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Imaging Signal Recovery Using Neural Network Priors Under Uncertain Forward Model Parameters

Xiwen Chen, Wenhui Zhu, Peijie Qiu, Abolfazl Razi

TL;DR

The paper tackles inverse imaging problems under forward-model parameter uncertainty by introducing Moment-Aggregation (MA), a framework that aggregates over a set of candidate forward parameters during training of a neural-network prior. MA constructs a moment-wise, gradient-stopped aggregation loss that preserves convexity and converges to the same optimum as if the precise parameter were known. Theoretical guarantees accompany proof-of-concept experiments in compressive sensing and phase retrieval, showing MA achieves near-upper-bound reconstruction quality across multiple datasets while offering computational efficiency over training multiple models. This approach holds practical promise for robust imaging in medical and scientific applications where forward-model calibration is imperfect or drift-prone.

Abstract

Inverse imaging problems (IIPs) arise in various applications, with the main objective of reconstructing an image from its compressed measurements. This problem is often ill-posed for being under-determined with multiple interchangeably consistent solutions. The best solution inherently depends on prior knowledge or assumptions, such as the sparsity of the image. Furthermore, the reconstruction process for most IIPs relies significantly on the imaging (i.e. forward model) parameters, which might not be fully known, or the measurement device may undergo calibration drifts. These uncertainties in the forward model create substantial challenges, where inaccurate reconstructions usually happen when the postulated parameters of the forward model do not fully match the actual ones. In this work, we devoted to tackling accurate reconstruction under the context of a set of possible forward model parameters that exist. Here, we propose a novel Moment-Aggregation (MA) framework that is compatible with the popular IIP solution by using a neural network prior. Specifically, our method can reconstruct the signal by considering all candidate parameters of the forward model simultaneously during the update of the neural network. We theoretically demonstrate the convergence of the MA framework, which has a similar complexity with reconstruction under the known forward model parameters. Proof-of-concept experiments demonstrate that the proposed MA achieves performance comparable to the forward model with the known precise parameter in reconstruction across both compressive sensing and phase retrieval applications, with a PSNR gap of 0.17 to 1.94 over various datasets, including MNIST, X-ray, Glas, and MoNuseg. This highlights our method's significant potential in reconstruction under an uncertain forward model.

Imaging Signal Recovery Using Neural Network Priors Under Uncertain Forward Model Parameters

TL;DR

The paper tackles inverse imaging problems under forward-model parameter uncertainty by introducing Moment-Aggregation (MA), a framework that aggregates over a set of candidate forward parameters during training of a neural-network prior. MA constructs a moment-wise, gradient-stopped aggregation loss that preserves convexity and converges to the same optimum as if the precise parameter were known. Theoretical guarantees accompany proof-of-concept experiments in compressive sensing and phase retrieval, showing MA achieves near-upper-bound reconstruction quality across multiple datasets while offering computational efficiency over training multiple models. This approach holds practical promise for robust imaging in medical and scientific applications where forward-model calibration is imperfect or drift-prone.

Abstract

Inverse imaging problems (IIPs) arise in various applications, with the main objective of reconstructing an image from its compressed measurements. This problem is often ill-posed for being under-determined with multiple interchangeably consistent solutions. The best solution inherently depends on prior knowledge or assumptions, such as the sparsity of the image. Furthermore, the reconstruction process for most IIPs relies significantly on the imaging (i.e. forward model) parameters, which might not be fully known, or the measurement device may undergo calibration drifts. These uncertainties in the forward model create substantial challenges, where inaccurate reconstructions usually happen when the postulated parameters of the forward model do not fully match the actual ones. In this work, we devoted to tackling accurate reconstruction under the context of a set of possible forward model parameters that exist. Here, we propose a novel Moment-Aggregation (MA) framework that is compatible with the popular IIP solution by using a neural network prior. Specifically, our method can reconstruct the signal by considering all candidate parameters of the forward model simultaneously during the update of the neural network. We theoretically demonstrate the convergence of the MA framework, which has a similar complexity with reconstruction under the known forward model parameters. Proof-of-concept experiments demonstrate that the proposed MA achieves performance comparable to the forward model with the known precise parameter in reconstruction across both compressive sensing and phase retrieval applications, with a PSNR gap of 0.17 to 1.94 over various datasets, including MNIST, X-ray, Glas, and MoNuseg. This highlights our method's significant potential in reconstruction under an uncertain forward model.
Paper Structure (10 sections, 1 theorem, 14 equations, 7 figures, 2 tables, 1 algorithm)

This paper contains 10 sections, 1 theorem, 14 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. Method
  5. Preliminaries
  6. Moment-Aggregation Training Framework
  7. Experiment
  8. Standard CS problem
  9. Applications in Phase Retrieval
  10. Discussion and Conclusion

Key Result

Theorem 1

At each moment, a loss has the following format is an MA loss: where Here, $H(i; F(\boldsymbol{x}; \theta_1),\cdots, F(\boldsymbol{x}; \theta_{n_c}) )$ is the function to calculate the weight for each candidate based on the surrogate qualities at each moment (i.e. $\omega_i$ will be updated at each iteration). Stopping gradient means when the neural network performs

Figures (7)

  • Figure 1: The illustration of the Moment-Aggregation (MA) framework for IIPs with a neural network that considers the effect of all possible candidate parameters of the forward model simultaneously. MA loss is constructed after every forward propagation (we call this time point a "moment") and then is used to update parameters in backward propagation. Left: The losses by candidate forward model parameters, and one of them is the precise parameter. Their labels are unknown (i.e. the precise or not precise) during training. Right: The loss at different moments by MA. The loss is moment-wise convex$/$smooth, and the overall training can achieve the global minima as reconstruction using the precise parameter.
  • Figure 2: The typical workflow of IIPs. First, a forward model is applied to a signal to obtain the measurement. Then the measurement is used to reconstruct the original signal via machine learning (ML) or deep learning (DL) algorithms.
  • Figure 3: Using CS-DIP to reconstruct $\boldsymbol{x}_0$ with measurement $\boldsymbol{y}$. Left: The signal is successfully reconstructed under the forward model with the precise parameter while failing under a wrong parameter. Right: Our method can successfully reconstruct $\boldsymbol{x}_0$ by optimizing under a set of candidate parameters (we assume one of them is close to the precise parameter).
  • Figure 4: Left: Derivative of $y$ w.r.t $a_1$ without gradient stopping. Right: Derivative with gradient stopping.
  • Figure 5: Comparison of different reconstruction strategies in Left: MNIST dataset when the $m$ is 200. Right: X-ray dataset when $m$ is 2000 (total pixel is 65536).
  • ...and 2 more figures

Theorems & Definitions (5)

  • definition 1
  • definition 2
  • Theorem 1
  • Remark 1
  • proof