# Evidential Semantic Mapping in Off-road Environments with Uncertainty-aware
Bayesian Kernel Inference

Junyoung Kim∗, Junwon Seo∗, Jihong Min This work was supported by the Agency
for Defense Development Grant funded by the Korean Government in 2024.Junyoung
Kim, Junwon Seo, and Jihong Min are with the Agency for Defense Development,
Republic of Korea {junyoung.kimv, junwon.vision<EMAIL_ADDRESS>authors contributed equally to this work.Our project website can be found at
https://bit.ly/EvSemMap

###### Abstract

Robotic mapping with Bayesian Kernel Inference (BKI) has shown promise in
creating semantic maps by effectively leveraging local spatial information.
However, existing semantic mapping methods face challenges in constructing
reliable maps in unstructured outdoor scenarios due to unreliable semantic
predictions. To address this issue, we propose an evidential semantic mapping,
which can enhance reliability in perceptually challenging off-road
environments. We integrate Evidential Deep Learning into the semantic
segmentation network to obtain the uncertainty estimate of semantic
prediction. Subsequently, this semantic uncertainty is incorporated into an
uncertainty-aware BKI, tailored to prioritize more confident semantic
predictions when accumulating semantic information. By adaptively handling
semantic uncertainties, the proposed framework constructs robust
representations of the surroundings even in previously unseen environments.
Comprehensive experiments across various off-road datasets demonstrate that
our framework enhances accuracy and robustness, consistently outperforming
existing methods in scenes with high perceptual uncertainties.

##  I INTRODUCTION

Robotic mapping, which aims to construct an explicit environmental model using
sensor measurements, is an essential component in various robotic systems.
Conventional approaches, such as occupancy grid mapping [1, 2], construct 3D
voxel maps by estimating the occupancy state of each voxel using sensor
measurements intersecting with the voxel. However, due to their independent
voxel assumptions, these approaches often produce discontinuous and sparse
maps, especially when dealing with sparse sensor data like LiDAR [3].
Consequently, recent works have focused on enhancing the geometric
completeness of occupancy estimation by leveraging local spatial information
[3, 4]. Specifically, the adoption of Bayesian Kernel Inference (BKI) [5] has
enabled the efficient utilization of neighboring measurements, resulting in
more reliable continuous maps [4].

Semantic mapping has significantly improved the ability of autonomous systems
to comprehend their surroundings across various robotic applications [6, 7, 8,
9, 10, 11, 12]. By estimating voxel-wise semantic states, it provides a richer
understanding of the environments. Building upon the advancements in
continuous mapping, S-BKI [13] extended BKI into $3$D continuous semantic
mapping. Notably, recent semantic mapping approaches depend on the inference
of Deep Neural Networks (DNNs) [14, 15, 16, 17, 18]. These networks derive
semantic predictions from sensor measurements, which are then integrated into
the mapping process. As a result, the performance of semantic mapping is
highly dependent on the accuracy of semantic predictions generated by DNNs.

Figure 1: In perceptually challenging unstructured outdoor environments,
considering semantic uncertainty from the segmentation network is beneficial
to produce more accurate semantic maps by prioritizing confident predictions.
By incorporating the semantic uncertainty of the network, our uncertainty-
aware semantic BKI mapping framework can produce reliable and accurate 3D
semantic maps in off-road scenes.

However, the predictions from DNNs often suffer from unreliability [19, 20],
compromising the mapping performance, especially in off-road settings. The
diverse appearances and unstructured nature of off-road environments, combined
with illumination changes that increase intra-class variation, lead to
unreliable operation of DNNs [21, 22]. Moreover, the common approach of
compressing probabilistic predictions from DNNs into one-hot vectors [13, 23,
24] results in the loss of valuable information, further exacerbating mapping
performance. Consequently, semantic mapping in perceptually challenging off-
road environments frequently leads to unreliable outputs. To address this
challenge, it is crucial to consider the uncertainty of semantic predictions
from DNNs to construct semantic maps robustly, underscoring the necessity for
advanced, uncertainty-aware mapping techniques, as illustrated in Fig. 1.

In this work, we propose an uncertainty-aware semantic BKI mapping framework
for robust deployments in off-road environments with high perceptual
uncertainties. To achieve this goal, we incorporate the uncertainty of
semantic predictions into the mapping process, thereby improving the
reliability and accuracy of the map. To estimate the semantic uncertainty, we
adopt Evidential Deep Learning [25], known for its reliable uncertainty
estimates. Then, probabilistic semantic predictions are seamlessly integrated
into our mapping process, enabling precise representations of highly uncertain
environments. Furthermore, the mapping process is adapted to incorporate
semantic uncertainty by not only filtering out uncertain predictions but also
reinforcing the effect of confident predictions. Through validation with off-
road datasets, including RELLIS-3D and our own off-road datasets, we verify
that our method enhances the accuracy and reliability of semantic mapping
compared to existing methods.

