Order-Complexity Aesthetic Assessment

• OCAM is a quantitative framework that defines aesthetic quality as the balance between structural order and informational complexity based on Birkhoff’s M = O/C.
• It employs domain-specific methodologies—such as symmetry and entropy analysis in images and harmonic and redundancy measures in music—to objectively predict human preference.
• The modular framework supports applications from automated aesthetic assessment and guided content generation to aesthetic-aware recommendation in UI layouts.

Order–Complexity Aesthetic Assessment Model (OCAM) encompasses a set of quantitative frameworks for evaluating the aesthetic quality of artifacts—most notably images and music—by reducing the perceptual assessment of "beauty" or "interest" to explicit measures of structural order and informational complexity. These frameworks integrate and generalize the Birkhoffian paradigm ($M = O/C$; beauty as the balance between order and complexity) across several domains, leveraging both domain-specific feature engineering and statistical learning to achieve objective, reproducible judgments. Contemporary OCAMs are distinguished by their modularity, their empirical connection to human perceptual responses (typically via large-scale crowd-sourced or expert ratings), and their adaptability to both symbolic and continuous representations.

1. Theoretical Foundations

The foundational principle of OCAM is George David Birkhoff’s aesthetic measure $M = O/C$, where $O$ represents "order" (regularity, symmetry, harmony) and $C$ represents "complexity" (variety, unpredictability, entropy). This formulation is motivated by the empirical observation that human aesthetic preference frequently exhibits an inverted-U dependency on complexity: both extremely simple (fully regular) and highly complex (random or chaotic) artifacts tend to be perceived as less aesthetically appealing than those which present an intermediate balance of order and complexity. This principle has been established in the visual domain (Lakhal et al., 2019) and has guided subsequent work in computational aesthetics for music (Jin et al., 2023, Jin et al., 2023, Jin et al., 13 Feb 2024).

OCAMs instantiate this principle by quantifying $O$ and $C$ via features adapted to the structure of the target domain (e.g., symbol variety and symmetry in patterns, harmony and entropy in music), and combining them through linear or non-linear scoring rules calibrated against empirical preference data.

2. Domain-Specific Methodologies

A. Visual Patterns and Images

For regular $n \times n$ arrays of graphical elements, "order" is quantified via the prevalence and hierarchical persistence of symmetries (rotations, reflections) and the diversity of element types across multiple spatial scales. "Complexity" is measured by counting the number of distinct subpatterns or subblocks at varying levels of granularity—reflecting the combinatorial richness of the pattern (Klinger et al., 2011).

OCAM in visual analysis often applies additional statistical complexity measures, including:

• Shannon entropy ($H$) of pixel/value histograms,
• Fractal dimension ($d_f$) via box-counting,
• Radially averaged Fourier spectrum slopes ($\alpha$), and
• Compression-based proxies for Kolmogorov complexity via file-size ratios ($\tau$) (Lakhal et al., 2019).

A key methodological elaboration is the use of geometric coarse-graining to suppress perceptual noise and isolate mid-scale structures. Here, the salient "structural complexity" ($\tau_{\rm cg}$) is computed on images that have been locally averaged and discretized (e.g., into tri-level grayscale), with empirical studies demonstrating that preference ratings peak when $\tau_{\rm cg}$ achieves intermediate values.

B. Music (Symbolic and Audio)

Order–Complexity models of musical aesthetic quality directly adapt Birkhoff’s equation to the musical domain (Jin et al., 2023, Jin et al., 2023, Jin et al., 13 Feb 2024). Order ($O$) is decomposed into harmony (e.g., intervallic dyad distributions, chord-progression structure, dynamic-metric alignment), and symmetry (self-similarity metrics, skewness of pitch/rhythm/dynamics). Complexity ($C$) aggregates measures of entropy (pitch/rhythm histograms, spectral flux) and redundancy (Kolmogorov complexity estimates via lossless compression, autocorrelation).

The standard OCAM formula, as instantiated in recent work, reads: $$M = \frac{\omega_1 H + \omega_2 S + \theta_1}{\omega_3 C + \omega_4 R + \theta_2}$$ where $H$ = harmony, $S$ = symmetry, $C$ = chaos/entropy, $R$ = redundancy, and $\omega_i$, $\theta_j$ are regression-calibrated coefficients (Jin et al., 13 Feb 2024).

Model pipelines involve extraction of fine-grained features (using music21, jSymbolic, librosa, Monkey’s Audio, etc.), aggregation by logistic or linear regression into the four aesthetic dimensions, and final scoring/sorting for assessment or recommendation.

C. User-Interface Layouts

In the assessment of web or UI layouts, the Self-Developed Aesthetics Measurement Application (SDA) operationalizes order as the arithmetic mean of five geometric/layout attributes: balance, equilibrium, symmetry, sequence, and rhythm. Complexity is then simply their complement, $1 - \text{Order}$, on a unit scale (Zain et al., 2011). This approach emphasizes interpretability and computational efficiency.

