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Quantifiying Doubt and Confidence in Image "Deconvolution"
Enhanced Imaging Multiscale Methods for Diffuse Emission and/or Model Fitting
A. Connors, D. van Dyk, J.Chiang, A. Roy, R. Izem, CHASC; (D. Esch, M.Karovska)


What's the Problem ?
True Sky: Model Sky:

Sources: Point; Compact or Broad Diffuse (SNR, Clouds); Other (Dark Matter??)

5

10

15 20 25 30


Real Instrument:

Model Instrument:

Instruments: Exposure; Effective Area (e.g. ARF) ; Spatial Response (e.g. PSF); Energy Response (e.g. RMF) ... (All with Calibration Uncertainties)


* Data: Poisson, Other complications :
Real Data, Exposure:

Model Expected R ate:

0

2

4

6

8

10

100000

20000

CGRO/EGRET Data from HEASARC; Model from GALPROP collaboration

5

10

15 20 25 30


What's the Problem? Looks Easy!
1. 2. 3. 4. 5. MODEL the SKY MODEL the INSTRUMENT RESPONSE MODEL any BACKGROUND PREDICT the EXPECTED RATE COMPARE EXPECTED RATE to DATA: Search through parameter-space Using Likelihood framework 6. PROBLEMS: "Unknown" (i.e. no physics model) components; What's a "Good-Enough-Fit" (e.g. Poisson)? Correlations among (say) neighboring pixels Significance of unknown (maybe irregular) features? Searching larger and larger parameter spaces


Our Solution - Plain but Tedious!
1. We CRACKED the GOODNESS-OF-FIT, etc . (for Poisson, but method can work for any distribution). 2. We MODEL a MISMATCH (between data and expected rate; analogous to a residual) with a FLEXIBLE MODEL (like Multi-scale) 3. We FIT enhanced model (physics-model+flexible-model) 3.1 Using MCMC to explore the parmeter space; 3.2 Using MEAN as the "best estimate" 3.3 (Allows for calibration uncer tainty see H. Lee poster 41.15)


Our Solution - Plain but Tedious, 2:
4. We QUANTIFY the MISMATCH and get SIGNIFIC ANCE of UNKNOWN FEATURES by: 4.1 ANALYZE the INTERESTING (usually, real) DATA 4.2 SIMULATING samples from the NULL (physics-model) 4.3 ANALYZE SIMULATIONS from NULL: I.E. in exactly the same way as for the INTERESTING DATA 4.4 NOW one has many MCMC SAMPLES of each. Use a few key SUMMARIES like: TOTAL COUNTS in flexible Multi-Scale component; Norm, or SCALE FACTOR, of physics-based model NOTE: Because of intrinsic correlations among pixels inherent in most multi-scale or flexible models , pixel-by-pixel significances won't work in a simple way.


Our Solution - Plain but Tedious, 3:
5. COMPARE EXPECTED RATE to DATA: Search parameter-space; Using Likelihood framework 5.1 We COMPARE the distributions of the SUMMARIES (e.g.Total MS Counts; scale factor) for NULL and INTERESTING Data. What is the probability of `overlap' ? THIS IS OUR GOODNESS-OF-FIT TEST. 5.2 We RANK these SUMMARY STATISTICS. We take SLICES through the tails to get (say) the +/-5% limits on flux and position. THIS IS OUR FEATURE SIGNIFICANCE TEST. 5.3 PROBLEM: Currently takes a lot of time to do this nicely (On the order of a day for a 128x128 sky image, on G4).


ASIDE: MULTI-SCALE FOR POISSON- called "Multiplicative Multiscale Innovation Models"

MMI Slides courtesy of R.Willett, SAMSI 2006
Timmermann & Nowak, 1999 Kolaczyk, 1999


ASIDE: MULTI-SCALE FOR POISSON- called "Multiplicative Multiscale Innovation Models"

MMI Slides courtesy of R.Willett, SAMSI 2006
Timmermann & Nowak, 1999 Kolaczyk, 1999


ASIDE: MULTI-SCALE FOR POISSON- called "Multiplicative Multiscale Innovation Models"

MMI Slides courtesy of R.Willett, SAMSI 2006
Timmermann & Nowak, 1999 Kolaczyk, 1999


ASIDE: MULTI-SCALE FOR POISSON- called "Multiplicative Multiscale Innovation Models"

MMI Slides courtesy of R.Willett, SAMSI 2006
Timmermann & Nowak, 1999 Kolaczyk, 1999