##  II RELATED WORK

###  II-A 3D Semantic Mapping

Among various 3D geometric representations such as Gaussian [26], surfel [14],
and mesh [7], voxel-based 3D model is commonly used in semantic mapping [6, 8,
15, 16, 10, 11, 12]. Since traditional voxel-based mapping techniques assume
the statistical independence of each voxel, each sensor measurement is used to
update only a restricted subset of voxels. As a result, sparse LiDAR
measurements lead to the construction of discontinuous maps [3]. Therefore,
methods have been proposed to relax the voxel independence assumption by
incorporating local spatial information, demonstrating improved performance
[3, 4].

GPOctoMap [3] introduces Gaussian Process (GP) to account for local spatial
correlations, relaxing the voxel independence assumption. However, the
computational cost $\mathcal{O}(n^{3})$ with $n$ being the number of sensor
measurements makes it prohibitive for real-time applications. To address this
gap, Bayesian Kernel Inference (BKI) [5] has been introduced as an efficient
approximation to GP [4]. This significantly reduces the computational
complexity to $\mathcal{O}(\log n)$, facilitating real-time updates and
efficient integration of local spatial information into the mapping process.
This efficient BKI was then adopted in S-BKI [13] for semantic mapping,
extending the capability to include semantic information within the geometric
model.

While traditional semantic mapping systems have utilized Conditional Random
Field (CRF) [27, 7, 6, 28, 8, 29], subsequent improvements have increasingly
favored the usage of Deep Neural Networks (DNNs) to incorporate semantic
information into maps [14, 15, 16, 17, 18]. The semantic segmentation results
obtained from DNNs at each time step are commonly utilized for the incremental
update of the semantic map. After S-BKI [13] introduces the BKI-based
continuous mapping into the semantic mapping field, various approaches are
proposed to improve the accuracy of continuous semantic mapping. ConvBKI [24]
proposes a trainable version of S-BKI using depth-wise convolution,
encouraging the kernel to learn class-specific geometry. Meanwhile, SEE-CSOM
[23] is proposed to address the challenges of overinflation and inefficiency
by considering the label inconsistency of the voxels. However, the uncertainty
of semantic predictions is yet to be introduced in continuous semantic
mapping.

###  II-B Uncertainty Estimation of Deep Neural Networks

Uncertainty estimation [19, 20], considered an important aspect of deploying
DNNs in safety-critical applications, is a process of estimating the
uncertainty of the predictions from DNNs. The most widely used methods are
sampling-based approaches [30], requiring multiple inferences for each input.
For example, Monte Carlo (MC) dropout [31] performs inference multiple times
with a stochastic dropout layer, and Deep Ensemble [32] uses an ensemble of
networks to obtain multiple predictions to estimate uncertainty. Therefore,
they are unsuitable for resource-constrained real-time applications [33].
Thus, methods that enable uncertainty estimation with a single forward pass,
such as Evidential Deep Learning (EDL) [25], have been proposed.

EDL framework estimates uncertainty by modeling a second-order Dirichlet
distribution based on the Dempster-Shafer Theory of Evidence [34]. The network
is trained to compute parameters for the Dirichlet distribution from which the
uncertainty of prediction is calculated. Since it can be easily applied to
existing DNNs, it has been widely adopted in various fields for uncertainty
estimation [35, 36, 37, 38].

##  III METHODS

Figure 2: Overview pipeline of our uncertainty-aware semantic BKI framework.
With an evidential segmentation network trained by EDL, input data is
processed to derive continuous semantic probability and uncertainty. These 3D
semantic points are then integrated into the semantic map through Bayesian
updates using the uncertainty-aware BKI, resulting in a dependable semantic
map and variance map in uncertain off-road environments.

In this section, we present our uncertainty-aware semantic mapping framework.
First, we provide an overview of the semantic BKI mapping process. Then, we
incorporate the EDL framework into DNNs for semantic segmentation to estimate
semantic uncertainty. Lastly, we introduce our uncertainty-aware BKI
framework, which enhances the performance of semantic mapping by leveraging
the predictive uncertainty of semantic predictions. Our entire framework is
illustrated in Fig. 2.

###  III-A Semantic Bayesian Kernel Inference

In semantic BKI frameworks, the mapping objective is to estimate the
probability distribution $\bm{\theta}_{*}=[\theta_{*}^{1},...,\theta_{*}^{C}]$
across $C$ semantic classes for arbitrary query point $x_{*}$, where
$\sum^{C}_{c=1}\theta_{*}^{c}=1$ and $\theta_{*}^{c}\in[0,1]$. The
$\bm{\theta}_{*}$ is then used to predict semantic label $\mathbf{y}_{*}$
based on the Categorical distribution $p(\mathbf{y}_{*}|\bm{\theta}_{*})$.
Thus, the semantic points $\mathcal{D}=\\{x_{i},\mathbf{y}_{i}\\}^{N}_{i=1}$
is utilized to estimate the posterior distribution
$p(\bm{\theta}_{*}|x_{*},\mathcal{D})$, where $x_{i}$ is the coordinates of
the $i$-th observation and $\mathbf{y}_{i}$ represents the corresponding
semantic label. In practice, the posterior distribution is estimated only for
the center of voxels during the mapping for efficiency, though queries on
arbitrary locations can still be processed.

