Endo-G²T: Unified Geometric & Algebraic Framework

• Endo-G²T is a family of constructs unifying computer vision, differential geometry, and representation theory to enable robust, geometry-guided processing.
• It introduces a novel 4D Gaussian Splatting pipeline using geo-guided prior distillation, time-embedded Gaussian fields, and keyframe-constrained streaming for enhanced reconstruction accuracy.
• Extensions include applying intrinsic torsion analysis in G₂-structures and universal deformation in modular representation theory, demonstrating broad and practical applications.

Endo-G$^{2}$T denotes a family of technical constructs spanning computer vision, differential geometry, and representation theory, each unified by the theme of geometry- or endomorphism-guided structures. The term appears in key contexts: as a geometry-guided temporally aware training scheme for dynamic 3D reconstruction ("Endo-G$^{2}$T" in 4D Gaussian Splatting for endoscopy (Liu et al., 26 Nov 2025)), as a canonical torsion endomorphism in $G_2$-structure geometry ("Endo-G$_2$T" for intrinsic torsion-induced maps (Niedzialomski, 2020)), and as a universal deformation theme for endo-trivial modules in modular representation theory (Bleher et al., 2016). This article focuses on the rigorous details underlying the most prominent instantiations, centering on the computer vision architecture and its mathematical and algebraic analogs.

1. Geometry-Guided Temporally Aware 4D Gaussian Splatting (Endo-G$^{2}$T)

Endo-G$^{2}$T refers to a training methodology for time-embedded 4D Gaussian Splatting (4DGS) tailored to dynamic endoscopic video scenes (Liu et al., 26 Nov 2025). The pipeline comprises three synergistic modules: geo-guided prior distillation, time-embedded Gaussian fields, and keyframe-constrained streaming. This scheme stabilizes geometry in environments with complex view-dependent reflectances, occlusions, and dynamic topology.

• Geo-Guided Prior Distillation (GPD): Anchors the reconstructed geometry by distilling confidence-gated monocular depth priors into rendered depth, employing scale-invariant log losses and depth-gradient losses under a warm-up-to-cap schedule. This soft scheduling mitigates early geometric drift and precludes overfitting to unreliable depth signals, allowing supervision to ramp up gently before plateauing.
• Time-Embedded Gaussian Field (TEGF): Extends standard 3D Gaussian primitives into space–time (XYZT), parameterizing each primitive at time $t$ by its center $\mu_i(t)\in\mathbb{R}^3$, scale $S_i(t)\in\mathbb{R}^3$, rotation $R_i(t)\in SO(3)$ (updated via minimal rotor operators), opacity $\alpha_i(t)\in(0,1)$, and spherical harmonic color coefficients $c_i(t)$. The covariance is given by $\Sigma_i(t) = R_i(t) S_i^2(t) R_i(t)^\top$. Temporal coherence is enforced via regularizers on opacity entropy (favoring crisp $\alpha_i(t)\sim\{0,1\}$) and local velocity smoothness among spatiotemporal neighbors.
• Keyframe-Constrained Streaming (KCS): Frames are partitioned into keyframes (stride $w$) and candidates, with full optimization and densification/pruning at keyframes (subject to a global Gaussian budget $G_\text{max}$), and lightweight image-space updates at candidate frames. This design achieves throughput and long-term stability by anchoring geometry periodically and capping point cloud size.

2. Geo-Guided Prior Distillation: Losses and Scheduling

Geo-guided supervision in Endo-G$^{2}$T utilizes externally supplied monocular depth priors with per-pixel confidence masks to generate valid pixel sets. Two central loss components operate on this subset:

• Scale-Invariant Log Depth Loss (SILog): For min-max normalized depths $\tilde{D}$, $\tilde{D}^*$, $$g(p) = \log(\tilde{D}(p) + \epsilon) - \log(\tilde{D}^*(p) + \epsilon)$$, the scale-invariant loss is

$$\mathcal{L}_\text{SILog} = 10\cdot\sqrt{\operatorname{Var}_p[g(p)] + \beta (\operatorname{Mean}_p[g(p)])^2}.$$

• Depth Gradient Loss: Enforces local geometric consistency,

$$\mathcal{L}_\text{grad} = \frac{1}{|\Omega_v|}\sum_{p\in\Omega_v} \big(\|\nabla_x \hat{D}(p) - \nabla_x D^*(p)\|_1 + \|\nabla_y \hat{D}(p) - \nabla_y D^*(p)\|_1\big).$$

Both losses are annealed via the warm-up-to-cap schedule, where the relative weight ramps linearly over iterations $t$, up to $T_\text{warm}$, capped at $w_\text{max}$, ensuring stable geometry emergence.

