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CAR(1)
Process described by

- relaxation time b - mean value of the process, 2/2 - variance (t) - white noise with mean = 0 and variance = 1

Initial value

Stochastic component


Likelihood

Example


Low Counts X-ray Images
How to describe and quantify significance of irregular structures? Radio-band image showing a linear jet structure X-ray Image with a hint of a jet

Many empty pixels with 0 counts


Statistical Model
· · photon counts distributed on a grid of pixels Two Poisson processes to represent: ­ known aspects of the image - baseline component · background, point sources, diffuse structures ­ Unknown image features - added component · Diffuse emission, extension, a jet

Perfect detector Includes PSF
Pi,j - PSF blurring, 0j, 0
1j,

- intensity in each pixel j

A - detector efficiency
Connors& Van Dyk 2007 Stein et al 2014

due to baseline and added components

1 - expected photon counts


Simulation Study with LIRA
LIRA - Low-counts Image Reconstruction and Analysis

Baseline Model

Null Simulation

Posterior Mean

PSF
Observation McKeough et al 2014

Posterior Mean


Results

X-ray Data

Multiscale Component From LIRA


Significance of Structures

Structure in regions 2, 3, 4, 5 p-value of < 0.01

Blue - posterior density for the alternative model given the data Dashed black - posterior density under alternative given the simulated data under the null

Stein et al 2015