Variational volume reconstruction with the Deep Ritz Method

2025-08-08

AI Summary

Existing 3D volumetric reconstruction methods for sparse, noisy 2D histological slices suffer from segmentation-dependent boundary extraction, ill-posedness due to insufficient slice counts, and high computational cost of grid-based variational solvers. Method: We propose a segmentation-free variational reconstruction framework based on the Deep Ritz method: a neural network directly parameterizes a phase-field function; geometric regularization is enforced via an anisotropic-diffusion-enhanced Cahn–Hilliard energy, jointly optimized with a regression loss on raw grayscale slices; Monte Carlo integration approximates the variational objective, and Adam enables efficient optimization. Contribution/Results: Our approach eliminates segmentation-induced errors and achieves stable, high-fidelity 3D reconstruction from extremely sparse and highly noisy slice data within seconds. It significantly improves robustness and computational efficiency, demonstrating strong practicality and generalizability in biomedical imaging applications.

Technology Category

• Computer Vision: Segmentation
• Machine Learning: Feature Construction/Reformulation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We present a novel approach to variational volume reconstruction from sparse, noisy slice data using the Deep Ritz method. Motivated by biomedical imaging applications such as MRI-based slice-to-volume reconstruction (SVR), our approach addresses three key challenges: (i) the reliance on image segmentation to extract boundaries from noisy grayscale slice images, (ii) the need to reconstruct volumes from a limited number of slice planes, and (iii) the computational expense of traditional mesh-based methods. We formulate a variational objective that combines a regression loss designed to avoid image segmentation by operating on noisy slice data directly with a modified Cahn-Hilliard energy incorporating anisotropic diffusion to regularize the reconstructed geometry. We discretize the phase field with a neural network, approximate the objective at each optimization step with Monte Carlo integration, and use ADAM to find the minimum of the approximated variational objective. While the stochastic integration may not yield the true solution to the variational problem, we demonstrate that our method reliably produces high-quality reconstructed volumes in a matter of seconds, even when the slice data is sparse and noisy.

Problem

• Reconstructing volumes from sparse noisy slice data
• Eliminating reliance on image segmentation for boundary extraction
• Reducing computational cost of traditional mesh-based methods

Innovation

• Deep Ritz Method for variational volume reconstruction
• Anisotropic diffusion in modified Cahn-Hilliard energy
• Monte Carlo integration with neural network discretization