ASIDE: MULTI-SCALE FOR POISSON- called "Multiplicative Multiscale Innovation Models" X
0,0,0 0,0,0

(0)

(1)

(2)

(3)

1,0,0 X

1,0,0

1,1,0 X

1,1,0

1,0,1 X

1,0,1

1,1,1 X

1,1,1

MMI Slides courtesy of R.Willett, SAMSI 2006
Timmermann & Nowak, 1999 Kolaczyk, 1999

· Recursively subdivide image into squares · Let {} denote the ratio between child and parent intensities · Knowing {} Knowing {} · Estimate {} from empirical estimates based on counts in each partition square


REMINDER OF TRICKS: (Using our MMI, Enhanced EMC2) * Match Models to Physics: Multiply, Add; SO Quantify Difference: Multiscale + Scale-Factor*(Null) * Get Uncertainties by Embedding in MCMC; SO Many Samples of Images * Compare to Null Simulations: Low-Dim (2+) Summar y


imulation Study #1: ULL - Model Only
[of MultiScale Results]

[Simulated from Model]


Nothing (Null Hypothesis Log10(Baseline Scale Factor)
+ ++ ++ ++ + + + ++ ++ ++ + ++ + +++++++++ + + + +++ ++ ++++++++++++ + + +++ ++++ ++ + + + + + ++ ++ + + + + ++ +++ + ++++++++++++++++++ + +++++++++++ ++ + ++++++++++++ + +++++++++++++++++ + + ++ ++++++ + + ++++++++++++++++++++ + + ++++++++++++++++++ ++ + + ++++++++++++++++++ + + ++++++++++++++++ + + + +++++++++++++++ + + ++ +++++++++++++++++ +++++ +++++++++++++++++ + + + + ++++++++++++++++ ++ + ++++++++++ + + + + + + ++++++++++++++++ + +++++ +++++ ++++ ++++++++++++++++++++ + + +++++++++++++ +++++++ + ++ + + ++++++++++ + ++++++++++++ +++ +++ ++++++++++++++ ++ + + + + ++++++++++++++++++++++++++ + +++++++++++++++++++++++++ +++ + ++ +++++++++++++++++ + + ++ + +++++++++++++++++++ + + + + ++++++++++++++++++++ ++++++++++ + + ++++ +++ + ++ + + + +++++ ++++++++++++++ ++++++++++++++++++ + + +++++++++++++++++++++ + + +++++ + + +++++++++++++++++ ++ + + +++ + +++++++++++++++++++ ++ + ++++++++++ + +++++++++++++++++ + + ++++++++++++++++++++++++ ++ +++++++++++++++++++++++++++ + + +++ ++++ ++++++++++++ + + + ++ ++++ ++++++ + + +++++ + + + + + +++ ++++++++++++++++++++++++++ ++++ ++ +++ + ++ + + +++++++++++++++++++++ +++ +++++++++++++ ++++++++++++++ + ++ ++++ +++++++++++++ + +++ + + + + ++ + + +++++++++ + ++ ++++ ++++++ ++++++++++++++ + +++++ + ++ + ++ +++++++ +++ ++ ++ ++++++++++++++++++ +++++++++++++++++++++++ + ++ ++++ + +++ +++++++++++++++++ + + + ++ ++ + + + + +++ +++ +++ + + +++ + +++ +++++ +++ ++++++ ++ + ++++ +++++ + ++ + + ++++ ++++++++++++++++++++++ + +++ ++++ + + ++ +++ + ++++++++++++++++++++++ +++++++++++++ + +++++++++++++++++ + + ++++++++++++ + ++ + +++ +++++++ + + ++ +++ + + ++ ++++++ ++++++ + +++ ++ +++++++++++++++ ++ + + + +++++++++++++++++ ++++++++++++ + +++ +++++ + + + +++++++ + + + ++++ + ++ + + ++ + +++++++++++++++ + + +++++++ +++++ + + ++ ++++++++++++++ + + ++ + + ++ ++ ++++ ++++++++ ++ + ++++++++++ + ++ ++ ++++++++ + + ++ ++ ++ +++ ++ + + ++ +++++ + + ++ + ++ ++++ + ++ + ++ + ++ + ++ + ++ + + ++ + + + ++ ++++ + + + ++ -3 -2 -1 0 1

-0.010

0.000

0.010

Log10(Expected Total MS Counts)


Nothing (Null Hypothesis)
250

Frequency

0 -2

50

100

150

200

-1

0

1

2

3

Log10(Expected Total MS Counts)


