11institutetext: ShanghaiTech University, Shanghai, China 22institutetext:
Shanghai Engineering Research Center of Intelligent Vision and Imaging,
Shanghai, China
22email<EMAIL_ADDRESS>
# Dual-level Adaptive Self-Labeling for Novel Class Discovery in Point Cloud
Segmentation

Ruijie Xu 11** Chuyu Zhang Both authors contributed equally. Code is available
at Github.11 Hui Ren 11 Xuming He 1133

###### Abstract

We tackle the novel class discovery in point cloud segmentation, which
discovers novel classes based on the semantic knowledge of seen classes.
Existing work proposes an online point-wise clustering method with a
simplified equal class-size constraint on the novel classes to avoid
degenerate solutions. However, the inherent imbalanced distribution of novel
classes in point clouds typically violates the equal class-size constraint.
Moreover, point-wise clustering ignores the rich spatial context information
of objects, which results in less expressive representation for semantic
segmentation. To address the above challenges, we propose a novel self-
labeling strategy that adaptively generates high-quality pseudo-labels for
imbalanced classes during model training. In addition, we develop a dual-level
representation that incorporates regional consistency into the point-level
classifier learning, reducing noise in generated segmentation. Finally, we
conduct extensive experiments on two widely used datasets, SemanticKITTI and
SemanticPOSS, and the results show our method outperforms the state of the art
by a large margin.

###### Keywords:

Novel class discovery Point clouds semantic segmentation Long-tailed learning

##  1 Introduction

Point cloud segmentation is a core problem in 3D perception [19] and
potentially useful for a wide range of applications, such as autonomous
driving and intelligent robotics [21, 29]. Recently, there has been tremendous
progress in semantic segmentation of point clouds due to the utilization of
deep learning techniques [16, 17]. However, current segmentation methods
primarily focus on a closed-world setting where all the semantic classes are
known beforehand. As such it has difficulty in coping with open-world
scenarios where both known and novel classes coexist, which are commonly seen
in real-world applications.

For open-world perception, a desirable capability is to automatically acquire
new concepts based on existing knowledge [15]. While there has been much
effort into addressing the problem of novel class discovery for 2D or RGBD
images [25, 14, 10, 41], few works have explored the corresponding task for 3D
point clouds. Only recently, Riz et al. [28] propose an online point-wise
clustering method for discovering novel classes in 3D point cloud
segmentation. To avoid degenerate solutions, their method relies on an equal
class-size constraint on the novel classes. Despite its promising results,
such a simplified assumption faces two key challenges: First, the distribution
of novel classes in point clouds is inherently imbalanced due to the different
physical sizes of objects and the density of points. Imposing the equal-size
constraint can be restrictive, causing the splitting of large classes or the
merging of smaller ones. In addition, point-wise clustering tends to ignore
the rich spatial context information of objects, which leads to less
expressive representation for semantic segmentation.

To tackle the above challenges, we propose a dual-level adaptive self-labeling
framework for novel class discovery in point cloud segmentation. The key idea
of our approach is two-fold: 1) We design a novel self-labeling strategy that
adaptively generates high-quality imbalanced pseudo-labels for model training,
which facilitates clustering novel classes of varying sizes; 2) To incorporate
semantic context, we develop a dual-level representation of 3D points by
grouping points into regions and jointly learns the representations of novel
classes at both the point and region levels. Such a dual-level representation
imposes additional constraints on grouping the points likely belonging to the
same category. This helps in mitigating the noise in the generated
segmentation.

Specifically, our framework employs an encoder to extract point features for
the input point cloud and average pooling to compute representations of pre-
computed regions. Both types of features are fed into a prototype-based
classifier to generate predictions across both known and novel categories for
each point and region. To learn the feature encoder and class prototypes, we
introduce a self-labeling-based learning procedure that iterates between
pseudo-label generation for the novel classes and the full model training with
cross-entropy losses on points and regions. Here the key step is to generate
imbalanced pseudo labels, which is formulated as a semi-relaxed Optimal
Transport (OT) problem with adaptive regularization on class distribution.
Along with the training, we employ a data-dependent annealing scheme to adjust
the regularization strength. Such a design prevents discovering degenerate
solutions and meanwhile enhances the model flexibility in learning the
imbalanced data distributions.

To demonstrate the effectiveness of our approach, we conduct extensive
experiments on two widely-used datasets: SemanticKITTI [3] and SemanticPOSS
[26]. The experimental results show that our method outperforms the state-of-
the-art approaches by a large margin. Additionally, we conduct comprehensive
ablation studies to evaluate the significance of the different components of
our method. The contributions of our method are summarized as follows:

  1. 1.

We propose a novel adaptive self-labeling framework for novel class discovery
in point cloud segmentation, better modeling imbalanced novel classes.

  2. 2.

We develop a dual-level representation for learning novel classes in point
cloud data, which incorporates semantic context via augmenting the point
prediction with regional consistency.

  3. 3.

Our method achieves significant performance improvement on the SemanticPOSS
and SemanticKITTI datasets across nearly all the experiment settings.

##  2 Related Work

##### Point cloud semantic segmentation.

Point cloud semantic segmentation has attracted much attention in recent years
[7, 38, 42, 22]. While previous methods have made significant progress, their
primary focus is on closed-world scenarios that heavily rely on annotations
for each class and cannot address open-world challenges. In contrast, we aim
to develop a model to discover novel classes in 3D open-world scenarios. In
the context of point cloud representation learning, incorporating spatial
context is pivotal for enhancing representation learning. Several works [24,
39] introduce a hierarchical representation learning strategy that leverages
regions as intermediaries to connect points and semantic clusters. Unlike
them, we develop a dual-level learning strategy that concurrently learns to
map points and regions to semantic classes. Thanks to the learning of region-
level representation, our method is less sensitive to the local noises in
point clouds. Moreover, we cluster regions into semantic classes by an
imbalance-aware self-labeling algorithm instead of simple K-Means.

##### Novel class discovery.

The majority of research on Novel Class Discovery (NCD) has focused on
learning novel visual concepts in the 2D image domain via designing a variety
of unsupervised losses on novel class data or regularization strategies [15,
40, 10, 34, 37, 13]. Among them, EUMS [41] addresses novel class discovery in
semantic segmentation, employing a saliency model for clustering novel
classes, along with entropy ranking and dynamic reassignment for clean pseudo
labels. More relevantly, Zhang et al. [36] consider the NCD task in long-
tailed classification scenarios, and develop a bi-level optimization strategy
for model learning. It adopts a fixed regularization to prevent degeneracy,
imposing strong restrictions on learned representations, and a complex dual-
loop iterative optimization procedure. In contrast, we propose an adaptive
regularization strategy, which is critical for the success of our self-
labeling algorithm. Moreover, our formulation leads to a convex pseudo-label
generation problem, efficiently solvable by a fast scaling algorithm [8, 6]
(see Appendix A for detailed comparisons). Perhaps most closely related to our
work is [28], which explored the NCD problem for the task of point cloud
semantic segmentation. Assuming a uniform distribution of novel classes, they
develop an optimal-transport-based self-labeling algorithm to cluster novel
classes. However, the method neglects intrinsic class imbalance and spatial
context in point cloud data, often leading to sub-optimal clustering results.

##### Optimal transport for pseudo labeling.

Unlike naive pseudo labeling [20], Optimal Transport (OT) [33, 27]-based
methods allow us to incorporate prior class distribution into pseudo-labels
generation. Therefore, it has been used as a pseudo-labels generation strategy
for a wide range of machine learning tasks, including semi-supervised learning
[18, 31, 30], clustering [1, 4, 35], and domain adaptation [11, 5, 23].
However, most of these works assume the prior class distribution is either
known or simply the uniform distribution, which is restrictive for NCD. By
contrast, we consider a more practical scenario, where the novel class
distribution is unknown and imbalanced, and design a semi-relaxed OT
formulation with a novel adaptive regularization.

##  3 Method

In this section, we first introduce the problem setup of novel class discovery
for point cloud segmentation and an overview of our method in Sec.3.1. We then
describe our network architecture, including dual-level representation of
point clouds in Sec.3.2. Subsequently, we present in detail our adaptive self-
labeling framework for model learning that discovers the novel classes in
Sec.3.3. Finally, we introduce our strategy to estimating the number of novel
classes in Sec.3.4.

###  3.1 Problem Setup and Overview

For the task of point cloud segmentation, the novel class discovery problem
aims to learn to classify 3D points of a scene into known and novel semantic
classes from a dataset consisting of annotated points from the known classes
and unlabeled points from novel ones.