Initially, Bayes’ rule is applied to derive the posterior in closed form
equation:

$\displaystyle p(\bm{\theta}_{*}|x_{*},\mathcal{D})$ $\displaystyle\propto
p(\mathcal{D}|\bm{\theta}_{*},x_{*})p(\bm{\theta}_{*}|x_{*})$ (1)
$\displaystyle\propto\Big{[}\prod^{N}_{i=1}\underbrace{p(\mathbf{y}_{i}|x_{i},\bm{\theta}_{*},x_{*})}_{\text{extended
likelihood}}\Big{]}\underbrace{p(\bm{\theta}_{*}|x_{*})}_{\text{prior}}.$

Then, BKI [5] is applied as it provides the relationship between the extended
likelihood and the likelihood. In [5], the maximum entropy distribution $g$
satisfying $D_{\text{KL}}(g||f)\leq\rho(x_{*},x)$ is proven to be $g(y)\propto
f(y)^{k(x_{*},x)}$ with a kernel function
$k:\mathcal{X}\times\mathcal{X}\rightarrow[0,1]$, where
$D_{\text{KL}}(\cdot||\cdot)$ is the Kullback-Leibler divergence and
$\rho(\cdot,\cdot)$ is a smoothness bound. Substituting $g$ for the extended
likelihood and $f$ for the likelihood, we can obtain the following
proportional relationship:

$\displaystyle p(\mathbf{y}_{i}|x_{i},\bm{\theta}_{*},x_{*})\propto
p(\mathbf{y}_{i}|\bm{\theta}_{*})^{k(x_{*},x_{i})}.$ (2)

By combining (1) and (2), we have:

$\displaystyle
p(\bm{\theta}_{*}|x_{*},\mathcal{D})\propto\Big{[}\prod^{N}_{i=1}p(\mathbf{y}_{i}|\bm{\theta}_{*})^{k(x_{*},x_{i})}\Big{]}p(\bm{\theta}_{*}|x_{*}).$
(3)

As $\mathbf{y}_{i}$ represents the one-hot encoded semantic label and the
Categorical likelihood $p(y_{i}|\bm{\theta}_{*})$ is used in the frameworks, a
Dirichlet prior is adopted to enable incremental Bayesian inference since it
is the conjugate prior distribution. An initial Dirichlet prior
$Dir(C,\bm{\alpha}_{0})$ over $\bm{\theta}_{*}$ is parameterized by
hyperparameters $\bm{\alpha}_{0}=[\alpha_{0}^{1},...,\alpha_{0}^{C}]$.
Applying the definition of the prior and the likelihood, (3) becomes:

$\displaystyle p(\theta_{*}|x_{*},\mathcal{D})$
$\displaystyle\propto\Big{[}\prod^{N}_{i=1}\bigl{(}\prod^{C}_{c=1}(\theta_{*}^{c})^{y_{i}^{c}}\bigr{)}^{k(x_{*},x_{i})}\Big{]}\prod^{C}_{c=1}(\theta_{*}^{c})^{\alpha_{0}^{c}-1}$
(4)
$\displaystyle\propto\prod^{C}_{c=1}(\theta_{*}^{c})^{\alpha_{0}^{c}+\sum^{N}_{i=1}k(x_{*},x_{i})y_{i}^{c}-1}.$

Consequently, a Dirichlet posterior $Dir(C,\bm{\alpha}_{t})$ at time $t$ can
be calculated based on the posterior at time $t-1$. This update process for
the map cell can be recursively executed as:

$\displaystyle\alpha^{c}_{t}={\alpha_{t-1}^{c}+\sum^{N}_{i=1}k(x_{*},x_{i})y^{c}_{i}},$
(5)

and the expectation and variance of the posterior are calculated as:

$\displaystyle S_{t}=\sum^{C}_{c=1}\alpha_{t}^{c},\ \
\mathbb{E}[\alpha_{t}^{c}]=\frac{\alpha_{t}^{c}}{S_{t}},\ \
\mathrm{Var}[\alpha_{t}^{c}]=\frac{\alpha_{t}^{c}(S_{t}-\alpha_{t}^{c})}{S_{t}^{2}(S_{t}+1)}.$
(6)

It is worth noting that the variance of the predicted label
$\mathrm{Var}[\alpha_{t}^{\psi}]$ is utilized as an uncertainty measure of the
map cell in [13, 24], where
$\psi=\mathrm{argmax}_{c}\mathbb{E}[\alpha_{t}^{c}]$.

For the kernel function $k(x_{*},x_{i})$ in (5), it is desired to prioritize
information from nearby points and ignore information from the points that are
too distant. Among the functions with these characteristics, the sparse kernel
[39] is commonly used to ensure smoothness and reduce computational complexity
[4, 13, 23]:

$\displaystyle k(x_{*},x_{i})$ $\displaystyle=k^{\prime}(d,l,\sigma_{0})$ (7)
$\displaystyle=\underset{d<l}{\mathbbm{1}}\>\sigma_{0}\Big{[}\frac{2+\cos(2\pi\frac{d}{l})}{3}(1-\frac{d}{l})+\frac{1}{2\pi}\sin(2\pi\frac{d}{l})\Big{]},$

where $\mathbbm{1}$ is the indicator function, $d=||x_{*}-x_{i}||$ is the
distance between $x_{*}$ and $x_{i}$, $l>0$ is the length-scale
hyperparameter, and $\sigma_{0}$ is a kernel scaling hyperparameter. As the
value of the sparse kernel decreases smoothly as the distance increases up to
$l$, the surrounding size of space considered by the BKI framework can be
scaled by adjusting $l$.

