DAWN-SI: Data-Aware and Noise-Informed Stochastic Interpolation for Solving Inverse Problems

2024-12-06

arXiv.org

AI Summary

This work addresses ill-posed inverse problems in medical imaging and geophysics, aiming to robustly reconstruct parameters from noisy, undersampled observations while quantifying solution uncertainty. We propose a data-aware and noise-aware stochastic interpolation (SI) framework: for the first time, it explicitly incorporates observed data and noise level into the modeling of a time-varying velocity field, enabling end-to-end differentiable inverse solving; it further supports dynamic noise adaptation and task-specific training. Compared to pretrained diffusion models, our method achieves significant improvements in reconstruction accuracy on image deblurring and tomographic reconstruction tasks—particularly under low signal-to-noise ratios and severe undersampling—while maintaining strong robustness. Moreover, it simultaneously delivers high-fidelity reconstructions and principled modeling of multimodal solution uncertainty.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Inverse problems, which involve estimating parameters from incomplete or noisy observations, arise in various fields such as medical imaging, geophysics, and signal processing. These problems are often ill-posed, requiring regularization techniques to stabilize the solution. In this work, we employ $ extit{Stochastic Interpolation}$ (SI), a generative framework that integrates both deterministic and stochastic processes to map a simple reference distribution, such as a Gaussian, to the target distribution. Our method $ extbf{DAWN-SI}$: $ extbf{D}$ata-$ extbf{AW}$are and $ extbf{N}$oise-informed $ extbf{S}$tochastic $ extbf{I}$nterpolation incorporates data and noise embedding, allowing the model to access representations about the measured data explicitly and also account for noise in the observations, making it particularly robust in scenarios where data is noisy or incomplete. By learning a time-dependent velocity field, SI not only provides accurate solutions but also enables uncertainty quantification by generating multiple plausible outcomes. Unlike pre-trained diffusion models, which may struggle in highly ill-posed settings, our approach is trained specifically for each inverse problem and adapts to varying noise levels. We validate the effectiveness and robustness of our method through extensive numerical experiments on tasks such as image deblurring and tomography.

Problem

• Estimates parameters from noisy, incomplete observations.
• Enhances robustness in ill-posed inverse problems.
• Adapts to varying noise levels for accurate solutions.

Innovation

• Flow Matching integrates deterministic processes for distribution mapping
• Data and noise embedding enhances robustness in noisy scenarios
• Time-dependent velocity field enables accurate solutions and uncertainty quantification