Arbitrary-resolution Super-resolution (ARSR)

• ARSR is the process of generating high-resolution images or data from low-resolution inputs at any user-defined scale by learning continuous, resolution-invariant mappings.
• ARSR methods employ implicit neural decoders, continuous field techniques like Gaussian splatting, and transformer-based models to enable seamless zooming and adaptive reconstruction.
• ARSR integrates frequency-aware losses and adaptive training strategies to achieve state-of-the-art fidelity and efficiency across applications such as computer vision and medical imaging.

Arbitrary-resolution Super-resolution (ARSR) is the computational task of reconstructing high-resolution (HR) representations of images, volumes, or scientific data from low-resolution (LR) inputs at arbitrary, user-specified output resolutions. Unlike traditional super-resolution models restricted to a fixed set of integer scaling factors (e.g., ×2, ×4), ARSR systems are designed to allow inference at any real-valued, possibly non-integer scaling—enabling seamless zooming, adaptive resampling, and augmentation for diverse applications spanning computer vision, scientific computing, and medical imaging.

1. Foundational Principles and Problem Formulation

The ARSR task is formally posed as learning a continuous (or resolution-invariant) mapping

$\mathcal{F} : \mathcal{X} \rightarrow \mathcal{Y}$

where $\mathcal{X}$ and $\mathcal{Y}$ are typically Banach spaces of functions or images defined over potentially different spatial domains, and the solution is evaluated on arbitrary discretizations, i.e., the grids corresponding to the user’s choice of target resolution. This operator learning perspective generalizes the classical SR paradigm to arbitrary grids and sizes (Luo et al., 2024).

Core ARSR architectures are constructed to be scale-agnostic, either by directly parameterizing a function that predicts output at any input query location (implicit representations) or by generating explicit continuous fields from which arbitrary-resolution images can be sampled efficiently (e.g., Gaussian or “Brane” splatting) (Federico et al., 28 Jun 2026, Chen et al., 12 Jan 2025, Peng et al., 9 Mar 2025). Compared to scale-specific SR, ARSR imposes additional algorithmic and computational constraints:

• Networks must generalize gracefully across both interpolation and significant extrapolation ranges of scaling.
• Primitive upsampling modules (e.g., pixel-shuffle) must be replaced by mechanisms supporting continuous, high-fidelity reconstruction.
• Data augmentations, losses, and architectures must be robust to the ill-posedness introduced by supporting a wide, continuous family of target scales.

2. Architectures and Methodologies

2.1 Implicit Neural Representations (INR) and Coordinate-based Decoders

A dominant approach involves learning an implicit decoder $f_\theta$ that, given an LR feature code and a continuous query coordinate (and possibly a scale parameter), predicts the HR pixel intensity at that location: $$I_\text{HR}(q; s) = f_\theta\left(\varphi(\mathbf{X}_\text{LR}), q, s\right)$$ where $\varphi(\cdot)$ is a feature encoder (Tsai et al., 20 Nov 2025, Wu et al., 2021, Li et al., 2024). The decoder may be an MLP (as in LIIF or variants), or employ more sophisticated modulations and positional encoding for spatial expressivity. In medical and scientific ARSR, this philosophy naturally extends to 3D or even higher-dimensional grids (Wu et al., 2021, Zhang et al., 2023).

2.2 Gaussian Splatting and Kernel Field Models

Explicit continuous field methods construct an intermediate representation as a sum of localized analytical kernels—typically anisotropic Gaussians, or generalized “Brane” primitives with polynomial/Hermite basis expansion: $f_c(x, y) = \sum_{n=1}^N G_n(x, y)$ with each $G_n$ parameterized by feature-conditioned means, covariances, color/opacity, and, in advanced cases, higher-order coefficients capturing local structure (Federico et al., 28 Jun 2026, Chen et al., 12 Jan 2025, Peng et al., 9 Mar 2025). These fields are rasterized into any grid size via efficient (often CUDA-based) rasterization, circumventing the need for per-pixel neural querying.

2.3 Adaptive Resource Allocation and Hierarchical Structures

Recent advancements focus on allocating model capacity non-uniformly by predicting the number and fidelity of continuous primitives as a function of local image complexity. For instance, QuADA-GS adaptively densifies Gaussian primitives using a neural routing architecture and enforces a global resource budget via structure-tensor-driven targets. Neighbor communication among primitives in these non-grid topologies is achieved via Hierarchical Pointer Convolution, enabling efficient O(1) kernel lookup (Federico et al., 28 Jun 2026).

Hierarchical and Laplacian frameworks (e.g., ASDN) pre-compute a finite set of discrete-scale reconstructions, then interpolate between levels for fine-grained resolution control, using recursive strategies for extreme upscaling (Shen et al., 2020).

2.4 Transformer and State-space Paradigms

Transformer-based ARSR injects scale-awareness and coordinate queries into global self-attention or cross-attention modules—either as explicit plugins (e.g., ARIS (Zhou et al., 2022)) or as dynamic backbones with adaptive routing for computational efficiency (TADT (Xu et al., 2024)). The S³Mamba framework replaces traditional CNN/Transformer bottlenecks with modulated scalable state-space models, offering linear-complexity modeling of long-range dependencies at arbitrary scales (Xia et al., 2024).