Formally, we consider a training set of 3D scenes, where each scene comprises
two parts: 1) an annotated part of the scene
$\\{(x^{s}_{n},y^{s}_{n})\\}^{N}_{n=1}$, which belongs to the known classes
$C^{s}$ and consists of original point clouds along with the corresponding
labels for each point; 2) an unknown part of the scene
$\\{(x^{u}_{m})\\}^{M}_{m=1}$, which belongs to the novel classes $C^{u}$ and
does not contain any label information. These two sets $C^{s}$ and $C^{u}$ are
mutually exclusive, i.e., $C^{s}\cap C^{u}=\emptyset$. Our goal is to learn a
point cloud segmentation network that can accurately segment new scenes in a
test set, each of which includes both known and novel classes.

To tackle the challenge of discovering novel classes in point clouds, we
introduce a dual-level adaptive self-labeling framework to learn a
segmentation network for both known and novel classes. The key idea of our
method includes two aspects: 1) utilizing the spatial smooth prior of point
clouds to generate regions and developing a dual-level representation that
incorporates regional consistency into the point-level classifier learning; 2)
generating imbalance pseudo-labels with a novel adaptive regularization. An
overview of our framework is depicted in Fig.1.

Figure 1: Our method starts with two views of the same point cloud ($x$ and
$\hat{x}$) and clusters the points into corresponding regions. Then, we
extract individual point features via a forward pass and calculate regional
representations by averaging the point features within each region. Next, we
make predictions $p$ by the classifier $h$, and generate pseudo labels $y$ for
unlabeled points and regions using our novel adaptive self-labeling algorithm
(generating pseudo-labels does not involve gradients). Lastly, we exchange the
pseudo labels between the two views and update the model accordingly.

###  3.2 Model Architecture

We adopt a generic segmentation model architecture consisting of a feature
encoder for the input point cloud and a classifier head to generate the point-
wise class label prediction. Note that to capture both known and novel
classes, our feature encoder is shared by all the classes $C^{s}\cup C^{u}$
and the output space of our classifier head also includes known and novel
classes. Below we first introduce our feature representation and encoder,
followed by the classifier head.

##### Dual-level Representation.

Instead of treating each point independently, we exploit the spatial
smoothness prior to 3D objects in our representation learning. To this end, we
adopt a dual-level representation of point clouds that describes the input
scene at different granularity. Specifically, given an input point cloud
$\mathbf{X}$, we first use a backbone network $f_{\theta}$ to compute a point-
wise feature $\mathbf{Z}^{p}=\\{\mathbf{z}_{i}^{p}\\}$, where
$\mathbf{z}_{i}^{p}\in\mathbb{R}^{D\times 1}$. In this work, we employ
MinkowskiUNet [7] for the backbone. In addition, we cluster points into
regions based on their coordinates and then compute regional features by
average pooling of point features. Concretely, during training, we first
utilize DBSCAN [9] to generate $K_{i}$ regions,
$\mathcal{R}=\\{r_{k}\\}^{K_{i}}_{k=1}$, for unlabeled point in sample $i$,
and calculate the regional features as follows,

$\\{r_{k}\\}^{K_{i}}_{k=1}\leftarrow\text{DBSCAN}(\\{x^{u}_{i}\\}^{M}_{i=1}),\quad\mathbf{z}^{r}_{k}=\text{AvgPool}\\{{\mathbf{z}_{i}^{p}}|\mathbf{z}_{i}^{p}=f_{\theta}(x_{i}^{u}),\;x_{i}^{u}\in
r_{k}\\},$ (1)

where $\mathbf{z}^{r}_{k}$ is the feature of region $r_{k}$. Such a dual-level
representation allows us to enforce regional consistency in representation
learning.

##### Prototype-based Classifier.

We adopt a prototype-based classifier design for generating the point-wise
predictions. Specifically, we introduce a set of prototypes for known and
novel classes, denoted as
$h=[h^{s},h^{u}]\in\mathbb{R}^{D\times(|C^{s}|+|C^{u}|)}$, and $D$ denotes the
dimension of the last-layer feature. For each point or region, we compute the
cosine similarity between its feature and the prototypes, followed by Softmax
to predict the class probabilities. Here we use the same set of prototypes for
the points and regions, which enforces a consistency constraint within each
region and results in a more compact representation for each class.

###  3.3 Adaptive Self-labeling Framework

To handle class-imbalanced data, we propose an adaptive self-labeling
framework that dynamically generates imbalanced pseudo-labels. To this end, we
adopt the following loss function for the known and novel classes,

$\mathcal{L}=\mathcal{L}_{s}+\alpha\mathcal{L}_{u}^{p}+\beta\mathcal{L}_{u}^{r},$
(2)

where $\mathcal{L}_{s}$ is the cross-entropy loss for known classes,
$\mathcal{L}_{u}^{p}$ is point-level loss and $\mathcal{L}_{u}^{r}$ is region-
level loss for novel classes. $\alpha$ and $\beta$ are weight parameters. For
the novel classes, we first generate pseudo-labels for points and regions by
solving a semi-relaxed Optimal Transport problem and then adopt the cross-
entropy loss with the generated labels. The pseudo-code of our algorithm is
shown in Appendix B and below we will focus on our novel pseudo-label
generation process.

Function  _Self-Labeling(_$-\log\mathbf{P}$ , $\gamma$, $\epsilon$_)_

$\mathbf{K}=\exp(\log\mathbf{P}/\epsilon),\quad
f\leftarrow\frac{\gamma}{\gamma+\epsilon}$

$\mathbf{\mu},\mathbf{\nu}\leftarrow\frac{1}{M}\mathbf{1}_{M},\frac{1}{|C^{u}|}\mathbf{1}_{|C^{u}|}$
//Marginal distribution

$\mathbf{b}_{0}\leftarrow\mathbf{1}_{|C^{u}|}$ //Initialize $\mathbf{b}$

while  _$\|\mathbf{b}_{t+1}-\mathbf{b}_{t}\| <1e-4$_ do

$\mathbf{a}\leftarrow\frac{\mathbf{\mu}}{\mathbf{K}\mathbf{b}_{t}}$

$\mathbf{b}_{t+1}\leftarrow\left(\frac{\mathbf{\nu}}{\mathbf{K}^{\top}\mathbf{a}}\right)^{f}$

end while

$\mathbf{Q}\leftarrow
M\text{diag}(\mathbf{a})\mathbf{K}\text{diag}(\mathbf{b})$

return $\mathbf{Q}$

end

Algorithm 1 Semi-relaxed Optimal Transport Algorithm

##### Imbalanced Pseudo Label Generation.

The pseudo-labels generation for balanced classes can be formulated as an
optimal transport problem as follows [1, 36]:

$\displaystyle\min_{\mathbf{Q}}\frac{1}{M}\langle{\mathbf{Q}},-\log\mathbf{P}^{u}\rangle_{F},\quad\text{s.t.
}{\mathbf{Q}}\mathbf{1}_{|C^{u}|}=\mathbf{1}_{M},\mathbf{Q}^{\top}\mathbf{1}_{M}=\frac{M}{|C^{u}|}\mathbf{1}_{|C^{u}|},$
(3)

where $\mathbf{Q}\in\mathbb{R}^{M\times|C^{u}|}$ are the pseudo labels of
unlabeled data, $<,>_{F}$ is Frobenius inner product and $\mathbf{P}^{u}$ are
the output probabilities of the model. For imbalanced point cloud data, we
relax the second constraint on the class sizes in Eq. 3, which leads to a
parameterized semi-relaxed optimal transport problem as below:

$\displaystyle\min_{{\mathbf{Q}}}\mathcal{F}_{u}({\mathbf{Q}},\gamma$
$\displaystyle)=\frac{1}{M}\langle{\mathbf{Q}},-\log\mathbf{P}^{u}\rangle_{F}+\gamma
KL(\frac{1}{M}{\mathbf{Q}}^{\top}\mathbf{1}_{M},\frac{1}{|C^{u}|}\mathbf{1}_{|C^{u}|})$
$\displaystyle\text{s.t.
}{\mathbf{Q}}\in\\{{\mathbf{Q}}\in\mathbb{R}^{M\times|C^{u}|}|{{\mathbf{Q}}}\mathbf{1}_{|C^{u}|}=\mathbf{1}_{M}\\},$
(4)

where $\gamma$ is a weight coefficient for balancing the constraint on cluster
size distribution in the second term. We further add an entropy term
$-\epsilon\mathcal{H}(\frac{1}{M}\mathbf{Q})$ to Sec. 3.3 and for any given
$\gamma$, this entropic semi-relaxed OT problem can be efficiently solved by
fast scaling algorithms [8, 6]. Algorithm 1 outlines the optimization process,
and further details are provided in Appendix A.

