Volume-DROID SLAM Framework

• Volume-DROID is a SLAM framework that integrates real-time trajectory estimation with learned volumetric mapping to produce dense, semantically annotated 3D maps.
• It utilizes CNN-based dense feature extraction, recurrent pose and depth inference, and dense bundle adjustment to achieve robust performance.
• The system operates at low latency (~200 ms per frame) on commodity GPUs while processing monocular, stereo, or RGB-D inputs and fusing 2D segmentation into 3D volumes.

Volume-DROID is a real-time Simultaneous Localization and Mapping (SLAM) framework that integrates Differentiable Recurrent Optimization-Inspired Design (DROID-SLAM) for accurate trajectory estimation with learned volumetric mapping via Convolutional Bayesian Kernel Inference (ConvBKI). Processing monocular, stereo, or RGB-D camera inputs, Volume-DROID simultaneously estimates the robot’s camera trajectory and generates a dense, semantically annotated 3D volumetric map in real time, with open-source implementation available in Python (Stratton et al., 2023).

1. Pipeline Architecture and Data Flow

Volume-DROID ingests a stream of images or video frames that may be monocular, stereo, or RGB-D. The processing sequence is as follows:

• Dense Feature Extraction: Each input frame $I_i$ passes through a dense CNN $f_\theta$, yielding per-pixel feature maps $F_i$.
• Correlation-Volume Construction: For frame pairs $(i, j)$ in the active keyframe window, a 4D correlation volume is constructed as $C_{ij}(u, v) = \langle F_i(u), F_j(v) \rangle$ with a lookup operator $L_r$ for arbitrary locations.
• Recurrent Pose and Depth Inference (DROID-SLAM):
  • Initialize per-pixel optical flow $p^0_{ij}(u)$ and inverse depth $d^0_i(u)$.
  • For $t = 0, \ldots, T-1$:
  • Lookup and encode correlation context: $c^t_{ij}(u) = L_r(C_{ij}, p^{t}_{ij}(u))$, $$F^t_{ij}(u) \leftarrow \mathrm{CNN}(c^t_{ij}(u), p^{t}_{ij}(u), d^t_i(u))$$
  • ConvGRU update: $$h^{t+1}_{ij}(u) = \mathrm{ConvGRU}( h^t_{ij}(u), F^t_{ij}(u) )$$
  • Predict flow correction $\Delta p^t_{ij}(u)$ and confidence $w^t_{ij}(u)$.
  • Dense Bundle Adjustment: Collect Jacobians and residuals to solve a single damped Gauss–Newton optimization for poses $\{T_i\}$ and depths $\{d_i\}$.
• Point Cloud Generation: Valid for RGB-D frames. Depth pixels $u$ from frame $i$ are back-projected to world coordinates $x_i(u)$, using camera intrinsics $K$ and pose $T_i$.
• 2D-to-3D Semantic Lifting: An off-the-shelf 2D semantic segmentation network computes per-pixel class probability vectors $s_i(u)$, which are inherited by 3D points $x_i(u)$.
• Volumetric Assimilation (ConvBKI): The environment is discretized into a 3D voxel grid. For each voxel $v$, semantic measurement histograms $M_v$ are accumulated, and Bayesian updates are performed using a learned 3D kernel.
• Output/Visualization: Resultant trajectory $\{T_i\}$ is emitted as ROS TF frames. The 3D voxel grid with semantic data is visualized in RViz.

2. DROID-SLAM Optimization Formulation

DROID-SLAM operationalizes dense joint optimization for pose and depth as follows (notation adapted from Yi et al. (Teed et al., 2021)):

• Photometric Residual:

$$r^I_{ij}(u) = I_j(\pi(T_j^{-1} T_i \pi^{-1}(u, d_i(u)))) - I_i(u)$$

• Geometric (Inverse-Depth) Residual:

$$r^G_{ij}(u) = d_j(\pi(T_j^{-1} T_i \pi^{-1}(u, d_i(u)))) - d_i(u)$$

• Total Loss:

$$E(\{T\}, \{d\}) = \sum_{(i, j) \in \text{Edges}} \sum_{u \in \Omega} w^I_{ij}(u) \, \rho_I(r^I_{ij}(u)) + w^G_{ij}(u) \, \rho_G(r^G_{ij}(u))$$

where $\rho$ are robust penalty functions (e.g., L₁ or Huber), and $w^{I,G}$ are learned confidences.

• Gauss–Newton Update:

$(H + \lambda D) \, \delta = -g$

$x \leftarrow x^0 + \delta$

where $H = J^T W J$ and $g = J^T W r(x^0)$. Updates are realized via a “Dense Bundle Adjustment” layer integrated with a ConvGRU for recurrent refinement.

