Improving Ground Based Telescope Focus through Joint Parameter Estimation. Maj J. Chris Zingarelli USAF AFIT/ENG

Transcription

1 Iproving Ground Based Telescope Focus through Joint Paraeter Estiation Maj J Chris Zingarelli USAF AFIT/ENG Lt Col Travis Blake DARPA/TTO - Space Systes Dr Stephen Cain USAF AFIT/ENG Abstract-- Space Surveillance Telescope (SST) is a Defense Advanced Research Projects Agency (DARPA) progra designed to facilitate the detection of space debris in earth s orbit In order to achieve optial perforance, focusing of the can be conducted by retrieving phase inforation in the iage to deterine the aount of defocus and then oving the irrors aial to shift the focal point One of its unique features is that operates with a echanical shutter that s speed restricts the to collecting long eposure iagery Long eposure iagery s consequently averages the atosphere, which creates a point spread function (PSF) which can iic one that contains fied aberrations such as focus and spherical error The average atosphere asks the static aberrations of the in the iage affecting the ability to achieve an optial focus This paper will eplore the joint estiation of the focus and the atospheric seeing paraeter The Craer-Rao lower bounds for variance are coputed to facilitate an understanding of the joint estiation proble These bounds will serve to deonstrate how the average atospheric transfer function akes sensing a focus error ore difficult in the presence of noise The views epressed are those of the author and do not reflect the official policy or position of the Departent of Defense or the US Governent INTRODUCTION The Departent of Defense recently fielded an f/ Mersenne-Schidt called the Space Surveillance Telescope (SST) that advances the United States space situational awareness (SSA) capability SSA directly supports the stated US National Space Policy to (p)reserve the Space Environent the United States shall develop, aintain, and use space situational awareness inforation fro coercial, civil, and national security sources to detect, identify, and attribute actions in space that are contrary to responsible use and the long-ter sustainability of the space environent [] The critical technology that enables SST s 6 deg field of view (FOV) caera is its unique curved charged coupled device (CCD) shown in Fig A high speed shutter was developed for the caera with a iniu eposure tie of s, considerably longer than the s typically associated with a short eposure iage [] Fig SST s 6 deg field of view caera and high speed echanical shutter

2 Currently, one of the ain challenges to optiizing SSTs perforance is to reduce the point spread function (PSF) through focus and alignent The piels in the CCD are however, the piels canbe being binned due to the epected atospheric blurring Fig shows the variations of full width half aiu (FWHM) blur spot in ters of 3 binned piels over 3 onths Fig Variations in the Full Width Half Maiu (FWHM) of SST s Point Spread Function (PSF) easured by binned piels Ideally, the PSF FWHM would be within one 3 binned piel Reducing the PSF is possible by accurately deterining the aount of focus error (and other aberrations) in the iage of a calibration star then adjusting the focus and alignent to reduce the blur spot size However due to the long eposures, to find unbiased estiates of the aberrations fro the star iage the atospheric effects ust be included in the odel TELESCOPE MODEL The odel developed in this section follows a siilar odel eployed for analysis of the Hubble Space Telescope [3] The is considered a linear shift invariant syste where the ipulse response of the syste will be the PSF The light propagating fro the distance point source (ie a star) is assued to be teporally incoherent Therefore, according to Goodan, the iage irradiance in the detector plane, i, is the convolution of point source irradiance, ( ), with s PSF [] The paraeters used in the odel are listed in Tab Tab Telescope Paraeters Model Paraeter Value Center Wavelength n Telescope Pupil/Obscuration Diaeter 3/8 Telescope Effective Focal Length 3 CCD Piel Pitch µ Star Irradiance per Frae ~^ photons Background Irradiance per Frae 3 photons Grid Size ^,, of the is defined by its annular aperture shown in Fig 3a, where u and vare coordinates in the pupil plane Wave front error caused by defocus is introduced into the pupil function using the Zernike polynoial for defocus, [] The pupil function, Au v () ( u, v ) 36( u v ) 73

