FIRM: Federated Image Reconstruction using Multimodal Tomographic Data

TL;DR

The paper addresses robust tomographic image reconstruction from dispersed multimodal data under privacy constraints by formulating a joint constrained least-squares problem with a global multimodal coupling $w_N = \sum_{i=1}^{N-1} c_i w_i$. It introduces FIRM, a federated algorithm that replaces expensive server projections with lightweight vector operations and demonstrates a foundational connection to the quadratic-penalty method, enabling an adaptive, sublinear convergence rate; it also presents FIRM$^+$ as an augmented-Lagrangian extension. Theoretical results establish descent, convergence to KKT points, and rates $O(1/\epsilon^2)$ (with potential $O(1/\epsilon)$ under favorable conditions), while experiments show substantial computational gains (roughly 52x faster than FedPGD) and improved reconstruction quality from multimodal data compared to unimodal baselines, especially in low-noise regimes. This work enables privacy-preserving, scalable, and accurate multimodal tomographic reconstruction across distributed facilities, with practical implications for synchrotron and medical imaging contexts where data centralization is impractical.

Abstract

We propose a federated algorithm for reconstructing images using multimodal tomographic data sourced from dispersed locations, addressing the challenges of traditional unimodal approaches that are prone to noise and reduced image quality. Our approach formulates a joint inverse optimization problem incorporating multimodality constraints and solves it in a federated framework through local gradient computations complemented by lightweight central operations, ensuring data decentralization. Leveraging the connection between our federated algorithm and the quadratic penalty method, we introduce an adaptive step-size rule with guaranteed sublinear convergence and further suggest its extension to augmented Lagrangian framework. Numerical results demonstrate its superior computational efficiency and improved image reconstruction quality.

Figures (8)

• Figure 1: Illustration of the discrete tomographic geometry.
• Figure 1: The overview of FIRM. At each round $t$, $N$ agents (represented by blue circles) send their local solutions $\{v^t_i\}_{i=1}^N$ computed by the formulation \ref{['pgd_1']} to the server (represented by a red box). Then the server conducts a set of vector operations as in \ref{['our_approach']}. The resulting $w^{t+1}_i$ is then sent to the agent $i \in [N]$ which will be used as an initial point for local training in the next round.
• Figure 1: The ground-truth images.
• Figure 1: Visualizations of the tomographic multimodal datasets.
• Figure 1: Improvement achieved by PGD over PLSQR.

Theorems & Definitions (24)

• Remark 3.1: Less computation load at the server compared to FedPGD
• Proposition 3.2
• Proof 1
• Lemma 3.3
• Proposition 3.4
• Proof 2
• Remark 3.5
• Definition 3.6
• Definition 3.7
• Lemma 3.8