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Posterior-Variance-Based Error Quantification for Inverse Problems in Imaging

Dominik Narnhofer, Andreas Habring, Martin Holler, Thomas Pock

TL;DR

The paper tackles the problem of obtaining reliable per-pixel error bounds in Bayesian imaging inverse problems without strong distributional assumptions. It introduces a novel framework that regresses reconstruction error onto posterior variance estimates and uses conformal prediction to calibrate pixelwise confidence quantiles, yielding distribution-free coverage guaranteed by finite data. A key practical contribution is the Unadjusted Langevin Primal-Dual Algorithm (ULPDA) that enables sampling from non-smooth posterior energies, allowing the method to accommodate non-convex or non-differentiable priors as well as learned priors in a black-box fashion. Across denoising and MRI experiments with TV, FoE, and TDV priors, the approach demonstrates tight, reliable error quantiles and competitive coverage, highlighting its potential for informing clinical and high-stakes imaging decisions.

Abstract

In this work, a method for obtaining pixel-wise error bounds in Bayesian regularization of inverse imaging problems is introduced. The proposed method employs estimates of the posterior variance together with techniques from conformal prediction in order to obtain coverage guarantees for the error bounds, without making any assumption on the underlying data distribution. It is generally applicable to Bayesian regularization approaches, independent, e.g., of the concrete choice of the prior. Furthermore, the coverage guarantees can also be obtained in case only approximate sampling from the posterior is possible. With this in particular, the proposed framework is able to incorporate any learned prior in a black-box manner. Guaranteed coverage without assumptions on the underlying distributions is only achievable since the magnitude of the error bounds is, in general, unknown in advance. Nevertheless, experiments with multiple regularization approaches presented in the paper confirm that in practice, the obtained error bounds are rather tight. For realizing the numerical experiments, also a novel primal-dual Langevin algorithm for sampling from non-smooth distributions is introduced in this work.

Posterior-Variance-Based Error Quantification for Inverse Problems in Imaging

TL;DR

The paper tackles the problem of obtaining reliable per-pixel error bounds in Bayesian imaging inverse problems without strong distributional assumptions. It introduces a novel framework that regresses reconstruction error onto posterior variance estimates and uses conformal prediction to calibrate pixelwise confidence quantiles, yielding distribution-free coverage guaranteed by finite data. A key practical contribution is the Unadjusted Langevin Primal-Dual Algorithm (ULPDA) that enables sampling from non-smooth posterior energies, allowing the method to accommodate non-convex or non-differentiable priors as well as learned priors in a black-box fashion. Across denoising and MRI experiments with TV, FoE, and TDV priors, the approach demonstrates tight, reliable error quantiles and competitive coverage, highlighting its potential for informing clinical and high-stakes imaging decisions.

Abstract

In this work, a method for obtaining pixel-wise error bounds in Bayesian regularization of inverse imaging problems is introduced. The proposed method employs estimates of the posterior variance together with techniques from conformal prediction in order to obtain coverage guarantees for the error bounds, without making any assumption on the underlying data distribution. It is generally applicable to Bayesian regularization approaches, independent, e.g., of the concrete choice of the prior. Furthermore, the coverage guarantees can also be obtained in case only approximate sampling from the posterior is possible. With this in particular, the proposed framework is able to incorporate any learned prior in a black-box manner. Guaranteed coverage without assumptions on the underlying distributions is only achievable since the magnitude of the error bounds is, in general, unknown in advance. Nevertheless, experiments with multiple regularization approaches presented in the paper confirm that in practice, the obtained error bounds are rather tight. For realizing the numerical experiments, also a novel primal-dual Langevin algorithm for sampling from non-smooth distributions is introduced in this work.
Paper Structure (33 sections, 4 theorems, 57 equations, 21 figures, 4 tables, 3 algorithms)

This paper contains 33 sections, 4 theorems, 57 equations, 21 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Works
  4. Regularization and Posterior Sampling
  5. Uncertainty Quantification
  6. Conformal Prediction
  7. Theoretical Results
  8. Preliminaries
  9. The Proposed Method
  10. Application to Bayesian Imaging
  11. Toy example in 1D
  12. Algorithms for the Computation of Posterior Expectation and Variance
  13. Langevin Algorithms for Differentiable Energies
  14. Primal-Dual Langevin Algorithm for Non-differentiable Energies
  15. Belief Propagation
  16. ...and 18 more sections

Key Result

Lemma 1

Let $Y_1, Y_2,\dots,Y_{N+1}$ real valued random variables with $N\in\mathbb{N}$ a random sample size. Assume that $Y_1, Y_2,\dots,Y_{N+1}$ are i.i.d. with respect to the probability measure $\mathrm{P}[\;.\;|\; N=n]$ for every $n\in\mathbb{N}$, $n\geq 1$. Then for any $q\in (0,1)$ and $n\geq 1$, where $\hat{Y}_q = Y_{(\lceil(N+1)q\rceil)}$ is the empirical q-quantile of $Y_1, Y_2,\dots,Y_{N+1}$.

Figures (21)

  • Figure 3.1: Left: Joint distribution $\mathrm{p}(x,z)$ with marginal distributions $\mathrm{p}(x)$, $\mathrm{p}(z)$ and posterior distributions $\mathrm{p}(x|z)$. Right: Posterior expectation $\mathbb{E}[X|Z=z]$ and posterior variance $t=\mathop{\mathrm{Var}}\limits[X|Z=z]$ as functions of $z$ as well as the conditional distribution of the error $p(s|t)$ for specific instances of $t$. As in later usage, posterior variance and error are already presented in logarithmic scaling.
  • Figure 3.2: Joint log-distribution $\log\mathrm{p}(s,t)$ (left) and cumulative conditional distribution over error (right). The gray dashed line indicates the conditional 0.9 quantile.
  • Figure 3.3: Empirical joint log-distribution $\log\mathrm{p}(s,t)$ (left) and empirical cumulative conditional distribution over error (right). The dashed gray line indicates the exact conditional 0.9 quantile of the distribution, the red lines indicates the respective estimated conformalized quantile.
  • Figure 4.1: Mean absolute difference of MMSE and posterior variance estimated using ULPDA for different values of $\tau$ and ULA with Huber TV ($\mathop{\mathrm{TV}}\limits_h$) compared to BP as function of the number of samples with discretization threshold $\Delta=\frac{1}{1024}$.
  • Figure 5.1: Comparison of ULPDA-sampling results and BP reconstruction results for denoising with $\sigma=15/255$. The ULPDA reconstructions were obtained for a primal step size of $\tau={5}\mathrm{e}{-5}$ and 50k iterations (${8}\mathrm{e}{-4}$${3.5}\mathrm{e}{-2}$, $0$$0.01/{1}\mathrm{e}{-3}$ for MMSE and variance difference respectively).
  • ...and 16 more figures

Theorems & Definitions (12)

  • Lemma 1
  • Remark 1
  • proof
  • Lemma 2
  • proof
  • Remark 2
  • Proposition 1
  • proof
  • Remark 3
  • Corollary 1
  • ...and 2 more