3 waves Noralized intensity The aount of focus error is captured by scaling ( uv, ) with a Zernike coefficient for defocus, Z error( u, v) Z ( u, v) () An iage of ( uv, ) scaled by a wave coefficient is shown in Fig 3b The aberrations are then represented in the pupil plane, u, by the generalized pupil function u Au j error u P ep (3) The PSF is then coputed as h P u e opt u P j u u e u e j u ju u u P jz u ju jz u ju u u * H H H, A u e e A u e e () where, H is the field in the detector plane [] The s PSF with waves of defocus is shown in Fig 3c where is a piel coordinate in the detector plane The large aount of defocus causes the PSF to have an annular shape - A(u ) (u ) h opt () 3 (a) Fig 3 (a) Pupil function used to odel SST (b) Zernike polynoial for defocus with Z waves (c) Telescopes PSF with Z waves of focus error (b) (c) For a ore coplete odel odel, the effect of the finite square piels, a, is included in the PSF where the transfer function for the piels and respectively are represents as the following digital Fourier transfors, F, H piel u F rect( a), and () u hopt Hopt F ( ), therefore (6) h F H u H u (7) opt piel Iages observed by SST have been easured to be shot noise doinated, so the iage data, d ( ), is considered to be Poisson and has a ean value that is equal to the irradiance of light in that piel []

4 i The odel for the iage irradiance centered on the optical ais is E d (8) (9) i ( ) h ( ) B It includes additional ters to account for the background light, B, and the total photons eitted fro the star per integration tie, The joint distribution of the iage data is represented by the Poisson probability ass function, the associated log likelihood equation is i d e i Pd, () d ( )! ln Pd L Z i dln i ln d! () 3 CRAMER-RAO LOWER BOUND (CRLB) FOR VARIANCE The Craer-Rao lower bounds provides a theoretical lower liit of variance for estiates of the Zernike coefficient for defocus, Ẑ [3] The bounds in Fig illustrate that standard deviations of Ẑ on the order of waves are possible even in the presence of long eposure atosphere To deterine the CRLB for estiates of the Zernike coefficient for defocus the Fisher inforation, J( Z), is coputed via the following calculation [6] L Z J ( Z) E, Z () where, the CRLB for the variance of Ẑ is defined as var( Zˆ ) J Z (3) The first and second derivative of the log likelihood function Eq () respectively are The resulting Fisher inforation is d h L Z, and () Z i Z h h L Z d d () Z i Z i Z J Z The derivative of the PSF with respect defocus is L Z h ( E ) Z i Z (6)

5 * h H H * H H piel u Z Z Z The derivative of the wavefront in the detector plane with respect to (wrt) Z is F H (7) H Z u jz u j u j u A u e e jz u jf u A u e (8) Thus, recalling that for arbitrary variables a and b; j a jb a jb b I a jb derivative of the PSF to be leading to the first h Z jz u jz u * j -F u A u e H F u A u e H F H piel u jz u * I F u A u e H F H piel u (9) The resulting Fisher inforation for the optical syste containing focus error is u * F F H piel () I jz J Z u A u e H i Z where JZ is plotted as the purple line with star data arkers in Fig Because SST uses a shutter, with an integration tie greater than s, an accepted odel for that atosphere is a long-eposure atospheric transfer function, which given by Goodan as [] fu 3 H at u ep 3 () r In Eq () is the ean wavelength, f is the focal length, and r is the atospheric seeing paraeter The total PSF is then coputed as h F H u H u H u () total opt piel at Saples of the three transfer functions are shown in Fig (a-c) to illustrate how the piels and atosphere reduce special frequency content of the diffraction liited s optical transfer function and due to it Fourier transfor relationship broaden the PSF The -ais is shown in ters of the spatial frequency, u, divided by the cutoff frequency,, u for the annular pupil function As the focus error increases H increasingly liit the spatial resolution of the further broadening the PSF opt u begins to

