1. Details: Gridding, Weight Functions, the W-term
Fundamentals of Radio Interferometry (Sections 5.3, 5.4, 5.5)
Griffin Foster
SKA SA/Rhodes University
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2. Array Visibility Sampling
Visibilities are regularly sampled in time, but not in the UVW-space (irregular sampling).
The Fourier transform of irregularly sampled data requires use of the generic Discrete Fourier Transform (DFT).
For an N x N image from M (M ~ N2) visibility samples this requires N2 x M computational operations.
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3. Array Visibility Sampling
The Fourier transform of regularly sampled data can be done with a Fast Fourier Transform (FFT).
In order to transform an irregularly sampled signal to regularly sampling a gridding operation must be performed.
For an N x N image from N x N visibility samples this requires 2N log (N) computational operations.
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4. VLA Antenna Layout
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5. VLA UV Coverage
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6. VLA UV Coverage
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7. VLA UV Coverage
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8. VLA UV Coverage
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9. VLA UV Coverage
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10. Mapping Irregular Sampling to a Regular Grid
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11. Mapping Irregular Sampling to a Regular Grid
Continuous Visibility Function
Array Sampling Function
Gridding Convolution Kernel
Gridding Operator (Shah)
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12. De-gridding De-gridding Convolution Kernel Array Sampling Function
Gridded Visibilities
De-gridding Convolution Kernel
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13. Mapping Irregular Sampling to a Regular Grid
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14. Image Size and Resolution
Given the the resolution
and the desired field of view
the number of pixels in the image (the image size) is
For a given image resolution and image size the uv domain resolution/grid size is:
And the number of pixels is unchanged Nu = Nl, Nv = Nm.
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15. Anti-Aliasing Filter Effects
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16. Anti-Aliasing Filter Effects
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17. Window Functions
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18. The Dirty Image
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19. The Dirty Image
For simplicity, let us assume the gridding operation is perfect, the regularly sampled visibilities VS are then the product of the observed visibility function Vobs and the array sampling function S.
Using the van Cittert-Zernike approximate relation, the dirty image ID is the Fourier transform of the observed visibilities
By the convolution theorem
The Fourier transform of the sampling function is the point spread function (PSF) and the Fourier transform of the visibility function is the true sky image.
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20. KAT-7 Sampling Function
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21. KAT-7 Sampling Function
What happens when a position in the visibility space is sampled multiple times during an observation?
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22. Weighting Function: Generalization of the Sampling Function
R (reliability): reliability of a given sample
T (taper): matched filter function
D (density): normalization of redundant sampling
S (array sampling): uv-coverage sampling
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23. 1-D Weight Functions
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24. Natural Weighting: maximizes sensitivity at the cost of resolution.
Density Functions
Natural Weighting: maximizes sensitivity at the cost of resolution.
Uniform Weighting: maximizes resolution at the cost of sensitivity.
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25. Weight Example: True Sky
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26. Weight Example: Natural and Uniform Weighting
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27. Weight Example: Natural and Uniform Weighting
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28. Robust (Briggs') Weighting
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29. Taper Functions
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30. Matched Filtering
Given a known signal in noise, in order to maximize the signal to noise of the signal a matched filter is used.
The matched filter is the time-reverse of the input signal (in 1-D), this amounts to a cross-correlation in all dimensions.
SNR Maximized Signal
Noisy signal
Matched Filter
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31. Matched Filters
True Signal
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32. True Signal in a Noisy System
Matched Filters
True Signal in a Noisy System
Matched Filter
Cross-Correlation
Filtered Signal
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33. True Signal in a Noisy System
Un-Matched Filters
True Signal in a Noisy System
Un-Matched Filter
Cross-Correlation
Filtered Signal
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34. Matched Filters
True Signal
Noisy Signal
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35. Matched Filters Filter Filter Filter Filtered Signal Filtered Signal
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36. NASSP 2016

37. The more complete van Cittert-Zernike Theorem
Our 2-D Fourier relation between the visibility function and the sky is an approximate form of the van Cittert-Zernike theorem:
A more complete version of van Cittert-Zernike is:
Unfortunately, this is not a 2-D Fourier relation.
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38. Approximations to reduce to a 2D Fourier transform
Small Angle/Narrow Field of View Approximation:
From positional astronomy, the sky instensity is distributed on the celestial (unit) sphere such that for any position (l,m,n):
If we are only interested in a small area on the sky then the extant of (l,m) is small:
Then the van Cittert-Zernike theorem can be approximated as:
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39. Approximations to reduce to a 2D Fourier transform
Delay term/w-term Approximation:
If our visibility sampling is approximately on a plane, or again we are only interested in a narrow field of view then the so-called w-term
can be seen as a constant delay term, with a simple correction w=0.
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40. W-term Approximation
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41. Simple Array Layouts
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42. Simple Array Layouts
An array is coplanar if there exists (approximately) a 2-D plane all visibility measurement lie on in the 3-D visibility space.
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43. Coplanar W-term
For a coplanar array there exists a w-plane in which (l, m) coordinates can be linearly transformed to (l', m')
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44. Projection Effects
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45. In the small angle approximation we can Taylor expand the w-term:
Non-coplanar Sampling
Unfortunately, few modern arrays are coplanar...so approximation become the name of the game.
In the small angle approximation we can Taylor expand the w-term:
Change in Phase →
Change in Position
Phase factors
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46. W-Term Effects: Uncorrected
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47. W-Term Effects: Corrected
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48. Correcting Non-coplanar Wide-field Effects
Full 3D Fourier Transform: the 2D Fourier transform is used for computational efficiency, but a 3D transform is a 'more correct' transform.
Snapshot Imaging: when imaging for short time periods a non-coplanar array can be well approximated as coplanar.
Facet Imaging: form many small, narrow field images and stitch them together to form a single wide field image.
W-Projection: apply convolutional filters to the visibilities to project down to a single w-plane.
W-Stacking: form multiple images at different w-planes and stack the resulting images in the image domain.
See Section 5.5
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