Technical Summary: Stochastic Signed Distance Processes (SSDP)

Problem Statement

Multi-view surface reconstruction is a fundamental challenge in computer vision, with applications ranging from content creation to robotics. While Signed Distance Fields (SDFs) offer a continuous, resolution-independent implicit representation suitable for this task, optimizing them via rendering presents significant difficulties.

Direct surface rendering, which finds the first ray-surface intersection and propagates gradients via implicit differentiation, suffers from discontinuous visibility. This results in sparse gradients, often necessitating additional silhouette supervision (foreground masks) to facilitate optimization. To address this, recent methods like NeuS and VolSDF introduced SDF-based volume rendering. These approaches map SDF values to local attenuation coefficients, enabling differentiable volume rendering without silhouette supervision.

However, existing volume rendering formulations rely on an exponential transport interpretation. This assumes that the hazard rate (probability of intersection) at any point can be summarized by a local attenuation coefficient independent of past survival history. While valid for independent particle models (Boolean-Poisson), this assumption fails for general opaque solids with spatially correlated geometry, where the first-passage-time density depends on the history of the ray's path. Consequently, current methods may not fully capture the probabilistic nature of ray-surface interactions in complex geometries.

Methodology

The authors propose Stochastic Signed Distance Processes (SSDP), a framework that reformulates SDF-based volume rendering as probabilistic surface rendering.

1. Stochastic Signed Distance Processes (SSDP)

Instead of treating the SDF as a deterministic field, SSDP models the SDF along each ray $r(t)$ as a stochastic process $S_r(t)$:
$$S_r(t) = \mu_r(t) + R_r(t)$$

• $\mu_r(t)$: A deterministic mean field optimized to approximate the SDF.
• $R_r(t)$: A stochastic residual process modeled as a time-inhomogeneous Ornstein–Uhlenbeck (OU) process. This process is parameterized by view-dependent neural fields ($\kappa$ and $\tau$) to capture ray-wise stochastic dynamics.

This formulation induces a first-passage-time distribution for each ray, where the first-passage time $H_r$ is defined as the first time the process $S_r(t)$ crosses zero (intersects the surface).

2. Probabilistic Surface Rendering

The pixel color $C_r$ is modeled as a mixture distribution induced by the random first-passage time $H_r$:
$$f_{C_r}(c_r) = \mathbb{E}_{H_r}[f_{C_r|H_r}(c_r | H_r)]$$
The expected color is approximated by summing over discrete sampling intervals $[t_i, t_{i+1}]$:
$$f_{C_r}(c_r) \approx \sum_{i=0}^{N-1} f_{C_r|H_r}(c_r | t_i) P(t_i < H_r \leq t_{i+1}) + \text{Background Term}$$
Here, $P(t_i < H_r \leq t_{i+1})$ represents the first-passage probability (opacity) for the interval.

3. Derivation of First-Passage Probability

The core technical contribution is the derivation of the first-passage probability $P(B_i | A_i)$ for each interval, where $A_i$ is the event that the ray has not intersected the surface up to $t_i$, and $B_i$ is the event of intersection within $(t_i, t_{i+1}]$.

• Bayesian Filtering: The authors derive the posterior distribution of the SDF value at the sample point, conditioned on the ray having survived (not intersected) up to that point. This involves propagating the survival-conditioned distribution via Bayesian filtering.
• Zero-Crossing Probability: The probability of crossing zero within an interval is decomposed into:
  1. Downward crossing ($B^\downarrow_i$): The ray ends inside the surface ($S_{i+1} \leq 0$).
  2. Upward crossing ($B^\uparrow_i$): The ray enters and exits the surface within the interval.
     The probability of $B^\uparrow_i$ is approximated using the reflection principle applied to a Brownian bridge, derived from the Dambis–Dubins–Schwarz theorem.

4. Negative-Absorbing Approximation

The full Bayesian formulation requires sequential computation along each ray, preventing parallelization. To enable efficient parallel rendering, the authors introduce a negative-absorbing approximation. This assumes that once the SDF process $S(t)$ becomes negative, it never returns to a positive value (no sign reversion).

• This simplifies the survival condition to $S_i > 0$ and the crossing condition to $S_{i+1} \leq 0$.
• This removes the need for sequential Bayesian filtering, allowing the opacity calculation to be parallelized across samples, achieving efficiency comparable to standard volume rendering.

5. Relationship to Prior Work

The paper demonstrates that NeuS arises as a special case of SSDP. In the limit where the OU process becomes perfectly correlated (zero variance in the transition kernel), the SSDP formulation reduces exactly to NeuS's opacity formulation. This establishes SSDP as a natural generalization of NeuS that accounts for ray-wise covariance structures.

Key Contributions

1. Reformulation: The paper reformulates SDF-based volume rendering as probabilistic surface rendering, modeling pixel colors as mixture distributions induced by random first ray-surface intersections.
2. SSDP: Introduction of Stochastic Signed Distance Processes, modeling the SDF along rays as stochastic processes to induce first-passage-time distributions.
3. First-Passage Derivation: Derivation of first-passage probabilities for sampling intervals using Bayesian filtering and the reflection principle, alongside a practical parallelizable approximation.
4. Theoretical Unification: Proof that NeuS is a special case of the proposed formulation.
5. Empirical Validation: Demonstration of improved surface reconstruction and uncertainty quantification on DTU and MobileBrick datasets.

Experimental Results

The method, implemented as SSDP-Facto (using the NeuS-Facto sampling scheme for fair comparison), was evaluated against NeuS-Facto and OaV-Facto (Objects as Volumes).

• Surface Reconstruction: On the DTU and MobileBrick datasets, SSDP-Facto consistently outperformed baselines in Chamfer Distance (CD) under both masked (visual hull) and unmasked (viewing frustum) evaluation protocols. It effectively reduced spurious surfaces outside the visual hull.
• Uncertainty Quantification: SSDP-Facto achieved superior performance in Continuous Ranked Probability Score (CRPS), Mean Absolute Error (MAE), and Sharpness, indicating a higher quality learned first-passage-time distribution.
• Ablation Studies:
  • Negative-Absorbing Approximation: Removing the approximation (using full Bayesian filtering) improved uncertainty metrics slightly (e.g., ~0.1 mm CRPS improvement) but doubled training time. The approximation was found to be a stable and effective inductive bias for reconstruction.
  • Loss Function: Using the Negative Log Marginal Likelihood (NLML) loss directly optimized the probabilistic objective, yielding substantial improvements over the standard expectation loss used in prior works.

Significance and Claims

The paper claims that by moving away from the restrictive exponential transport assumption (local attenuation independent of history) and adopting a first-passage perspective for spatially correlated geometry, SSDP provides a more faithful probabilistic model for surface rendering.

The authors state that their method improves both surface reconstruction accuracy and uncertainty quantification compared to controlled baselines. They highlight that while the full Bayesian formulation offers theoretical completeness, the negative-absorbing approximation provides a practical trade-off, enabling efficient parallel rendering without significant loss in reconstruction quality. The work clarifies the theoretical link between existing volume rendering methods and stochastic processes, positioning SSDP as a generalization of NeuS.