6 H opt H piel H at - u /u - u /u - u /u (a) (b) (c) Fig (a) Telescope Models Optical Transfer Function (OTF) with Z (b) Piels Transfer Function (c) Atospheric Transfer Function with r 8c By including the effects of the atosphere in the PSF, the iage intensity odel in Eq (9) becoes The eleents of the Fisher inforation atri, total (3) i ( ) h ( ) B total J Z J Z, r, I J Z r J r () are calculated in order to deterine the CRLB for variance of Ẑ in the presence of an average atosphere Using the log likelihood function in Eq () and taking the second derivative of Eq () wrt r & Z L Z, r d htotal d htotal htotal, () Zr itotal Zr itotal Z r L Z, r d htotal d htotal, and r itotal r itotal r (6), L Z r d htotal d htotal (7) Z itotal Z itotal Z Because E d i total the eleents of the Fisher inforation atri are L Z, r htotal htotal J Z, r E, (8) Zr itotal z r J r L Z htotal E r itotal r J Z, and (9) L Z htotal E Z itotal z (3) Then the derivatives of the PSF are

7 CRLB * htotal H H * F F H H H ath Z Z Z jz u * F F I F u A u e H H ath piel, and htotal 73 F H H H 8 r r 3 3 optu3 piel at piel (3) The CRLB for variance is coputed by inverting the Fisher inforation atri var( Zˆ ) cov( Zˆ, r ) I cov( ˆ Z, r ) var( r ) (3) The resulting CRLB for the standard deviation, CRLB, of Ẑ is plotted in Fig for cases with and without an average atosphere present Atospheric turbulence increases CRLB and as r decrease the effect of the atosphere on the bound increase In addition as Z decrease the lower bound increases Therefore, estiation of Z should becoe ore inaccurate as the aount of defocus decreases In addition, as r decreases the CRLB uniforly increases indicating that ore turbulent atospheres will increase the variance on estiates of defocus r = 8 [c] r = [c] r = [c] No At Z Fig Craer-Rao Lower Bound of the standard deviation, CRLB, of estiates for the defocus paraeter, Zˆ CRLB are shown for cases with no atosphere in the odel and increasing atospheric seeing by changing r STAR SIMULATIONS Stars were siulated as syste ipulses, ( ), and then the effects of the atosphere,, defocus, piilation, background light and star intensity were introduced using Eq (3) The analog iages of a star with and without atospheric effects are shown in Fig 6 (a) & (b) The piilated iages of those sae stars are picture in Fig 6 (c) & (d) Shot noise is siulated in the stars using a Poisson rando nuber generator the training data, d ( ), evaluate the perforance of the estiators describes in Sec 8

8 Photons Photons Photons Photons Z = 8; No at Z = 8; r = 8c = (a) = (b) 3 3 Z = 8; No at 3 3 = (c) Fig 6 (a) Siulated analogue iage of a star with Z 8 waves and no atosphere (b) Siulated analog iage of a star with Z 8 waves and an average atosphere where r 8 c (c) Siulated digital iage of a star with Z 8 waves and no atosphere (d) Siulated digital iage of a star with Z 8 waves and an average atosphere where r 8 c PARAMETER ESTIMATION The ethod of least squares (LS) estiation was used to estiate the Zernike coefficient for defocus, Ẑ fro the siulated star data The LS ethod is used because of coputer precision challenges encountered in the aiu likelihood estiation approach due to the large background noise levels inside the log-likelihood function In addition, the LS ethod does not require any paraetric assuptions aking the estiator ore robust to changes in the noise statistics [] The intensity odels fro Eqs (9) & (3) are used to define the su of squares 3 3 Z = 8; r = 8c 3 = (d) Q i ( ) h ( ) B, and (33) Qtotal itotal ( ) htotal ( ) B (3) In order to estiate defocus, the unknown photons per iage,, ust also be estiated fro the data by take the derivative of Eq (33) and (3) and setting the equal to zero to get the generalized function i Bh h (3) Deterining Ẑ without accounting for the atosphere is found by the single paraeter estiate Z Zˆ arg in Q, (36)