In this work, we propose a novel adaptive regularization strategy that adjusts
the weight $\gamma$ according to the progress of model learning, significantly
improving pseudo-label quality. Details of our strategy will be illustrated
subsequently.

##### Adaptive Regularization Strategy.

The objective Sec. 3.3 aims to strike a balance between the distribution
represented by model prediction $\mathbf{P}^{u}$ and the uniform prior
distribution. A large $\gamma$ tends to prevent the model from learning a
degenerate solution, e.g. assigning all the samples into a single novel class,
but it also restricts the model’s capacity to learn the imbalanced data. One
of our key insights is that the imbalanced NCD learning requires an adaptive
strategy for setting the value of $\gamma$ during the training. Intuitively,
in the early training stage where the model performance is relatively poor, a
larger constraint on ${\mathbf{Q}}^{\top}\mathbf{1}_{M}$ is needed to prevent
degenerate solutions. As the training progresses, the model gradually learns
meaningful clusters for novel classes, and the constraint should be relaxed to
increase the flexibility of pseudo-label generation.

To achieve that, we develop an annealing-like strategy for adjusting $\gamma$,
inspired by the ReduceLROnPlateau method that reduces the learning rate when
the loss does not decrease. Here we employ the KL term in Sec. 3.3 as a guide
for decreasing $\gamma$, as the value of the KL term reflects the relationship
between the distribution of pseudo labels and the uniform distribution.
Specifically, our formulation for the adaptive regularization factor is as
follows:

$\gamma_{t+1}=\lambda\gamma_{t},\text{\ if
$KL(\frac{1}{M}{\mathbf{Q}}^{\top}\mathbf{1}_{M},\frac{1}{|C^{u}|}\mathbf{1}_{|C^{u}|})$
$\leq\rho$ consecutively for $T$ iter.}$ (5)

where $\rho,\lambda$, $T$ and $\gamma_{0}$ are hyperparameters. Compared to
typical step decay and cosine decay strategies, our adaptive strategy is aware
of the model learning process and allows for more flexible control of $\gamma$
based on the characteristics of the input itself.

##### Hyperparameter Search.

To search the values of our hyperparameters, we design an indicator score that
can be computed on the training dataset. Specifically, our indicator
regularizes the total loss in Eq. 2 with a KL term that measures the distance
between the distribution of novel classes and the uniform distribution.
Formally, the indicator is defined as follows:

$\mathcal{I}=\mathcal{L}+\gamma
KL(\frac{1}{M}{\mathbf{Q}}^{\top}\mathbf{1}_{M},\frac{1}{|C^{u}|}\mathbf{1}_{|C^{u}|}),$
(6)

where $\gamma$ is obtained by Eq. 5. Empirically, this indicator score
provides a balanced evaluation of the model’s performance in the known and
novel classes.

###  3.4 Estimate the number of novel classes

To deal with realistic scenarios, where the number of novel classes ($C^{u}$)
is unknown, we extend the classic estimation method [32] in NCD to point
clouds semantic segmentation for estimating $C^{u}$. Specifically, we extract
representation from a known-class pre-trained model for training data, define
a range of possible total class counts ( $|C^{s}|\textless|C_{all}|\textless$
max classes), and apply Kmeans to cluster the labeled and unlabeled point
clouds across different $|C_{all}|$. Then, we evaluate the clustering
performance of known classes under different $|C_{all}|$, and select
$|C_{all}|$ with the highest clustering performance as the estimated
$|C_{all}|$.

##  4 Experiments

###  4.1 Experimental setup

##### Dataset.

We perform evaluation on the widely-used SemanticKITTI [12, 2, 3] and
SemanticPOSS [26] datasets. The SemanticKITTI dataset consists of 19 semantic
classes, while the SemanticPOSS dataset contains 13 semantic classes. Both
datasets have intrinsic class imbalances. For a fair comparison with existing
works [28], we divide the dataset into 4 splits and select one split as novel
classes, while treating the remaining splits as the known classes.
Additionally, to assess the effectiveness of our method under more challenging
conditions, we further split the SemanticPOSS dataset into two parts,
selecting one part as novel classes. The dataset details are provided in
Appendix C.

##### Evaluation Metric.

Following the official guidelines in SemanticKITTI and SemanticPOSS, we
conduct evaluations on sequences 08 and 03, respectively. These sequences
contain both known and novel classes. For the known classes, we report the IoU
for each class. Regarding the novel classes, we employ the Hungarian algorithm
to initially match cluster labels with their corresponding ground truth
labels. Subsequently, we present the IoU values for each of these novel
classes. Additionally, we calculate the mean of columns across all known and
novel classes.

##### Implementation Details.

We follow [28] to adopt the MinkowskiUNet-34C [7] network as our backbone. For
the parameters in DBSCAN, we set the min_samples to a reasonable value of 2,
and select an epsilon value of 0.5, ensuring that 95% of the point clouds are
included in the region branch learning process. A detailed analysis of DBSCAN
is included in Appendix J. For the input point clouds, we set the voxel size
as 0.05 and utilize the scale and rotation augmentation to generate two views.
The scale range is from 0.95 to 1.05, and the rotation range is from -$\pi/20$
to $\pi/20$ for three axes. We train 10 epochs and set batch size as 4 for all
experiments. The optimizer is Adamw, and the initial learning rate is 1e-3,
which decreases to 1e-5 by a cosine schedule. For the hyperparameters, we set
$\alpha=\beta=1$ and fix $\lambda$ at 0.5. We choose $T=10$ and $\rho=0.005$
based on the indicator mentioned in Sec. 3.3 and analyze them in the ablation
study. Both the point- and region-level self-labeling algorithms employ the
same parameters. All experiments are conducted on a single NVIDIA A100.

Table 1: The novel class discovery results on SemanticPOSS dataset. ‘Number’ denotes the number of points. ‘Full’ denotes the results obtained by supervised learning. The gray values are the novel classes in each split. Split |  Method |  bike |  build. |  car |  cone. |  fence |  grou. |  pers. |  plants |  pole |  rider |  traf. |  trashc. |  trunk |  Novel |  Known |  All
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| Full | 45.0 | 83.3 | 52.0 | 36.5 | 46.7 | 77.6 | 68.2 | 77.7 | 36.0 | 58.9 | 30.3 | 4.2 | 14.4 | - | - | 48.5
0 | EUMS | 25.7 | 4.0 | 0.6 | 16.4 | 29.4 | 36.8 | 43.8 | 28.5 | 13.1 | 26.8 | 18.2 | 3.3 | 16.9 | 17.4 | 21.5 | 20.3
NOPS | 35.5 | 30.4 | 1.2 | 13.5 | 24.1 | 69.1 | 44.7 | 42.1 | 19.2 | 47.7 | 24.4 | 8.2 | 21.8 | 35.7 | 26.6 | 29.4
Ours | 46.3 | 51.5 | 6.0 | 35.7 | 48.5 | 83.0 | 67.9 | 53.1 | 35.5 | 59.3 | 31.0 | 2.8 | 15.5 | 48.4 | 38.0 | 41.2
1 | EUMS | 15.2 | 68.0 | 28.0 | 24.0 | 11.9 | 75.1 | 36.0 | 74.5 | 26.9 | 48.6 | 26.0 | 5.6 | 23.1 | 21.0 | 40.0 | 35.6
NOPS | 29.4 | 71.4 | 28.7 | 12.2 | 3.9 | 78.2 | 56.8 | 74.2 | 18.3 | 38.9 | 23.3 | 13.7 | 23.5 | 30.0 | 38.2 | 36.4
Ours | 31.5 | 83.2 | 48.7 | 25.4 | 23.9 | 77.3 | 53.1 | 77.1 | 32.5 | 57.3 | 35.0 | 9.3 | 18.0 | 36.2 | 46.4 | 44.0
2 | EUMS | 40.1 | 69.5 | 27.7 | 13.5 | 34.9 | 76.0 | 54.7 | 75.6 | 5.3 | 39.2 | 7.8 | 8.5 | 11.9 | 8.3 | 44.0 | 35.7
NOPS | 37.2 | 71.8 | 29.7 | 14.6 | 28.4 | 77.5 | 52.1 | 73.0 | 11.5 | 47.1 | 0.5 | 10.2 | 14.8 | 9.0 | 44.2 | 36.0
Ours | 45.3 | 82.8 | 49.8 | 28.4 | 46.3 | 76.7 | 66.2 | 77.2 | 10.9 | 58.4 | 18.6 | 7.3 | 8.2 | 12.6 | 53.8 | 44.3
3 | EUMS | 41.2 | 70.7 | 28.1 | 4.3 | 38.3 | 76.7 | 38.3 | 75.4 | 25.8 | 34.3 | 28.3 | 0.4 | 24.4 | 13.0 | 44.7 | 37.4
NOPS | 38.6 | 70.4 | 30.9 | 0.0 | 29.4 | 76.5 | 56.0 | 71.8 | 17.0 | 31.9 | 26.2 | 1.0 | 22.6 | 10.9 | 43.9 | 36.3
Ours | 45.5 | 82.9 | 47.7 | 0.0 | 45.1 | 77.8 | 66.3 | 77.7 | 34.3 | 49.1 | 35.6 | 4.0 | 15.3 | 17.7 | 52.8 | 44.7
Table 2: Results on splits of SemanticPOSS dataset with more severe imbalance. The gray values are the novel classes in each split. NOPS is based on its released code. Split |  Method |  bike |  build. |  car |  cone. |  fence |  grou. |  pers. |  plants |  pole |  rider |  traf. |  trashc. |  trunk |  Novel |  Known |  All
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| Full | 45.0 | 83.3 | 52.0 | 36.5 | 46.7 | 77.6 | 68.2 | 77.7 | 36.0 | 58.9 | 30.3 | 4.2 | 14.4 | - | - | 48.5
0 | NOPS | 37.4 | 22.9 | 8.1 | 0.0 | 30.3 | 78.9 | 4.8 | 72.9 | 1.0 | 42.9 | 25.8 | 9.2 | 9.6 | 7.7 | 42.5 | 26.5
Ours | 46.0 | 26.1 | 27.5 | 2.8 | 46.9 | 77.6 | 35.0 | 77.8 | 0.2 | 58.6 | 30.5 | 3.2 | 0.0 | 15.3 | 48.7 | 33.2
1 | NOPS | 6.1 | 71.3 | 35.6 | 21.2 | 3.1 | 42.9 | 44.5 | 26.0 | 24.4 | 0.7 | 0.6 | 0.1 | 24.8 | 11.4 | 37.0 | 23.2
Ours | 26.3 | 82.0 | 51.4 | 18.0 | 10.4 | 40.0 | 67.5 | 32.5 | 31.2 | 0.0 | 6.3 | 0.0 | 11.7 | 16.5 | 44.5 | 29.4