3. Batched Point Cloud and Semantic Projection

RGB-D pixel data is back-projected efficiently:

• Camera Intrinsics:

$$K = \begin{pmatrix} f_x & 0 & c_x \ 0 & f_y & c_y \ 0 & 0 & 1 \end{pmatrix}, \quad K_h = \begin{pmatrix} K & 0 \ 0 & 1 \end{pmatrix}$$

• Back-projection:

$x_{cam} = z \cdot K^{-1} [u, v, 1]^T$

$x_{world} = T_i^{-1} [x_{cam}; 1]$

• Vectorized Form:

$$X = (T_i^{-1} K_h^{-1}) \, U \odot [z_1 \ldots z_N]$$

where $U$ gathers all pixel coordinates and inverse depths.

Per-pixel class-probabilities $s_i(u)$ (from the segmentation network) are retained with each 3D point $x_i(u)$. This allows efficient mapping from 2D semantic predictions to 3D spatial annotations in the voxel grid.

4. Convolutional Bayesian Kernel Inference (ConvBKI)

The semantic measurement $M_v \in \mathbb{R}^C$ for voxel $v$ is assembled by summing class-probability vectors for all points falling within $v$:

$M_v = \sum_{i, u: x_i(u) \in v} s_i(u)$

ConvBKI implements a learned Bayesian update—parametrized by a 3D convolutional kernel $K(\tau)$—over the histogram field. In log-probability,

$$\log P_v^{new} = \log P_v^{old} + \sum_{\tau \in \Omega_k} K(\tau) \cdot M_{v + \tau}$$

where $\Omega_k$ denotes a selected voxel neighborhood (e.g., $5 \times 5 \times 5$ window). The updated voxel probabilities are normalized:

$$P_v^{new} \leftarrow \mathrm{softmax}( \log P_v^{new} )$$

This ensures robust integration of spatial context and measurement uncertainty.

5. Semantic Segmentation and Fusion in Volumetric Mapping

Volume-DROID leverages off-the-shelf 2D segmentation networks (e.g., dilated ConvNets) to produce softmax class-probabilities per pixel. Through back-projection, these vectors are assigned to corresponding 3D points. The ConvBKI operation fuses all such per-voxel semantic measurements in a Bayesian fashion, continuously refining per-voxel class posteriors. The prior for each voxel is initialized as uniform and updated as new frames arrive, yielding consistent probabilistic semantic labeling in the 3D map.

6. System Implementation and Real-Time Performance

The software stack is built in Python with PyTorch, utilizing ROS Noetic for messaging and RViz for visualization. The system is containerized using Docker, including a GUI interface via noVNC.

Critical operations—feature extraction, correlation computation, ConvGRU recurrent updates, and 3D convolutions for ConvBKI—are executed on the GPU. Batched point cloud computation leverages large matrix multiplications for high throughput. Asynchronous ROS callbacks, with separate “SLAM” and “mapping” threads, decouple image ingestion from optimization and mapping. Keyframe window size is limited (8–12 frames) and FP16 precision is used for efficiency.

Observed performance using a server with 4×RTX 3090 GPUs includes:

• DROID-SLAM inference: ≈10 Hz
• Point cloud and ConvBKI update: ≈5 Hz (on a $200 \times 200 \times 100$ voxel grid)
• End-to-end pipeline: $\sim$5 Hz, $<200$ ms latency per frame
• Volumetric mapping introduces $<5\%$ overhead in compared to DROID-SLAM alone once ConvBKI is fully initialized on GPU.

7. Quantitative Evaluation and Accuracy

On the TartanAir “neighborhood” subset (using ground-truth segmentation as a stand-in for real 2D networks), measured metrics are:

Metric	Value
Absolute Trajectory Error (ATE)	0.01755 m
Relative Pose Error (RPE)	0.003769 m / 0.06087 rad (translation/rotation)
KITTI sub-trajectory score	0.01088 m / 0.002345 rad
Semantic-map accuracy	Patchy (due to class-mapping and synthetic segmentation)
Runtime (4 × RTX 3090)	~5 Hz overall, <200 ms latency per frame

These results suggest competitive real-time localization and mapping performance, with semantic map quality likely to improve when using a trained 2D segmentation model (Stratton et al., 2023).

---

Volume-DROID exemplifies a tightly coupled integration of learned recurrent Gauss–Newton SLAM algorithms and differentiable volumetric mapping. The mathematical foundations—joint photometric/geometric registration, efficient batched 3D point cloud generation, and convolutional Bayesian semantic updates—enable accurate and low-latency SLAM with semantic labeling on commodity GPU hardware.