Given sequential point clouds of input frames, DELO applies DGCNN as backbone encoder to obtain a point-wise feature embedding \(\phi_{\mathcal{Y}}\) and \(\phi_{\mathcal{X}}\). Then it simultaneously aligns the frames using the Partial Optimal Transport plan for LiDAR Descriptor Matching (POT-LDM), and estimates the Predictive Uncertainty for the Evidential Pose Estimation (PU-EPE). With the help of PU estimates pose-graphs are refined. (b) This part depicts a sub-map between frame 1 to 300 of KITTI test sequence-10 with all outputs from DELO.

The proposed DELO applies frame-to-frame LiDAR descriptor matching (LDM) using partial optimal transport (POT) [POT, Optimal transport, old and new, Sinkhorn OT]

1. A neural network for frame-to-frame LiDAR odometry (LO) estimation using Partial Optimal Trasport.
2. A network component that joitly learns predictive uncertainty (PU) for Evidential Pose Estimation (EPE).
3. Learned PU helps in classifying under-confident, confident, and over-confident odometry estimation. The uncertainty quantification (UQ) over the predicted transformations helps in setting boundary conditions for any downstream pose-based decision making tasks.

Deep Evidential LiDAR Odometry estimation network is comprised of three components:

1. POT-LDM: Sharp Correspondence Matching
2. PU-EPE: Evidential Pose Estimation.
3. Pose Graph Optimization.

POT-LDM: Sharp Correspondence Matching PU-EPE: Evidential Pose Estimation Pose-Graph Optimization

Local sharp matching through differentiable partial optimal transport module reduces the matching cost between the attention maps \(\Phi_{\mathcal{X}}\) and \(\Phi_{\mathcal{Y}}\).

The POT-LDM network component can inherently learn the data uncertainty (i.e., aleatoric uncertainty), but it cannot automat- ically learn the model's predictive uncertainty (i.e., epis- temic uncertainty) [21]. The PU-EPE network component is trained jointly with the POT-LDM to estimate the model's confidence in the LiDAR odometry predic- tions at different frames, i.e., \(\mathbf{T}_f , \mathbf{T}_{f+o}, \mathbf{T}_{f+2o}, ..,\) when the frame-gap is \(o\).

The odometry predictions from POT-LDM \([.., \mathbf{T}_f , \mathbf{T}_{f+o}, \mathbf{T}_{f+2o}, . . .]\) from POT-LDM and the epistemic uncertainty measures\([.., Var [\mu]_{f} , ..]\) from PU-EPE for all frames \(f\) in a given sub-sequence \(\mathcal{S}\) are all streamed in for pose-graph optimization. We employ GTSAM for pose-graph optimization by matching the LO inference with LiDAR sensor rate.

Results of different approaches for LiDAR odomety on KITTI [18] dataset are quantified by RRE, RTE metrics. The sequences \(07-10\) that are used for testing or inference, are not seen during training the network of the supervised approaches [ 24, 41, 42, 2 ] and ours.
Black/Gray: The best and second best entries are underlined and marked in bold black and gray colors
+: Denotes the error metrics are reported from from PWCLONet [41]
PWCLO\(^{x}\): The superscript \(x\) means \(x * 1024\) number of input points are used for PWCLONet [41].

This work was partially funded by the project DECODE (01IW21001) of the German Federal Ministry of Education and Research (BMBF) and by the Luxembourg National Research Fund (FNR) under the project reference C21/IS/15965298/ELITE/Aouada.

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