###  III-B Evidential Semantic Segmentation

To utilize the semantic uncertainty of predictions in the mapping process, we
apply the EDL framework [25] into existing segmentation networks to estimate
its uncertainty reliably. The output of the networks for $i$-th input is
regarded as an evidence vector $\mathbf{e}_{i}=[e_{i}^{1},...,e_{i}^{C}]$ by
replacing the last $\text{SoftMax}(\cdot)$ of the network into $\exp(\cdot)$
[40].

From the evidence vector $\mathbf{e}_{i}$, the uncertainty $u_{i}$ is
estimated as $u_{i}=C/\hat{S}_{i}$, where
$\hat{S}_{i}=\sum^{C}_{c=1}(e_{i}^{c}+1)$, enabling deterministic uncertainty
estimation with a single-pass operation. Then, the semantic probability
distribution $\mathbf{p}_{i}=[p_{i}^{1},...,p_{i}^{C}]$ can be modeled as
$p_{i}^{c}=\alpha_{i}^{c}/\hat{S}_{i}$ by the Dirichlet distribution with
concentration parameters of EDL
$\bm{\hat{\alpha}}_{i}=[\hat{\alpha}_{i}^{1},...,\hat{\alpha}_{i}^{C}]$, where
$\hat{\alpha}_{i}^{c}=e_{i}^{c}+1$.

We adopt the loss from [25, 37] to train the evidential segmentation network
with EDL. The following evidential loss encourages collecting semantic
evidence of its corresponding one-hot label $\mathbf{y}_{i}$ from the training
dataset $\mathcal{D}_{\text{sem}}=\\{x_{i},\mathbf{y}_{i}\\}^{N}_{i=1}$ for
each pixel $x_{i}$:

$\displaystyle\mathcal{L}_{\text{EDL}}(x_{i},y_{i})=\sum^{C}_{c=1}y_{i}^{c}(\log(\hat{S}_{i})-\log(\hat{\alpha}_{i}^{c})).$
(8)

While the EDL loss in (8) can theoretically encourage minimizing $\hat{S}_{i}$
and maximizing the concentration parameter of ground-truth class
$\hat{\alpha}_{i}^{c}$, an additional regularization term is commonly utilized
to penalize incorrect evidence on non-ground-truth class. The most widely used
KL regularization term [25] is adopted as follows:

$\displaystyle\mathcal{L}_{\text{reg}}(x_{i})=D_{\text{KL}}\bigl{[}\text{Dir}\left(\mathbf{p}_{i}|\bm{\tilde{\alpha}}_{i}\right)\
||\ \text{Dir}\left(\mathbf{p}_{i}|\langle 1,...,1\rangle\right)\bigr{]},$ (9)

where
$\bm{\tilde{\alpha}}_{i}=[\tilde{\alpha}_{i}^{1},...,\tilde{\alpha}_{i}^{C}]$
is the Dirichlet parameters after the removal of correct evidence from
$\bm{\hat{\alpha}_{i}}$ to penalize the misleading evidence only:

$\displaystyle\tilde{\alpha}_{i}^{c}=$ $\displaystyle\begin{cases}1&\text{if
}y_{i}^{c}=1\\\ \hat{\alpha}_{i}^{c}&\text{if }y_{i}^{c}=0\\\ \end{cases}.$
(10)

The overall loss for our EDL segmentation network is

$\displaystyle\mathcal{L}=\mathcal{L}_{\text{EDL}}+\lambda_{\text{kl}}\mathcal{L}_{\text{reg}},$
(11)

where $\lambda_{\text{kl}}$ is the hyperparameter.

Once the evidential model is trained to minimize the loss in (11), the
probability $\mathbf{p}_{i}$ and its uncertainty $u_{i}$ is reliably estimated
for each input $x_{i}$. By incorporating these estimates into the mapping
process, we can improve the performance of semantic mapping even in
perceptually challenging environments such as off-road settings.

###  III-C Uncertainty-aware Mapping Extension

Since the knowledge from the evidential semantic segmentation networks cannot
be fully utilized through a squeezed one-hot prediction $\mathbf{y}_{i}$, we
generalize the semantic points
$\mathcal{D}=\\{x_{i},\mathbf{y}_{i}\\}^{N}_{i=1}$ to
$\mathcal{D^{\prime}}=\\{x_{i},\mathbf{p}_{i},u_{i}\\}^{N}_{i=1}$, where
$\mathbf{p}_{i}$ and $u_{i}$ are provided from the evidential semantic
segmentation network. Accordingly, the likelihood
$p(\mathbf{p}_{i}|\bm{\theta}_{*})$ is also generalized as the Continuous
Categorical [41] by

