MSIQ: Moment-based Scale-Invariant Quality Measure for Single Image Super-Resolution

Technical Summary: MSIQ – Moment-Based Scale-Invariant Quality Measure for Single Image Super-Resolution

Problem Statement
The assessment of Single Image Super-Resolution (SISR) results remains a methodological challenge. Standard full-reference metrics, such as PSNR and SSIM, primarily evaluate pixel-wise errors or local structural similarities but fail to explicitly assess the preservation of global geometric structure. Furthermore, these metrics require the reference image ($I_{GT}$) and the super-resolved image ($I_{SR}$) to be resized to the same dimensions before comparison. This "forced resizing" introduces external interpolation errors that are unrelated to the SISR model's performance, thereby contaminating the evaluation. While perceptual metrics (e.g., LPIPS, DISTS) align better with human vision, they often lack geometric specificity, failing to distinguish between geometric deformations (e.g., anisotropic scaling, shear) and non-geometric artifacts (e.g., JPEG compression). The paper addresses the need for a diagnostic tool that evaluates geometric fidelity without requiring forced resizing and can distinguish geometric anomalies from other distortions.

Methodology
The proposed solution, MSIQ (Moment-based Scale-Invariant Quality), is a full-reference metric that compares images in the space of normalized central geometric moments rather than pixel space.

1. Mathematical Foundation: The method utilizes geometric moments of order $(p, q)$, defined as integrals (or sums for discrete images) of pixel intensities weighted by spatial coordinates. To achieve invariance, the authors employ normalized central geometric moments ($\nu_{pq}$), which are invariant to translation and uniform scaling in the continuous model.
2. Descriptor Construction: For an image $I$, a moment descriptor $M_N(I)$ is constructed as an ordered set of non-trivial normalized central moments up to a fixed order $N$. The paper fixes $N=4$, utilizing 12 informative moments (excluding trivial zero-order and first-order terms).
3. Metric Definition: MSIQ is defined as the distance between the moment descriptors of the ground-truth image ($I_{GT}$) and the super-resolved image ($I_{SR}$). Two variants are proposed:
   • MSIQ$_{RMSE}$: The root mean square error between corresponding moment components.
   • MSIQ$_{W}$: A weighted version that reduces the influence of higher-order moments, which are more sensitive to discretization noise.
4. Key Properties:
   • Resizing-Free: Since moments are scale-invariant, $I_{GT}$ and $I_{SR}$ can be compared directly at their native resolutions, eliminating interpolation errors.
   • Model-Free: The calculation is purely analytical and does not rely on pre-trained neural networks.
   • Geometric Specificity: The metric is designed to be invariant to uniform scaling but sensitive to anisotropic deformations, shear, rotation, and perspective distortions.

Key Contributions
The paper outlines four primary contributions:

1. Conceptual Framework: It proposes evaluating SISR results by verifying the integrity of scale-invariant moment descriptors, shifting the comparison from pixel space to a feature space that eliminates methodological dependence on forced resizing.
2. Analytical Measure: It introduces MSIQ as a deterministic, model-free metric that quantifies the stability of an image's moment-geometric structure under scale transformations.
3. Geometric Selectivity: Experimental validation demonstrates that MSIQ possesses high "geometric specificity," effectively separating geometric deformations from non-geometric artifacts like JPEG compression, a capability lacking in standard pixel-based and perceptual metrics.
4. Methodological Distinction: The paper distinguishes between two metric characteristics: tracking ability (the capacity to monotonically track increasing distortion strength) and geometric specificity (the ability to distinguish geometric distortions from non-geometric ones). It argues that high tracking ability does not imply high geometric specificity.

Experimental Results
The authors conducted four series of experiments to validate MSIQ:

• Scale Invariance: MSIQ remained stable (values $\approx 10^{-7}$ to $10^{-6}$) under uniform scaling without resizing. In contrast, PSNR and SSIM showed significant instability (median spread of 5.28 dB for PSNR) when forced resizing was applied, with extreme values occurring in degenerate cases.
• Geometric Specificity: In controlled tests involving anisotropic scaling, shear, rotation, and perspective distortion versus JPEG compression, MSIQ demonstrated a geometric-to-JPEG response ratio ($RM$) of approximately 71 at moderate degradation ($\lambda=0.20$) and 161 at weak degradation ($\lambda=0.05$). Conversely, perceptual metrics like DISTS showed an $RM$ near 1, failing to distinguish geometric errors from compression.
• SISR Algorithm Validation: When applied to outputs from both classical interpolators and Deep Neural Network (DNN) models (e.g., EDSR, FSRCNN), MSIQ maintained high geometric specificity ($RM \approx 122$) and architectural stability (variations < 5% between algorithm classes).
• Benchmark Validation: On standard datasets (Set5, Set14, Urban100), MSIQ preserved its geometric specificity ($RM \approx 100$), confirming its robustness across diverse image content.

Significance and Claims
The paper positions MSIQ not as a replacement for general-purpose metrics (PSNR, SSIM, LPIPS) but as a complementary diagnostic tool. Its significance lies in addressing a specific niche: the objective control of structural integrity and geometric fidelity.

The authors claim that MSIQ is particularly critical for domains where geometric correctness is a priority over perceptual naturalness, such as medical imaging, remote sensing, and technical visualization. In these fields, preserving object proportions and spatial organization is essential, and standard metrics may fail to detect subtle geometric anomalies that are invisible to pixel-wise error measures or perceptual models. The paper concludes that MSIQ fills a gap in the evaluation paradigm by providing a scale-invariant, resizing-free, and geometry-specific assessment of SISR reconstruction quality.