PaperlayerView exact revision on arXiv

Source-linked AI summary

Local SVD-Entropy Maps as a Complementary Structural Representation for Full-Reference and No-Reference Image Quality Assessment

Andrei Velichko, Petr Boriskov

arXiv:2609.27959v1eess.IVcs.CVcs.MM

TL;DR

Conventional image-domain similarity may not capture all perceptually relevant manifestations of degradation. This paper evaluates direct local SVD entropy as a structural representation for full-reference and no-reference image quality assessment. The results show that it complements conventional SSIM rather than replacing it, with complementary information also observed without a pristine reference.

  • Problem

    Conventional image-domain similarity does not capture all perceptually relevant manifestations of degradation, motivating complementary representations.

    Source: §1 · p. 2

  • Method

    The paper introduces an interpretable structural representation based on direct local two-dimensional SVD-entropy maps and evaluates it across full-reference and no-reference quality assessment.

    Source: §1 · p. 4

  • Results

    HSVD-SSIM is weaker than SSIM for standalone full-reference prediction but provides complementary information when combined with SSIM and in no-reference prediction.

    Source: §1 · p. 5, §7 · p. 20

  • Takeaways & Limitations

    HSVD entropy is best understood as a complementary spectral-complexity channel, not a replacement for conventional SSIM.

    Source: §7 · p. 20

    Generalization to authentic mixed distortions has not been established; KADID severity levels are only a surrogate for successive degradation states.

    Source: §6 · p. 19

Abstract

from arXiv · show

We investigate a local spectral-complexity representation for perceptual image quality assessment (IQA) based on Shannon entropy of singular values computed directly from two-dimensional image patches. For each $3\times3$-pixel grayscale patch, SVD is applied directly and the normalized singular-value entropy defines one HSVD-map value. The construction requires neither flattening nor delay embedding, uses no boundary padding, and is invariant to $90^{\circ}$ rotations and mirror reflections at the local-descriptor level. A nested salt-and-pepper experiment on Lena separates absolute similarity to a clean reference from sensitivity to an additional degradation step. HSVD-SSIM responds more strongly to local corruption and retains a larger neighboring-state response at severe noise levels. Validation on all 10,125 distorted KADID-10k images shows that HSVD-SSIM is weaker than conventional SSIM as a standalone full-reference metric (SRCC $0.450$ vs. $0.619$), but complementary when combined with it: grouped cross-validation increases SRCC from $0.618$ to $0.659$, with a bootstrap 95\% confidence interval of $[0.036,0.046]$ for the gain. In a no-reference experiment, adding HSVD-derived single-image descriptors improves the best nonlinear model from SRCC $0.528$ to $0.575$ (95\% CI $[0.033,0.061]$) and also improves prediction of quality changes between neighboring distortion states. These results support direct local SVD entropy as an interpretable structural channel that complements conventional image-domain similarity and remains informative without a pristine reference.

1 Introduction

The paper introduces direct local two-dimensional SVD entropy as an interpretable structural channel for image quality assessment, testing its standalone, complementary, change-sensitivity, and no-reference value. It distinguishes absolute quality prediction from sensitivity to incremental degradation and evaluates both with a Lena experiment and KADID-10k.

Motivation

  • Absolute quality prediction and sensitivity to incremental changes are distinct: a representation can respond strongly to an added local degradation without being the best standalone predictor of absolute perceptual quality.The study examines these properties separately, including neighboring image states that are already severely degraded.Source: §1 · pp. 2, 4

Methodological progression

  • Unlike earlier entropy mapping that used circular-kernel traversal and sequence embedding, direct patch SVD removes scan-order design and derives rotation and reflection invariance from matrix algebra.The paper does not claim a measured speed advantage over the earlier implementation because the algorithms were not benchmarked under identical conditions.Source: §1 · pp. 2–3

Study design

  • Four tests assess standalone full-reference prediction, neighboring-state degradation sensitivity, complementarity with conventional SSIM, and no-reference prediction using HSVD-derived single-image statistics.KADID-10k supplies 81 source images, 25 distortion types, five severity levels, and subjective annotations for 10,125 distorted images; the nested salt-and-pepper Lena experiment exposes the representations’ physical response.Source: §1 · p. 4

Contributions

  • The proposed HSVD map computes normalized Shannon entropy from singular values of valid 3 × 3 patches, without flattening, delay embedding, or artificial boundary padding.Its local descriptor is invariant to right-angle rotations and mirror reflections by matrix algebra.Source: §1 · pp. 3–4
  • HSVD-SSIM is weaker than conventional SSIM as a standalone full-reference metric, but supplies statistically stable complementary information when combined with SSIM and adds useful no-reference information.The study also reports improved recovery of perceptual quality changes from independently predicted image-quality scores.Source: §1 · p. 5

2 Method

The method computes normalized singular-value entropy directly on local image patches, forming HSVD maps for comparison with SSIM or feature extraction without a pristine reference. The local descriptor is invariant to right-angle rotations, mirror reflections, and uniform multiplicative scaling under the stated conditions.

