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Joint Signal Recovery and Uncertainty Quantification via the Residual Prior Transform

Yao Xiao, Anne Gelb

TL;DR

The paper tackles inverse problems where fixed, variability-specific priors (such as TV or fixed-order differencing) fail to capture signals with spatially varying smoothness, especially under multimodal and noisy measurements. It reformulates the residual transform as a sparsity-promoting prior within a hierarchical sparse Bayesian learning framework, enabling principled uncertainty quantification and joint recovery across modalities via MAP estimation and MMV extensions. Empirical results on synthetic 1D/2D signals and real SAR/traffic data show that the residual prior reduces staircasing and yields sharper, more faithful reconstructions while providing credible intervals for reliability. The work highlights an adaptive variance mechanism that is heightened at jumps and subdued in smooth regions, and suggests future extensions to higher dimensions and alternative operator pairings for broader applicability.

Abstract

Conventional priors used for signal recovery are often limited by the assumption that the type of a signal's variability, such as piecewise constant or linear behavior, is known and fixed. This assumption is problematic for complex signals that exhibit different behaviors across the domain. The recently developed {\em residual transform operator} effectively reduces such variability-dependent error within the LASSO regression framework. Importantly, it does not require prior information regarding structure of the underlying signal. This paper reformulates the residual transform operator into a new prior within a hierarchical Bayesian framework. In so doing, it unlocks two powerful new capabilities. First, it enables principled uncertainty quantification, providing robust credible intervals for the recovered signal, and second, it provides a natural framework for the joint recovery of signals from multimodal measurements by coherently fusing information from disparate data sources. Numerical experiments demonstrate that the residual prior yields high-fidelity signal and image recovery from multimodal data while providing robust uncertainty quantification.

Joint Signal Recovery and Uncertainty Quantification via the Residual Prior Transform

TL;DR

The paper tackles inverse problems where fixed, variability-specific priors (such as TV or fixed-order differencing) fail to capture signals with spatially varying smoothness, especially under multimodal and noisy measurements. It reformulates the residual transform as a sparsity-promoting prior within a hierarchical sparse Bayesian learning framework, enabling principled uncertainty quantification and joint recovery across modalities via MAP estimation and MMV extensions. Empirical results on synthetic 1D/2D signals and real SAR/traffic data show that the residual prior reduces staircasing and yields sharper, more faithful reconstructions while providing credible intervals for reliability. The work highlights an adaptive variance mechanism that is heightened at jumps and subdued in smooth regions, and suggests future extensions to higher dimensions and alternative operator pairings for broader applicability.

Abstract

Conventional priors used for signal recovery are often limited by the assumption that the type of a signal's variability, such as piecewise constant or linear behavior, is known and fixed. This assumption is problematic for complex signals that exhibit different behaviors across the domain. The recently developed {\em residual transform operator} effectively reduces such variability-dependent error within the LASSO regression framework. Importantly, it does not require prior information regarding structure of the underlying signal. This paper reformulates the residual transform operator into a new prior within a hierarchical Bayesian framework. In so doing, it unlocks two powerful new capabilities. First, it enables principled uncertainty quantification, providing robust credible intervals for the recovered signal, and second, it provides a natural framework for the joint recovery of signals from multimodal measurements by coherently fusing information from disparate data sources. Numerical experiments demonstrate that the residual prior yields high-fidelity signal and image recovery from multimodal data while providing robust uncertainty quantification.
Paper Structure (17 sections, 74 equations, 12 figures, 2 algorithms)

This paper contains 17 sections, 74 equations, 12 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The hierarchical Bayesian model
  4. Sparsity-promoting hierarchical learning
  5. MAP estimation
  6. Exploiting joint sparsity from multiple measurements
  7. Uncertainty Quantification
  8. Edge detection
  9. Jump function approximation
  10. A local approach to edge detection
  11. A global approach to edge detection
  12. The residual prior transform
  13. MAP estimate error analysis
  14. Numerical results
  15. Joint Recovery of Multimodal One-dimensional Signals
  16. ...and 2 more sections