$\displaystyle
p(\mathbf{p}_{i}|\bm{\theta}_{*})\propto\prod^{C}_{c=1}(\theta_{*}^{c})^{p_{i}^{c}}.$
(12)

As the Dirichlet distribution is also the conjugate prior of the Continuous
Categorical, we can similarly derive the posterior distribution from (3):

$\displaystyle p(\bm{\theta}_{*}|x_{*},\mathcal{D}^{\prime})$
$\displaystyle\propto\Big{[}\prod^{N}_{i=1}\bigl{(}\prod^{C}_{c=1}(\theta_{*}^{c})^{p_{i}^{c}}\bigr{)}^{k(x_{*},x_{i})}\Big{]}\prod^{C}_{c=1}(\theta_{*}^{c})^{\alpha_{0}^{c}-1}$
(13)
$\displaystyle\propto\prod^{C}_{c=1}(\theta_{*}^{c})^{\alpha_{0}^{c}+\sum^{N}_{i=1}k(x_{*},x_{i})p_{i}^{c}-1}.$

Hence, the formula for the recursive update of the Dirichlet posterior
parameters in (5) becomes

$\displaystyle\alpha^{c}_{t}={\alpha_{t-1}^{c}+\sum^{N}_{i=1}k(x_{*},x_{i})p^{c}_{i}}.$
(14)

Building upon the extension with the Continuous Categorical, the uncertain
off-road environments can be accurately represented through semantic points
compared to utilizing one-hot squeezed segmentation results, thereby enhancing
semantic mapping performances.

Furthermore, we devise an uncertainty-aware adaptive kernel to prioritize
reliable semantic predictions based on their semantic uncertainty. The kernel
function in (7) is refined to incorporate uncertainty estimates of semantic
predictions into the mapping process as follows:

$\displaystyle k(x_{*},x_{i})=$
$\displaystyle\begin{cases}k^{\prime}(d,l\cdot\beta e^{1-\gamma
u_{i}},\sigma_{0})&\text{if }u_{i}\leq\mathrm{U}_{\text{thr}}\\\ 0&\text{if
}u_{i}>\mathrm{U}_{\text{thr}},\end{cases}$ (15)

where $\beta$ and $\gamma$ are hyperparameters. This uncertainty-aware kernel
can enhance semantic mapping accuracy and enable more reliable calculations of
the variance of map cells.

Firstly, semantic predictions with excessive uncertainty are excluded based on
the uncertainty threshold $\mathrm{U}_{\text{thr}}$, which is dynamically
calculated to filter out top-$\tilde{u}$% uncertain points. When the
segmentation network lacks sufficient evidence for an input, the semantic
prediction is disregarded in the mapping process. Consequently, this
thresholding can prevent untrustworthy predictions from overriding accurate
ones, improving the robustness against outliers in environments with
perceptual uncertainties.

Then, the length of the sparse kernel is adjusted based on uncertainty to
enhance the local influence of reliable predictions. The kernel length
undergoes an exponential decrease in response to higher uncertainty, while the
kernel length is increased for confident predictions. Increasing the kernel
length for reliable predictions dynamically expands the surrounding space
considered by the kernel, thereby reinforcing the local influence of reliable
ones during BKI. This ensures that confident semantic predictions contribute
more significantly during the recursive update of posterior parameters,
enabling reliable and consistent mapping in environments with perceptual
uncertainties.

##  IV EXPERIMENTS

In this section, we validate that our uncertainty-aware semantic BKI mapping
framework effectively enhances semantic mapping performance in off-road
settings characterized by varied visual appearances and perceptual
uncertainties. Using both publicly available off-road datasets and our own
dataset, we verify that our framework can improve semantic mapping
performance. Then, we show that our framework can operate reliably in unseen
environments using a pre-trained semantic segmentation network, facilitating
robust mapping across various off-road settings. Lastly, ablation studies are
performed to assess the validity of each component of our framework on its
overall performance.

###  IV-A Datasets

We utilize two widely-used off-road datasets, RELLIS-3D [42] and RUGD [43].
RELLIS-3D contains RGB images and LiDAR point clouds with dense semantic
annotations and precise robot poses, providing comprehensive information for
semantic mapping. The dataset consists of $5$ sequences, and we conduct a
$5$-fold evaluation where four sequences are designated for training the
segmentation network and one for evaluation. RUGD dataset provides RGB images
with pixel-wise semantic annotations. As it does not provide geometric
information, it is only used for network training.

To further validate our method across diverse off-road environments with
varying visual appearances, we collect our dataset (OffRoad) using our
platform equipped with an OS1-128 LiDAR and front-view RGB camera, accompanied
by accurate robot poses recovered by SLAM [44]. This dataset is obtained
through high-speed navigation and captured under varied conditions, including
different seasons, lighting conditions, and terrain types, making reliable 3D
mapping more challenging. For training and evaluation, RGB images are manually
annotated by experts, and the evaluation dataset contains distinct trajectory
sequences not present in the training dataset. The semantic classes of all
datasets are remapped to seven categories, as shown in Table II.

