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The LIRA-Ising Model: Estimating the boundaries of irregularly shaped X-ray sources

Kathryn McKeough, Vinay L. Kashyap, Aneta Siemiginowska, David A. Van Dyk, Shihao Yang, Xiao-Li Meng, Brendan Martin, Andreas Zezas

TL;DR

The paper tackles boundary delineation for irregular, extended X-ray sources in low-count, PSF-blurred images. It introduces LIRA-Ising, a three-step Bayesian procedure that (1) uses LIRA for multiscale reconstruction of the added component, (2) imposes an Ising-based cohesion prior to identify a cohesive source region via pixel indicators, and (3) selects a boundary by marginalizing over the posterior of the region. This yields a boundary with quantified per-pixel uncertainty through posterior probability maps and boundary estimates validated on simulations and applied to Chandra jets PKS J1421-0643 and 0730+257. The approach cleanly combines a proven multiscale reconstruction with a spatial-cohesion prior, enabling objective, data-driven morphology analyses in sparse X-ray imaging and enhancing downstream astrophysical inference.

Abstract

Mapping the boundary of an extended source is a key step in the study of its morphology. The background contamination and statistical fluctuations of typical astronomical images make this a challenging statistical task, particularly for X-ray images with low surface brightness. We develop a three-step Bayesian procedure to identify the boundaries of irregularly shaped sources. We first apply a Bayesian multiscale reconstruction algorithm known as LIRA to obtain posterior pixelwise probability distributions of the source intensity that properly account for known structures, astrophysical background, and the effect of the telescope point spread function. Next, we adopt an Ising model to group pixels with similar intensities into cohesive regions corresponding to background and source. Finally, the boundary is derived on the basis of the most likely aggregation of pixels into the source region. Because the overall model combines LIRA and the Ising model, we call it LIRA-Ising. We verify the proposed method using a set of simulation studies. We then apply it to the Chandra X-ray Observatory images of two high redshift quasars, PKS J1421-0643 and 0730+257, to determine the extent and morphology of X-ray jets. Our method shows a uniform X-ray surface brightness of PKS J1421-0643 jet, and identifies knotty structure in the X-ray jet of 0730+257.

The LIRA-Ising Model: Estimating the boundaries of irregularly shaped X-ray sources

TL;DR

The paper tackles boundary delineation for irregular, extended X-ray sources in low-count, PSF-blurred images. It introduces LIRA-Ising, a three-step Bayesian procedure that (1) uses LIRA for multiscale reconstruction of the added component, (2) imposes an Ising-based cohesion prior to identify a cohesive source region via pixel indicators, and (3) selects a boundary by marginalizing over the posterior of the region. This yields a boundary with quantified per-pixel uncertainty through posterior probability maps and boundary estimates validated on simulations and applied to Chandra jets PKS J1421-0643 and 0730+257. The approach cleanly combines a proven multiscale reconstruction with a spatial-cohesion prior, enabling objective, data-driven morphology analyses in sparse X-ray imaging and enhancing downstream astrophysical inference.

Abstract

Mapping the boundary of an extended source is a key step in the study of its morphology. The background contamination and statistical fluctuations of typical astronomical images make this a challenging statistical task, particularly for X-ray images with low surface brightness. We develop a three-step Bayesian procedure to identify the boundaries of irregularly shaped sources. We first apply a Bayesian multiscale reconstruction algorithm known as LIRA to obtain posterior pixelwise probability distributions of the source intensity that properly account for known structures, astrophysical background, and the effect of the telescope point spread function. Next, we adopt an Ising model to group pixels with similar intensities into cohesive regions corresponding to background and source. Finally, the boundary is derived on the basis of the most likely aggregation of pixels into the source region. Because the overall model combines LIRA and the Ising model, we call it LIRA-Ising. We verify the proposed method using a set of simulation studies. We then apply it to the Chandra X-ray Observatory images of two high redshift quasars, PKS J1421-0643 and 0730+257, to determine the extent and morphology of X-ray jets. Our method shows a uniform X-ray surface brightness of PKS J1421-0643 jet, and identifies knotty structure in the X-ray jet of 0730+257.
Paper Structure (19 sections, 38 equations, 4 figures, 1 table)

