the formula, x,y XI (1) where I is a two dimensional image funtion with M M pixels, i and j take values 1,,.M. In the ase of WCoG, a weighting funtion (generally Gaussian) is used to take the advantage of the shape of the spot. x,y I XIW () IW 1 ( )x ( )y,(w yx ) exp x y where, σ is the spread of the spot. 010 International Journal of Computer Appliations ( ) IWCoG works on the same priniple as WCoG, but with a differene that it is an iterative method where the enter of the weighting funtion is shifted to the position of the estimated entroid in the previous iteration. In IWC, the weighting funtion is defined as powers of intensity, W(x, y) = {I(x, y)} p, p is real and positive. CoG performs best in high Signal to Noise Ratio (SNR) onditions. WCoG is generally used in the losed loop AO where the shift in the spot is small. This tehnique shows a signifiant improvement over CoG in the presene of readout and bakground noise. IWCoG an be used effetively in losed as well as open loop AO systems but this tehnique is time onsuming and has problems in iterative onvergene. IWC performs best in photon noise limited onditions [10]. The phase values are related to the measured slopes through a linear system of equations, Ax=b, where A is the oeffiient matrix, x is a vetor ontaining phase values to be alulated and b represents the measured slope values [11]. The oeffiient matrix is a sparse matrix in AO ase and hene sparse matrix methods are generally used for wavefront reonstrution from slope values [1]. There are many fators that ause wavefront reonstrution errors in AO like sensor alignment, sensor disretization, detetor pixelization, detetor SNR, detetor readout noise, frame jitter, ross talk between lenslets, bakground light and sintillations effets due to large intensity flutuations (also magnified while using a Laser Guide Star (LGS) as referene star), random spot wobbling during the exposure time and many other random effets [13]. It was shown theoretially and experimentally that the sintillations an greatly degrade the orretion ability of AO systems [14]. Sintillations depend mainly on the Fried parameter whih indiates the strength of turbulene and the variane of intensity [15]. (3) A simple denoising tehnique based on Zernike reonstrution that greatly improves the wavefront reonstrution auray was used in this paper [16]. In this noise removal method the spot patterns orresponding to individual subapertures of a SHS are reonstruted using Zernike polynomials by alulating Zernike moments. The reonstruted images are thresholded and the resultant images are used for entroid estimation. Sine the noise features arising due to bakground, readout are of low spatial extent, they will not be found on the images reonstruted using less number of Zernike moments. The sintillation effets or rather prominent events omparable to the atual spot image will sustain even after following the Zernike reonstrution and thresholding proess. In this paper we propose a Gaussian Pattern Mathing (GPM) algorithm to improve the entroid detetion auray. In this method we take the advantage of the shape of the spot pattern. As a first step, we identify the features. Seondly, we seletively eliminate the features that do not have a irular shape. Finally we reount the number of features, f N and if it is greater than one, we use the fat that the intensity distribution from the feature entroid drops like a Gaussian. This algorithm an be further improved by inluding wavefront predition algorithms [17]. The methodology used for simulating the SHS spot pattern in the presene of readout noise, photon noise and bakground noise is desribed in setion. Zernike reonstrutor based denoising tehnique is presented in setion 3 with illustrations and graphs. Setion 4 desribes the basi idea behind the Gaussian pattern mathing algorithm and the design steps involved to minimize faulty reognition. The results obtained by applying Monte Carlo simulations on various entroiding algorithms are presented in setion 5 and the derived onlusions are presented briefly in setion 6..SIMULATION OF SPOT PATTERN Photon noise, bakground noise and detetor readout noise are three major noise soures in AO. In astronomial adaptive optis system, a Shak Hartmann spot pattern is greatly influened by photon noise whih is aused due to the low photon ount. Photon noise an be minimized by using a LGS or a bright Natural Guide Star (NGS) as a referene star. Bakground noise ours beause of the finite sky brightness and other unwanted light soures. Readout noise is an unavoidable noise aused by the detetor. We inluded these effets in our simulations. The following steps were followed to simulate the spot pattern. A two dimensional Gaussian funtion was simulated on an array of size M M pixels with ontrol over the position at whih the spot enter lies and the spread of the spot. Sample ideal spot images are shown in Fig.. Figure Gaussian spots with a spread, σ taking values (a) 0.5 (b) 1.0 () 1.5 (d).0 (e).5 (image enter is the spot enter) A photon noise limited image is then simulated as a seond step. Photon noise is an intensity dependent 33