###  IV-B Experimental Setup

Implementation Details In all experiments, semantic segmentation is conducted
on RGB images to leverage their richer semantic information, and the 2D
semantic predictions are projected onto paired 3D point clouds. We adopt
DeepLabV3 [45] with the EDL framework for semantic segmentation, and the
network is trained with the loss in (11) for $100$ epochs. For each dataset,
the network is separately trained with the training subset. During the
training, the hyperparameter $\lambda_{\text{kl}}=0.5\times(\tau/100)$ is
progressively increased, where $\tau$ is the current training epoch.

TABLE I:  Hyperparameters for the semantic mapping experiments.

Dataset | Resolution | $\alpha_{0}^{c}$ | $l$ | $\beta$ | $\gamma$ | $\tilde{u}$
---|---|---|---|---|---|---
RELLIS-3D | 0.2m | 1.0 | 0.2 | 0.5 | 10.0 | 10%
OffRoad | 0.3m | 1.0 | 0.3 | 1.0 | 3.0 | 30%

The mapping hyperparameters are summarized in Table I. In both datasets, the
kernel scaling parameter $\sigma_{0}$ is set to $1.0$, to ensure the validity
of the BKI framework [5]. To evaluate the performance of semantic mapping
techniques, we first generate 3D ground truths by projecting 2D semantic
labels onto a 3D point cloud. Evaluations are then conducted by querying the
semantic map for each point location of the 3D ground truth and assessing the
accuracy of the obtained semantic labels.

Evaluation Metrics Intersection over Union ($\mathrm{IoU}$) metric is employed
to evaluate the semantic mapping performance. The $\mathrm{IoU}$ for the
$c$-th class is computed as:

$\displaystyle\mathrm{IoU}_{c}=\frac{\text{TP}_{c}}{\text{TP}_{c}+\text{FP}_{c}+\text{FN}_{c}},$
(16)

where $\text{TP}_{c},\text{FP}_{c}$, and $\text{FN}_{c}$, represent the number
of true positive, false positive, and false negative queries from the semantic
map, respectively. The calculation is restricted only to queries on occupied
cells, excluding any queries where the semantic map identifies them as
unoccupied. Then, the mean $\mathrm{IoU}$ ($\mathrm{mIoU}$) is calculated by
averaging the $\mathrm{IoU}$ scores for all classes to evaluate the overall
performance. Moreover, we also report voxel semantic accuracy to measure the
collective accuracy of queries, which is defined as:

$\displaystyle\mathrm{Acc}=\frac{\sum_{c=1}^{C}\text{TP}_{c}}{\sum_{c=1}^{C}(\text{TP}_{c}+\text{FP}_{c})}.$
(17)

Comparison Methods We perform comparative analyses with existing 3D semantic
mapping methods. S-CSM and S-BKI [13] are adopted as baselines for discrete
and continuous semantic mapping, respectively. We also include results from
ConvBKI [24], which utilizes a learnable class-wise kernel. The learned kernel
is derived from a pre-trained weight trained with RELLIS-3D. Lastly, we
compare the results of SEE-CSOM [23], which incorporates a simple class
entropy during the mapping process.

###  IV-C Quantitative and Qualitative Results

TABLE II: Quantitative results on RELLIS-3D and our OffRoad dataset. Semantic
classes not present in the evaluation dataset are excluded when calculating
$\mathrm{mIoU}$ and denoted with a dash (-). Our method shows superior
performance compared to other 3D continuous semantic mapping methods.

Dataset | Method |  puddle |  object |  paved |  unpaved |  dirt |  grass |  vegetation |  $\mathrm{mIoU}$ [%] |  $\mathrm{Acc}$ [%]
---|---|---|---|---|---|---|---|---|---|---
RELLIS 3D | S-CSM [13] | 29.6 | 34.9 | 52.9 | - | 18.4 | 77.2 | 73.9 | 47.8 | 84.3
2-11 | S-BKI [13] | 27.0 | 28.5 | 53.8 | - | 13.4 | 76.4 | 73.8 | 45.5 | 83.7
| ConvBKI [24] | 30.6 | 22.4 | 45.5 | - | 6.3 | 69.6 | 68.4 | 40.5 | 79.7
| SEE-CSOM [23] | 27.7 | 33.4 | 53.4 | - | 6.7 | 76.4 | 74.0 | 45.3 | 83.6
| Ours | 29.7 | 34.6 | 55.8 | - | 12.1 | 76.7 | 73.8 | 47.1 | 84.0
OffRoad | S-CSM [13] | - | 43.2 | - | 72.3 | - | 52.3 | 91.5 | 64.8 | 89.7
2-11 | S-BKI [13] | - | 38.0 | - | 68.6 | - | 49.9 | 90.9 | 61.9 | 89.0
| ConvBKI [24] | - | 34.7 | - | 61.2 | - | 36.8 | 91.3 | 56.0 | 88.2
| SEE-CSOM [23] | - | 41.4 | - | 72.3 | - | 52.3 | 91.8 | 64.4 | 90.1
| Ours | - | 44.2 | - | 76.4 | - | 57.9 | 92.5 | 67.8 | 91.3

Figure 3: Qualitative results of 3D semantic mapping methods. Compared to
others, our method generates reliable and accurate maps that preserve semantic
details while excluding noisy predictions. In RELLIS-3D, only our method
reconstructs grass consistently (First row), and dirt roads and puddles in
detail (Second row). In our OffRoad dataset, our method accurately
reconstructs the boundaries of unpaved roads, grass, and vegetation, compared
to others.