3. Quantification and Implementation

OCAMs are algorithmically defined by feature-extraction matrices and explicit aggregation rules. Key computational steps, exemplified for music, include:

• Feature Extraction: Compute histograms, entropy, harmonic relations, dynamic and metrical features via symbolic and/or audio representations.
• Dimensional Aggregation: Use (typically) logistic regression to assemble basic musical or perceptual features into four scalar dimensions corresponding to $O$ (harmony $H$, symmetry $S$) and $C$ (chaos $C$, redundancy $R$).
• Scoring: Apply a Birkhoff-type quotient (with learned weights/offsets) to combine $O$ and $C$ into the final aesthetic score.
• Normalization: When needed, features are min-max or z-score normalized within the dataset prior to regression or scoring layers.
• Model Calibration: Coefficient learning is performed using cross-entropy or softmax losses, with gradient descent on labeled datasets (typically human/AI labels), and ablation studies quantifying the contribution of each module to performance.

Algorithmic pipelines are straightforwardly reproducible, with all formulas, regression strategies, and feature definitions provided in explicit detail in the primary sources.

4. Empirical Validation and Human Preference

OCAM validity is established via large-scale human subject experiments—crowd-sourced pairwise comparisons for images (Lakhal et al., 2019), listening tests for music (Jin et al., 2023, Jin et al., 2023, Jin et al., 13 Feb 2024). These studies have consistently demonstrated:

• A preference maximum for intermediate complexity/structural complexity (inverted-U response in $S(x)$ vs. $x$ for various $x$, such as entropy, fractal dimension, algorithmic complexity).
• High correlation (Pearson $r > 0.9$ in vision; similar metrics in music) between OCAM’s structural complexity measure ($\tau_{\rm cg}$ for images, $M$ for music) and subjective ratings.
• Effective discrimination between human, AI-generated, and mechanically rendered content, with model accuracies in the range of 90–92% for classification among these types.

Ablation studies within OCAMs repeatedly emphasize the centrality of harmony and structural order to perceptual aesthetic quality, with significant performance drops upon their omission.

5. Applications in Assessment, Generation, and Recommendation

OCAMs have been deployed in the following contexts:

• Automated Aesthetic Assessment: Predicting subjective appeal of novel patterns, images, or music in generative or evaluative workflows.
• Guided Generation: Serving as differentiable objective functions for optimizing content generation models (e.g., leading AI music generators to higher harmony/symmetry and lower chaos/redundancy outputs).
• Aesthetic-aware Recommendation: Integrating $M$ as an additional embedding in transformer-based recommendation architectures (CL4SRec), enabling playlist curation by aesthetic score and user-adjustable filtering by aesthetic criteria (Jin et al., 13 Feb 2024).
• Human Performance Evaluation: Yielding interpretable subscore breakdowns for assessment and educational feedback in musical performance.
• Interface and Visualization Design: Objective ranking and optimization for spatial layouts and UI arrangements.

6. Limitations and Prospective Extensions

OCAM frameworks, while effective, face several limitations:

• Limited Modelling of Creativity/Novelty: Current formulations do not explicitly capture innovativeness, stylistic surprise, or emotional valence, focusing instead on regularity/entropy balances.
• Global versus Personalized Weights: Most implementations feature globally learned $\omega$ and $\theta$ coefficients; adaptation to individual user taste has not yet been demonstrated at scale.
• Domain-Specific Confinement: While the theoretical framework is generic, practical applications require domain- and modality-specific feature engineering.
• Neglect of Higher-Order Semantics: Aspects such as narrative, phrase-level structure, or deep semantic coherence (especially in music and visual storytelling) are only partially or indirectly addressed.

Emerging directions involve:

• Incorporation of surprise/novelty terms via information-theoretic divergences.
• Multi-objective optimization in recommender systems, enabling dynamic user trade-offs between popularity and aesthetic score.
• Extension of OCAMs to polyphonic music, full orchestration, and raw audio modalities.
• Inclusion of neurophysiological or affective response features (e.g. EEG coupling, sentiment analysis) as direct or indirect contributors to $O$ and $C$.

7. Comparative Overview of Model Structure

A cross-domain tabular summary is provided for clarity:

Domain / Model	Order ($O$) Features	Complexity ($C$) Features	Scoring Function
Image Patterns (Klinger et al., 2011)	Multi-scale symmetry, element variety	Subblock diversity, entropy	2D attributes, weighted sums
Images (OCAM) (Lakhal et al., 2019)	Structural complexity (coarse-grained)	Shannon entropy, fractal dim., Fourier slope, compression	Inverted-U empirical fit
UI Layouts (SDA) (Zain et al., 2011)	Balance, equilibrium, symmetry, sequence, rhythm	$1-$Order	Simple linear mean
Symbolic Music (Jin et al., 2023)	Harmony (interval, chords), symmetry (self-sim.)	Entropy (pitch/rhythm), Kolmogorov	Weighted quotient
Music Performance (Jin et al., 2023, Jin et al., 13 Feb 2024)	Harmony, symmetry	Chaos, redundancy	Weighted quotient

This unified representation elucidates that, irrespective of medium, OCAMs instantiate the central empirical and theoretical insight that perceived aesthetic value is maximized by an optimal balance of structural regularity and statistical (or algorithmic) complexity.