This paper contains 19 sections, 38 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model & Inference
  3. LIRA: Low-count Image Reconstruction and Analysis
  4. Motivating Example: An Extragalactic Jet
  5. Adding Spatial Cohesion to the LIRA Model
  6. Steps 1 and 2: Fitting the LIRA-Ising Model for Spatial Cohesion
  7. Implementing Steps 1 and 2
  8. Step 3: Estimating the Extended Source Region Boundary under the LIRA-Ising Model
  9. Validation
  10. Simulation with a Soft Boundary
  11. Simulation with a Hard Boundary
  12. Illustrating the method with Chandra data
  13. Conclusion
  14. The Joint Posterior Distribution Under the LIRA-Ising Model
  15. Gibbs Sampler for the Probabilistic Classification of Pixels
  16. ...and 4 more sections

Figures (4)

  • Figure 1: Estimating the boundary of a jet in the Chandra observation of the $z = 3.69$ quasar PKS J1421-0643 (ObsID 7873). (a) $\hat{\mathbb E}(\tilde{\Lambda} \mid Y)$, the posterior mean of the LIRA added component, i.e., the estimated expected deconvolved counts attributed to the jet, excluding the central galaxy. (b) $\hat{\mathbb E}(Z \mid Y)$, the map of the probability that each pixel is association with the extended (jet) source. (c) The best-fit boundary of the jet overlaid on the original Chandra X-ray image. (d) The best-fit boundary of the jet overlaid on a radio image of PKS J1421-0643 (VLA 4.9 GHz; from McKeough2016) (As the radio image can be zero or negative, in panel (d) we plot $\log_{10}(\max(\hbox{radio},0) + 0.001)$.) The color scale shows the radio measurement units in Jy/beam, beam size of $0.\arcsec66 \times 0.\arcsec38$ at PA=$-20.^o7$.
  • Figure 2: Simulation study with a soft boundary. Rows correspond to the the four extended source settings (from top to bottom: Gaussian with variance equal to 4, 8, and 16 pixels, and no extended source). Columns correspond to noise level (from left to right 0.01, 0.1 and 1 expected counts per pixel). Each panels shows the simulated counts and their fitted boundaries. The fitted boundaries are plotted as solid cyan curves and are compared with the $2\sigma$ and $3\sigma$ dashed green contours of the Gaussian distributions used to simulate the extended sources.
  • Figure 3: Simulation study with a hard boundary. Columns correspond to the three extended source settings (from left to right: squares with 4, 8, and 16 pixels to the side, centered in the field, each with a brightness of 1 count pixel$^{-1}$). Rows correspond to noise level (from top to bottom 0.01, 0.1 and 1 expected counts per pixel). Each panel shows the simulated counts and their fitted boundaries (solid cyan). The boundaries are compared with the the true edges of the square extended sources (dashed green).
  • Figure 4: Identifying hot spots in a jet in the Chandra observation of a quasar $0730+257$ (ObsID 10307). (a) $\hat{\mathbb E}(\tilde{\Lambda} \mid Y)$, the posterior mean of the LIRA added component, i.e., the estimated expected deconvolved counts attributed to the jet, excluding the central galaxy. (b) $\hat{\mathbb E}(Z \mid Y)$, the map of the probability that each pixel is association with the extended (jet) source. (c) The best-fit boundary of the jet overlaid on the original X-ray image. (d) The best-fit boundary of the jet overlaid on a 8.7 GHz VLA radio image of $0730+257$ from McKeough2016. The radio measurement units are Jy/beam, with the circular beam size of 0.35.