Directional TSDF (DTSDF) in 3D Mapping

Updated 8 November 2025

Directional TSDF is a volumetric field representation that assigns independent TSDF values per voxel for specific canonical directions, improving accuracy in complex geometries.
It prevents surface bleeding by fusing distance, color, and confidence data per direction, enabling robust reconstruction of thin, multi-sided, or overlapping surfaces.
The approach integrates analytic and neural methods, enhancing applications in real-time SLAM, novel view synthesis, and high-fidelity 3D mapping.

The Directional Truncated Signed Distance Function (DTSDF) designates a category of volumetric field representations that augment the classic Truncated Signed Distance Function (TSDF) with orientation-aware structure. DTSDFs explicitly encode signed distance or surface information in a finite set of directions per voxel or query, rather than collapsing all geometry into a single scalar per location. This directional awareness enables accurate modeling, rendering, and tracking of scenes with thin, multi-sided, or overlapping surfaces, which confound standard TSDF approaches. Methodologies, empirical results, and mathematical formulations for DTSDFs span from hand-engineered voxel grids with axis-aligned channels, through deep learning-based differentiable variants, to analytic parallels in phase-space transport models.

1. Conceptual Foundation: From TSDF to Directional TSDF

Classic TSDFs encode, for each voxel at position $\mathbf{x}$ , the signed distance $d$ (truncated to some interval $[-\tau, \tau]$ ) to the closest observed surface, along with optional weight and color accumulators: $\Phi(\mathbf{x}) = (d, w_d, \mathbf{c}, w_c)$ This construction fails when two or more surfaces with distinct orientation intersect the truncation region of a voxel—especially in the thin-structure or double-sided observation regime. As a result, opposing or skewed surfaces mutually overwrite each other's evidence (the so-called "bleeding" or "thinning" effect), corrupting mesh extraction and update consistency.

DTSDFs resolve this by maintaining, in each voxel, a bank of independent TSDF values, each associated with a discrete canonical direction, typically the six major axes: $\{X^+, X^-, Y^+, Y^-, Z^+, Z^-\}$ . Distance, color, and confidence are updated and rendered per direction, using assignment, fusion, and rendering weights derived from the surface normal of depth observations and viewing geometry. This ensures that surfaces with distinct orientations do not destructively interfere during fusion or raycasting.

2. Mathematical Formulation and Fusion Schemes

In the canonical formulation (Splietker et al., 2019, Splietker et al., 2021, Splietker et al., 2023), the DTSDF $\Phi^d$ augments the domain: $\Phi^d(\mathbf{p}) = (\Phi_D(\mathbf{p}))_{D \in \text{Directions}}$ Measurements are assigned to direction slots according to the surface normal $\mathbf{n}$ —for example, via

$w_D(\mathbf{n}) = \langle \mathbf{n}, \mathbf{v}_D \rangle$

where $\mathbf{v}_D$ is the canonical direction vector, and $w_D(\mathbf{n})$ thresholds or blends the update to one or more compatible directions, often with soft weighting over an angular sector $\theta$ : $w_{\textrm{dir}^D}(\mathbf{n}) = \min\left(\max\left(\frac{1 - \arccos \langle \mathbf{n}, \mathbf{v}^D \rangle}{2\theta - \frac{\pi}{4}}, 0\right), 1\right)$ The SDF update at voxel $\mathbf{x}$ for a measurement $(\mathbf{p}, \mathbf{n})$ is given by the point-to-plane form: $d = \frac{1}{\tau} \langle \mathbf{p} - \mathbf{x}, \mathbf{n} \rangle$ Fusion weights may further incorporate measurement confidence, view angle, and proximity to the surface.

Color fusion is performed per-direction, with weights adjusted for the surface–voxel distance and observation confidence: $w_c = w_d \left(1 - \min\left(1, \frac{\|\mathbf{p} - \mathbf{x}\|}{\tau}\right)\right)$ This explicit per-direction storage prevents color bleeding and loss of definition at sharp edges.

3. Rendering, Tracking, and Mesh Extraction

Rendering in a DTSDF volume requires combining per-direction SDFs to synthesize a single depth (and color) value per camera ray. This is achieved via view-dependent weighting and zero-crossing search:

For each camera ray, per-voxel SDFs are weighted using their direction membership, surface orientation with respect to the viewing direction, and measurement confidence:

$w_{\textrm{comb}}^D = w_{\textrm{dir}^D}(\mathbf{n}^D) \cdot \langle \mathbf{n}^D, -\mathbf{r} \rangle \cdot w_d^D$

The raycasting procedure constructs a temporary, view-aligned TSDF by appropriately fusing directional channels, enabling efficient integration with classic tracking methods (ICP, frame-to-model).

Mesh extraction (e.g., marching cubes) for DTSDFs extends the standard isosurface extraction to accommodate multiple direction slots per voxel. Each direction maintains its own sign configuration, and consistency constraints—such as intra- and inter-directional filtering, voting, and regularization—are imposed to guarantee geometrically coherent surfaces, especially when multiple faces intersect within the same voxel or edge.