Simulation Study #1: NULL - Model Only


Simulation Study #2: Bright Discontinuous Extra Component
[of MultiScale Results]

[Simulated from Model+Bright Extra]


Bright Discontinuous Unknown Log10(Baseline Scale Factor)
0.025 ++ + + + + + + ++ + + + +++ + ++++ + + ++ ++ + + + + ++ ++ + ++ + + + + + ++ + + + ++ + ++ + + + + + ++ ++ + +++++ + ++ + ++++++++++++ ++ + ++ ++++++++ +++ + +++ +++++++++ ++ + +++ + ++ + +++ +++ ++ ++++++ + + ++ + ++ ++ + ++ + + ++ +++++++++++ ++ ++ + +++++++++ + ++ +++++++++++++++ +++++ + + + ++ ++++++ +++ ++ ++++ ++ + ++ + ++++ ++ ++ + + ++++++++++++ ++ +++++ + + +++++++++ + + + + + + +++++++++++ + + + + + ++++++++++ + + + +++++++++++ + ++ + + ++++++ +++ + + +++++++++ ++++++ + ++ ++ +++ +++++ + ++ + + + + + + + + + +++++++++ + ++ ++++++++++++++ + + + ++ ++ + + ++ ++++++++++++ +++ + + ++ ++++ + + ++ + ++ +++ +++ +++ +++ ++++ ++ + ++ +++++++++++++++ + + + ++ + + + + + +++++ ++ + +++ ++ + + +++ + + + ++++++ + + + +++ + + + ++ + ++ ++ + + + + + ++ + +++++ ++ + ++ + + + +

0.015

+

!0.005

0.005

+ 3.34 3.36 3.38 3.40 3.42

Log10(Expected Total MS Counts)


Null (.) vs Bright Unknown (+) Log10(Baseline Scale Factor)
0.03 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + !0.01 !3

0.01

0.02

!2

!1

0

1

2

3

Log10(Expected Total MS Counts)


Simulation Study #2: Bright Discontinuous Extra Component


Simulation Study #3: Faint Model MisMatch

[Simulated from Model+Faint Extra]


Faint Model Mis-Match Log10(Baseline Scale Factor)
0.040 + + + + + + + ++ + + + ++ + + + + + ++ + ++ +++ + ++ + ++ + + ++++ ++ + ++ + ++ + +++++ + + + + ++ + + ++ ++ +++++ ++ ++ + ++ + + ++ ++++ +++ + + + + + + +++ +++++ +++ + + + ++++++++ + + +++ + ++++ +++ ++++ +++ + +++ + + ++ ++++ +++++++ + ++++ ++++ ++++++ + + + + +++ + + +++ +++++++++++ + ++ + ++++ +++++ + ++ + ++++ + +++ ++++++ +++ + ++ +++++++ + + + + + +++++++++++++++++ ++++ ++ + ++++++++ + + + ++++ ++++++++++ + + + ++ ++++++++++ + + ++ ++ ++ + ++++++++++++++++ +++++++ + ++++++++++++ + ++++++++ + ++ + + ++ + ++++++++ + ++++ + ++++ + + + + + + +++++ ++++++ +++++ ++ + ++ +++ +++ ++ + +++ ++ ++++++++ + ++ + + + +++++ +++++++++ + +++ + + + ++ + ++ ++++ +++++ ++++ + +++++ + + + + ++ + +++++++++ + + ++ ++ + +++ + + + ++ + +++ + + ++ ++ ++ + ++ + + + + + + ++ + + +++ + + +++++++ + + + ++ + + + +++ + ++ + + + ++ +++++ ++ + + ++ + + + ++ + + + ++ + + ++ + + ++ + + + + -2 -1 0 1

0.030

+ +

0.020

+

Log10(Expected Total MS Counts)


!ull %s Faint Model Mis!Matc1
250

Frequency

0 !2

50

100

150

200

!1

0

1

2

3

Log10(Expected Total MS Counts)


Simulation Study #3: Faint Model MisMatch


Special Thanks To:
NSF and SAMSI 2006 Special Topics in AstroStatistics NASA and AISR Python Tools for AstroStatistics CHASC: http://hea-www.harvard.edu/AstroStat/
Quick Reference: See Statistical Challenges in Modern Astronomy IV, Proceedings, Connors and van Dyk, "How To Win With Non-Gaussian Data: Poisson Goodness-of-Fit"
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