2004 January 27 Mathematical Challenges of Using Point Spread Function Analysis Algorithms in Astronomical Imaging Mighell 1

Mathematical Challenges of Using Point Spread Function Analysis

Algorithms in Astronomical Imaging

Kenneth John MighellNational Optical Astronomy Observatory

Mathematical Challenges of Astronomical Imaging January 26-30, 2004Institute for Pure and Applied Mathematics @ University of California, Los Angeles

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Outline

• Motivation• Image Formation Process & Challenges• Performance Model: Photometry• Performance Model: Astrometry• Analytical PSFs• Numerical PSFs• Ugly (real) PSFs• Future Challenges

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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)

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Image Formation Process

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Under-Sampled Point Spread Functions

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original sampling 3x3 supersampled 3x3 supersampled linear stretch linear stretch logarithmic stretch

HST WFPC2: WFC F555W 0.10 arcsecond pixels

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Non-Uniform Pixel Response Function

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Photometric Error: NIC3 F160W

white=0.12 mag excess, black=0.09 mag deficit

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Point Spread Function▼ ▼ Photon Distribution Function

▲ Detector Response Function

Volume ▲

▲

Normalized PSF

standard deviation▲

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parameter vector ▲▲measurement error

▲data ▲model

correction vector ▲

measurement error ▲

▲Hessian matrix

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▲ ▲ ▲ ▲Each column forms an imageEach column forms an image

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Derivatives of the PSF

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Systematic Error: Poor Sky Measurement

sky too lowsky too low

sky too highsky too high

sky just rightsky just right

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▲ ≡ 1 if critically sampled

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Analytical Derivatives

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◄◄550%0%

◄◄99.699.6%%

◄◄7575%%

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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.

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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 The implementation methodology, however, is significantly different.methodology, however, is significantly different. The position partial derivatives of numerical PSFs can be determined using numerical differentiation numerical differentiation techniquestechniques on the numerical PSF. Numerical experiments have shown that the following five-point differentiation formula works well with numerical PSFs:

Numerical Derivatives

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Analytical PSFs:Analytical PSFs: Just compute the PSF at the desired location in the observational model.

Numerical PSFs:Numerical PSFs: Take the reference numerical PSF and shift itshift it to the desired location using a perfect 2-d using a perfect 2-d interpolation functioninterpolation 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?

SolutionSolution Use the following 21-pixel-wide damped sinc function:Use the following 21-pixel-wide damped sinc function:

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Why does this work so well?

Photon NoisePhoton Noise▲and That’s A Good Thing®

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Problem:Problem: Negative values caused by shifting an undersampled PSF Negative values caused by shifting an undersampled PSF

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Solution:Solution: Use Use supersampled supersampled Point Spread FunctionsPoint Spread Functions

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Why did it work with faint stars?

One need lots of photons to properly sample the higher spatial frequencies of a PSF!

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Ugly (real) PSFs?

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James Webb Space Telescope James Webb Space Telescope

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Now consider a realistic IR detector …

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^ 44% of the pixel is ^ 44% of the pixel is dead!dead!

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Non-Uniform Pixel Response Functions

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Excellent IR stellar Excellent IR stellar photometry should be photometry should be possible with accurate possible with accurate knowledge of the Pixel knowledge of the Pixel Response FunctionResponse Function

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Calibration Errors

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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

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This work is supported by a grant from the Applied Information Applied Information Systems Research Program (AISRP)Systems Research Program (AISRP) of the Office of Space Office of Space ScienceScience of the of the National Aeronautics and Space Administration National Aeronautics and Space Administration (NASA)(NASA) [ [NRA 01-OSS-01: Interagency Order No. S-13811-G] .