# Boosting Semi-Supervised Object Detection in Remote Sensing Images with
Active Teaching

Boxuan Zhang, Zengmao Wang, and Bo Du,  This work was supported in part by the
National Natural Science Foundation of China under Grants 62271357, the
Natural Science Foundation of Hubei Province under Grants 2023BAB072, the
Fundamental Research Funds for the Central Universities under Grants
2042023kf0134.(Corresponding author: Zengmao Wang.) Boxuan Zhang is with the
School of Computer Science, Wuhan University, Wuhan 430072, China(e-mail:
zhangboxuan1005@gmail.com). Zengmao Wang and Bo Du are with the National
Engineering Research Center for Multimedia Software, School of Computer
Science, Institute of Artificial Intelligence, and the Hubei Key Laboratory of
Multimedia and Network Communication Engineering, Wuhan University, Wuhan
430072, China (e-mail<EMAIL_ADDRESS>gunspace@163.com).

###### Abstract

The lack of object-level annotations poses a significant challenge for object
detection in remote sensing images. To address this issue, active learning and
semi-supervised learning techniques have been proposed to enhance the quality
and quantity of annotations. Active learning focuses on selecting the most
informative samples for annotation, while semi-supervised learning leverages
the knowledge from unlabeled samples. In this paper, we propose a novel active
learning method to boost semi-supervised object detection for remote sensing
images with a teacher-student network, called SSOD-AT. The proposed method
incorporates a RoI Comparison module (RoICM) to generate high-confidence
pseudo-labels for Regions of Interest (RoIs). Meanwhile, the RoICM is utilized
to identify the top-K uncertain images. To reduce redundancy in the top-K
uncertain images for human labeling, a diversity criterion is introduced based
on object-level prototypes of different categories using both labeled and
pseudo-labeled images. Extensive experiments on DOTA and DIOR two popular
datasets demonstrate that our proposed method outperforms state-of-the-art
methods for object detection in remote sensing images. Compared with the best
performance in the SOTA methods, the proposed method achieves 1% improvement
at most cases in the whole active learning.

###### Index Terms:

Active learning(AL), semi-supervised object detection(SSOD), teacher-student
framework, remote sensing.

##  I Introduction

Object detection in remote sensing images is a crucial task to identify the
objects with their locations [1]. In recent years, deep learning has shown
promising results in object detection for remote sensing images. However, the
success of deep learning approaches heavily relies on large-scale datasets
with accurately labeled data, which are typically annotated by human experts.
Unlike image classification, object detection requires object-level labels
that include both bounding box coordinates and object categories. This
annotation process is more laborious and time-consuming. Additionally, remote
sensing images often contain objects with various orientations and scales [2],
[3], [4], [5], [6], which further complicates the labeling process. Hence,
although the collection of remote sensing images are to be faster and easier
with the advancements of remote sensing technology, the availability of
labeled images for object detection is usually limited [7], [8], [9].

Semi-supervised learning (SSL) and active learning (AL) are two promising
techniques in machine learning to address the problem with limited labeled
images. SSL usually attempts to exploit the unlabeled data with a limited
amount of labeled data by assuming the consistency between the feature
distribution of unlabeled data and labeled data. For the semi-supervised
object detection(SSOD) method, most of them are developed based on a teacher-
student network, which introduces a secondary model (teacher) to guide the
training of the primary model (student). The teacher network uses weakly
augmented labeled data to generate high-quality pseudo-labels for the student
network [10]. The student network then is optimized by using the pseudo-labels
as supervised information.

Different from SSL, AL generally focuses on the labeled data, which are mainly
collected by selecting the most informative samples from the unlabeled data
for human experts labeling[11]. The query criterion is the core technique in
AL methods, i.e. uncertainty and diversity[12], [13]. Yuan et.al[14] proposes
MI-AOD, an instance-level uncertainty-based method which highlights the
informative instances while filtering out noisy ones to select the most
informative images for detector training. CALD[15] not only gauges individual
information for sample selection but also leverages mutual information to
alleviate unbalanced class distribution, thus ensuring the diversity of
selected samples. MOL[16] introduces a temporal consistency-based instance
selection strategy for discovering reliable foreground objects, reducing the
risk of background interference.

