Pore-Scale Keypoints Trajectory Analysis

• Pore-scale keypoints trajectory is a method for tracking localized features in porous media to analyze fluid transport and deformation dynamics.
• It employs experimental imaging and computational frameworks, using techniques like micro-CT velocimetry and dynamic 3D Gaussian splatting for precise data extraction.
• The approach enables evaluation of mixing, dispersion, and tortuosity, with applications that span geologic flows and facial motion reconstruction.

Pore-scale keypoints trajectory refers to the time-resolved tracking of spatially localized features—termed "keypoints"—within or upon the boundaries of porous media. These trajectories serve as direct, high-fidelity samplers of Lagrangian fluid transport, sub-millimeter geometric motion, or surface deformation at the pore scale. Key applications range from experimental particle velocimetry in geological flows, to sub-millimeter skin tracking in the context of dynamic facial reconstruction. Both physical flow systems and computational tracking frameworks leverage the extraction and analysis of pore-scale keypoint trajectories to characterize mixing, dispersion, tortuosity, and deformation phenomena with precise, data-driven metrics.

1. Theoretical Foundations: Lagrangian Framework and Keypoint Identification

Pore-scale particle trajectories are modeled within a Lagrangian CTRW (Continuous-Time Random Walk) formulation, in which the porous medium is construed as a network of "conducts" of length $\ell_v$ with Poiseuille flow profiles (Puyguiraud et al., 2021). Keypoints along these trajectories—such as pore entry, intra-pore mixing, and pore exit—are associated with discrete spatial events:

• Pore entry: Upstream node where a particle enters a conduct.
• Mixing/smoothing region: Location within the conduct where transverse molecular diffusion samples the velocity field, smoothing out intra-pore contrasts.
• Exit (absorbing boundary): Downstream node marking the end of the particle's intra-conduct transition.

Mathematically, the trajectory evolves as:

$s_{n+1} = s_n + \ell_v \ t_{n+1} = t_n + \tau_n$

where $\tau_n$ is the stochastic transition time derived from advection–diffusion, with the trajectory represented by $s(t) = s_{n(t)}$.

In facial tracking (PoreTrack3D (Li et al., 2 Dec 2025)), a pore-scale keypoint is defined as a sub-millimeter feature detected by the PSIFT operator and lifted to 3D via ray-casting against a high-resolution mesh:

$$P_{i} = C_{\mathrm{world}} + s_i^* d_{\mathrm{world},i}$$

with $s_i^*$ determined by mesh intersection.

2. Experimental Particle Trajectory Extraction in Porous Media

In physical systems, pore-scale keypoint trajectories are derived from time-resolved experiments such as X-ray micro-CT velocimetry (Bultreys et al., 2022). The workflow consists of:

• Prescan and segmentation: High-resolution micro-CT scan (6 μm voxel size), segmentation of pore space.
• Dynamic imaging: Laboratory EMCT with 11.8 μm voxel size at 35 s frame intervals.
• Particle detection: Crocker–Grier local-maxima in fluid-masked volumes, subvoxel centroid localization in $5\times5\times5$ neighborhoods.
• Trajectory linking: Nearest-neighbor and predictive association, with track length filtering ($N_\mathrm{min}=20$ frames).
• Coordinate conversion: $x = i \cdot s$, $y = j \cdot s$, $z = k \cdot s$.
• Velocity estimation: Finite-difference, $v_n = \frac{\Delta \mathbf{r}_n}{\Delta t}$.

Keypoints are identified as the particle positions themselves; relationships to pore geometry are established post hoc via tortuosity calculation and proximity analyses relative to pore throats.

3. Mathematical Characterization of Trajectory Statistics

Three fundamental mixing mechanisms govern pore-scale keypoint trajectories in flows (Puyguiraud et al., 2021):

• Velocity-contrast smoothing: Diffusion across streamlines yields an advection–diffusion transition-time PDF $\psi_0^*(\lambda|v)$ in Laplace space. Mean times interpolate between advective ($\tau_v$) and diffusive ($\tau_D$) regimes.
• Path tortuosity: Diffusion increases the geometric length of trajectories, quantified by a Péclet-dependent tortuosity $\chi$ that satisfies $$v_\infty = \ell_v / (\chi \langle \tau \rangle) = \bar{u}$$.
• Maximum transition time constraint: Diffusive sampling limits $\tau \leq \mathcal{O}(\ell_v^2/D_m)$, introducing exponential cutoffs in otherwise power-law transition-time PDFs.