2.1 Direct two-dimensional local SVD entropy

  • Direct SVD of valid 3×3 grayscale patches yields an HSVD map of normalized singular-value entropy without boundary padding.A locally rank-one patch has entropy near zero; more evenly distributed singular values produce higher entropy.Source: §2.1 · p. 5

2.2 Rotation and reflection invariance of the local descriptor

  • Local HSVD values are invariant to 90° rotations, mirror reflections, and nonzero multiplicative scaling, but not additive offsets or arbitrary nonlinear contrast changes.Rotating an entire image rotates its map correspondingly; this does not make comparisons invariant to misregistration between images.Source: §2.2 · p. 6

2.3 Image-domain and entropy-domain SSIM

  • Conventional SSIM is evaluated on aligned grayscale images, while HSVD-SSIM applies the same SSIM operator to their HSVD maps.Source: §2.3 · p. 6

2.4 No-reference HSVD descriptors

  • No-reference prediction combines image-domain intensity, gradient, and texture features with analogous HSVD-map statistics, excluding reference identity, distortion type, and distortion level.Source: §2.4 · pp. 6–7

3 Experimental design

The experiments use a controlled nested-noise sequence and four KADID-10k studies to examine standalone full-reference quality, neighboring-level sensitivity, full-reference complementarity, and no-reference prediction. KADID scores retain their native higher-is-better orientation, and all images are converted to grayscale/luminance for analysis.

3.1 Illustrative nested salt-and-pepper-noise experiment

  • The illustrative experiment applies nested salt-and-pepper corruption to Lena, comparing each noisy state with the clean reference and each consecutive pair of states.A single pixel-location permutation makes each higher-density corruption contain the preceding level’s damaged pixels; densities run from 5% to 100% in 5% steps, isolating the effect of each added step.Source: §3.1 · p. 7

3.2 KADID-10k data and protocol

  • 10,125 distorted KADID-10k images span 81 references, 25 distortion types, and five levels; grayscale/luminance images use native higher-is-better dmos scores.Each distorted image received 30 ratings on a five-point impairment scale; the official dmos range is [1, 5], with larger values indicating higher visual quality.Source: §3.2 · p. 7
  • Study I correlates standalone SSIM and HSVD-SSIM scores for each distorted image against subjective scores using SRCC, PLCC, and Kendall correlation.For each distorted image D and reference R, the compared scores were Simg(R, D) and SHSVD(R, D).Source: §3.2 · p. 7
  • Study II evaluates metric-change sensitivity across 8,100 adjacent-level pairs and relates change magnitude to absolute DMOS differences.Adjacent nominal levels were paired sequentially within each reference/distortion chain; 1,000 cluster-bootstrap replicates resampled references while preserving their pairs.Source: §3.2 · p. 8
  • Study III predicts absolute DMOS from SSIM, HSVD-SSIM, or both using reference-grouped cross-validation and cluster-bootstrap uncertainty estimates.Evaluation used 10 repetitions of five-fold cross-validation grouped by reference, keeping all 125 distorted variants of each reference together in training or testing; uncertainty used 2,000 reference-level cluster resamples.Source: §3.2 · p. 8
  • Study IV predicts absolute DMOS from Image-only, HSVD-only, or Combined features with ridge regression and HistGradientBoosting, then uses out-of-fold predictions for adjacent-level changes.Both model families used five repetitions of five-fold cross-validation grouped by reference; reference-level bootstrap resampling retained all images from each sampled reference.Source: §3.2 · p. 8

4 Results

Across KADID-10k, HSVD-derived information complements conventional SSIM in full-reference prediction and improves no-reference prediction, although HSVD-SSIM is weaker than SSIM alone as a standalone metric. Nested-noise and neighboring-level results show stronger responses to additional local structural changes, while illustrative cases show HSVD-SSIM can distinguish images with nearly matched SSIM.