Figures (12)

  • Figure 1: Graphical representation of \ref{['alg:MMV_GSBL']} for two ($L=2$) measurement and parameters vectors, $\bm{y}_1, \bm{y}_2$ and $\bm{x}_1, \bm{x}_2$, respectively. Shaded (plain) circles indicate observed (hidden) random variables, respectively. The arrows represent the conditional dependencies among the random variables. The likelihood \ref{['eq:model2pdflikelihood']} bridges the parameter vectors $\bm{x}_1,\bm{x}_2$ and the measurement vectors $\bm{y}_1,\bm{y}_2$; The joint-sparsity-promoting prior \ref{['eq:model2pdfhypertheta']} links the shared hyperparameter vector $\boldsymbol{\theta}_\mathrm{MMV}$ to $\bm{x}_1, \bm{x}_2$. Using shared Gamma distributed hyperparameter vector $\boldsymbol{\theta}_\mathrm{MMV}$ (instead of separate ones for $\bm{x}_1, \bm{x}_2$) results in $\Phi \bm{x}_1$ and $\Phi \bm{x}_2$ sharing the support.
  • Figure 1: (top) The red lines show the measurements, $\bm y_l$, $l = 1,2,3$ as given by \ref{['eq: data_acquisition']} for corresponding $f_l(s)$ in \ref{['ex: example_1d']}. (top-left) Noisy data for the piecewise smooth signal $f_1$, measured using the identity matrix $F_1 = I_n$ and contaminated by additive noise at an $\text{SNR}=5$dB. (top-middle) Blurred data for signal $f_2$, using the Gaussian blur operator $F_2=\tilde{F}$ with psf parameter $\gamma=0.03$, as defined in \ref{['eq:bluroperator']}, and contaminated by additive noise at an $\text{SNR}=20$dB. (top-right) Undersampled data for signal $f_3$, obtained with the operator $F_3=HI_n$ with undersampling ratio $r=0.3$, and contaminated by additive noise at an $\text{SNR}=20$dB. The dashed black lines represent the corresponding discretized true underlying signals, $\bm f_l$, $l=1,2,3$. (middle row) MAP estimates obtained by \ref{['alg:GSBL']}. (bottom) Corresponding $\log_{10}E^{\text{abs}}_j$, $j=1,\dots,n$, given by \ref{['eq: abs_err']}.
  • Figure 2: The MAP estimates for (left) $f_1$, (middle) $f_2$, and (right) $f_3$ and 99% credible intervals for the recovered signals, using the local differencing prior transform (top) $\Phi = T_n^p$ and the residual prior transform (bottom) $\Phi = R_{n,\zeta}^p$.
  • Figure 3: (top) The red lines show the measurements in \ref{['eq: data_acquisition']} that correspond to $f_{4}(s), f_5(s)$ and $f_6(s)$ in \ref{['ex: example_1d_const']}. The multimodal data collections are described in \ref{['fig:trueSignals']}. (middle row) MAP estimates obtained by \ref{['alg:GSBL']} from data $\bm y$. (bottom) Corresponding $\log_{10}E^{\text{abs}}_j$, $j=1,\dots,n$, given by \ref{['eq: abs_err']}.
  • Figure 4: The multimodal measurement scenarios are consistent with those described in \ref{['fig:trueSignals']}, but here the recovery focuses exclusively on $f_3$. (top and third rows) MAP estimates obtained by \ref{['alg:GSBL', 'alg:MMV_GSBL']} from data $\bm y$. (second and bottom rows) Corresponding $\log_{10}E^{\text{abs}}_j$, $j=1,\dots,n$ of the top/third row solutions given by \ref{['eq: abs_err']}.
  • ...and 7 more figures

Theorems & Definitions (9)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 3.1
  • Remark 3.2
  • Example 1
  • Example 2
  • Example 3
  • Remark 4.1