Forensic Manifold Analysis

• Forensic manifold analysis is a quantitative, geometry-based methodology that models high-dimensional forensic evidence, from 3D fracture surfaces to neural network activations, to enable precise classification and authentication.
• It employs spectral decomposition and manifold metrics such as intrinsic dimensionality, curvature, and feature selectivity to capture and compare relevant physical and learned features.
• The method integrates statistical modeling with matrix-variate distributions and Bayesian decision rules, yielding near-perfect match probabilities and enhancing evidence transparency in forensic tasks.

Forensic manifold analysis is a quantitative, geometry-based methodology for the statistical and mechanistic interrogation of forensic evidence. It flows from the mathematical modeling of high-dimensional surfaces or feature distributions—either physical (fracture surfaces in materials) or learned (activations in neural networks)—to support classification, authentication, and interpretability in forensic tasks. Two principal research domains are exemplified: physical evidence comparison via 3D microscopy and fracture mechanics (Dawood et al., 2021), and interpretable neural forensics for synthetic media detection (Sahoo et al., 25 Dec 2025).

1. Manifold Modeling in Forensic Evidence Comparison

3D confocal laser microscopy enables the representation of fracture surfaces as height manifolds $z = f(x, y)$, sampled on a dense regular grid (1024×1024 pixels over 640 µm×640 µm field). Physical alignment and pre-processing (least-squares tilt removal, median filtering of outliers) structure each scanned base or tip fracture into a mosaic of six overlapping subdomains, $\{\Omega_k\}$, capturing the spatial continuity of topological features and eliminating image registration ambiguities. This multi-patch manifold structure directly encodes the material’s distinctive fracture topology, extended to silicone replicas via standardized casting (Dawood et al., 2021).

In neural forensic applications, the feature manifold at a chosen network layer $L$ is a parametric set $$\mathcal{M}_A^{(L)} = \{\mathbf{x}_L(I(A,p))\,|\,p \in [0,1]\}$$ corresponding to network activations under a controlled forensic artifact $A$ at each severity $p$. Sampling at discrete levels produces an empirical curve in activation space $\mathbb{R}^{D_L}$, where $D_L$ is the layer’s dimensionality (Sahoo et al., 25 Dec 2025).

2. Spectral and Geometric Characterization

Spectral decomposition is fundamental for physical manifolds: each subdomain $f_k(x, y)$ is windowed (2D Hann, 10% taper) and Fourier transformed, $F_k(u, v)$. Radial frequencies $\omega = \sqrt{u^2 + v^2}$ partition the surface features into bands corresponding to physically relevant length scales—specifically, “low” frequency bands where features exceed two grain diameters ($d_g \approx 25$–$35 µm$). Band-limited correlations between image pairs, stabilized by the Fisher-Z transform, identify specimen-specific signal beyond self-affine scaling noise (Dawood et al., 2021).

For neural feature manifolds, manifold geometry is investigated through three metrics:

• Intrinsic dimensionality ($d_{\mathrm{int}}$): PCA estimates, thresholded to explain $95\%$ variance.
• Average curvature ($C$): Computed via the $L_2$ norm of discrete second differences $\Delta^2_t$ in activation trajectories, detecting nonlinear encoding behaviors.
• Feature selectivity ($S$): Layer-average of the absolute Pearson correlations between latent feature activations and artifact severity, quantifying linear tuning to forensic manipulations (Sahoo et al., 25 Dec 2025).

3. Statistical Modeling and Decision Rules

Physical manifold-based forensic comparison aggregates cross-band, cross-patch spectral correlations into a $2 \times 6$ data matrix $X$. The matrix-variate-$t$ distribution $X \sim t_{2, 6, \nu}(M, \Sigma, \Psi)$ models both row (band) and column (patch overlap) covariance structures—specifically, AR(1) correlations for overlapping subdomains. All model parameters ($\mu$, $\Sigma$, $\rho$) are fit via EM across controlled training sets. Bayes’ rule then computes posterior match probabilities: $$P(\mathrm{match}|X) = \frac{\pi_1 f_1(X)}{\pi_1 f_1(X) + (1 - \pi_1) f_2(X)}$$ with classification threshold at $P=0.5$. Reported classification results for steel fracture and replica pairs achieve $100\%$ accuracy; posterior probabilities for true matches exceed $0.9996$, indicating maximal discriminative power in the modeled bands and mosaic topology (Dawood et al., 2021).

For neural systems, sparse autoencoder (SAE) bottleneck layers extract concentrated latent features before manifold analysis. Feature activation counts and selectivity metrics facilitate downstream classification based on artifact encoding, enabling interpretable “forensic scores” and mechanistic explanations for detector output (Sahoo et al., 25 Dec 2025).

4. Fracture Mechanics, Artifact Encoding, and Individuality

Physical fracture manifolds exhibit unique characteristics above critical wavelengths ($2 d_g$), where correlated surface heights lose fractal scaling and encode specimen-specific topology. Selection of comparison bands with $\lambda > 50 µm$ isolates those features most governed by fracture mechanics and least confounded by self-affine noise. Silicone replicas retain topological fidelity down to $\approx 20 µm$, well below the matching threshold for most alloys, confirming protocol robustness for real-world forensic tasks (Dawood et al., 2021).

In deepfake forensics, the geometry of internal activation manifolds reveals how network layers encode synthetic artifacts. Early layers show low intrinsic dimension and selectivity, reflecting weak forensic sensitivity; mid-late layers develop multi-dimensional, nonlinear encodings ($d_{\mathrm{int}} \approx 3$–$4$, $C>0$, $S \approx 0.5$); penultimate layers collapse to single logit-like features, enabling transparent identification of artifact type and severity (Sahoo et al., 25 Dec 2025). This suggests a convergent strategy for mechanistic forensic analysis across physical and neural evidence domains.

5. Practical Implementation and Workflow

A summary of principal workflows is given below.

Domain	Manifold Construction	Feature Extraction	Statistical Model
Physical Evidence (Dawood et al., 2021)	3D height map + 6-overlap mosaic	Fourier bands ($\lambda > 50\,µm$), Fisher-Z transform	Matrix-variate-$t$ (AR(1) overlap), Bayes matching
Neural Forensics (Sahoo et al., 25 Dec 2025)	Feature activation curve ($p$-sweep)	SAE latent codes, curvature, selectivity	PCA dimension, curvature ($C$), selectivity ($S$)

For physical evidence, tasks include acquisition, topological pre-processing, spectral correlation, statistical fitting, and Bayesian decision-making. Neural forensic workflow involves targeted artifact generation, SAE encoding, manifold metrics computation, and selection of interpretable latent features.

6. Implications, Transparency, and Applications

Forensic manifold analysis provides quantitative confidence in evidence matching, with discriminative power traceable to domain-specific physical or algorithmic properties. In fracture analysis, the approach enables rigorous statistical comparison and certifies replication fidelity, supporting judicial standards for evidence admissibility (Dawood et al., 2021). In neural forensics, manifold analysis advances model interpretability, allowing identification of key latent features responsible for artifact detection, guiding architecture refinement and real-time, artifact-specific explanations (Sahoo et al., 25 Dec 2025).

A plausible implication is that manifold-based diagnostics unify methodologies across physical and learned evidence domains, yielding principled, mechanistically transparent workflows. Forensic manifold analysis thus underpins both robust evidence authentication and the interpretability of forensic decision systems, relevant to materials science, computational forensics, and synthetic media detection.