We note that SSL usually relies on the labeled data to exploit the unlabeled
data for task learning, while AL aims to label the most informative samples
from the unlabeled data with a query criterion. Therefore, it is a natural
consideration to combine AL and SSL together to improve the performance of
object detection with limited labeled RSIs. In fact, many methods have been
developed for SSOD with AL in natural images. Lv et al. [17] proposed a novel
semi-supervised active salient object detection (SOD) method that utilized a
salient encoder-decoder with an adversarial discriminator to select the most
representative data. Mi et al. [18] introduced the method ”Active Teacher”,
which involves data initialization through AL using a teacher-student-based
SSOD approach. However, these methods usually ignore the redundancy in the
RoIs, resulting in that more images are selected to improve the performance
for SSOD.

In this paper, we propose a method Semi-Supervised Object Detection with
Active Teaching, termed as SSOD-AT, for object detection in remote sensing
images using a teacher-student network. Remote sensing images often contain
objects with large scale variations and high density, which is easy to produce
noisy RoIs. To mitigate the impact of these noisy RoIs, we introduce a RoI
Comparison Module (RoICM) that compares the RoIs generated by the teacher
network and the student network. In our method, when the RoIs exhibit
consistent predictions between the teacher and student networks, they are
assigned pseudo-labels for semi-supervised training. Conversely, for RoIs with
divergent predictions, we utilize them to calculate the uncertainty of the
image based on the teacher network’s predictions. To remove the redundancy
from the queried images, a diversity is designed by introducing a global class
prototype to ensure the diverse between the current query image and selected
images. Finally, we integrate the uncertainty and diversity as the query score
to select the most valuable images for human labeling. The main contributions
of this article can be summarized as follows:

  * •

A novel method to boost semi-supervised object detection with active learning
is proposed for remote sensing images based on the teacher-student network.
The proposed method can provide both confident pseudo-labels and informative
images.

  * •

A RoI comparison module(RoICM) is introduced by comparing the RoIs generated
by teacher and student network. It can effectively alleviate the influence of
noisy RoIs for semi-supervised learning and improve the ability of active
learning to evaluates the uncertainty of images.

  * •

The proposed method further incorporates the global class prototype for the
diversity of selected images. The combination of the two sampling strategies
maximizes the effectiveness of AL process.

##  II Methodology

Figure 1: Overview of SSOD-AT framework for three stages. Semi-Supervised
Object Detection(SSOD): Using limited label set to initialize the parameters
of Teacher-Student framework. Active Learning(AL): Select the top-N valuable
samples for labeling. Label set Augmentation(Oracle): Using the active
selected samples to augment the label set. Repeat the preceding procedures to
train the Teacher-Student framework.

The proposed SSOD-AT method is shown in Fig.1. We introduce an iterative
strategy to train Teacher-Student network with active learning for semi-
supervised object detection. In SSOD-AT, a RoI Comparison Module is introduced
by producing confident pseudo-labels for the ROIs and providing candidate ROIs
for active learning to measure the uncertainty of the images. Meanwhile, a
diversity criterion is designed based on the Global Class Prototype to remove
the redundancy with the queried images. In this section, we will introduce the
details of the RoI Comparison Module(RoICM) and the active sampling strategy.

###  II-A RoI Comparison Module(RoICM) for Uncertainty selection

Given an unlabeled image $x_{i}^{u}$, we first input it into both the Teacher
and Student networks to obtain their respective predictions of RoIs. These
RoIs are then fed into the RoICM for comparison. RoICM employs a set of
comparison rules that ensure the highest degree of accuracy and precision in
the comparison process: If the RoIs predicted by both the Teacher and Student
networks have consistent classes, we add $x_{i}^{u}$ to the set of images with
consistent class predictions $I_{consis}^{u}$. We deem the RoIs of
$x_{i}^{u}\in I_{consis}^{u}$ predicted by the Teacher network to be reliable
and proceed to pseudo-label the image, allowing to feed it into the Student
network for the next stage of training. If the RoIs predicted by both the
Teacher and Student networks have different classes, we add $x_{i}^{u}$ to the
set of images with different class predictions $I_{diff}^{u}$. This decision
is predicated on the fact that $x_{i}^{u}\in I_{diff}^{u}$ is considered to
have a higher level of uncertainty for the detection network, thereby
increasing its overall annotation value. Thus, they should be included in the
active learning selection sequence for human labeling.