For Eulerian speed PDFs with low-velocity power laws, short-time transition PDFs exhibit heavy tails $\psi(t) \propto t^{-2-\alpha}$, inducing anomalous dispersion ($\sigma^2(t) \propto t^{2-\alpha}$) before Fickian recovery at $t\gg\tau_D$.

4. Dynamic 3D Gaussian Splatting for Facial Keypoint Trajectories

In PoreTrack3D (Li et al., 2 Dec 2025), pore-scale keypoint trajectories on faces are reconstructed and analyzed by dynamic 3D Gaussian splatting methods. A time-series is encoded as $M$ Gaussians $\{\mathcal{G}_{i,t}\}$, with:

• Center $\mu_{i,t} \in \mathbb{R}^3$
• Covariance $\Sigma_{i,t} \in \mathbb{R}^{3 \times 3}$
• Color $c_{i,t} \in \mathbb{R}^3$
• Weight $\alpha_{i,t} \in \mathbb{R}^+$

Volume rendering along ray $r(u) = o + ud$ accumulates colors:

$$C_{\mathrm{rend}}(r) = \sum_{i=1}^M w_{i,t}(r) c_{i,t}$$

with weights $w_{i,t}(r)$ derived from the integrals over Gaussian densities.

Optimization implements a composite loss:

$$\mathcal{L} = \mathcal{L}_{\mathrm{photo}} + \mathcal{L}_{\mathrm{reg}}$$

Structural mesh priors (GSAvatars) substantially enhance tracking accuracy and trajectory survival, as shown in benchmark results.

Key Performance Metrics for Facial Pore-scale Keypoint Trajectories

Method	Median Trajectory Error (mm)	Positional Accuracy (\%)	Survival Rate (\%)
GSAvatars	0.18	50–70	70–90
Dyn3DGS	0.23	40–45	50–60
4DGS	0.35	15–25	15–35
Dyn3DGS-robot	0.63	5–10	5–25

Mesh-anchored Gaussian splatting yields the highest positional accuracy and survival rates.

5. Quantitative Error Analysis and Validation

Both experimental velocimetry (Bultreys et al., 2022) and facial tracking (Li et al., 2 Dec 2025) frameworks employ rigorous error quantification:

• Localization error: $$e_\mathrm{loc} = \|\mathbf{r}_\mathrm{detected} - \mathbf{r}_\mathrm{true}\|$$
• Velocity error: $e_v = |v_\mathrm{detected} - v_\mathrm{true}|$, $$e_\mathrm{rel} = \frac{|v_\mathrm{detected} - v_\mathrm{true}|}{|v_\mathrm{true}|}$$
• Directional error: $$\theta_\mathrm{err} = \arccos\left(\frac{\mathbf{v}_\mathrm{detected} \cdot \mathbf{v}_\mathrm{true}}{\|\mathbf{v}_\mathrm{detected}\|\|\mathbf{v}_\mathrm{true}\|}\right)$$
• RMSE and bias
• Survival rate: Fractional persistence of trajectory within defined error thresholds.

For simulation validation, statistics include detection probability versus particle radius and 90% bounds on error metrics.

6. Integrated Significance and Future Directions

Pore-scale keypoints trajectory analysis provides a mechanistic link between microstructural geometry, transport processes, and macroscopic dispersion. In fluid systems, extracting keypoints along trajectories enables direct reconstruction of velocity PDFs, tortuosity, mixing statistics, and dispersion coefficients without empirical fitting (Puyguiraud et al., 2021). In facial 3D dynamic tracking (Li et al., 2 Dec 2025), high-fidelity trajectory datasets at pore-scale resolution push accuracy limits for non-rigid surface motion analysis.

Limitations persist in both contexts: facial tracking suffers attrition of coherent pore-scale tracks, even for structurally regularized models. Physical experiments remain constrained by detection sensitivity and imaging throughput. Future work is directed towards dynamic priors for fine-scale motion, richer subject and geometry diversity in datasets, and fusion of simulation and experimental modalities for validation and inversion of pore-scale transport models.

A plausible implication is that pore-scale keypoints trajectory frameworks are extensible to a wider class of porous and deformable structures, with potential applications in subsurface hydrology, catalysis, microfluidics, and biomedical imaging, contingent on advances in feature detection, temporal resolution, and modeling expressiveness.