###  4.2 Results

##### SemanticPOSS Dataset.

As presented in Tab. 1, our approach exhibits significant improvements in
novel classes over the previous method across all four splits. Specifically,
we achieve an increase of 12.7% and 6.2% in split 0 and 1, respectively. It is
worth noting that the fully supervised upper bounds for novel classes in split
0 and 1 are 72.7% and 53.3%, respectively, and the performance gaps have been
significantly reduced. In the more challenging split 2 and split 3, we observe
gains of 3.6% and 4.7%, respectively. The corresponding upper bounds for these
splits are 26.9% and 33.2%, indicating their increased difficulty compared to
splits 0 and 1. On average, we achieve an IoU of 30.2% for novel classes
across all four splits, outperforming NOPS (22.5%) by 7.7%. In addition, we
provide a detailed comparison with NOPS on head, medium, and tail classes in
Appendix D, as well as under a more comparable setting that applies our
training strategy to NOPS in Appendix E.

To further verify that our method can alleviate the imbalanced problem, we
divided the SemanticPOSS dataset into two splits, creating a more severe
imbalance scenario that poses a greater challenge for clustering novel
classes. As shown in Tab. 2, on novel classes, our method outperforms NOPS
significantly on both splits, with a margin of 7.6% on split 0 and 5.1% on
split 1. In particular, for the novel classes, we observe that our improvement
mainly stems from the medium classes, such as person and bike. It is worth
noting that NOPS employs extra training techniques, such as multihead and
overclustering, whereas we use a simpler pipeline without needing them,
further demonstrating our effectiveness.

##### SemanticKITTI Dataset.

The results in Tab. 3 demonstrate our superior performance compared to
previous methods on different splits. Specifically, we achieve significant
improvements of 8.6%, 3.3%, and 3.6% on splits 0, 1, and 2, respectively, for
novel classes. The supervised upper bounds for these splits are 82.0%, 42.4%,
and 39.6%, respectively. In split 3, our results are slightly higher than NOPS
by 0.2%, possibly due to the scarce presence of these novel classes in split
3. On average across all four splits, our approach achieves an IoU of 27.5%,
surpassing NOPS (23.4%) by 4.1% on novel classes.

##### Visualization Analysis.

Additionally, in Fig. 2, we perform visual comparisons on the results between
NOPS and our method, and it is evident that our method shows significant
improvements compared to NOPS. Specifically, as shown in the first row of Fig.
2, NOPS produces noisy predictions due to uniform constraints, mixing medium
classes (e.g., building) and tail classes (e.g., car). In the second and third
rows of Fig. 2, NOPS often confuses between medium and head classes, such as
building and plants, as well as parking and car. In contrast, our method
achieves better results for both datasets due to adaptive regularization and
dual-level representation learning, generating high-quality imbalanced pseudo
labels. More visual comparisons for additional splits are provided in the
Appendix K.

Table 3: The novel class discovery results on the SemanticKITTI dataset.
‘Full’ denotes the results obtained by supervised learning. The four groups
represent the four splits in turn, and the gray values are the novel classes
in each split.

Method bi.cle b.clst build. car fence mt.cle m.clst oth-g. oth-v. park. pers.
pole road side2. terra. traff. truck trunk veget. Novel Known All Full 2.9
55.4 89.5 93.5 27.9 27.4 0.0 0.9 19.9 35.8 31.2 60.0 93.5 77.8 62.0 39.8 50.8
53.9 87.0 - - 47.9 EUMS 5.3 40.0 15.8 79.2 9.0 16.9 2.5 0.1 11.4 14.4 12.7
29.2 42.6 26.1 0.1 10.3 47.4 37.9 38.4 24.6 21.1 23.1 NOPS 5.6 47.8 52.7 82.6
13.8 25.6 1.4 1.7 14.5 19.8 25.9 32.1 56.7 8.1 23.8 14.3 49.4 36.2 44.2 37.1
26.5 29.3 Ours 5.5 51.1 74.6 92.3 29.8 22.8 0.0 0.0 23.3 24.8 27.7 59.7 41.4
22.5 23.6 39.3 43.6 51.1 66.4 45.7 33.7 36.8 EUMS 7.5 42.4 80.0 76.8 8.6 19.6
1.4 0.6 12.0 14.1 14.0 40.7 86.3 66.5 56.3 12.0 44.8 20.9 72.4 24.2 37.1 35.6
NOPS 7.4 51.2 84.5 50.9 7.3 28.9 1.8 0.0 22.2 19.4 30.4 37.6 90.1 72.2 60.8
16.8 57.3 49.3 85.1 25.4 46.2 40.7 Ours 3.7 57.4 89.2 56.5 17.3 20.3 0.0 0.0
20.0 30.6 34.8 60.6 93.2 77.6 62.0 38.7 56.9 39.2 86.7 28.7 50.1 44.5 EUMS 8.3
50.8 83.0 88.1 17.9 2.8 2.3 0.2 3.2 25.4 25.0 20.2 88.3 71.0 57.9 8.6 27.2
38.4 77.0 12.4 42.2 36.6 NOPS 6.7 49.2 86.4 90.8 23.7 2.7 0.6 1.9 15.5 29.5
27.9 36.4 90.3 73.4 61.2 17.8 10.3 46.2 84.3 16.5 48.0 39.7 Ours 3.6 54.2 88.9
93.3 28.4 10.2 0.0 0.9 9.6 33.4 32.2 36.1 92.7 77.4 62.2 10.7 34.2 51.7 86.9
20.1 50.4 42.5 EUMS 4.0 2.5 80.1 87.2 16.8 14.0 15.0 0.3 14.1 20.8 6.8 37.6
86.8 66.5 55.3 16.2 40.6 38.4 76.2 7.1 43.4 35.7 NOPS 2.3 27.8 86.0 89.9 23.1
24.5 2.9 3.1 18.2 30.1 16.3 39.9 90.7 73.5 61.0 17.4 49.8 44.0 83.2 12.4 49.0
41.2 Ours 2.6 32.5 88.7 93.3 28.1 24 0.1 1.0 23.7 35.6 15.3 59.8 93.2 77.6
61.4 37.8 56.6 52.1 86.7 12.6 54.6 45.8

Figure 2: Visualization comparison between Our method and NOPS on the
SemanticPOSS and SemanticKITTI datasets. In the first and second rows,
compared to NOPS, our method achieves much better segmentation for the
‘Building’, significantly reducing confusion with tail classes (such as ‘Car’)
or medium classes (like ‘Plants’). In the third row, our approach generates
better segmentation for ‘Parking’ and ‘Car’.

