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Inferring Object Boundaries and their Roughness with Uncertainty Quantification

Babak Maboudi Afkham, Nicolai André Brogaard Riis, Yiqiu Dong, Per Christian Hansen

TL;DR

The paper develops a goal-oriented Bayesian framework to infer object boundaries and their roughness from imaging data, representing boundaries with a periodic radial function and controlling regularity via a Whittle–Matérn prior on a fractional differentiability parameter. A KL-type expansion introduces an auxiliary variable 𝑈 that decouples from the roughness hyperparameter S, enabling efficient sampling through FFT-based mappings 𝔽 and 𝔽^{-1}. The approach is validated on EEG data fitting, limited-angle X-ray CT, and image inpainting, providing not only boundary estimates but also quantified uncertainties for both the boundary and its roughness, even under noise and data incompleteness. The results show robustness to noise types and levels and demonstrate the method’s ability to capture realistic boundary irregularities while offering principled uncertainty quantification. The work suggests extensions to higher dimensions and alternative hierarchical priors, broadening applicability to diverse inverse problems where boundary regularity is informative.

Abstract

This work describes a Bayesian framework for reconstructing the boundaries that represent targeted features in an image, as well as the regularity (i.e., roughness vs. smoothness) of these boundaries.This regularity often carries crucial information in many inverse problem applications, e.g., for identifying malignant tissues in medical imaging. We represent the boundary as a radial function and characterize the regularity of this function by means of its fractional differentiability. We propose a hierarchical Bayesian formulation which, simultaneously, estimates the function and its regularity, and in addition we quantify the uncertainties in the estimates. Numerical results suggest that the proposed method is a reliable approach for estimating and characterizing object boundaries in imaging applications, as illustrated with examples from X-ray CT and image inpainting. We also show that our method is robust under various noise types, noise levels, and incomplete data.

Inferring Object Boundaries and their Roughness with Uncertainty Quantification

TL;DR

The paper develops a goal-oriented Bayesian framework to infer object boundaries and their roughness from imaging data, representing boundaries with a periodic radial function and controlling regularity via a Whittle–Matérn prior on a fractional differentiability parameter. A KL-type expansion introduces an auxiliary variable 𝑈 that decouples from the roughness hyperparameter S, enabling efficient sampling through FFT-based mappings 𝔽 and 𝔽^{-1}. The approach is validated on EEG data fitting, limited-angle X-ray CT, and image inpainting, providing not only boundary estimates but also quantified uncertainties for both the boundary and its roughness, even under noise and data incompleteness. The results show robustness to noise types and levels and demonstrate the method’s ability to capture realistic boundary irregularities while offering principled uncertainty quantification. The work suggests extensions to higher dimensions and alternative hierarchical priors, broadening applicability to diverse inverse problems where boundary regularity is informative.

Abstract

This work describes a Bayesian framework for reconstructing the boundaries that represent targeted features in an image, as well as the regularity (i.e., roughness vs. smoothness) of these boundaries.This regularity often carries crucial information in many inverse problem applications, e.g., for identifying malignant tissues in medical imaging. We represent the boundary as a radial function and characterize the regularity of this function by means of its fractional differentiability. We propose a hierarchical Bayesian formulation which, simultaneously, estimates the function and its regularity, and in addition we quantify the uncertainties in the estimates. Numerical results suggest that the proposed method is a reliable approach for estimating and characterizing object boundaries in imaging applications, as illustrated with examples from X-ray CT and image inpainting. We also show that our method is robust under various noise types, noise levels, and incomplete data.
Paper Structure (15 sections, 1 theorem, 32 equations, 16 figures, 2 algorithms)

This paper contains 15 sections, 1 theorem, 32 equations, 16 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. A Goal-Oriented Bayesian Approach
  3. The Hierarchical Prior
  4. Algorithms for the mappings $\mathcal{F}$ and $\mathcal{F}^{-1}$
  5. Numerical Experiments
  6. Data Fitting
  7. Influence of $m$
  8. Influence of the noise level
  9. Comparison with BMS
  10. Limited-Angle X-Ray Computed Tomography (CT)
  11. Test on a phantom from the prior
  12. Test on a gear phantom
  13. Comparison with multi-step method
  14. Image Inpainting
  15. Conclusions

Key Result

Proposition 1

Let $G$ and $H$ be independent and standard normal distributed real-valued random variables, then the random variables $(G+H)/\sqrt{2}$ and $(H - G)/\sqrt{2}$ are independent and standard normal distributed.

Figures (16)

  • Figure 1: Samples from the random variable $\boldsymbol{V}|S$ for a fixed realization of $\mathbf u$ and four values of $s$.
  • Figure 2: The influence of discretization levels on the reconstructed signals. The plots show the posterior means for the signal $\boldsymbol{v}$, and the uncertainties are represented as 95% HDI.
  • Figure 3: The influence of discretization levels on the roughness parameter $s$. The violin plots show the posterior distributions.
  • Figure 4: The influence of the noise level $r$ on the posterior statistics for $\boldsymbol{v}$. The uncertainty is represented as 95% HDI.
  • Figure 5: The influence of the noise level $r$ on the posterior statistics for $s$. The violin plots show the posterior distribution $\pi_{S|\boldsymbol{Y}}$.
  • ...and 11 more figures

Theorems & Definitions (3)

  • Remark 1
  • Proposition 1
  • Remark 2