3 noise. In this step, eah of the pixel values is replaed by a randomly hosen value from a Poisson distribution whose mean is the atual pixel value. The atual spot and the photon noise limited spot images are shown in Fig. 3. The pixel values were amplified for better visualization. Bakground noise whih is aused due to unwanted light soures is generally random. This an be simulated by generating pseudo Gaussian random numbers. Choosing a suitable SNR, the maximum bakground intensity is adjusted. The effet of bakground noise on photon noise limited spot image with SNR=1.33 is shown in Fig. 4. Figure 3 Poisson Noise limited image (b) in omparison with an ideal Gaussian spot (a) 010 International Journal of Computer Appliations ( ) The denoising tehnique of spot images onsists of two steps. Firstly the images are represented in terms of a finite number of Zernike polynomials. The proess of onversion of the image in terms of Zernike polynomials is performed by the Zernike reonstrutor. Thresholding is performed on the reonstruted images to obtain a nearly noise free image. 3.1Zernike Reonstrutor Zernike polynomials are a set of ontinuous orthogonal irular polynomials defined over the unit disk. Sine they form a omplete set of orthogonal polynomials, any two dimensional funtion, I(x, y) an be represented as a proper linear ombination of this basis set. (I,) x y az(, xy) i i where, Z i(x, y) represents Zernike polynomials and the oeffiients a i s standing before Zernike polynomials are image weights alled Zernike moments. Sine Zernike polynomials are mathematially ompliated funtions, the alulation of Zernike moments of images is tedious. In this paper, we used a very fast and nearly aurate method suggested by Hosny to ompute the Zernike moments [18]. The denoising ability varies with the number of Zernike orders used for representing the image. The reonstruted noisy spot pattern images are shown in Fig. 5. It is advised to use more number of Zernike moments sine the entral Gaussian feature an be aptured better during reonstrution. i (5) Figure 4 (a) Photon noise limited (b) Bakground noise superimposed on photon noise limited image, SNR=1.33 Detetor readout noise is projeted as random flutuations about the atual intensity value. This noise beomes ritial during short exposure time sale imaging, at low light level onditions and faint bakground noise levels. This is due to the detetor imperfetion in signal detetion. The impat of readout noise in low light level ondition on the ideal Shak Hartmann spot pattern is similar to the effet of a uniform bakground noise. The spot pattern hene obtained after addition of the above desribed noise is used for further analysis. They are used in the Monte Carlo simulations for testing the entroiding algorithms. In the simulations the user has ontrol over the atual position of the spot. The Centroid Estimation Error (CEE) an be evaluated using the formula, Figure 5 Representation of images using Zernike moments (a) Ideal spot image (b) Noisy image. Zernike reonstruted image using () 10 (d) 15 (e) 0 (f) 5 (g) 30 moments. 3.Thresholding The image reonstruted using the above method is then thresholded to remove the higher order aberrations (low spatial extent events) on the spot pattern. It an be observed by a omparison of Fig. 5 and Fig. 5g. that lower spatial sale events will be reonstruted only if further more orders of Zernike polynomials are used. The images in Fig. 5-g are thresholded and shown in Fig. 6. CEE x x yy (4) where, (x *, y * ) is the estimated entroid and (x, y ) is the atual entroid. Effetively, CEE measures the distane of the estimated entroid from the atual position of the spot. 3.ZERNIKE RECONSTRUCTOR + THRESHOLDING Figure 6 Images formed after Zernike reonstrution and thresholding Figure 5b. using (a) 10 (b) 15 () 0 (d) 5 (e) 30 Zernike moments 34

