Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Warm-basis Method for Bridging Learning and Iteration: a Case Study in Fluorescence Molecular Tomography

Ruchi Guo, Jiahua Jiang, Bangti Jin, Wuwei Ren, Jianru Zhang

TL;DR

This work addresses the depth-resolution challenge in fluorescence molecular tomography by proposing WB-IPM, a warm-basis iterative projection method that integrates a data-driven warm basis from an Attention U-Net with a majorization-minimization projection framework. WB-IPM decomposes the solution as $\boldsymbol{x}=c\hat{\boldsymbol{x}}_{nn}+\boldsymbol{z}$ and solves in the orthogonal complement via the Augmented Flexible Golub-Kahan (AFGK) process, yielding a principled two-space refinement that preserves useful network information while enabling stable correction. Theoretical error guarantees show the reconstruction error depends on the angle between the true solution and the NN output, the noise level, and the chosen regularization; a weaker angle-based loss is proposed to reduce training effort without sacrificing downstream accuracy. Numerical and experimental results in FMT demonstrate that WB-IPM achieves superior depth localization and overall reconstruction accuracy compared with pure learning or iterative methods, with robust performance under noise and distribution shifts, making it a practical approach for large-scale inverse problems.

Abstract

Fluorescence Molecular Tomography (FMT) is a widely used non-invasive optical imaging technology in biomedical research. It usually faces significant accuracy challenges in depth reconstruction, and conventional iterative methods struggle with poor $z$-resolution even with advanced regularization. Supervised learning approaches can improve recovery accuracy but rely on large, high-quality paired training dataset that is often impractical to acquire in practice. This naturally raises the question of how learning-based approaches can be effectively combined with iterative schemes to yield more accurate and stable algorithms. In this work, we present a novel warm-basis iterative projection method (WB-IPM) and establish its theoretical underpinnings. The method is able to achieve significantly more accurate reconstructions than the learning-based and iterative-based methods. In addition, it allows a weaker loss function depending solely on the directional component of the difference between ground truth and neural network output, thereby substantially reducing the training effort. These features are justified by our error analysis as well as simulated and real-data experiments.

A Warm-basis Method for Bridging Learning and Iteration: a Case Study in Fluorescence Molecular Tomography

TL;DR

This work addresses the depth-resolution challenge in fluorescence molecular tomography by proposing WB-IPM, a warm-basis iterative projection method that integrates a data-driven warm basis from an Attention U-Net with a majorization-minimization projection framework. WB-IPM decomposes the solution as and solves in the orthogonal complement via the Augmented Flexible Golub-Kahan (AFGK) process, yielding a principled two-space refinement that preserves useful network information while enabling stable correction. Theoretical error guarantees show the reconstruction error depends on the angle between the true solution and the NN output, the noise level, and the chosen regularization; a weaker angle-based loss is proposed to reduce training effort without sacrificing downstream accuracy. Numerical and experimental results in FMT demonstrate that WB-IPM achieves superior depth localization and overall reconstruction accuracy compared with pure learning or iterative methods, with robust performance under noise and distribution shifts, making it a practical approach for large-scale inverse problems.

Abstract

Fluorescence Molecular Tomography (FMT) is a widely used non-invasive optical imaging technology in biomedical research. It usually faces significant accuracy challenges in depth reconstruction, and conventional iterative methods struggle with poor -resolution even with advanced regularization. Supervised learning approaches can improve recovery accuracy but rely on large, high-quality paired training dataset that is often impractical to acquire in practice. This naturally raises the question of how learning-based approaches can be effectively combined with iterative schemes to yield more accurate and stable algorithms. In this work, we present a novel warm-basis iterative projection method (WB-IPM) and establish its theoretical underpinnings. The method is able to achieve significantly more accurate reconstructions than the learning-based and iterative-based methods. In addition, it allows a weaker loss function depending solely on the directional component of the difference between ground truth and neural network output, thereby substantially reducing the training effort. These features are justified by our error analysis as well as simulated and real-data experiments.

Paper Structure

This paper contains 15 sections, 2 theorems, 48 equations, 13 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. MM approach
  4. Attention U-Net for producing warm basis
  5. Network prediction and iteration
  6. A Warm-basis iterative projection method
  7. Space decomposition
  8. AFGK iterative method \newlabelsec:AFGK0
  9. Analysis of the WB-IPM \newlabelsec:assessment0
  10. Numerical experiment
  11. Forward problem and data generation \newlabelsec:U-Net0
  12. Simulation results \newlabelsec:numerical_results0
  13. Experimental results \newlabelsec:exp_results0
  14. Conclusion
  15. Efficient QR update of $\boldsymbol{Z}_k$ \newlabelsec:eff-QR0

Key Result

Lemma 4.1

\newlabellem_DB0 Given any vector $\boldsymbol{v}$, there holds where $\theta_+$ is the smallest non-zero eigenvalue of $\boldsymbol{D}^{-1}\widetilde{\boldsymbol{B}}$.

Figures (13)

  • Figure 1: (a) Schematic illustration of FMT system: a laser beam scans the tissue sample from the bottom to excite fluorescent inclusions and emit fluorescence, and emitted photons propagate to the top surface and are collected by a camera. (b) Numerical simulation setup for FMT using a slab phantom: a $55\times55$ detector array (blue patches) is placed on the top surface to record photon intensity, while a $10\times10$ array of laser sources (red dots) illuminates the sample from the bottom surface.
  • Figure 1: Overview of the Attention U-Net for generating warm basis. The network adopts an encoder-decoder structure with attention-enhanced skip connections. The encoder extracts multiscale features through convolutional layers with ReLU, batch normalization, and max-pooling, while the decoder reconstructs the fluorescence distribution by integrating encoder features via self-attention blocks.
  • Figure 1: Relative reconstruction error of the experimental case for fixed $\alpha$ ($\alpha\in\{0.1,10,200,500\}$) and an iteration-adaptive choice $\alpha_k$ via WGCV (red triangles; $\alpha_{30}=155.4$). A small $\alpha$ achieves errors comparable to the WGCV schedule, whereas a large $\alpha$ over-regularizes and progressively degrades accuracy.
  • Figure 1: Three simulated cases: reconstructions from Attention U-Net, fHybr, and WB-IPM, with slices along the $z$-axis. fHybr yields wrong results in the first and fourth rows.
  • Figure 2: Comparison of a warm start method with fHybr iteration and WB-IPM still with fHybr iteration for numerical (top) and experimental (bottom) cases. Warm start risks degrading prior prediction, while WB-IPM ensures robustness and accuracy.
  • ...and 8 more figures

Theorems & Definitions (5)

  • Lemma 4.1
  • Proof 1
  • Theorem 4.2
  • Proof 2
  • Remark 4.3