4. Extensions: Bitmask and Neural Directional SDFs

Recent frameworks reinterpret the basic DTSDF paradigm for advanced efficiency or representational power.

Bitmask-based encoding: DB-TSDF (Maese et al., 24 Sep 2025) achieves constant-per-frame update times by fusing LiDAR points with directional, precomputed bitmask kernels. Each voxel stores a 32-bit mask encoding truncated L1 distance per direction, a sign bit, and a hit counter, all updated via simple bitwise AND operations. Directional kernels are indexed according to sensor-relative azimuth and elevation, enabling high-resolution and efficient CPU-based volumetric mapping, albeit with higher absolute memory usage.

Neural direction-aware SDFs: The Signed Directional Distance Function (SDDF) (Zobeidi et al., 2021) generalizes the notion of DTSDF to a continuous function $h(\mathbf{p}, \boldsymbol{\eta})$ , mapping a query position and direction to the signed distance to the surface along that direction. Crucially, SDDFs enforce a linearity constraint along the ray direction: $\nabla_{\mathbf{p}} h(\mathbf{p}, \boldsymbol{\eta})^\top \boldsymbol{\eta} = -1$ This enables training from unstructured distance measurements (not full meshes), confers analytical guarantees on prediction confidence, and reduces input dimension. SDDF-based models support direct query of distance in arbitrary directions, facilitating shape interpolation, completion, and direct novel view inference.

5. Impact on Reconstruction Quality and Practical Applications

Empirical comparisons across benchmarks (SCANNET, ICL-NUIM, TUM RGB-D, 7SCENES, Mai City, Newer College) consistently show that DTSDF-based methods:

Reduce mesh error metrics such as RMSE and Chamfer-L1, often by factors of 2–5 $\times$ compared to standard TSDFs at coarse voxel sizes (Splietker et al., 2019, Maese et al., 24 Sep 2025).
Eliminate the “bleeding” artifact at thin or opposing surfaces.
Preserve color sharpness at surface boundaries by decoupling fusion among directions.
Enable robust, drift-resistant tracking in SLAM and mapping pipelines, with improvements in relative pose error (RPE), particularly for thin, multi-sided, or revisited geometry (Splietker et al., 2021, Splietker et al., 2023).
Increase map reusability after remapping or overwriting, as independent directional channels avoid destructive fusion across revisitations.
Scale to high-resolution maps in CPU-only settings due to constant per-scan update mechanisms (Maese et al., 24 Sep 2025).

Memory and compute overheads scale linearly with the number of stored directions, but are manageable for practical values (six axis-aligned slots or limited angular bins).

6. Relation to Phase-Space and Analytic Directional Models

The structure of DTSDFs links closely to phase-space transport models in mathematical physics. Direction-preserving discretizations for ray densities in vibro-acoustic analysis (Chappell et al., 2021) similarly decompose energy or surface densities at each boundary location into channels indexed by a finite, globally consistent direction set. This formalism guarantees directionality preservation across spatial domains and interfaces, mirroring the channelized update and rendering logic of DTSDFs in geometric vision. Both paradigms exploit the sparsity and orthogonality of channel assignments to achieve computational and representational efficiency in the presence of strong directional dependence.

7. Limitations, Variants, and Contemporary Extensions

DTSDFs incur higher memory usage relative to single-channel TSDFs as multiple direction slots must be maintained and updated selectively per voxel. Marching cubes and mesh extraction algorithms require substantial extension to extract multiple potentially overlapping surfaces per cube while ensuring topological coherence.

Variants such as FAWN (Sokolova et al., 17 Jun 2024) enforce alignment of predicted normals to canonical axes via explicit regularization in neural TSDF predictors, leveraging 3D semantic masks for floors and walls, but differ from DTSDF proper in restricting orientation to a priori scene classes rather than allocating channels per se.

Recent research directions include:

Continuous parameterizations over directions (beyond fixed angular bins).
Learning-based fusion and rendering strategies in neural fields.
Applications to shape completion under partial observations, and direct distance view synthesis without explicit mesh construction (Zobeidi et al., 2021).
Integration with uncertainty quantification and probabilistic mapping frameworks.

Method / Variant	Representation per Voxel	Primary Strength	Typical Use Case
TSDF	Scalar signed distance	Efficient for simple geometry	Standard dense mapping, KinectFusion
DTSDF (multi-axis)	Multiple signed distances (per direction)	Thin surface, edge preservation	Mapping thin/facing surfaces
Bitmask DTSDF	Bitmasks per direction, hit counting	Fast CPU, LiDAR, real-time maps	Large-scale, high-res on CPU
Neural SDDF	Continuous (pos., direction) $\to$ distance	Direct novel view synthesis	Shape completion, multi-view fusion
FAWN	Directional loss on normals	Axis-aligned scene regularization	Indoor room mapping (walls, floors)

DTSDFs constitute a generalization of distance function-based scene representations, augmenting traditional scalar fields to capture orientation-localized surface information. This directional structuring addresses key failure cases in surface fusion and rendering, particularly for scenes with multi-sided, thin, or complex geometry, and supports advanced applications in real-time SLAM, high-fidelity mapping, and data-driven 3D perception.