The quantitative results with RELLIS-3D and OffRoad are presented in Table II.
Our method shows superior performance among continuous mapping methods in both
datasets. While the discrete semantic mapping approach, S-CSM, exhibits better
performance in RELLIS-3D, this is primarily because the evaluation is
conducted only on occupied voxels from discontinuous maps. In our dataset,
characterized by a higher level of sparsity and perceptual uncertainty, our
method surpasses S-CSM while also constructing a continuous map. Methods
adopting label inconsistency during the mapping (SEE-CSOM) lead to enhanced
performance in off-road datasets since otherwise inconsistent predictions can
be negatively propagated during BKI. Our method shows better results than SEE-
CSOM, implying the relevance of EDL for quantifying semantic uncertainty and
effectively utilizing this uncertainty through the adaptive kernel. Moreover,
ConvBKI leads to even poorer performance, suggesting that pre-trained kernels
for specific semantic classes would result in inaccuracies in uncertain
unstructured environments.

Fig. 3 provides a qualitative comparison of 3D semantic maps generated by
different methods. The first and second rows display results from RELLIS-3D,
while the subsequent rows show results from our OffRoad. Semantic mapping
methods that do not incorporate perceptual uncertainty tend to produce noisy
semantic maps. Our method achieves more accurate semantic map generation by
conducting uncertainty quantification and integrating this uncertainty into
the mapping process.

###  IV-D Robust Semantic Mapping in Off-road

We show that our uncertainty-aware 3D semantic mapping method can be robustly
deployed in various off-road settings. Given that robots operating in off-road
scenes frequently encounter unseen surroundings, we simulate deployments in
unseen environments. To assess the capability of our method to perform
robustly in a zero-shot manner, we utilize a pre-trained semantic segmentation
network trained on distinct datasets. Specifically, the semantic segmentation
network is trained on four different datasets: our off-road dataset (OffRoad),
a subset of our off-road dataset (OffRoad-P), RUGD, and RELLIS-3D. Then, the
performance of semantic mapping is evaluated without adaptation on our off-
road dataset.

TABLE III: Quantitative results on our OffRoad dataset using semantic
segmentation networks pre-trained on different datasets.

Training Data | OffRoad | OffRoad-P | RUGD | RELLIS-3D
---|---|---|---|---
Method | $\mathrm{mIoU}$ | $\mathrm{Acc}$ | $\mathrm{mIoU}$ | $\mathrm{Acc}$ | $\mathrm{mIoU}$ | $\mathrm{Acc}$ | $\mathrm{mIoU}$ | $\mathrm{Acc}$
S-CSM [13] | 64.8 | 89.7 | 60.4 | 87.0 | 41.2 | 71.1 | 21.9 | 65.9
1-9 S-BKI [13] | 61.9 | 89.0 | 57.7 | 86.1 | 39.4 | 71.0 | 22.0 | 65.7
ConvBKI [24] | 56.0 | 88.2 | 52.4 | 85.1 | 39.3 | 71.2 | 20.8 | 64.2
SEE-CSOM [23] | 64.4 | 90.1 | 59.2 | 86.9 | 42.1 | 71.4 | 21.6 | 66.4
Ours | 67.8 | 91.3 | 61.0 | 87.7 | 42.0 | 73.1 | 21.8 | 66.4

The quantitative results are summarized in Table III. Our method outperforms
other methods, even when utilizing segmentation models trained with different
training datasets. This highlights the robustness of our approach, which
maintains stable performance under variations in the training data
distribution. Despite using a subset of the training data (OffRoad-P), which
introduces higher uncertainty in semantic prediction, our method remains
superior to others. Additionally, methods integrating label inconsistency into
the mapping process (SEE-CSOM) exhibit robust performance. Nevertheless, our
method performs better than SEE-CSOM, suggesting the capability to more
effectively mitigate untrustworthy predictions and handle predictive
uncertainties by incorporating EDL and uncertainty-aware BKI. Due to the
substantial domain gap with the evaluation dataset, the performance gap
observed in experiments with RELLIS-3D trained models is less evident than
others.

Figure 4: Zero-shot semantic mapping results on our OffRoad dataset using a
semantic segmentation network pre-trained on RUGD. Our method robustly
constructs semantic maps despite prediction uncertainties in unseen
environments, whereas other methods struggle to produce clear maps.

Fig. 4 shows the semantic mapping results using the segmentation network
trained on RUGD. While other methods struggle to reconstruct accurate semantic
maps or produce discontinuous maps, our method successfully creates a clear,
continuous map with unambiguous boundaries. SEE-CSOM fails to generate fully
continuous semantic maps, indicating inefficacy in reliably addressing
temporal inconsistency and uncertainties in semantic predictions.
Additionally, ConvBKI, which requires dense 3D ground truths for learning the
class-wise adaptive kernel, shows limited adaptability to unseen environments
due to potential overfitting. In contrast, our method consistently produces
continuous maps with clear boundaries without requiring additional 3D ground
truth.