Assisted by RoICM, for $x_{i}^{u}\in I_{consis}^{u}$, we calculate the KL
divergence based on the predicted class distributions by the Teacher and
Student networks:

$D_{kl}=\frac{1}{n_{b}^{i}}\sum_{j=1}^{n_{b}^{i}}\sum_{k=1}^{N_{c}}p_{t}^{i}\left(b_{j},c_{k}\right)\frac{\log
p_{t}^{i}\left(b_{j},c_{k}\right)}{\log p_{s}^{i}\left(b_{j},c_{k}\right)}$
(1)

where $n_{b}^{i}$ represents the number of proposal bounding boxes generated
by the Teacher network after NMS and confidence threshold filtering, $N_{c}$
is the number of instance categories and $p^{i}\left(b_{j},c_{k}\right)$ is
the prediction probability of the k-th category by the network.
$b^{{}^{\prime}}_{j}$ and $b_{j}$ are the bounding boxes predicted by student
and teacher network respectively. A lower $D_{kl}$ indicates more reliable
pseudo-labels with a higher weight in the total loss as follows:

$\mathcal{L}_{det}^{ts}\left(\mathcal{D}_{l},\mathcal{D}_{u}\right)=\mathcal{L}_{det}^{sup}\left(\mathcal{D}_{l}\right)+\exp(-D_{kl})\cdot\lambda_{u}\cdot\mathcal{L}_{det}^{unsup}\left(\mathcal{D}_{u}\right)$
(2)

Instead, the level of uncertainty for $x_{i}^{u}\in I_{diff}^{u}$ is
calculated by the predicted category distribution of the Teacher network as
follows:

$\cdot
S_{unc}^{i}=-\frac{1}{n_{b}^{i}}\sum_{j=1}^{n_{b}^{i}}\sum_{k=1}^{N_{c}}p_{t}^{i}\left(b_{j},c_{k}\right)\log
p_{t}^{i}\left(b_{j},c_{k}\right)$ (3)

###  II-B Global Class Prototype for Diversity selection

We have established a global prototype for each class, which serves as the
basis for ensuring the diversity of the selected image categories. Given a
labeled image $x_{i}^{l}$, we obtain a set of RoI features
$\mathrm{F}_{i}^{gt}$ from the ground-truth bounding boxes generated by the
RoI Head of the Student Detector in each training stage:

$\mathrm{F}_{i}^{gt}=\left\\{\left(f_{i,j}^{gt},y_{i,j}^{gt}\right)\right\\}$
(4)

where $f_{i,j}^{gt}$ denotes the RoI feature of the j-th ground-truth bounding
box, $y_{i,j}^{gt}\in C$ is its class label while $C$ is the set of total
instance categories. We obtain a local prototype for each class by calculating
the average of the RoI features by class:

$v_{k}=\left\\{\begin{matrix}&\frac{{\textstyle\sum_{i,j}}f_{i,j}^{gt}\mathbbm{1}\left(y_{i,j}^{gt}=k\right)}{{\textstyle\sum_{i,j}}\mathbbm{1}\left(y_{i,j}^{gt}=k\right)}&\sum_{i}\mathbbm{1}\left(y_{i,j}^{gt}=k\right)>0\\\
&\textbf{0}&\sum_{i}\mathbbm{1}\left(y_{i,j}^{gt}=k\right)=0\end{matrix}\right.$
(5)

where $v_{k}$ is the local prototype of k-th class in $C$, 0 denotes the zero
vector and $\mathbbm{1}\left(y_{i,j}^{gt}=k\right)$ is defined as follows:

$\mathbbm{1}\left(y_{i,j}^{gt}=k\right)=\left\\{\begin{matrix}&1&\text{ if
}y_{i,j}^{gt}=k\\\ &0&\text{ if }y_{i,j}^{gt}\neq k\end{matrix}\right.$ (6)

The global class prototype is updated using the EMA algorithm[19], with the
local class prototype serving as a reference:

$g_{k}=\alpha g_{k}+(1-\alpha)v_{k}$ (7)

where $g_{k}$ is the global prototype of k-th class in $C$, $\alpha$ denotes
the hyper-parameter that is typically closed to 1. By applying semi-supervised
training on a small initial set of labeled data, we are able to obtain the
initial global class prototype, which serves as a foundation for ensuring the
diversity and effectiveness of the subsequent active learning process.