4 Figure 7 Performane of IWCoG method in the presense of noise before and after noise removal algorithm This method is advantageous in the presene of very high noise level onditions as shown in Fig. 7. At SNR<0.45, the denoising proedure in onjugation with CoG method performs better than IWCoG with 8 iterations [16]. This helps is overoming the onvergene and speed problems assoiated with IWCoG. 010 International Journal of Computer Appliations ( ) 4.Shape identifiation Most features do not have a irular shape as shown in Fig. 8. The irularity or the extent of the feature being irular is measured for eah of the features. In the proposed algorithm, the entroid position was omputed for a single feature loally and the distane from the entriod at whih intensity beomes zero is measured. This distane is alled the radius parameter. The radius parameter is estimated at different angles (0-360) from the feature entroid position. We define irularity, C of a single feature as the inverse of the variane of the radius parameter omputed over different angles from the entroid position. For an ideal irular feature, the variane is zero and hene the irularity is infinity. A lower utoff for this parameter is hosen to eliminate features that are not lose to a irular shape. This shape identifiation proess was applied to Fig 8() and is shown in Fig. 9. It is possible to eliminate all the noise features in this step at better SNR and applying a suitable feature size thresholding. 3.3Erroneous spot pattern after thresholding At high noise onditions, the thresholded Zernike reonstruted images lead to multiple features as shown in Fig. 8. This is due to the fat that at high noise level onditions there an be large sale features (sales omparable to the size of the spot) whih might not be removed even after the denoising proedure is implemented. Figure 8 Erroneous threholding at SNR=1 for perentage thresholding of (a) 80 (b) 70 () 60 4.GAUSSIAN PATTERN MATCHING To use this erroneous spot for aurate entroiding, we take the advantage of the fat that the final spot formed at the foal plane of a lens must assume an Airy pattern whih an be approximated to a Gaussian like struture. The pattern mathing algorithm is implemented in three steps: Feature reognition Shape identifiation Profile identifiation 4.1Feature reognition In this step, the features on the spot pattern image are ounted and identified. Feature reognition an be performed by using many existing pattern reognition algorithms. In this paper, we used a simple Hough peak identifiation method to detet the features and number them in the order of peak height. To eliminate small sale features whih may arise due to unavoidable sintillations, we imposed threshold onditions on the size of the features. The number of features varied from 30 to 1 from very low SNR to high SNR. Figure 9 Shape Identifiation: (a) Erroneous spot pattern image (b) Spot pattern after removal of small sale features and elimination of non-irular features 4.3Profile identifiation In the previous step, the shape of the spot was used for seletive elimination. There an be ases of large noise features that are irular in shape as shown in Fig. 10. In the profile identifiation step, the intensity profile is used to hoose the atual spot feature. The number of features are reounted and re-identified after going through the shape identifiation step. The intensity fall off from the entroid position of individual features is measured and pattern mathing was performed with a standard one dimensional Gaussian funtion, G(x) = exp[-(x-x 0) /σ ]. In this proess, the features that do not follow a Gaussian like struture are eliminated. A omparison of the atual spot line profile with the noise feature line profiles is shown in Fig. 11. The sum of squares of errors was used as a statisti to measure the total deviation of the atual values from the fitted values. Figure 10 Comparably large irular features 35

5 010 International Journal of Computer Appliations ( ) Figure 11 Comparison of Gaussian fitted line profile for atual spot and a noisy feature 5.COMPUTATIONAL RESULTS Monte Carlo simulations were performed using the spots simulated as desribed in setion. Zernike Reonstrutor + Thresholding algorithm is implemented as desribed in setion 3. The images formed after thresholding are given as input to the Gaussian pattern mathing (GPM) algorithm, where a single feature is seleted from a set of features erroneously left on the spot pattern after thresholding. Simulations were run on 100 randomly simulated noisy spot pattern images (64 64) with the spot oupying 8 8 pixels. The performane of CoG and IWC before and after applying GPM algorithm on images proessed using thresholded Zernike reonstrution is shown in Fig. 1. Figure 13 Performane of GPM on WCoG and IWCoG WCoG and IWCoG perform better even without the appliation of GPM algorithm at low SNR and very small shift in the spot as shown in Fig. 13. But for large shift in the spots, the performane of WCoG is worst in the absene of GPM algorithm [10]. The performane of WCoG an be enhaned by applying GPM algorithm