1. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 1 Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical Imaging Kenneth John Mighell National Optical Astronomy Observatory Mathematical Challenges of Astronomical Imaging January 26-30, 2004 Institute for Pure and Applied Mathematics @ University of California, Los Angeles

2. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 2 Outline Motivation Image Formation Process & Challenges Performance Model: Photometry Performance Model: Astrometry Analytical PSFs Numerical PSFs Ugly (real) PSFs Future Challenges

3. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 3

4. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 4

5. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 5

6. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 6

7. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 7 Skill Sets for Precision Stellar Photometry and Astrometry Image Analysis (how bright is the star? where is it located?) Error Analysis (what is the error of those measurements?) Numerical Analysis (non-linear least-squares minimization) Computational Methods (precise & robust computations) Signal Processing –Low Signal vs. High Noise (finding the needle in the haystack) Detector Technology –Solid-State Physics (conversion process of photons to electrons) –Fabrication Techniques (how detectors made and read) Optics (blurring due to telescope & camera optics) Atmospheric Physics –Turbulence Theory (blurring due to the atmosphere) Stellar Physics (photon statistics & pulsation theory)

8. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 8 Image Formation Process Fruchter & Hook 2000 (ADASS IX) SKY  TELESCOPE  DETECTOR WFPC2

9. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 9 Under-Sampled Point Spread Functions Lauer 1999, PASP, 111, 1434 1.7 arcsecond original sampling 3x3 supersampled 3x3 supersampled linear stretch linear stretch logarithmic stretch HST WFPC2: WFC F555W 0.10 arcsecond pixels

10. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 10 Non-Uniform Pixel Response Function Fruchter & Hook 2000 (ADASS IX)Lauer 1999, PASP, 111, 1434 Photometric Error: NIC3 F160W white=0.12 mag excess, black=0.09 mag deficit

11. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 11 Point Spread Function▼▼ Photon Distribution Function ▲ Detector Response Function Volume ▲ ▲ Normalized PSF standard deviation▲

12. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 12 parameter vector ▲ ▲measurement error ▲data▲model correction vector ▲ measurement error ▲ ▲Hessian matrix

13. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 13

14. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 14

15. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 15 ▲ ▲ Each column forms an image

16. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 16 Derivatives of the PSF

17. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 17

18. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 18

19. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 19 Systematic Error: Poor Sky Measurement sky too low sky too high sky just right

20. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 20

21. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 21

22. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 22

23. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 23 ▲ ≡ 1 if critically sampled

24. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 24

25. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 25

26. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 26 Analytical Derivatives

27. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 27 ◄50% ◄99.6% ◄75%

28. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 28

29. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 29 Why does this work so well? We analyzed the volume integral of the photon distribution function rather than sample it in the middle of a pixel. But remember that moderation in all things is good. Failure is inevitable with extreme undersampling when all of the light from a star can fall within a single pixel; photometry gets diluted and astrometry suffers as information about the location of the star within the pixel is lost.

30. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 30 The implementation methodology, however, is significantly different. numerical differentiation techniques The mathematics of determining the position partial derivatives of the observational model with respect to the x and y direction vectors is exactly the same with analytical or numerical PSFs. The implementation methodology, however, is significantly different. The position partial derivatives of numerical PSFs can be determined using numerical differentiation techniques on the numerical PSF. Numerical experiments have shown that the following five-point differentiation formula works well with numerical PSFs: Numerical Derivatives

31. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 31 Analytical PSFs: Analytical PSFs: Just compute the PSF at the desired location in the observational model. Numerical PSFs:shift itusing a perfect 2-d interpolation function Numerical PSFs: Take the reference numerical PSF and shift it to the desired location using a perfect 2-d interpolation function. OK… but how is that done in practice? Note: The 2-d sinc function is separable in x and y. How does one move a PSF? Solution Use the following 21-pixel-wide damped sinc function: Use the following 21-pixel-wide damped sinc function:

32. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 32

33. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 33 Why does this work so well? Photon Noise ▲and That’s A Good Thing®

34. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 34

35. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 35 Problem: Negative values caused by shifting an undersampled PSF

36. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 36 Solution: Use supersampled Point Spread Functions

37. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 37 Why did it work with faint stars? One need lots of photons to properly sample the higher spatial frequencies of a PSF!

38. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 38 Ugly (real) PSFs?

39. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 39 James Webb Space Telescope

40. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 40 Next Generation Space Telescope 8-m TRW-concept 8-m TRW-concept 0.0128 arcsec/pixel  =31.05 PSF by John Krist contours: 90%, 50%, 10%, 1%, 0.1% of peak

41. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 41

42. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 42 Now consider a realistic IR detector …

43. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 43 ^ 44% of the pixel is dead! Note: The dead zone is centered at (0.625,0.625) Non-Uniform Pixel Response Functions

44. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 44 Excellent IR stellar photometry should be possible with accurate knowledge of the Pixel Response Function

45. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 45 Calibration Errors

46. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 46

47. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 47 Challenges of PSF Extraction highly variable PSF within the field of view too few stars which are not bright enough from undersampled observations that are poorly dithered with –significant Charge Transfer Efficiency variations –variable diffusion –loss of photons due to charge leakage and may possibly be nonlinear at <1% level

48. 2004 January 27Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical ImagingMighell 48 Applied Information Systems Research Program (AISRP)Office of Space Science of the National Aeronautics and Space Administration (NASA) [ This work is supported by a grant from the Applied Information Systems Research Program (AISRP) of the Office of Space Science of the National Aeronautics and Space Administration (NASA) [NRA 01-OSS-01: Interagency Order No. S-13811-G].