For unlabeled images, we only adopt $x^{u}\in I_{diff}^{u}$ to measure their
diversity and update the global class prototype, which is due to the fact that
this part of the images already has a certain level of uncertainty and thus is
more representative in unlabeled set. First, to make the updating process of
the global class prototype smoother, we sort $x_{i}^{u}\in I_{diff}^{u}$ from
smallest to largest uncertainty values. As described in Section 2.2.1, the
weak augmented unlabeled image $T_{w}\left(x_{j}^{u}\right)$ is first fed into
the Teacher network, which subsequently generates a set of proposal bounding
boxes $b_{j}$. $b_{j}$ is then input into the RoI head of the Student network
to obtain its RoI feature $f_{i,j}^{pgt}$. To measure the similarity between
$f_{i,j}^{pgt}$ and $g_{k}$, we adopt cosine similarity as a metric:

$\operatorname{sim}\left(f_{i,j}^{pgt},g_{k}\right)=\frac{f_{i,j}^{pgt}\left(g_{k}\right)^{T}}{\left\|g_{k}\right\|\cdot\left\|f_{i,j}^{pgt}\right\|}$
(8)

For each RoI feature $f_{i,j}^{pgt}$ , the class $c$ in the global class
prototype with the highest similarity score can be identified, denoted as
$\max_{k\in N_{c}}\operatorname{sim}\left(f_{i,j}^{pgt},g_{k}\right)$. If the
similarity score falls below a certain threshold $s$, it suggests that the
target object might belong to a novel class that is not yet present in the
global class prototype. Consequently, the global class prototype must be
updated using equations (12) and (14).

To ensure the diversity of selected samples, we need to suppress instances
with high similarity to the global class prototype. Therefore, we can
calculate the diversity score $S_{div}^{i}$ of the unlabeled image
$x_{i}^{u}\in I_{diff}^{u}$:

$S_{div}^{i}=1-\frac{1}{n_{b}^{i}}\sum_{j=1}^{n_{b}^{i}}\max_{k\in
N_{c}}\operatorname{sim}\left(f_{i,j}^{pgt},g_{k}\right)$ (9)

Higher $S_{div}^{i}$ means more dissimilar the image is to other images,
indicating that it may represent a novel class or a new aspect of a known
class. Finally, we use L-p normalization to combine the two metrics as the
final selection score of the unlabeled image $x_{i}^{u}$:

$S_{sel}^{i}=\sqrt[p]{\left(S_{unc}^{i}\right)^{p}+\left(S_{div}^{i}\right)^{p}}$
(10)

##  III Experiment And Analysis

###  III-A Datasets and Experiments Designation

To extensively evaluate the proposed framework, two representative and public
datasets in remote-sensing images, known as DOTA[20] and DIOR[21] were
employed in our experiments. DOTA[20] contains 2806 aerial images from
different sensors and platforms with crowd sourcing and 188,282 instances,
covered by 15 common object categories. DIOR[21] consists of 23,463 images and
192,472 instances that are manually labeled with axis-aligned bounding boxes,
covering 20 object categories. In our experiments, we focus on the task of
detection with horizontal bounding boxes (HBB for short). We divide both of
the datasets into training set and test set, with a ratio of 2:1.

Similar to the previous works for SSOD[22], [18], each training set is
randomly divided into the labeled set and the unlabeled set. The labeled set
is initialized with 5% images from the training set of DOTA and 2.5% of DIOR.
For evaluation, we adopt mAP (50:95)[23] as the basic comparison metric. In
SSOD-AT, we adopt Faster-RCNN with ResNet-50 as the basic detection network.
The batch size for training is set to 32, which consists 16 labeled and 16
unlabeled images via random sampling. For the batch size $h$ of active
sampling, we selected the 5% samples from the unlabeled data in DOTA and the
2.5% in DIOR for human labeling in each iteration.