4.1 Nested noise: absolute similarity and incremental sensitivity

  • At 5% nested corruption, clean-reference SSIM was 0.334 while HSVD-SSIM was 0.137; at high noise, HSVD retained a larger neighboring-state response.Across the five high-noise transitions, mean dissimilarity was 0.0330 for image-domain SSIM and 0.1131 for HSVD, approximately 3.42 times larger; no human opinion scores were collected for this illustrative experiment.Source: Figure 3 · p. 10, §4.1 · p. 10, Figure 4 · p. 11

4.2 KADID-10k Study I: HSVD-SSIM is not a standalone replacement for SSIM

  • Across 10,125 distorted KADID images, conventional SSIM outperformed HSVD-SSIM in global SRCC (0.619 vs. 0.450) and for 23 of 25 distortion types.This establishes HSVD-SSIM as a complementary representation rather than a standalone replacement for SSIM.Source: Table 3 · p. 11, §4.2 · p. 11

4.3 KADID-10k Study II: neighboring degradation levels

  • For 8,100 adjacent-level pairs, HSVD-change SRCC with |∆DMOS| was 0.1338 versus 0.1115 for SSIM-change; the difference’s 95% CI was [+0.0134, +0.0304].Correlations remained modest, so the result supports incremental sensitivity rather than strong standalone perceptual prediction.Source: Table 4 · p. 12, §4.3 · p. 12

4.4 Illustrative KADID cases with nearly matched conventional SSIM

  • In two post-hoc examples with nearly identical SSIM values, HSVD-SSIM differed substantially and matched the subjective quality ordering.For I22.png, SSIM was 0.934090 and 0.934098 while HSVD-SSIM was 0.6695 and 0.9036; for I18.png, SSIM was 0.933774 and 0.933802 while HSVD-SSIM was 0.5785 and 0.8411.Source: §4.4 · p. 12, Figure 5 · p. 13

4.5 KADID-10k Study III: full-reference complementarity

  • SRCC rose from 0.618 (SSIM-only) to 0.659 (SSIM+HSVD), a +0.0414 gain; the grouped-fold bootstrap 95% CI for ΔSRCC was [+0.0363, +0.0462].The two FR scores are strongly related; HSVD acts as a correction channel containing residual information not represented by SSIM alone.Source: Table 5 · p. 13, §4.5 · pp. 13–14

4.6 KADID-10k Study IV: no-reference prediction

  • For no-reference HGB prediction, Combined features raised SRCC from 0.5277 (Image-only) to 0.5748; the gain’s bootstrap 95% CI was [+0.0332, +0.0611].No pristine image, distortion identity, type, or level was used as a predictive feature; Combined also improved Ridge SRCC from 0.4157 to 0.4884.Source: Table 6 · p. 14, §4.6 · p. 14

5 Discussion

The discussion presents local SVD entropy as a structural cue that responds to some degradation changes and adds information to conventional IQA, but does not replace it. Its value depends on the task and distortion, and the proposed applications require further validation.

5.1 Sensitivity to change is not the same as absolute perceptual quality

  • HSVD-SSIM retains a larger response than image-domain SSIM when comparing neighboring degraded states, although it falls faster against the clean Lena reference.The KADID neighboring-level experiment is a sensitivity test, not an alternative absolute-IQA benchmark; low correlations with |∆DMOS| show that pairwise similarity alone does not predict subjective change magnitude.Source: §5.1 · p. 16

5.2 What local SVD entropy measures

  • HSVD measures the shape of a patch’s normalized singular-value distribution: concentration in one direction yields low entropy, while a more even distribution raises it.Scaling a nonzero patch leaves HSVD unchanged, suppressing amplitude information; different degradations can alter the spectrum differently, so HSVD is an additional structural channel rather than a complete quality measure.Source: §5.2 · p. 16

5.3 Full-reference complementarity rather than replacement

  • SRCC rises from 0.618 to 0.659 when HSVD complements SSIM, with grouped validation and bootstrap supporting a stable gain in residual image ordering.The gain in PLCC and RMSE is smaller than the gain in SRCC; HSVD is weaker than SSIM alone, and its negative conditional coefficient is interpreted as a correction effect, not an intrinsic association with quality.Source: §5.3 · pp. 16–17

5.4 Distortion dependence and the limits of a single structural cue

  • HSVD produced stronger neighboring-level correlation in 17 of 25 KADID distortion types, and combined no-reference models improved over Image-only features in 17 of 25 types.Its local-complexity cue responds to some distortion properties, but not uniformly to changes dominated by luminance, color, global geometry, or semantic appearance; further distortion-wise analysis is warranted.Source: §5.4 · p. 17

5.5 From traversal-designed entropy maps to direct 2-D matrix entropy

  • Direct 3 × 3-patch SVD removes sequence construction and delay embedding, with right-angle rotation and reflection invariance following from SVD at the local-descriptor level.This does not make HSVD-SSIM globally registration invariant, and local orientation invariance discards orientation-specific information, supporting use alongside image-domain or directional features.Source: §5.5 · pp. 17–18

5.6 Implications for no-reference IQA

  • HGB’s no-reference SRCC increases from 0.528 to 0.575, PLCC from 0.564 to 0.607, and RMSE decreases from 0.918 to 0.883 with HSVD-map statistics.The result supports quality-related information without a reference, not state-of-the-art performance against contemporary deep blind-IQA models; adjacent distortion levels only surrogate actual restoration trajectories.Source: §5.6 · p. 18

5.7 Potential applications

  • Proposed uses include restoration or enhancement monitoring, parameter selection when SSIM scores are similar, and no-reference comparison of before-and-after states.Stopping or convergence monitoring, remote sensing, scientific and medical imaging, compression control, and color-aware extensions are candidates for validation, not demonstrated applications; structural change does not necessarily mean perceptual improvement.Source: §5.7 · p. 18

6 Limitations and future work

The study’s findings are bounded by a fixed 3 × 3 grayscale descriptor, controlled KADID distortions, modest no-reference models, and computational and invariance tradeoffs. Future work should test multiscale and color variants, broader validation, stronger baselines, and real restoration trajectories.