2.5 Generative and Diffusion Models

Generative models extend ARSR to perceptual fidelity and uncertainty quantification. Self-cascaded diffusion models (e.g., CasArbi) decompose upsampling into a sequence of “atomic” stages (e.g., ×2 factors), employing coordinate-guided residual denoising at each phase and permitting coherent progressive refinement at any target resolution (Bang et al., 9 Jun 2025).

3. Loss Functions, Training, and Spectral Considerations

Loss design is critical for ARSR. Alongside traditional spatial L1/L2 losses, recent methods introduce frequency-aware or adaptive spectral priors:

• Frequency-domain loss priors derived from spectral resizing and Fourier-analysis dynamically reweight pixel or frequency contributions to match the distribution of lost and preserved information at different scales (Luo et al., 2024, Fang et al., 2022).
• Deep reinforcement learning can be employed to adaptively split feature recovery efforts according to statistical spectral analysis of LR–HR block pairs (Fang et al., 2022).
• Perceptual losses (adversarial, VGG, LPIPS) are selectively deployed, especially in face SR, for photorealism at extreme scales (Tsai et al., 20 Nov 2025, Tan et al., 2020).

Curriculum or mixed-distribution scale scheduling is widely used: networks are trained to randomly sample a scale per iteration from a continuous range (uniform in [1.1,4.0], etc.), or to sequence training from easy/integer scales to arbitrary ones to improve stability (Zhang et al., 2023, Bang et al., 9 Jun 2025).

Correction for real-world degradations employs dual-level or deformable implicit representations: global appearance embeddings compensate photometric shifts, while pixel-wise predicted residuals handle spatially variant camera or ISP aberrations, supervised via dedicated branches (Li et al., 2024).

4. Evaluation, Benchmarks, and Empirical Results

ARSR methods are evaluated on both canonical and real-world datasets across a range of in- and out-of-distribution scales:

• Standard vision sets: DIV2K, Set5, Set14, Urban100, BSD100, Manga109, and various real smartphone, hyperspectral, and thermal data (Xie et al., 7 Aug 2025).
• Medical imaging: MRI (fastMRI, IXI, BraTS), with metrics including PSNR, SSIM, and LPIPS, and, for downstream tasks, Dice/HD95 for segmentation (Wu et al., 2021, Zhang et al., 5 Aug 2025, Zhang et al., 2023).

Benchmarks confirm that recent adaptive splatting (RBS (Federico et al., 28 Jun 2026), QuADA-GS (Federico et al., 28 Jun 2026)), dual-level or frequency-domain ARSR, and state-space/Transformer backbones yield SOTA quality, particularly at non-integer or extreme upscaling factors, with competitive or superior runtime and resource usage compared to implicit methods (see summary tables within each work).

5. Efficiency, Scalability, and Practical Implications

ARSR methods diverge significantly in computational characteristics:

• Naïve INR-based decoders require per-pixel MLP query, incurring substantial latency at large HR outputs or numerous scales.
• Explicit continuous field methods (GS, Brane splatting, ContinuousSR) decouple field construction (single pass) from rendering, enabling ultra-fast ($\sim$1 ms/scale) rasterization at any resolution; dynamic densification further optimizes resource allocation (Federico et al., 28 Jun 2026, Peng et al., 9 Mar 2025).
• Transformer-based and state-space models with efficient routing or linear-complexity state updates (e.g., S³Mamba) reduce both memory and runtime by 2× compared to previous Transformer ARSR (Xia et al., 2024).
• Diffusion ARSR models employ cascaded/progressive inference and achieve strong distortion–perceptual trade-off with moderate runtime overhead by comparison to regression-based approaches (Bang et al., 9 Jun 2025).

Practical deployment is facilitated by the capacity of ARSR models to:

• Generalize to all scales in the training interval and beyond (even up to ×30), due to scale- and coordinate-conditioning.
• Achieve state-of-the-art fidelity (e.g., +0.5–0.9 dB PSNR over prior INR or meta-learning baselines).
• Support clinical/diagnostic workflows where arbitrary zoom is imperative and pretraining at every scale is infeasible (Zhang et al., 2023, Zhang et al., 5 Aug 2025).
• Integrate plug-and-play modules into legacy fixed-scale super-resolvers for immediate ARSR capability extension (Zhou et al., 2022, Wang et al., 2020).

6. Extensions, Limitations, and Research Directions

Active challenges and open questions in ARSR research include:

• Robustness to severely out-of-domain inputs, e.g., high anisotropy, new acquisition modalities, or nonstationary degradations.
• Model scalability beyond 2D to high-dimensional data (3D MRI, video, multispectral); development of efficient 3D field primitives and spatially-aware neural architectures remains ongoing (Wu et al., 2021, Zhang et al., 2023, Zhang et al., 5 Aug 2025).
• Further reduction of per-query overhead, especially for real-time interactive applications or processing large scientific datasets.
• Integration of stronger priors (generative or semantic), hybrid field/implicit/diffusion approaches, and geometric invariances (e.g., scale, rotation, shearing) to support fully general tasks (Xie et al., 7 Aug 2025).
• Extension to multitask pipelines—joint SR+segmentation, cross-modal or multimodal fusion, and global–local reasoning (Zhang et al., 5 Aug 2025, Li et al., 2024).
• Theoretical analysis of operator consistency and out-of-sample behavior, especially under hierarchical or adaptive primitive allocation frameworks.

A plausible implication is that ARSR will become a baseline capability in diverse computational pipelines—enabling continuous zoom, arbitrarily-precise resampling, and adaptive multiscale processing wherever digital imaging is required.