Structured 3D-SVD: A Practical Framework for the Compression and Reconstruction of Biological Volumetric Images

Singular Value Decomposition (SVD) underlies matrix factorisation tasks across computational imaging, with medical applications increasingly demanding real-time processing. Yet SVD algorithms are inherently sequential, constraining real-time GPU throughput and limit online deployment in clinical pipelines. This study introduces Quasi-SVD, a differentiable, fully parallelized matrix factorization framework for GPUs. Rather than enforcing orthogonality on both factors, it guarantees exact orthogonality for a single Lie-parameterized factor while recovering the remaining components through soft constraints, enabling efficient parallel decomposition without iterative singular-vector optimization. This asymmetric design, provably sufficient for valid factorisation, achieves reconstruction fidelity of SSIM = 0.89-0.94 and accelerates computation by 3-20x relative to cuSOLVER and randomised SVD, enabling throughput above 25 FPS. Performance is evaluated on two medical imaging tasks spanning complementary computational regimes: (1) spatio-temporal background subtraction for ultrasound localisation microscopy, requiring high-dimensional matrix separation, and (2) Mueller matrix polarimetry for neurosurgical tissue characterisation, requiring massive batch processing of small matrices. Across both regimes and multiple imaging instruments, the proposed framework demonstrates robust domain transfer and throughput exceeding 25 FPS at clinical matrix scales, a rate sufficient for live image-guided workflows that classical solvers cannot currently support in these settings. By prioritising downstream reconstruction fidelity over exact spectral recovery, Quasi-SVD makes structured matrix factorisation practical for real-time imaging.

Recent work has shown that implicit neural representations (INRs) can be trained to effectively compress structured and unstructured volume data, allowing for direct data querying with a reduced memory footprint. However, as existing INRs for unstructured volumes do not encode geometry, they require partial mesh storage for later sampling, limiting achievable compression. At the same time, novel view synthesis methods have shown that explicit collections of 3D Gaussians can be used to accurately visualize volume data. In this work, we introduce an explicit model for volume data compression based on 3D Gaussian primitives. We reinterpret collections of 3D Gaussians as an explicit representation of a scalar field and use a sampling strategy that reconstructs scalar values at spatial locations through weighted aggregation of intersecting Gaussians. We develop optimized CUDA-accelerated pipelines for structured and unstructured model sampling, loss functions that encourage accurate domain encoding by our models, and a novel sampling-error based densification strategy. Our explicit formulation naturally encodes domain geometry, eliminating the need for mesh storage in unstructured volumes and introducing significantly higher compression opportunities. Compared to existing INRs, we demonstrate that our explicit model achieves competitive reconstruction quality with significant training speedups on structured volumes, while markedly outperforming in all metrics on unstructured volumes.

We investigate volumetric reconstruction for compressive sensing light-sheet microscopy (CS-LSM), where fast volumetric imaging is achieved by encoding multiple axial planes into each camera exposure. To recover the underlying volume from highly multiplexed measurements, we propose a plug-and-play (PnP) framework that flexibly incorporates any user-specified denoiser into the reconstruction process. Building on a slice-based formulation, we further introduce an axial-coupled model that exploits correlations between adjacent slices to improve volumetric continuity. For efficient computation, we derive a Woodbury-based update for the data-consistency step in both the slice-based and axial-coupled formulations, and employ a Gauss-Seidel sweep for the denoising step in the axial-coupled model. Under a weakly convex regularization assumption, we establish subsequential convergence of the proposed algorithm. Experiments on synthetic and real zebrafish-heart data demonstrate that the proposed framework successfully recovers cellular structures from compressed measurements, and provide practical insights into the comparative performance of commonly used denoisers within the PnP framework under the CS-LSM setup.