6 Limitations and future work

  • All reported HSVD maps use a fixed 3 × 3 window, so systematic multiscale testing such as E3(X), E5(X), and E7(X) is a priority.Neighborhood size trades sharp local transitions against broader structural averaging; an earlier circular-kernel entropy study found analogous scale-dependent changes [33].Source: §6 · p. 19
  • The KADID analysis is grayscale, limiting sensitivity to chromatic distortions; future work should test luminance/chrominance channels, perceptual color spaces, or quaternion-SVD.Grayscale ensures conventional SSIM and HSVD use the same luminance representation; quaternion-SVD can test whether inter-channel singular structure adds information.Source: §6 · p. 19
  • KADID-10k uses controlled artificial distortions, so generalization to authentic mixed distortions and real restoration trajectories remains unestablished.Cross-database testing on natural in-the-wild IQA data and restoration datasets such as PIPAL [28] is needed; KADID severity levels are only a surrogate for successive degradation states.Source: §6 · p. 19
  • The ridge and HGB no-reference models test HSVD alongside interpretable statistics, not against the strongest deep BIQA systems; comparisons with blind-IQA baselines and learned representations remain needed.Future comparisons should preserve reference-grouped or cross-database validation to test whether HSVD adds value alongside stronger feature families.Source: §6 · p. 19
  • HSVD construction took approximately 0.33 s per distorted KADID image in the unoptimized Python implementation and is more computationally expensive than conventional SSIM.Vectorization, batched linear algebra, GPU execution, lookup or approximation strategies, and analytic treatment of small matrices may improve efficiency; identical-hardware timing is required before claiming a quantitative speed advantage over the earlier formulation.Source: §6 · pp. 19–20
  • HSVD’s rotation, reflection, and uniform-scaling invariance applies locally to individual patches, while HSVD-SSIM still requires spatial image registration.Orientation invariance may suppress useful directional information, motivating combinations with gradient orientation or anisotropic descriptors.Source: §6 · p. 20
  • Aggregate gains are statistically stable, but HSVD does not benefit every KADID distortion type, motivating class-wise ablation and feature attribution.The proposed analyses cover blur, noise, pixelation, texture-related, geometric, and chromatic distortions and could inform adaptive fusion that weights HSVD more when degradation suits its local-complexity cue.Source: §6 · p. 20
  • The two KADID examples in Fig. 5 illustrate complementarity rather than independently validating it; the statistical conclusions instead use all 10,125 images, grouped cross-validation, and reference-level bootstrap analysis.A prospective test should use real noisy/restored trajectories from multiple algorithms and settings, with independent subjective ratings, to assess whether FR and NR HSVD models identify the direction and magnitude of perceptual improvement for restoration monitoring and stopping criteria.Source: §6 · p. 20

7 Conclusion

Direct local SVD entropy provides a complementary structural channel for image quality assessment: it adds sensitivity to degradation beyond conventional image-domain similarity, but is not a replacement for SSIM.

7 Conclusion

  • The HSVD map uses direct SVD on valid two-dimensional patches without flattening or delay embedding, and its local values are invariant to right-angle rotations and reflections.The conclusion characterizes this representation as a complementary spectral-complexity channel rather than a replacement for conventional SSIM.Source: §7 · p. 20
  • HSVD-SSIM was weaker than SSIM for absolute FR prediction, but HSVD fusion improved grouped ranking, while HSVD-derived NR statistics improved quality prediction and neighboring-state change reconstruction.The controlled Lena experiment found stronger responses to local corruption and larger responses between already heavily degraded neighboring states.Source: §7 · p. 20
  • Local singular-value entropy captures degradation overlapping with, but not exhausted by, structural similarity; future applications require validation on authentic distortions and real restoration trajectories.Promising directions include fused FR/NR assessment, fine-grained monitoring of image-processing changes, and multiscale or color-aware models; computational optimization and stronger NR baselines are also needed.Source: §7 · p. 20

CRediT authorship contribution statement

Andrei Velichko led the paper’s conception, methods, analysis, and project administration, while Petr Boriskov contributed validation and manuscript work.

CRediT authorship contribution statement

Loading 2609.27959v1…