###  4.3 Ablation Study

##### Component Analysis.

To analyze the effectiveness of each component, we conduct extensive
experiments on split 0 of the SemanticPOSS dataset. Here we provide ablation
on three components, including Imbalanced Self-Labeling (ISL), Adaptive
Regularization (AR), and Region-Level Branch (Region). As shown in Tab. 4,
compared to baseline which employs equal-size constraints, imbalanced self-
labeling improves performance by 4.2%. The confusion matrix in Fig. 3
indicates that except for the highly-accurate class “ground", there is a
significant improvement in the head and medium classes. This phenomenon is
clearly depicted in Fig. 4, where the predictions of the baseline exhibit
noticeable noise. From the second and third rows of Tab. 4, the adaptive
regularization leads to a significant improvement of 8.2% in split0 and 4.5%
in overall splits. As shown in Fig. 3, adaptive regularization enhances the
quality of pseudo-labels for each class, especially for the head class
(plants). We also visualize the class distribution of pseudo-labels in
Appendix F, which shows adaptive regularization provides greater flexibility
than fixed regularization term. According to the third, fourth and last rows
of Tab. 4, the inclusion of the region-level branch leads to a 9.1%
improvement and an additional 4.2% improvement built upon the AR. In addition,
more experiments and analysis on prototype learning are included in Appendix
G. In Fig. 3, there’s a significant improvement in pseudo-labels for each
category, particularly for the tail class (car) and the head class (plants).
From Fig. 4, it is evident that the region-level branch can correct cases
where a single object is mistakenly labeled as multiple categories. Due to the
utilization of spatial priors, where closely-located points are highly likely
to belong to the same category, our region-level branch can correct
misclassifications by considering context from neighboring points, preventing
splitting a single object into multiple entities. Those experiments validate
the effectiveness of each component in our method.

Table 4: Ablation study on SemanticPOSS, focusing on novel classes. ISL and AR
denote imbalanced self-labeling and regularization. Region denotes region-
level learning. The last two columns represent the average mIoU for split 0
and across all splits.

|  |  | Split0 | Overall
---|---|---|---|---
ISL | AR | Region | Building | Car | Ground | Plants | Avg | Avg
|  |  | 21.6 | 2.7 | 76.6 | 26.1 | 31.8 | 20.9
$\checkmark$ |  |  | 27.6 | 3.1 | 81.2 | 32.1 | 36.0 | 23.9
$\checkmark$ | $\checkmark$ |  | 53.1 | 5.3 | 81.1 | 37.4 | 44.2 | 28.4
$\checkmark$ |  | $\checkmark$ | 41.9 | 9.3 | 83.6 | 45.6 | 45.1 | 26.9
$\checkmark$ | $\checkmark$ | $\checkmark$ | 51.5 | 6.0 | 83.0 | 53.1 | 48.4 | 30.2

Figure 3: Confusion Matrix, GT on the y-axis, Pseudo Label on the x-axis.
$(i,j)$ represents the % of GT in class $j$ assigned pseudo label $i$. We
categorize ‘plants’ and ‘ground’ as head classes, ‘building’ as medium, and
‘car’ as tail classes.

Figure 4: Visualization analysis. The introduction of ISL notably reduces the
misclassification between ‘Plant’ and ‘Car’. Then, the integration of AR
further mitigates the confusion between ‘Plant’ and ‘Building’. Ultimately,
the incorporation of Region component (Ours) effectively minimizes the mix-up
between ‘Plant’, ‘Car’, and ‘Building’. Table 5: Analysis of adaptive
regularization on SemanticPOSS dataset. GT denotes we directly assign the
ground truth distribution of cluster size.

$\gamma$ | 0.01 | 0.05 | 0.1 | 0.5 | 1 | 5 | $+\infty$ | GT | Adaptive
---|---|---|---|---|---|---|---|---|---
Split0 | 10.1 | 33.8 | 33.3 | 36.0 | 32.2 | 33.8 | 31.8 | 32.5 | 44.2

##### Estimate the number of novel classes.

For computational simplicity, we conduct experiments on splits 0 of the
SemanticPOSS dataset and randomly sample 800,000 points from all scenes to
estimate $|C^{u}|$. We set max classes to 50, which is an estimate of the
maximum number of new classes that might appear in a typical scene. The
estimated $|C^{u}|$ is 3, which is close to the ground truth value (GT is 4).
Finally, we conduct experiments with $|C^{u}|$ as 3. As Tab. 6 illustrated our
method still outperforms NOPS by a large margin.

Table 6: Comparison between Ours and NOPS with the estimated number of novel
classes in Split 0 of SemanticPOSS. The estimated $C^{u}$ is 3, and the ground
truth is 4.

Method | Building | Car | Ground | Plants | Avg
---|---|---|---|---|---
NOPS | 25.54 | 0.00 | 68.15 | 34.12 | 31.95
Ours | 64.05 | 0.00 | 82.22 | 67.63 | 53.47

##### Adaptive Regularization and Hyperparameters Selection.

To analyze the impact of adaptive regularization, we compare it with various
fixed regularization factors, as illustrated in Tab. 5. We notice that
employing a very small fixed $\gamma$, such as 0.05 as indicated in the table,
results in a weak prior constraint, and the model tends to learn a degenerate
solution where all samples are assigned to a single cluster. When the $\gamma$
increases to 0.5, the model achieves optimal results, but the increment
decreases when the $\gamma$ further increases. Compared with adaptive
$\gamma$, the optimal results of fixed $\gamma$ is nearly 8.2% lower,
demonstrating that the adoption of an adaptive $\gamma$ not only enhances the
model’s flexibility but also prevents any performance degradation.
Furthermore, we experiment with the setup adopting the GT class distribution
and substituting the KL constraint in Sec. 3.3 with an equality constraint.
Surprisingly, the results indicate that the GT class distribution constraint
is not the optimal solution for clustering imbalanced novel classes. At last,
in Fig. 7, we visualize the $\gamma$ curves for SemanticPOSS in four splits.
Split 0 exhibits the highest rate of change, followed by Split 1, while Splits
2 and 3 remain constant, indicating that our strategy is adaptive to each
dataset.

To further validate the effectiveness of adjusting $\gamma$ based on KL
divergence, we also compare it with typical step decay and cosine annealing
strategies. For the step decay, we set the initial $\gamma$ to 1 and decay it
by multiplying it with $\lambda$ every epoch. For the cosine annealing
approach, we also set the initial $\gamma$ to 1 and reduce it to the minimum
value (min $\gamma$). From the Tab. 8 and Tab. 8, we observe that the results
of simple step decay and cosine annealing are nearly 10% worse than adaptive
$\gamma$ (which is 44.2). We believe that these two typical strategies lack
flexibility compared to adaptive $\gamma$. They might not facilitate the
adaptive control of the $\gamma$ decay process based on the model learning
process.

To choose the hyperparameters $\rho$ and $T$ according to the indicator
outlined in Sec. 3.3, we conduct experiments for various values of $\rho$ and
$T$. The results are displayed in Tab. 10 and Tab. 10. Additionally, we plot
the indicator’s curve for each experiment in Fig. 7 and 7. The plots reveal
that when $\rho$ falls within the range of 0.01 to 0.005, and $T$ is set
between 5 and 20, the indicator value remains low while achieving a high novel
IoU. Those results demonstrate the efficiency of our hyperparameters selection
strategy and the robustness of our method.

Table 7: Step decay results

$\lambda$ | 0.1 | 0.3 | 0.5 | 0.7 | 0.9
---|---|---|---|---|---
Step decay | 34.0 | 34.2 | 32.9 | 34.6 | 33.3

Table 8: Cosine annealing results

min $\gamma$ | 0.1 | 0.05 | 0.01 | 0.005 | 0.001
---|---|---|---|---|---
Cosine annealing | 32.2 | 32.5 | 35.8 | 36.1 | 32.0

Table 9: The results for different $\rho$

$\rho$ | 0.05 | 0.01 | 0.005 | 0.001
---|---|---|---|---
Split0 | 10.08 | 45.84 | 44.21 | 33.11

Table 10: The results for different $T$

$T$ | 5 | 10 | 20 | 30
---|---|---|---|---
Split0 | 44.46 | 44.21 | 44.12 | 32.48

Figure 5: $\gamma$ variation

Figure 6: Selecting $\rho$

Figure 7: Selecting $T$

##### Limitations.

One limitation is our problem setup which follows [28] and only addresses
scenarios where unlabelled data constitutes novel classes. In contrast, a more
realistic open-world setting necessitates handling situations where both known
classes and novel classes lack labels. Nevertheless, we anticipate that our
method will establish a robust baseline and stimulate further research aimed
at addressing the challenges presented by practical open-world situations.

