1. A tunable weighting scheme for imaging extended sources
Ryan A. Loomis (Harvard-Smithsonian CfA)
Collaborators: K. Öberg, E. Price, S. Andrews, and D. Wilner
Harvard-Smithsonian Center for Astrophysics

2. Motivation ALMA Partnership + 2015
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maybe even explicitly mention brigg's thesis chapter (e.g.)
ALMA Partnership
Ryan A. Loomis
CALIM 2016

3. Motivation Andrews + 2016 ----- Meeting Notes (10/6/16 14:59) -----
maybe even explicitly mention brigg's thesis chapter (e.g.)
Andrews
Ryan A. Loomis
CALIM 2016

4. Motivation 50 AU Loomis + 2016 (in prep)
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maybe even explicitly mention brigg's thesis chapter (e.g.)
Loomis (in prep)
Ryan A. Loomis
CALIM 2016

5. Motivation Many resolved sources show sub-structure
SNR limitations may prevent imaging this sub-structure
Tapering increases SNR, but isn’t ideal, can wash out features
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maybe even explicitly mention brigg's thesis chapter (e.g.)
Ansdell
Ryan A. Loomis
CALIM 2016

6. Motivation Uniform weighting SNR robust v2 (FOV x3) SNR = 49 SNR = 87
Ryan A. Loomis
CALIM 2016

7. Definitions
Weights:
Briggs 1995
Ryan A. Loomis
CALIM 2016

8. Definitions Weights: Taper weights Briggs 1995 Ryan A. Loomis
CALIM 2016

9. Definitions Weights: Taper weights Density weights Briggs 1995
Ryan A. Loomis
CALIM 2016

10. Definitions Weights: Taper weights Density weights SNR weights = 1/σk2
Briggs 1995
Ryan A. Loomis
CALIM 2016

11. Definitions Weights: Dirty image: Taper weights Density weights
SNR weights = 1/σk2
Dirty image:
Briggs 1995
Ryan A. Loomis
CALIM 2016

12. Definitions Weights: Dirty image: Variance of the dirty image:
Taper weights
Density weights
SNR weights = 1/σk2
Dirty image:
Variance of the dirty image:
Briggs 1995
Ryan A. Loomis
CALIM 2016

13. Definitions Weights: Dirty image: Variance of the dirty image:
Taper weights
Density weights
SNR weights = 1/σk2
Dirty image:
Variance of the dirty image:
RMS thermal noise:
Briggs 1995
Ryan A. Loomis
CALIM 2016

14. Natural weighting Minimize (ΔID)2 using Lagrangian multiplier:
Thermal noise is lowest if Tk and Dk = 1, as wk = 1/σk2
Briggs 1995
Ryan A. Loomis
CALIM 2016

15. Thermal noise and SNR Normalized RMS noise:
SNR at a given point in the image:
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the weights for natural weighting encode no information
Briggs 1995
Ryan A. Loomis
CALIM 2016

16. ‘Natural’ weighting for a resolved source
Natural weighting is only ideal for a point source
Maximize SNR for an arbitrary source: Vk
TkDkwk /σk2 becomes: TkDkwk Re(Vk)/σk2
Gaussian tapering increases SNR of faint sources because the taper approximates a matched filter
TkDkwk |Vk|/σk2
Briggs 1995
Ryan A. Loomis
CALIM 2016

17. SNR natural weighting vs tapering
FFT
Ryan A. Loomis
CALIM 2016

18. SNR natural weighting vs tapering
FFT
Ryan A. Loomis
CALIM 2016

19. SNR natural weighting vs tapering
FFT
Taper will damp out high frequencies
Ryan A. Loomis
CALIM 2016

20. SNR natural weighting vs tapering
FFT
Ryan A. Loomis
CALIM 2016

21. SNR natural weighting vs tapering
Gaussian taper
Ryan A. Loomis
CALIM 2016

22. SNR natural weighting vs tapering
Ryan A. Loomis
CALIM 2016

23. SNR natural weighting vs tapering
FFT
+ noise
FFT
Ryan A. Loomis
CALIM 2016

24. SNR natural weighting vs tapering
Matched taper
Ryan A. Loomis
CALIM 2016

25. Robust weighting
Find a smooth knob between natural and uniform weighting through minimization
Briggs 1995
Ryan A. Loomis
CALIM 2016

26. Robust weighting
Briggs 1995
Ryan A. Loomis
CALIM 2016

27. Robust weighting
Briggs 1995
Ryan A. Loomis
CALIM 2016

28. Robust weighting
Briggs 1995
Ryan A. Loomis
CALIM 2016

29. A tunable SNR weighting
Can we find a way to connect SNR natural weighting with uniform weighting?
Minimize: ||ID||2 + ΔID / ID
Find that: Wpq /(S2 + 2σpq/|Vpq|)
Analogous to robust weighting, but tunes between uniform and SNR natural weighting
Ryan A. Loomis
CALIM 2016