9 Estiated Z Estiated Z FWHM PSF [piels] or by accounting for the atosphere with the joint estiator Zˆ arg in Qtotal rˆ Z, r (37) By conducting a nuerical grid search of realistic values for Z the paraeter estiates that the gives the LS solutions are nuerically deterined Finding Ẑ for ultiple iage fraes of siulated star data the results are used to deterine the saple ean and variance for the LS estiator To produce the plot in Fig 7a, training data, d ( ), is generated without shot noise and with focus errors ranging fro 3- waves in order to deterine the estiators biases Estiates of defocus using Eq (36) are ade on siulated stars with and without an average atosphere present The graph shows that when the siulated star data has an average atosphere, the single paraeter estiator has a defocus dependent bias In contrast, the results of joint estiator, Eq (37), on the sae siulated star data with an average atosphere present does not have a significant bias The joint estiator is used to estiate the defocus fro training data containing shot noise with the ean and standard deviation plotted in Fig 7b As the aount of defocus decreases the standard deviation increases significantly due to the narrowing of the PSF As the blur spot narrows less of the shape of the PSF can be discerned fro the star iages affecting the accuracy of the estiates of Z The saple standard deviation, s, of the joint paraeter estiate also plotted with their associated CRLB in Fig 7c The CRLB is not achieved by but, the standard deviation is below a wave until the blur spot becoes too sall s Single Est; No At Single Est; r = 8c Joint Est; r = 8c True Z E[Z ] PSF True Z CRLB s True Z (a) (b) (c) Fig 7 (a) The estiated defocus paraeter is deterined fro siulated star data with no noise present The blue X arks the single paraeter estiate for defocus without an atosphere present in the siulated star data The green circles are the single paraeter estiates of Z where r 8c in the siulated star data The red boes are the joint paraeter estiate of Z where r 8c in the siulated star data (b) The joint paraeter estiates fro the siulated stars with shot noise EZˆ and s are represented as blue dots and plotted as a function of the defocus The FWHM of the PSFs are plotted with the green asterisks as a function of defocus (c) The joint paraeter estiates are plotted as blue dots and the CRLB as red stars as a function of star defocus s

10 6 CONCLUSIONS The fact that the atosphere can affect estiation of the coefficient for the defocus polynoial deonstrates the need to account for the effects atosphere seeing in order to accurately estiate SSTs aberrations fro star iages Joint estiation of the atospheric seeing paraeter for the long eposure atosphere s and the coefficient for the defocus polynoial deonstrates an effective way to account for the atosphere when estiating aberrations fro iagery data Results of this work are proising in that there is no indication thus far that joint estiation of the Zernike coefficients for the static aberrations in SST can t be deterined and/or alleviated 7 FUTURE WORK The net steps in this research will be to epand the joint estiator to include the other aberrations present in SST data In addition, laboratory eperients can be conducted by iaging point sources in a diffraction liited then defocusing the and adding atospheric turbulence to further show that the algoriths developed for joint estiations of Z work Ultiate validation of the work will be to sharpen the SST caera iages by reducing the aberrations in the using the basic techniques developed herein 8 ACKNOLEGMENTS This aterial is based upon work sponsored by DARPA In addition, MIT Lincoln Laboratory inputs were instruental in proble identification MIT also provided the SST sensor subsyste iages and PSF chart in Sec 9 WORKS CITED [] U S Govnernent, "NATIONAL SPACE POLICY of the UNITED STATES of AMERICA," United States Govnernent, Washington DC, [] J W Goodan, Statistical Optics, New York: Wiley Interscience, 98 [3] J M J S T a S J Fienup, "Hubble Space Telescope characterized by using phase-retrieval algoriths," Applied Optics, vol 3, no, p 77767, 993 [] J W Goodan, Fourier Optics, Greenwood Village, CO: Roberts & Copany, [] R Noll, "Zernike polynoials and atospheric turbulence," J Opt Soc A, vol 66, pp 7-, 976 [6] S M Kay, Fundaentals of Statistical Signal Processing, Upper Saddle River, NJ: Prentice-Hall, [7] P H Kva and B Vidakovis, Nonparaetic Statistics with Apllications to Science and Engineering, Hoboken: John Wiley & Sons, Inc, 7