##  5 Conclusion

In this paper, we propose a novel dual-level adaptive self-labeling framework
for novel class discovery in point cloud segmentation. Our framework
formulates the pseudo label generation process as a Semi-relaxed Optimal
Transport problem and incorporates a novel data-dependent adaptive
regularization factor to gradually relax the constraint of the uniform prior
based on the distribution of pseudo labels, thereby generating higher-quality
imbalanced pseudo labels for model learning. In addition, we develop a dual-
level representation that leverages the spatial prior to generate region
representation, which reduces the noise in generated segmentation and enhances
point-level classifier learning. Furthermore, we propose a hyperparameters
search strategy based on training sets. Extensive experiments on two widely
used datasets, SemanticKITTI and SemanticPOSS, demonstrate the effectiveness
of each component and the superiority of our method.

#### Acknowledgments

This work was supported by National Science Foundation of China under grant
62350610269, Shanghai Frontiers Science Center of Human-centered Artificial
Intelligence, and MoE Key Lab of Intelligent Perception and Human-Machine
Collaboration (ShanghaiTech University).

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# Measurement of Low-Activity Uranium Contamination using Bayesian Statistical
Decision Theory

Hanan Arahmane<EMAIL_ADDRESS>Jonathan Dumazert  Eric Barat  Thomas
Dautremer  Frédérick Carrel  Nicolas Dufour  Maugan Michel  Université Paris-
Saclay, CEA, List, F-91120 Palaiseau, France  CEA-DAM, DIF, F-91297 Arpajon,
France

###### Abstract

Amongst the various technical challenges in the field of radiation detection
is the need to carry out accurate low-level radioactivity measurements in the
presence of large fluctuations in the natural radiation background, while
lowering the false alarm rates. Several studies, using statistical inference,
have been proposed to overcome this challenge. This work presents an advanced
statistical approach for decision-making in the field of nuclear
decommissioning. The results indicate that the proposed method allows to
adjust the confidence degree in the stationarity of the background signal. It
also ensures an acceptable tradeoff between the True Detection Rate (TDR), the
False Alarm Rate (FAR) and the response time, and is consistent with the
user’s requirements.

###### keywords:

Radioactivity Detection , Statistical Inference , Gamma-Ray Spectrometry

contents=  Submitted to the Proceedings of the African Conference on
Fundamental and Applied Physics Second Edition, ACP2021, March 7–11, 2022 —
Virtual Event

, scale=1,placement=top,opacity=1,color=black,position=3.25in,1.2in

##  1 Introduction

Accurate detection of radioactivity in low count rates is crucial for many
practical radiation-detection applications. The particular approach considered
here is that of statistical analysis that has recently provided new horizons
in the area of radioactive detection. It is one of the reliable and accurate
analysis for determining the most effective detection strategy with reasonable
operational requirements. In this paper, we aim to measure a low-activity
uranium contamination on concrete surfaces, with varying enrichment
encountered levels within a basic nuclear facility. Our surface contamination
limit was taken as $S_{A}=500~{}Bq/0.1~{}m^{2}$ [1]. In this context, we have
developed an advanced Bayesian method based on a Kibble bivariate gamma
distribution, referred to as the Bayesian Kibble Dirichlet (BKD) test. It
allows to adjust the confidence degree in the stationarity of the background
signal. In this perspective, the BKD test combines the properties of an
absolute and a relative hypothesis statistical test, that were both developed
with the same application at hand [1], and respectively referred to as the
Bayesian Absolute test by mixture of Multivariate Poisson laws (BAM) and the
Bayesian Relative test by mixture of Multinomial laws (BRM). Note that unlike
the BKD test, the modeling of the BAM test was based on the univariate gamma
distribution. The implementation of the Bayesian approach were based on a
priori vectors constructed from the coupling of experimental data acquired
within a basic nuclear facility using a high-purity germanium detector (HPGe),
as well as simulated data with Monte Carlo N-Particles 6 transport code
(MCNP6.1) [2]. The performance evaluation and characterization of the Bayesian
method were performed using classical receiver operating characteristic curves
(ROC) [3] with the study of the background signal variations effect.

##  2 Materials and Methods

###  2.1 Uranium gamma peaks of interest and background signal

Detection of low levels of uranium contamination is to determine its presence
by means of an HPGe detector. To do so, we used the main gamma-ray emission
lines from the decay chains of U-235 and U-238 [1]: (i) from the U-235 chain,
one main line at 185.7 keV, and an additional triplet at 143.8; 163.4; 205.3
keV, and (ii) from the U-238 chain, one main line at 1001.0 keV.

Regarding the background signal, it should be noted that the detection of a
low-surface activity uranium contamination on concrete surfaces, based on
gamma-ray emission, can be disturbed by the presence of the background
signatures, in particular K-40, and the decay chains of U-238 and Th-232. In
order to quantify this variation within nuclear facility that is subdivided
into three subunits (SUs), we performed the acquisitions over source-detector
distance $d=22.36~{}cm$ for a surface of $0.1~{}m^{2}$. Then, we calculated
the relative deviation between two separate spots inside each SU.

###  2.2 Simulation of the detector response

The performance evaluation of the Bayesian method needs prior knowledge of the
expected spectral response from a uranium contamination corresponding to the
minimum surface activities to be detected (MSAD) (i.e., decommissioning
criterion). MSAD used in this study is equivalent to $500~{}Bq/0.1m^{2}/2\pi$.
To this end, we followed three steps. First, we constructed a transfer
function relating the emission rates $(s^{-1})$ with the five gamma lines
identified above at this MSAD. Second, we built a numerical model of the HPGe
detector using the MCNP6.1, in order to estimate the spectral count rate
$(s^{-1})$ expected at its output from MSAD and enrichment level in U-235.
Finally, we simulated the U-235 enrichment levels of the spectral response of
the detector as well. It should be noted that each of the above stated SU has
average expected U-235 enrichment levels that were set to be $0.62$ wt
%$,2.03$ wt %, and$~{}6.84$ wt % . These enrichment levels are likely to
fluctuate around these average values. As a result, we used regular U-235
enrichment mass fractions corresponding to the natural ones ($\approx 0.7$ wt
%$),3$ wt %, and $8$ wt % for SU1, SU2, and SU3 respectively, in order to
construct tested spectra.

###  2.3 Bayesian Methodology

The construction of Bayesian statistical tests, either BAM, BRM and BKD needs
some prior Knowledge. The spectral signatures of both the background signal,
and an expected uranium contamination inside a nuclear facility, are the
priors retained in this study. We recall here that the prior spectra of the
background signal, and of an expected uranium contamination are respectively
provided by in-situ acquisitions, and Monte Carlo simulations based on
MCNP6.1.

The construction of the Bayesian test leading to the acceptance of one of the
two hypotheses $H_{0}$ or $H_{1}$ makes use of Bayes’ theorem as follows:

$\frac{P(H_{0}\mid\mathbf{m_{ref},m_{test})}}{P(H_{1}\mid\mathbf{m_{ref},m_{test}})}=\frac{P\mathbf{(m_{ref},m_{test}}\mid
H_{0})P(H_{0})}{P\mathbf{(m_{ref},m_{test}}\mid H_{1})P(H_{1})}$ (1)

where $m_{ref}$ and $m_{test}$ the multichannel countings, $P(H_{0})$ and
$P(H_{1})$ are priors on the respective likelihoods of hypotheses $H_{0}$ and
$H_{1}$.

