GSLR-Based Multi-dimensional Recovery

• The paper demonstrates that unsupervised GSLR-based recovery, utilizing adaptive Gaussian splatting, effectively reconstructs high-dimensional images with enhanced local detail.
• It leverages continuous, probabilistic low-rank models that capture spatial-spectral consistency, outperforming traditional discrete tensor decompositions.
• The approach supports various tasks such as inpainting, denoising, and 3D reconstruction, validated by strong quantitative benchmarks and multi-modal applications.

Unsupervised GSLR-based Multi-dimensional Image Recovery denotes a class of methodologies for reconstructing or enhancing high-dimensional image data from incomplete, noisy, or degraded measurements, using generalized structured low-rank (GSLR) models with no ground-truth supervision. These models replace classical fixed-basis or discrete tensor decompositions with adaptive, continuous, or probabilistic low-rank parameterizations, typically leveraging Gaussian splatting, Bayesian hierarchical priors, or implicit group-sparsity in latent representations. The core advances target capturing local high-frequency spatial/spectral structure, joint multi-dimensional consistency, and data-driven regularization, providing state-of-the-art recovery for tasks such as inpainting, denoising, demosaicking, and general inverse problems across domains including RGB/MSI images, medical imaging, and 3D multi-view geometry.

1. Foundations and Problem Setting

Multi-dimensional image recovery seeks to reconstruct data tensors $$\mathcal X \in \mathbb R^{n_1 \times n_2 \times \cdots \times n_d}$$ (e.g., color images, hyperspectral cubes, videos, or geometrically-consistent multi-view collections) from partial or indirect observations. Classical approaches exploit low-rank tensor structures (e.g., t-SVD, Tucker, CP), but impose limitations: (1) latent structures are parameterized on discrete meshgrids, losing flexibility in continuous domains, and (2) fixed transforms (e.g., DFT/DCT) cannot adaptively encode sharp, local variations.

Generalized structured low-rank (GSLR) recovery, especially in the unsupervised regime, extends this paradigm. GSLR frameworks impose continuous or adaptive low-rank priors—often via Gaussian splatting fields, learned function factorizations, or Bayesian hierarchies—by modeling the observed tensor $\mathcal{O}$, sampling mask $\mathcal{M}$, and constructing a data-fidelity objective together with regularizers that enforce spatial/spectral low-rankness, smoothness, or sparsity, all without reliance on ground-truth targets or paired data (Zeng et al., 18 Nov 2025, Luo et al., 2022, Glaubitz et al., 2022, Hong et al., 2021, Sultan et al., 2 Jan 2025).

2. Gaussian Splatting-based Low-Rank Representations

Gaussian splatting-based low-rank tensor representation (GSLR) replaces both the latent tensor and the transform basis with parameterizations based on continuous Gaussian primitives. This overcomes the limitations of discrete grid approximations and fixed spectral bases.

2D Gaussian Splatting (Latent Tensor):

A spatial field $\mathcal{A}(x,y)$ is modeled as the sum of $N$ anisotropic Gaussian kernels: $$\mathcal{A}(x,y) = \sum_{j=1}^{N} \exp\left(-\frac12([x,y]^T-\boldsymbol{\mu}_j)^T\boldsymbol{\Sigma}_j^{-1}([x,y]^T-\boldsymbol{\mu}_j)\right)\mathbf{c}_j$$ where each primitive is parameterized by mean $\boldsymbol{\mu}_j$, covariance $\boldsymbol{\Sigma}_j$, and feature vector $\mathbf{c}_j$.

1D Gaussian Splatting (Transform Matrix):

Each column $r$ of the transform matrix $\mathbf{T}(z,r)$ is defined by $K$ 1D Gaussian functions: $$\mathbf{T}(z,r) = \sum_{k=1}^K c_k^r \exp\left(-\frac{(z-\mu_k^r)^2}{2(\sigma_k^r)^2}\right)$$ allowing continuous, adaptive representation along the spectral (mode-3 fiber) dimension.

By expressing $\mathcal{X} = \mathcal{A} \times_3 \mathbf{T}$, this approach achieves a compact, highly adaptive model of multi-dimensional images, leading to strong preservation of local high-frequency content and robust completion from severely incomplete or corrupted data (Zeng et al., 18 Nov 2025).

3. Unsupervised Recovery and Optimization

Unsupervised GSLR-based recovery forms an optimization problem over the parameters of the Gaussian splatting primitives (means, covariance, weights) and possibly additional transform parameters, with no ground-truth supervision. The canonical objective is: $$\min_{\theta_{\mathcal{A}}, \theta_{\mathbf{T}}} \|\mathcal{M} \odot (\mathcal{O} - \mathcal{A} \times_{3} \mathbf{T})\|_F^2 + \lambda \sum_{i=1}^{R} \|\mathcal{A}_{[i]}\|_*$$ where the nuclear norm enforces low-rankness in each frontal slice, and $\lambda$ is a tunable parameter. Optimization is performed via stochastic gradient descent (Adam), leveraging the differentiability of all components. The method supports a variety of missing/irregular sampling settings (random, block, tube, or slice-missing), and the continuous splatting parameterization excels at reconstructing fine spatial and spectral details that elude conventional t-SVD approaches (Zeng et al., 18 Nov 2025).

