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The Formation of Astronomical Images

The Formation of Astronomical Images. Tod R. Lauer. Astronomical Images. A set of points at which a quantity is measured within a space. Pictures, spectra, IFU “data cubes” are all examples of images.

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The Formation of Astronomical Images

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  1. The Formation of AstronomicalImages Tod R. Lauer Tod R. Lauer (NOAO) July 19, 2010

  2. Astronomical Images • A set of points at which a quantity is measured within a space. Pictures, spectra, IFU “data cubes” are all examples of images. • In this talk we will focus on 2-D flux maps with regularly spaced samples, but this is only a subset of possible images. • To understand imaging, you need to understand how an object is represented by its image. Tod R. Lauer (NOAO) July 19, 2010

  3. Image Formation I = O  ( P   )  Ш + N Effective or total PSF Tod R. Lauer (NOAO) July 19, 2010

  4. The Image Source Tod R. Lauer (NOAO) July 19, 2010

  5. The Point Spread Function Tod R. Lauer (NOAO) July 19, 2010

  6. The Observed Image Tod R. Lauer (NOAO) July 19, 2010

  7. The Fourier Transform • The FT decomposes an image into a set of waves, with variable amplitude, phase, and direction. • This allows information of different spatial scales to be isolated, analyzed, and processed. • Convolution in Images space is multiplication in Fourier (and vice-versa). • Complete symmetry and complementarity with Space. • Reciprocal Fourier/Image space scale relationship. • Critical for understanding resolution, filtering, sampling, and on and on… Tod R. Lauer (NOAO) July 19, 2010

  8. The Fourier Transform Tod R. Lauer (NOAO) July 19, 2010

  9. The Fourier Domain Perspective Tod R. Lauer (NOAO) July 19, 2010

  10. And then add noise… Tod R. Lauer (NOAO) July 19, 2010

  11. Noise in the Fourier Domain Tod R. Lauer (NOAO) July 19, 2010

  12. Understanding Noise • Noise sets the limit on the photometric and structural information recovered from an image. • Noise limits the spatial resolution of features. Noise can mimic fine-scale features. • Separating noise from signal is the major task of image analysis. Tod R. Lauer (NOAO) July 19, 2010

  13. Filtering Noise Tod R. Lauer (NOAO) July 19, 2010

  14. Filtering in the Fourier Domain Tod R. Lauer (NOAO) July 19, 2010

  15. Filters and Measures • Filters isolate signal from noise, specific signals from other signals, interpolate images, etc. • All measurements from an image can be regarded as forms of filters. • Good filtering means understanding exactly how both noise and signal respond to the filter. Tod R. Lauer (NOAO) July 19, 2010

  16. Image Sampling • Accuracy of photometry, astrometry, etc. depends on good sampling. • Where possible, strive for “Nyquist-sampling,”which requires sampling at 2 highest spatial frequency present in image. • Roughly-speaking - two pixels per PSF FWHM. • No need to oversample! Harry Nyquist Tod R. Lauer (NOAO) July 19, 2010

  17. Sampling Tod R. Lauer (NOAO) July 19, 2010

  18. Sampling • Sampling draws values at discrete points from a continuous function - this includes the pixel kernel. • Samples = data pixels are pure delta functions. Distinguish data pixels from detector and display pixels. • Under-sampling beats sampling frequency against spatial frequencies, aliasing them as bogus lower or higher spatial frequency power. • Well-sampled data can be measured, interpolated, recast, etc. without resolution or photometric losses. • Under-sampled data contain intrinsic photometric errors and cannot be resampled or interpolated without incurring additional signal degradation. Tod R. Lauer (NOAO) July 19, 2010

  19. The Sinc, or Interpolating-Function • The assumption that the image is well-sampled and continuous implies the use of sinc(x). A “cutoff” box in the Fourier domain is sinc(x) in the image space. • Sinc(x) interpolates with no loss of resolution, smoothing, etc. • Any other interpolator is ad hoc. • Sinc is sensitive to artifacts, thus well-reduced images are required. • Sinc(x,y) is separable - it results from multiplying to 1-D functions. • Taper as needed. Tod R. Lauer (NOAO) July 19, 2010

  20. Display vs. Data Pixels Tod R. Lauer (NOAO) July 19, 2010

  21. Detector Pixel Shapes Tod R. Lauer (NOAO) July 19, 2010

  22. Aliasing in the Fourier Domain Well-Sampled Under-Sampled - Satellites Overlap Tod R. Lauer (NOAO) July 19, 2010

  23. 3X3 Sub-Sampled F555W WFC PSF Single Native PSF Tod R. Lauer (NOAO) July 19, 2010

  24. Fourier-Space 3X3 F555W WFC Image reconstruction Tod R. Lauer (NOAO) July 19, 2010

  25. Photometric Errors Due to Undersampling NIC3 J-Band Tod R. Lauer (NOAO) July 19, 2010

  26. Dithering to Fix Undersampled Data • Many cameras with large pixels produce undersampled data. • The pixel + PSF sets the resolution. • Shifting the camera by sub-pixel amounts recovers Nyquist-sampling given optical PSF + pixel kernal = total or effective PSF. • Dithering now standard on HST and other spacecraft - does require PSF stability over dither sequence. • Distinguish this dithering from large-scale dithering to mitigate detector defects and sensitivity variations. Tod R. Lauer (NOAO) July 19, 2010

  27. Dithering Tod R. Lauer (NOAO) July 19, 2010

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