Without any a priori information of the presence of a uranium contamination,
these probabilities are taken equal by default: $P(H_{0})$ = $P(H_{1})$ =
$0.5$. Moreover, we note that: $P(H_{1}\mid\mathbf{m_{ref},m_{test}})$ =
$1-P(H_{0}\mid\mathbf{m_{ref},m_{test}})$. As a result, by taking into account
these equalities in equation (1) the posterior probability of hypothesis
$H_{0}$, that will be ultimately compared to a decision threshold, can be
expressed as:

${P(H_{0}\mid\mathbf{m_{ref},m_{test})}}=\frac{1}{1+\kappa_{BAM,BRM,BKD}}$ (2)

with the Bayes factor expressed as:

$\kappa_{BAM,BRM,BKD}=\frac{P\mathbf{(m_{ref},m_{test}}\mid
H_{1})}{P\mathbf{(m_{ref},m_{test}}\mid H_{0})}$ (3)

From the last three equations, we can see that the construction of the
Bayesian tests is the comparison of $P(H_{0}\mid\mathbf{m_{ref},m_{test}})$
and $P(H_{1}\mid\mathbf{m_{ref},m_{test}})$ through the computation of the
Bayes factor $\kappa_{BAM,BRM,BKD}$.

It should be noted that the process of adjusting the properties of the BKD
test in order to retrieve the limiting behaviors of the absolute (BAM) and
relative (BRM) tests, relies essentially upon the parametrization of the BKD
test with respect to the correlation coefficient of Kibble law $\rho$ [4].
Therefore, the implementation of the BKD test requires the setting of a
parameter $\rho$ which governs the behavior of BKD between that of BRM ($\rho$
= $0$) and that of BAM ($\rho$ →$1$).

##  3 Results and Conclusions

In this section, we present the comparative performance study of the BKD test
with the absolute (BAM) and relative (BRM) ones by using a ROC-based analysis.
The scenarios involving these three settings of $\rho$ in both cases of a
stationary or non-stationary background signal, in amplitude as well as in
shape, yield the ROC curves in Figures 1 and 2.

These ROCs allow to identify explicit trends. From the figures, we can see
that the BKD test, whether the background signal is stationary and non-
stationary, has the same behavior as the relative BRM test (Figures 1(a) and
2(a)) in the case of $\rho$ = $0$, while it presents the properties of the
absolute BAM test in the case of $\rho$ →$1$ (Figures 1(b) and 2(b)).
Furthermore, the results showed that a reliable detection of uranium was
reachable with a significantly higher TDR/FAR tradeoff. However, we can
observe that the variation of the background signal, affected the detection
performance of the Bayesian tests, which is normal, due to an increase in the
statistical fluctuations in the spectra.

(a) $\rho$ = $0$

(b) $\rho$ → $1$

Figure 1: ROC curves for an enrichment level of $0.7$ wt%, an integration time
$t$ = $3000$ s, a stationary background signal, and $\rho$ : (a) $\rho$ = $0$,
(b) $\rho$ → $1$.

(a) $\rho$ = $0$

(b) $\rho$ → $1$

Figure 2: ROC curves for an enrichment level of $0.7$ wt%, an integration time
$t$ = $3000$ s, a non-stationary background signal, and $\rho$ : (c) $\rho$ =
$0$, (d) $\rho$ → $1$.

## References

  * [1] H. Arahmane, J. Dumazert, E. Barat, T. Dautremer, N. Dufour, F. Carrel, M. Michel, F. Lainé, A reliable absolute and relative bayesian method for nuclear decommissioning: Low-level radioactivity detection with gamma-ray spectrometry, IEEE Transactions on Instrumentation and Measurement 70 (2021) 1–18.
  * [2] C. J. Werner, et al., Mcnp users manual-code version 6.2, Los Alamos national laboratory (2017).
  * [3] C. E. Metz, Basic principles of roc analysis, in: Seminars in nuclear medicine, Vol. 8, Elsevier, 1978, pp. 283–298.
  * [4] N. Balakrishna, C. D. Lai, Bivariate gamma and related distributions, in: Continuous Bivariate Distributions, Springer, 2009, pp. 305–350.


11institutetext: 1 School of Electrical and Computer Engineering, National
Technical University of Athens, 157 80, Greece
2Department of Space, Earth and Environment, Chalmers University of
Technology, SE-412 96 Göteborg, Sweden
3 Centre for Fluid & Complex Systems, Coventry University, Priory St, Coventry
CV1 5FB, UK
4 Dept. of Physics, Aristotle University of Thessaloniki, 541 24,
Thessaloniki, Greece

# A statistical analysis approach of plasma dynamics in gyrokinetic
simulations of stellarator turbulence

A. D. Papadopoulos1 J. Anderson2 E-J. Kim3 M. Mavridis4 and H. Isliker4

##  1 Abstract

A geometrical method is used for the analysis of stochastic processes in
plasma turbulence. Distances between thermodynamic states can be computed
according the thermodynamic length methodology which allows the use of a
Riemannian metric on the phase space. A geometric methodology is suitable in
order to understand stochastic processes involved in e.g. order-disorder
transition, where a sudden increase in distance is expected. Gyrokinetic
simulations of Ion-Temperature-Gradient (ITG) mode driven turbulence in the
core-region of the stellarator W7-X, with realistic quasi-isodynamic
topologies are considered. In gyrokinetic plasma turbulence simulations
avalanches, e.g. of heat and particles, are often found and in this work a
novel method for detection is investigated. This new method combines the
Singular Spectrum Analysis algorithm and Hierarchical Clustering such that the
gyrokinetic simulation time series is decomposed into a part of useful
physical information and noise. The informative component of the time series
is used for the calculation of the Hurst exponent, the Information Length and
the Dynamic Time. Based on these measures the physical properties of the time
series is revealed.

##  2 The SSA & HC methods

The SSA analysis for 1D series is a special case of the 2D SSA method.
Therefore only the latter is presented. The Gene-data $Q\left(t,\rho\right)$
forms a matrix of size $N_{x}\times N_{y}$. A window $W$ of size $L_{x}\times
L_{y}$ is considered, running inside $Q\left(t,\rho\right)$, that is
$L_{x}\leq N_{x},~{}L_{y}\leq N_{y}$. In the two dimensional SSA, the
trajectory matrix is defined as
$\mathbf{X}=\left[X_{1},X_{2},\ldots,X_{K_{x}\cdot K_{y}}\right]$. It is a
matrix of size $L_{x}\cdot L_{y}\times K_{x}\cdot K_{y}$ where
$K_{x}=N_{w}-L_{w}+1$, $w=x,~{}y$ and its columns $X_{i}$ are vectorizations
(vec) of the $L_{x}\times L_{y}$ submatrices $W$,
$X_{k+(l-1)K_{x}}=\mbox{vec}\left(W\right)$, $W=X_{k,l}^{\left(L_{x}\times
L_{y}\right)}\equiv\left(Q_{i,j}\right)_{i=k,j=l}^{L_{x}+k-1,L_{y}+l-1}$. More
compactly $\mathbf{X}=\mathbf{T}_{2D}\left(Q\right)$, where $\mathbf{T}$ is
the embedding linear operator. It is assumed that the $x$-axis is oriented to
the bottom and the $y$-axis to the right of the $Q$ matrix and the origin is
the left upper corner.

The next step of the SSA analysis consists of the decomposition of the
trajectory matrix $\mathbf{X}$. In particular the Singular Value Decomposition
is applied on the trajectory matrix leading to the decomposition
$\mathbf{X}=\mathbf{X}_{1}+\mathbf{X}_{2},\ldots,\mathbf{X}_{d}$, where
$\mathbf{X}_{j}=U_{j}V_{j}^{T}\sqrt{\lambda_{j}}$, $j=1,2,\ldots,d$, the
$j^{\mbox{th}}$ singular vectors and value ($\lambda_{j}$) respectively and
$d$ the rank of $\mathbf{X}$. The $d$ values and corresponding matrices
$\mathbf{X}_{j}$ are grouped into $m$ sets:
$\\{1,2,\ldots,d\\}=\cup_{i=1}^{m}I_{i}$. The grouping of the indices,
$j=1,2,\ldots,d$, is achieved via the agglomerative HC method. In the
agglomerative HC method a hierarchy of clusters is produced following a
bottom-up approach. In particular each observation (index) forms its own
cluster and then observations are merged in an additive manner, moving up in
the hierarchy. Each observation corresponds to a particular reconstructed
matrix $\tilde{X}_{j}$ related to the $j^{\mbox{th}}$ component
$\mathbf{X}_{j}$, of the trajectory matrix $\mathbf{X}$. The observation
merging is based on a weighted distance matrix $d_{\mathbf{w}}\equiv
1-\rho_{\mathbf{w}}$, where
$\rho_{\mathbf{w}}\left(i,j\right)\equiv\left(\tilde{X}_{i},\tilde{X}_{j}\right)_{\mathbf{w}}/\left(\left\|\tilde{X}_{i}\right\|\left\|\tilde{X}_{i}\right\|\right),~{}\left\|\tilde{X}_{i}\right\|\equiv\sqrt{\left(\tilde{X}_{i},\tilde{X}_{i}\right)}_{\mathbf{w}},~{}\left(X,Y\right)_{\mathbf{w}}\equiv\sum_{l}w_{l}X^{T}_{l,l}Y_{l,l}$.
The distance between observations $i,~{}j$ is given by by
$\left(d_{\mathbf{w}}\right)_{i,j}$. In the first HC step, pairs of
observations $i$, $j$ form a cluster if
$i=\mbox{agr}_{i^{*}}\min\left(d_{\mathbf{w}}\right)_{i^{*},j}$ . In the
second HC step, since clusters have been formed, there must be a definition of
cluster distance. This is provided by the linkage clustering, for which there
are many choices, single-linkage, complete-linkage, average-linkage, Ward-
linkage, etc. The single-linkage clustering is used here, where clusters $A$,
$B$, are at a distance $\min\left(\left(d_{\mathbf{w}}\right)_{i,j},i\in
A,j\in B\right)$. The HC steps continue up-wards clustering until the
desirable number of clusters, $m$, are formed (the HC stopping criterion in
this work) or until the clusters are too far apart to merge. In this study
$m=3$ and we consider the classes (clusters), oscillations, trend and noise.
Eventually the noise part is removed.