30. Test case - ring Robust Gaussian taper SNR robust Ryan A. Loomis
CALIM 2016

31. Test case - ring Robust Gaussian taper SNR robust Ryan A. Loomis
CALIM 2016

32. Test case - ring Robust weighting Gaussian taper SNR robust
Ryan A. Loomis
CALIM 2016

33. Test case - ring Robust Gaussian taper SNR robust Ryan A. Loomis
CALIM 2016

34. Test case - disk Robust Gaussian taper SNR robust Ryan A. Loomis
CALIM 2016

35. Test case - disk Robust weighting Gaussian taper SNR robust
Ryan A. Loomis
CALIM 2016

36. Test case - disk Robust Gaussian taper SNR robust Ryan A. Loomis
CALIM 2016

37. Test case – Gaussian ring
Robust
Gaussian taper
SNR robust
Ryan A. Loomis
CALIM 2016

38. Test case – Gaussian ring
Robust weighting
Gaussian taper
SNR robust
Ryan A. Loomis
CALIM 2016

39. Test case – Gaussian ring
Robust
Gaussian taper
SNR robust
Ryan A. Loomis
CALIM 2016

40. Investigating Dk - ring
Robust weighting
Gaussian taper
SNR robust
Ryan A. Loomis
CALIM 2016

41. Investigating Dk – Gauss. ring
Robust weighting
Gaussian taper
SNR robust
Ryan A. Loomis
CALIM 2016

42. New formulation Previously had: Wpq 1/(S2 + 2σpq/|Vpq|)
Can we prevent the fast damping of long baselines?
Previously had: Wpq /(S2 + 2σpq/|Vpq|)
What about: Wpq /(S2 + 2σpq/|Vpq|S /S )
This is just a hunch – I don’t know the proper form or minimization condition. Ideas? 2 2
max
Ryan A. Loomis
CALIM 2016

43. Comparison - ring Robust Gaussian taper SNR robust SNR robust v2
Ryan A. Loomis
CALIM 2016

44. Comparison - ring
SNR robust
SNR robust v2
Ryan A. Loomis
CALIM 2016

45. Comparison - ring Robust Gaussian taper SNR robust SNR robust v2
Ryan A. Loomis
CALIM 2016

46. Comparison – Gaussian ring
Robust
Gaussian taper
SNR robust
SNR robust v2
Ryan A. Loomis
CALIM 2016

47. Comparison – Gaussian ring
SNR robust
SNR robust v2
Ryan A. Loomis
CALIM 2016

48. Comparison – Gaussian ring
Robust
Gaussian taper
SNR robust
SNR robust v2
Ryan A. Loomis
CALIM 2016

49. Synergy with super-uniform weighting
Robust
Gaussian taper
SNR robust
SNR robust v2
SNR robust v2 (2x FOV)
SNR robust v2 (3x FOV)
Ryan A. Loomis
CALIM 2016

50. Disk with multiple rings
Robust
Gaussian taper
SNR robust
SNR robust v2
Ryan A. Loomis
CALIM 2016

51. Disk with multiple rings
SNR robust
SNR robust v2
Ryan A. Loomis
CALIM 2016

52. Disk with multiple rings
Robust
Gaussian taper
SNR robust
SNR robust v2
Ryan A. Loomis
CALIM 2016

53. Disk with multiple rings (imaged)
Dirty image
SNR = 4
Gaussian Taper
SNR robust v2
SNR = 5
SNR = 5.5
Ryan A. Loomis
CALIM 2016

54. Real data example – AA Tau
Robust
Gaussian taper
SNR robust
SNR robust v2
Ryan A. Loomis
CALIM 2016

55. Real data example – AA Tau
SNR robust
SNR robust v2
Ryan A. Loomis
CALIM 2016

56. Real data example – AA Tau
Robust
Gaussian taper
SNR robust
SNR robust v2
Ryan A. Loomis
CALIM 2016

57. Realistic example – disks with rings
Uniform weighting
SNR robust v2 (FOV x3)
SNR = 49
SNR = 87
Ryan A. Loomis
CALIM 2016

58. SNR weighting – potential limitations
Beam pattern is more complicated, no longer close to a Gaussian; nasty shelves
Multi-source and asymmetric sources?
Deconvolution in high dynamic range cases likely more difficult due to shelves
Potentially problematic for line imaging
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potentially problematic for line imaging
Ryan A. Loomis
CALIM 2016

59. Summary and future
SNR weighting provides clear benefits over Gaussian tapering in the low SNR regime
Our tunable weighting scheme can often outperform robust weighting alone
What is the correct metric to minimize?
Implement for CASA?
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remove rings
Ryan A. Loomis
CALIM 2016