###  IV-E Ablation Studies

Figure 5: Visualization of variance maps. Through uncertainty-aware BKI, our
method generates more informative variance maps capable of reliably
quantifying semantic map uncertainty.

We conduct ablation studies on our uncertainty-aware BKI framework to evaluate
the impact of its key components described in Section III-C. Specifically, we
examine the contribution of the Continuous Categorical extension (Prob.) (14),
the adaptive kernel length (Adap.) (15), and the uncertainty thresholding
(Thr.) (15) to the overall performance.

To comprehensively evaluate the impact of each component, we further
investigate Voxel Occupancy Accuracy ($\mathrm{O.Acc}$), defined as the
proportion of queries with occupied voxels, measuring the geometric
completeness of semantic maps. Moreover, the Brier Score ($\mathrm{BS}$) is
calculated to evaluate the reliability of the variance map, which can be
interpreted as representing the uncertainty of the map cells. The confidence
of a map cell’s semantic estimation is defined as
$\mathrm{Conf}_{i}=1-\hat{\mathrm{Var}}[\alpha_{t}^{\psi}]$ by normalizing the
variance in (6) to $\hat{\mathrm{Var}}[\alpha_{t}^{\psi}]$. $\mathrm{BS}$ is
then defined as follows:

$\displaystyle\mathrm{BS}=\frac{1}{M}\sum^{M}_{i=1}(\mathbbm{1}_{\text{correct}}-\mathrm{Conf}_{i})^{2},$
(18)

where $M$ is the number of occupied map cells, and the indicator function
$\mathbbm{1}_{\text{correct}}$ is assigned $1$ if the semantic label of a map
cell $i$ is correctly estimated as the ground truth label of the cell and $0$
otherwise. This metric becomes lower if the confidence of correctly estimated
cells is high and the confidence of incorrectly estimated map cells is low.

The results are summarized in Table IV. In both datasets, the Continuous
Categorical extension and the uncertainty threshold improve $\mathrm{mIoU}$,
whereas the adaptive kernel length enhances the reliability of the uncertainty
measure with low $\mathrm{BS}$. Although the uncertainty threshold
significantly improves the mapping performances, it exacerbates geometric
completeness with decreased $\mathrm{O.Acc}$. Combining the adaptive kernel
length with the uncertainty threshold enhances both the semantic mapping
performance and the uncertainty’s reliability while maintaining geometric
completeness. The variance map illustrated in Fig. 5 shows that the variance
of our map holds more meaningful uncertainty information, as quantitatively
described with improved $\mathrm{BS}$.

TABLE IV:  Results of the ablation studies.

|  |  | RELLIS-3D | OffRoad
---|---|---|---|---
Prob. | Adap. | Thr. | $\mathrm{mIoU}$ | $\mathrm{Acc}$ | $\mathrm{O.Acc}$ | $\mathrm{BS}(\downarrow)$ | $\mathrm{mIoU}$ | $\mathrm{Acc}$ | $\mathrm{O.Acc}$ | $\mathrm{BS}(\downarrow)$
- | - | - | 45.5 | 83.7 | 96.7 | 35.7 | 61.9 | 89.0 | 99.7 | 39.7
✓ | - | - | 46.4 | 83.6 | 96.7 | 40.1 | 63.4 | 89.5 | 99.9 | 51.8
✓ | ✓ | - | 46.3 | 83.8 | 96.9 | 27.4 | 64.9 | 90.2 | 99.9 | 12.5
✓ | - | ✓ | 47.8 | 84.4 | 95.5 | 43.1 | 68.9 | 91.9 | 95.4 | 51.8
✓ | ✓ | ✓ | 47.1 | 84.0 | 96.9 | 24.1 | 67.8 | 91.3 | 98.2 | 12.5

##  V CONCLUSIONS

This paper presents an uncertainty-aware Bayesian Kernel Inference framework
for reliable semantic mapping in uncertain environments. To address the
challenges of high perceptual uncertainties, our framework incorporates
uncertainty of semantic predictions from DNNs into the mapping process.
Specifically, we leverage Evidential Deep Learning for semantic segmentation
and introduce the adaptive kernel and uncertainty threshold mechanism to
address the semantic uncertainty. Through extensive experiments, we
demonstrate the effectiveness of our framework in handling semantic
uncertainty, resulting in accurate and reliable semantic maps in perceptually
challenging unstructured outdoor environments. Our method is robust in unseen
environments with higher semantic uncertainty, constructing continuous
semantic maps with precise semantic boundaries. We believe our framework has
the potential to improve the accuracy and reliability of semantic mapping in a
variety of contexts with perceptual uncertainties beyond off-road
environments.

Future works include enhancing the real-time capability of our framework with
more efficient geometric models. Additionally, we intend to extend the
framework through uncertainty-aware multi-modal fusion, enabling effective
integration of information. Furthermore, we are interested in advanced
techniques for uncertainty calibration to ensure more reliable mapping.
Exploring the potential of our semantic map and its uncertainty for active
exploration also presents an interesting research avenue.

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