In related frameworks, unsupervised GSLR principles are manifested through:

• Tensor Function Factorizations: LRTFR parameterizes the data as $$f(x,y,z) = \mathcal C \times_1 f_x(x) \times_2 f_y(y) \times_3 f_z(z)$$, where each factor is a small MLP mapping coordinates to low-rank factors, enforcing continuous analytic structure beyond grid-based tensors (Luo et al., 2022).
• Score-based Generative Models: HGM constructs a score network in high-dimensional assist space and restores via annealed Langevin sampling and quadratic data-consistency updates, often involving multi-scale channel-copies or pixel-pooling to improve the statistical properties of score estimation (Hong et al., 2021).

4. Bayesian and Group-Sparsity GSLR Formulations

Bayesian GSLR models extend recovery with adaptive, hierarchical priors, naturally yielding uncertainty quantification and data-driven regularization.

Generalized Sparse Bayesian Learning: This framework posits a conditionally Gaussian prior on $x$, with precision hyperparameters driven by Gamma hyperpriors. The EM or coordinate-descent algorithm alternately updates the reconstruction, noise, and sparsity hyperparameters, thereby:

• Eliminating the need for hand-tuned regularization constants,
• Accommodating arbitrary linear forward models and sparsifying transforms,
• Providing full posterior distributions (mean and covariance) for uncertainty assessment.

Concretely,

$$p(x|\beta) \propto \det(B)^{1/2} \exp\left\{-\frac12 x^T R^T B R x\right\}$$

with $B = \mathrm{diag}(\beta_1, \dots, \beta_K)$ and $R$ a generic analysis operator (identity, finite differences, block transforms). Updates are closed-form and fully unsupervised, supporting robust multi-dimensional image recovery under noise, blur, and missing data (Glaubitz et al., 2022).

Group-sparsity in Latent Codes (DISCUS): In deep image prior frameworks such as DISCUS, a group-sparsity penalty is applied to the codes underlying a generative network mapping to image frames. This induces automatic discovery of a low-dimensional temporal/dynamic manifold, unifying deep priors with structured GSLR constraints (Sultan et al., 2 Jan 2025).

5. Multi-view, Geometric, and High-dimensional Extensions

Unsupervised GSLR principles extend powerfully to geometric and multi-view contexts previously dominated by MLP-based approaches.

LLGS and Multi-view Consistency: LLGS extends 3D Gaussian Splatting to low-light, multi-view enhancement and 3D reconstruction by introducing M-Color decomposition: per-Gaussian color is factorized into a view-independent material component and a view-dependent illumination component, enabling both enhancement and multi-view-consistent rendering. All model components are learned in an unsupervised, end-to-end way with Retinex- and sharpness-inspired priors, and the method outperforms NeRF-based and two-stage pipelines in both fidelity and efficiency for robotic and dark-scene modeling (Wang et al., 24 Mar 2025).

High-dimensional Assisted Reparameterizations: HGM employs high-dimensional tensor transformations (channel-copy, pixel-pooling) to improve the statistical efficiency of score-based generative models for color image restoration, directly improving the accuracy of unsupervised Langevin-based recovery and allowing theoretic guarantees via reduced intrinsic dimension in score estimation (Hong et al., 2021).

6. Empirical Benchmarks and Performance

Quantitative benchmarks across diverse modalities consistently demonstrate the superiority of unsupervised GSLR-based approaches over classical and deep learning baselines.

Methodology	Modality	PSNR (dB)	SSIM	Notes
GSLR (Zeng et al., 18 Nov 2025)	RGB/MSI Inpainting	21.68–43.0	0.57–0.99	Sharper local details, robust to slice/tube missing
LLGS (Wang et al., 24 Mar 2025)	Dark 3D MV Recovery	23.92	0.94	Multi-view, real-time, superior feature matching
HGM (Hong et al., 2021)	Demosaicking/Inpaint	38.8–40.9	0.97–0.98	Outperforms NCSN, Klatzer, Liu et al.
DISCUS (Sultan et al., 2 Jan 2025)	Dynamic MRI	−28.0 (NMSE)	0.978	Unsupervised, group sparsity, state-of-art in clinical/phantom MRI

Results show marked advantages in recovering local high-frequency content, spectral curve fidelity, geometric/temporal consistency, and training/inference efficiency over state-of-the-art alternatives (including CS, L+S, t-SVD, DIP, and classic deep supervision).

7. Limitations and Research Directions

Unsupervised GSLR-based multi-dimensional image recovery, while widely effective, maintains several limitations:

• Optimization hyperparameters (e.g., nuclear-norm weights, network width/depth, splatting kernel count) may require task-dependent tuning.
• Static-scene and fixed-illumination assumptions can restrict applicability; handling time-varying or dynamic scenes is an active area for temporal and self-supervised extensions (Wang et al., 24 Mar 2025, Zeng et al., 18 Nov 2025).
• MLP- or splatting-based parameterizations introduce model overhead relative to simplest analytic low-rank or Bayesian estimators.
• Interpretability of learned representations, especially in score-based or nonparametric settings, remains an open research topic.

Future directions include multi-modal/multi-spectral fusion, self-supervised transfer to dynamic/deformable scenes, joint semantic recovery, and further integration of data-driven low-rank, geometric, and uncertainty-quantified inference across scales from 2D signals to complex 3D and temporal data (Wang et al., 24 Mar 2025, Zeng et al., 18 Nov 2025, Luo et al., 2022).