3. Time-Embedded Gaussian Field: Parametrization and Regularization

Within TEGF, each Gaussian evolves in space–time according to:

• Center $\mu_i(t)$, scale $S_i(t)$, rotation $R_i(t)$, opacity $\alpha_i(t)$, color $c_i(t)$.
• Covariance $\Sigma_i(t) = R_i(t) S_i^2(t) R_i(t)^\top$.
• Motion: Per-primitive velocity updates for $\mu_i$, minimal rotor operators for $R_i$.
• Regularization:
  • Opacity entropy: $$\mathcal{L}_\text{ent} = -\frac{1}{N}\sum_i[\alpha_i\log\alpha_i + (1-\alpha_i)\log(1-\alpha_i)]$$.
  • Local velocity coherence: $$\mathcal{L}_\text{vel} = \frac{1}{N}\sum_{i} \frac{1}{|N_k(i,t)|}\sum_{j \in N_k(i,t)} \|( \mu_i(t) - \mu_i(t-\Delta t) ) - ( \mu_j(t) - \mu_j(t-\Delta t) ) \|_1$$,
  • where $N_k(i,t)$ are $k$ nearest neighbors in (XYZT) metric.

4. Keyframe-Constrained Streaming: Optimization and Stability

Frames $1\dots F$ are partitioned:

• Keyframes: $K = \{f: f \equiv 1 \bmod w\}$.
• Candidates: $C = \{1\dots F\}\setminus K$.

At each frame $t$, the active Gaussian count is capped: $|G_t| \leq G_\text{max}$. Keyframes receive full optimization plus densification and pruning (retaining budget), while candidate frames only undergo image-space fine-tuning. This periodic structure preserves accuracy and efficiency over long temporal horizons, demonstrably arresting drift and maintaining reconstruction fidelity.

5. Empirical Performance and Implementation Specifics

Experiments utilized EndoNeRF and StereoMIS-P1 datasets, with:

• Photometric supervision blending $\ell_1$ and SSIM, $\lambda_\text{dssim} \in [0,1]$.
• Adam optimizer, $lr=1.6 \times 10^{-3}$, RTX 4090, mixed precision PyTorch.
• Warm-up to $w_\text{max}$ by iteration $\sim$10K.
• Quantitative results (cutting, pulling on EndoNeRF; monocular on StereoMIS-P1):

Model	PSNR$\uparrow$	SSIM$\uparrow$	LPIPS$\downarrow$	FPS$\uparrow$
Endo-4DGS (Huang et al.)	36.165	0.959	0.039	100
ST-Endo4DGS (Li et al.)	39.290	0.973	0.016	123
Endo-G$^{2}$T (cutting)	40.080	0.982	0.007	148
Endo-G$^{2}$T (pulling)	38.290	0.970	0.016	148
StereoMIS-P1 (Endo-G$^{2}$T)	33.580	0.914	0.056	148

Endo-G$^{2}$T achieves up to $+0.79$ PSNR, $+0.009$ SSIM, and $-56\%$ LPIPS reduction relative to the strongest prior, at $20\%$ higher frame rate. Ablations confirm keyframe re-anchoring and strict global budget are critical to accuracy and throughput (Liu et al., 26 Nov 2025).

6. Endo-G$_2$T in Differential Geometry: Intrinsic Torsion Endomorphism

In the geometry of $G_2$-structures on $7$-manifolds, Endo-G$_2$T refers to the canonical endomorphism $T: TM \to TM$ induced by intrinsic torsion. For a positive $3$-form $\varphi \in \Omega^3(M)$ stabilizing $G_2$, the Levi-Civita connection's departure from preserving the $G_2$ structure is measured by $\xi$, a section of $T^*M \otimes (\mathfrak{so}(7)/\mathfrak{g}_2)$. Concrete parametrization:

• $T(X)$ defined by $\xi_X Y = Y \times T(X) = -T(X) \times Y$.
• Component index formula (Cabrera): $$T^i_j = \frac{1}{6} \varphi^{i k \ell} \nabla_j \varphi_{k \ell m} g^{m*}$$.
• Integral identity (Niedziałomski):

$$\int_M s\,d\text{vol} = \int_M [ -30\,\sigma_1(T) + i_0(T) + 60\,\sigma_2(T)] \, d\text{vol}$$

where $s$ is scalar curvature, $\sigma_1(T)$ and $\sigma_2(T)$ are elementary symmetric functions of $T$, $i_0(T)$ is a $G_2$-invariant quadratic form (Niedzialomski, 2020).

7. Endo-G$^{2}$T Themes in Modular Representation Theory: Endo-Trivial Modules

Endo-trivial modules ($V$) over group algebras $kG$, $k$ of char $p>0$, are those for which $\operatorname{Hom}_k(V,V)\cong k\oplus P$ with $P$ projective. For any such module, the universal deformation ring is $W[G^{\mathrm{ab},p}]$. Explicitly, for semidihedral and generalized quaternion $2$-groups, the universal deformation ring is $W[C_2\times C_2]$, and the universal module $U(G,V) = \widetilde V \otimes_W W[C_2\times C_2]$, with explicit lifts characterized for each indecomposable endo-trivial module (Bleher et al., 2016).

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Endo-G$^{2}$T thus identifies essential geometric and algebraic mechanisms—anchoring temporally consistent geometry in challenging video scenes, canonically encoding geometric torsion in $G_2$-structures, and universalizing deformation theory of endo-trivial modules through explicit group algebra constructions—across contemporary research in vision, geometry, and algebra.