and use it as a entroid pre-alulator. The weighting funtions for WCoG are evaluated from the pre-alulated entroid position. The improvement in the entroiding auray for WCoG when the spot shift is 5 pixels is shown in Fig. 14. The performane of IWCoG remains the same even for large shift in the spots. Problem arises for IWCoG only when large noise effets our very lose (loser by more than half-width of the spot) to the atual spot position. Sine large noise effets our randomly in time, it is advantageous to have GPM algorithm whih an nearly point out the entroid position prior to a more preise estimation using IWCoG. Figure 1 Improvement in CoG and IWC methods after noise removal The GPM algorithm an be used in many ways. It an be used diretly for entroid determination by applying CoG, IWC. Figure 14 Improved WCoG for 5 pixel shift in the spot 6.CONCLUSIONS Wavefront reonstrution auray is a ritial fator in high resolution ground based astronomial imaging through turbulene. Centroiding ontributes to most of the wavefront reonstrution error in Shak Hartmann sensor based adaptive optis system with readout, bakground noise and strong sintillations. Using an effetive denoising tehnique based on thresholded Zernike reonstrutor, the small sale noise features in the spot pattern images are removed. At very low signal to noise ratio, it was shown that taking the advantage of the shape of the spot an improve the entroiding auray of CoG, IWC, 36

6 WCoG and IWCoG algorithms. In the ase of laser guide star, an elongated spot an be used for pattern mathing. 7.REFERENCES [1] J. W. Hardy, Adaptive Optis for Astronomial Telesopes, Oxford University Press, New York, (1998). [] David R. Williams and Jason Porter, in Adaptive Optis for Vision Siene: Priniples, Praties, Design, and Appliations, (ed: J. Porter, et al.) (John Wiley & Sons, New Jersey, In Press) 3, 1 (006). [3] T. Weyrauh and M. A. Vorontsov, in Free-Spae Laser Communiations Priniples and Advanes Series: Optial and Fiber Communiations Reports, (ed: A. K. Majumdar and J. C. Riklin) (Springer) Vol., 47 (008). [4] R. K. Tyson, Priniples of Adaptive Optis. Aademi Press, New York, [5] MGuire, P. C., Sandler, D. G., Hart, M. L., Rhoadarmer, T. A., Adaptive Optis: Neural Network Wavefront Sensing, Reonstrution, and Predition, Sientifi Appliations of Neural Nets, Proeedings of the 194th W. E. Heraus Seminar, Clark, J. W., Lindenau, T., Ristig, M. L., Springer-Verlag Publishers, pp. 97, (1999). [6] A. Vyas, M. B. Roopashree, B. R. Prasad, in Proeedings of the National Conferene on Innovative Computational Intelligene & Seurity Systems, Sona College of Tehnology, Salem. Apr 3-4, 009. pp [7] S. Thomas, T. Fuso, A. Tokovinin,M. Niolle, V. Mihau, and G. Rousset, "Comparison of entroid omputation algorithms in a Shak-Hartmann sensor," Monthly Noties of the Royal Astronomial Soiety 371, (006). [8] K. L. Baker and M. M. Moallem, "Iteratively weighted entroiding for Shak- Hartmann wave-front sensors," Opt. Express 15, (007). [9] L. A. Poyneer, D. W. Palmer, K. N. LaFortune, and B. Bauman, "Experimental results for orrelation-based wave-front sensing," SPIE 5894, 58940N (005). [10] A. Vyas, M. B. Roopashree, B. R. Prasad, Performane of entroiding algorithms at low light level onditions in adaptive optis to appear in the Proeedings of the International Conferene on Advanes in Reent Tehnologies in Communiation and Computing, 009. [11] W.H. Southwell, Wave-front estimation from wave-front slope measurements, J. Opt. So. Am. 70, 998 (1980). [1] Walter J. Wild, Edward J. Kibblewhite, and Rodolphe Vuilleumier, "Sparse matrix wave-front estimators for adaptive-optis systems for large ground-based telesopes," Opt. Lett. 0, (1995). [13] B. C. Platt and R. Shak, "History and priniples of shakhartmann wavefront sensing." Journal of refrative surgery (Thorofare, N.J. : 1995), 17, 5, 001. [14] Lawton H. Lee, Gary J. Baker, and Robert S. Benson, "Corretability limitations imposed by plane-wave 010 International Journal of Computer Appliations ( ) sintillation in multionjugate adaptive optis," J. Opt. So. Am. A 3, (006). [15] V. V. Voitsekhovih, L. J. Sanhez, and V. G. Orlov, Effet of sintillation on adaptive opti systems Revista Mexiana de Astronomia y Astrofisia, 38, (00). [16] A. Vyas, M. B. Roopashree, B. R. Prasad, in Proeedings of the International Conferene on Advaned Computing, (ed: Gopinath Ganapathy) (Ma Millan) 303, (009). arxiv: v1 [17] A. Vyas, M. B. Roopashree, B. R. Prasad, in Proeedings of the International Conferene on Advaned Computing, (ed: Gopinath Ganapathy) (Ma Millan) 08, (009). arxiv: v1 [18] K. M. Hosny, Fast omputation of aurate Zernike moments, Journal of Real-Time Image Proessing 3, 97,