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QUANTIF Y ING, SUMMARIZING, AND REPRESENTING 'TOTAL' UNCERTAINTIES IN IMAGE ( AND SPECTRAL) 'DECONVOLUTION' A. Connors for `CHASC' or CBASC

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`The imme diate question arises as to the statistical significance of this feature... quantification of objectw ise significance (e.g., "this blob is significant at the n-sigma level") are difficult.' (Dixon et al. 1998 New Astronomy 3, 539)
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`The imme diate question arises as to the statistical significance of this feature... quantification of objectw ise significance (e.g., "this blob is significant at the n-sigma level") are difficult.' (Dixon et al. 1998 New Astronomy 3, 539)
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Tomographic Reconstruction: Comparing Examples (from Willett et al.)
True image Filtered back projection MLE reconstruction

Fessler's PWLS

Wedgelet reconstruction
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What's the significance of / uncertainty on features?


Talk Outline (Parallel pieces):
0. What/Why: Demos, Definitions 1. What/Why: Problem Definition:
1.1 Go o dness-of-fit an d feature- detection 1.2 Mismatch significance, shape error bars 1.3 All uncertainties: instrument, physics

2. How/Why: Histor y/Metho ds
2.1 Frequentist Multiscale, Bayesian Structure 2.2 DA/MCMC 2.3 Comparisons of Null (simulations) vs Data

3. Current Examples
Var ying signal to no ise: "E" an d Gamma-ray sky

4. Current Challenges
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How/Why: Histor y/Metho ds
* Putting Flexible/Multiscale `NP' mo dels * Together w ith parametrize d physics-base d mo dels * Full Bayesian Posterior framework * `Likeliho o dist' (Tanner); Priors ~ Complexity Penalty * Bayes allows Mo dularity: Data Augmentation, * Bayes allows complex, high- dimensions: MCMC

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Multiplicative Multiscale Innovation Models

Timmermann & Nowak, 1999 Kolaczyk, 1999


Multiplicative Multiscale Innovation Models

Timmermann & Nowak, 1999 Kolaczyk, 1999


Multiplicative Multiscale Innovation Models

Timmermann & Nowak, 1999 Kolaczyk, 1999


Multiplicative Multiscale Innovation Models

Timmermann & Nowak, 1999 Kolaczyk, 1999


Multiplicative Multiscale Innovation Models
0,0,0 X
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

·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 Timmermann & Nowak, 1999 Kolaczyk, 1999


Usual Equations fo r `True' Intensity, Instrument, Data:
S(l,b,e,t,) = Expecte d `True' Source Intensity (l,b,e,t,) = `True' Effective Area PSF(x,y | l,b,e,t,) = `True' instrument smearing (x,y,e,t,,,) = `True' Expecte d counts in detector D(x,y,e,t,,,) = measure d counts in detector (x,y,e,t,,,) = PSF(x,y|l,b,e,t,)@(l,b,e,t,)*S(l,b,e,t,) D (x,y,e,t,,,) ~ Po isson ( (x,y,e,t,,,) )

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Usual Equations fo r `Mo del' Intensity, Instrument, Data:
s(l,b,e,t,) = Expecte d `Mo del' Source Intensity (l,b,e,t,) = `Mo del' Effective Area psf(x,y | l,b,e,t,) = `Mo del' instrument smearing (x,y,e,t,,,) = `Mo del' Expecte d counts in detector D(x,y,e,t,,,) = measure d counts in detector (x,y,e,t,,,) = psf(x,y|l,b,e,t,)@(l,b,e,t)*s(l,b,e,t,) D (x,y,e,t,,,) ~ Po isson ( (x,y,e,t,,,) )

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Our Equations fo r `Mo del' Intensity, Instrument, Data:
s(l,b,e,t,) = Expecte d `Physics Mo del' Source Intensity m(l,b,e,t,,) = Expecte d Multiscale Source Counts = Smo othing Parameters for each scale = `Range' parameter for Hyper -priors on = `Scale Factor' for Physics Mo del (l,b,e,t,) = `Mo del' Effective Area psf(x,y | l,b,e,t,) = `Mo del' instrument smearing (x,y,e,t,,,) = `Mo del' Expecte d counts in detector D(x,y,e,t,,,) = measure d counts in detector (x,y,e,t,,,) = psf(x,y|l,b,e,t,)@ ( *(l,b,e,t)*s(l,b,e,t,) + m (l,b,e,t,,) )
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3. Mo derate Signal-To-No ise Examples:Gamma-Ray Sky:

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5

10

15

20

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3. Mo derate Signal-To-No ise Examples:Gamma-Ray Sky:

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3. Mo derate Signal-To-No ise Examples:Gamma-Ray Sky:

0

0.004

0.008

0.012

0.016

0.05

0.1

0.15

0.2 0.25 0.3 0.35 0.4 0.45
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