The reconstructed matrices $\tilde{X}_{j}$ are the result of antidiagonal
averaging [2] followed by inverse embedding, $\mathbf{T}_{2D}^{-1}$, applied
to the $\mathbf{X}_{i}$ matrices.

##  3 Evaluation of Information Length, Dynamic Time and Hurst Exponent

From SSA we obtain time subseries $\tilde{X}_{i}\left(\rho\right)$,
$i=1,\ldots,d$ of the various SSA components and it is possible to calculate
their dynamic time and information length. The dynamic time
$\tau\left(t,\rho\right)$ is a time scale over which the probability of the
$\tilde{X}_{i}\left(\rho\right)$ changes on average at time $t$ (denoted as
$p\left(\tilde{X}_{i}\left(\rho\right),t\right)$). The probability
$p\left(\tilde{X}_{i}\left(\rho\right),t\right)$ is estimated from a subset
(window) of $\tilde{X}_{i}\left(\rho\right)$ samples of length $W_{L}$
produced around the time index $t$. In particular, first the samples of
$\tilde{X}_{i}\left(\rho\right)$ are interpolated to equally spaced time
instances (separated by $dt$), then from the samples of each window $w$ of
size $W_{L}$ the $p\left(\tilde{X}_{i}\left(\rho\right),t\right)$ is
calculated at the time instant $t$ at the middle of the window’s time
interval. Then the window $w$ is moved (running window) by one time sample and
the $p\left(\tilde{X}_{i}\left(\rho\right),t+dt\right)$ is calculated at the
time instant $t+dt$, and so on. The calculation of
$p\left(\tilde{X}_{i}\left(\rho\right),t\right)$ is done based on the
histogram of $w$-samples. Usually the histogram-produced
$p\left(\tilde{X}_{i}\left(\rho\right),t\right)$ is non-smooth and a smoothing
Gaussian kernel is applied. Then $\tau_{i}\left(t,\rho\right)$ of the
$\tilde{X}_{i}\left(\rho\right)$ subsequence, is given by [4] :

$\tau_{i}\left(t,\rho\right)^{2}=1/\int
d\tilde{X}_{i}\left(\rho\right)\frac{1}{p\left(\tilde{X}_{i}\left(\rho\right),t\right)}\left(\frac{\partial
p\left(\tilde{X}_{i}\left(\rho\right),t\right)}{\partial t}\right)^{2}.$ (1)

From the dynamic time the information length , $L_{i}\left(t,\rho\right)$, can
be directly calculated [4]:

$L_{i}\left(t,\rho\right)=\int_{0}^{t}ds\frac{1}{\tau_{i}\left(s,\rho\right)}.$
(2)

(a) Dynamic time function of time samples $t$.

(b) Dynamic time as function of time samples $t$ and varying radius $\rho$.

Another quantity of interest is the Hurst exponent, $H$, of
$Q\left(\rho\right)$. In general $0<H<1$ but if $H=0.5$ the series is
considered random (uncorrelated), if $H>0.5$ the series has long term positive
autocorrelation meaning that high (low) values in series $Q\left(\rho\right)$
will have a higher probability of being followed by another high (low) value
whereas if $H<0.5$ in the long run with high probability high (low) values in
$Q\left(\rho\right)$ will have a higher probability of being followed by
another low (high) value. The Hurst exponent is calculated by the rescaled
range (RS) method as the exponent $H$ such that:
$\mathbb{E}\left[R\left(n\right)/S\left(n\right)\right]=Cn^{H}$ for
$n\rightarrow\infty$, where $C$ is a constant, $\mathbb{E}\left[\cdot\right]$
is the expected mean, $S\left(n\right)$ is the standard deviation of the
series
$Q_{1}\left(\rho\right),Q_{2}\left(\rho\right),\ldots,Q_{n}\left(\rho\right)$
and $R\left(n\right)$ is the range of the $n$ cumulative deviations from the
mean. That is
$R\left(n\right)=\max\left(Z_{1},Z_{s},\ldots,Z_{n}\right)-\min\left(Z_{1},Z_{s},\ldots,Z_{n}\right)$,
$Z_{j}=\sum_{i=1}^{j}\left(\tilde{X}_{i}-m\right)$,
$m=\left(\sum_{i=0}^{n}\tilde{X}_{i}\right)/n$. Then $H$ is calculated as the
slope of the line that fits the $\log\left(R(n)/S(n)\right)$ data as a
function of $\log\left(n\right)$.

##  4 Numerical Results

The analysis is performed using ion-heat flux data $Q\left(\rho\right)$ time
series from the gyrokinetic code GENE. The collected data are for normalized
magnetic flux radii $s=0.5$, normalized temperature gradient $R/L_{T}=1$ and
density gradient $R/L_{n}=0$. In addition the electrons are considered
adiabatic. The $Q\left(\rho\right)$ time series are analyzed with the 1D-SSA
($Q$ data averaged over the radius $\rho$) or with the 2D-SSA (non averaged
$Q$ data).

The dynamic time stores the instantaneous changes in the information length,
this is indicative of sudden changes depending on the present resolution, see
Figure 1a and 1b and [4]. In Figure 1a the dynamic time in the 1D case is
presented, this is average behaviour of the strongly fluctuating signal in
Figure 1b. However, the dynamic time exhibit rapid fluctuations and to have a
consistent change in dynamics a significant change in the information over a
short interval is needed. It is therefore suggested that only by analyzing the
information and the dynamic time in tandem, an indication of change in
dynamics can be detected. The information length is the additive effect of a
series of consistent fluctuations, see Fig 2a and 2b.

(a) Information length as function of time samples.

(b) Information length, as function of time samples, and varying radius
$\rho$.

A sudden increase in information length is found at around t = 5000 in the 1D
case. The information is almost linearly increasing until the change in the
dynamics is suggested by a sudden increase in information. In 2D a more varied
picture is found. At each radii the information length is calculated, which
then gives a almost linear increase in the information length for each radii.
It is observed that at certain radii there are sharp increases in information
length e.g. $\rho=0$ after a certain time similar to the 1D case.

Figure 3: Hurst exponent, $H\left(\rho\right)$ calculated by the 1D and 2D SSA
.

##  5 Discussion

The information length measures the distance between two consecutive states in
plasmas. It is thus believed that this measure could be used as indication
that some large structure is formed e.g. and is mediating significant
transport. In a previous effort the 1D model was developed however it was
deemed difficult to directly compare with larger events found in the
simulations thus a 2D version is now presented. It is seen that large
fluctuations in dynamic time, the instantaneous change in length Fig 1, can be
observed if they are consistent c.f. Hurst exponent Fig 3. It is found that at
certain radii and times there is a significant increase in information length.

##  6 Acknowledgement

AP was supported by the European Union via the Euratom Research and Training
Programme under Grant 101052200 EUROfusion through EUROfusion Consortium.

## References

  * [1] The GENE Development Team, http://genecode.org/
  * [2] N. Golyandina, V. Nekrutkin, A. Zhigljavsky, CHAPMAN $\&$ HALL/CRC, (2001).
  * [3] J. Anderson, et al, Phys. Plasmas 21, 122306 (2014). 122306 (2014).
  * [4] J. Anderson, et al, Physics of Plasmas 27, 022307 (2020).