TABLE I: Compared with State-of-the-Art Methods on the DOTA Dataset. The Metric is mAP(50:95). “Supervised” Refers to the Model Fed by Labeled Data Only. * is the Origin SSOD Model Fed by Our Active Sampled Data. $\Delta$: AP Gain to the Supervised Performance Method | DOTA
---|---
| L+5% | $\Delta$ | L+10% | $\Delta$ | L+15% | $\Delta$ | L+20% | $\Delta$ | L+25% | $\Delta$ | L+35% | $\Delta$ | L+45% | $\Delta$
Supervised | 26.62 | +0.00 | 32.48 | +0.00 | 36.71 | +0.00 | 40.84 | +0.00 | 43.83 | +0.00 | 47.41 | +0.00 | 52.16 | +0.00
CALD | 27.13 | +0.51 | 35.46 | +2.98 | 40.01 | +3.30 | 43.09 | +2.25 | 45.47 | +1.64 | 50.83 | +3.42 | 54.42 | +2.26
Unbiased-Teacher | 44.53 | +17.91 | 47.36 | +14.88 | 49.79 | +13.08 | 52.49 | +11.65 | 54.52 | +10.69 | 57.22 | +9.81 | 59.40 | +7.24
ILNet | 44.08 | +17.46 | 46.81 | +14.33 | 48.35 | +11.64 | 51.12 | +10.28 | 51.46 | +7.63 | 54.77 | +7.36 | 56.19 | +4.03
Unbiased-Teacher* | 44.31 | +17.69 | 49.21 | +16.73 | 52.19 | +15.48 | 53.95 | +13.11 | 55.55 | +11.72 | 57.65 | +10.24 | 59.88 | +7.72
ILNet* | 43.82 | +17.20 | 46.97 | +14.49 | 49.23 | +12.52 | 50.64 | +9.80 | 52.78 | +8.95 | 54.61 | +7.20 | 56.55 | +4.39
Active-Teacher | 44.27 | +17.65 | 45.31 | +12.83 | 46.84 | +10.13 | 48.57 | +7.73 | 51.27 | +7.44 | 56.97 | +9.56 | 60.46 | +8.30
SSOD-AT(Random) | 45.43 | +18.81 | 48.75 | +16.27 | 51.04 | +14.33 | 53.37 | +12.53 | 55.67 | +11.84 | 58.34 | +10.93 | 60.41 | +8.25
SSOD-AT(Ours) | 45.78 | +19.16 | 49.86 | +17.38 | 53.23 | +16.52 | 55.02 | +14.18 | 56.47 | +12.64 | 59.02 | +11.61 | 60.68 | +8.35
TABLE II: Compared with State-of-the-Art Methods on the DIOR Dataset. The Metric is mAP(50:95). “Supervised” Refers to the Model Fed by Labeled Data Only. * is the Origin SSOD Model Fed by Our Active Sampled Data. $\Delta$: AP Gain to the Supervised Performance Method | DIOR
---|---
| L+5% | $\Delta$ | L+7.5% | $\Delta$ | L+10% | $\Delta$ | L+12.5% | $\Delta$ | L+15% | $\Delta$ | L+20% | $\Delta$ | L+25% | $\Delta$
Supervised | 22.87 | +0.00 | 25.28 | +0.00 | 27.03 | +0.00 | 29.34 | +0.00 | 30.45 | +0.00 | 32.49 | +0.00 | 34.61 | +0.00
CALD | 24.10 | +1.23 | 26.96 | +1.68 | 28.85 | +1.82 | 30.93 | +1.59 | 32.21 | +1.76 | 34.57 | +2.08 | 36.45 | +1.84
Unbiased-Teacher | 43.25 | +20.38 | 44.48 | +19.20 | 45.65 | +18.62 | 46.86 | +17.52 | 47.20 | +16.75 | 48.24 | +15.75 | 49.03 | +14.42
ILNet | 43.91 | +21.04 | 44.55 | +19.27 | 45.69 | +18.66 | 46.61 | +17.27 | 47.20 | +16.75 | 47.65 | +15.16 | 48.63 | +14.02
Unbiased-Teacher* | 43.99 | +21.12 | 45.22 | +19.94 | 46.05 | +19.02 | 47.07 | +17.73 | 47.53 | +17.08 | 48.36 | +15.87 | 49.05 | +14.44
ILNet* | 43.77 | +20.90 | 45.07 | +19.79 | 46.06 | +19.03 | 46.90 | +17.56 | 47.38 | +16.93 | 47.88 | +15.39 | 48.61 | +14.00
Active-Teacher | 43.86 | +20.99 | 44.92 | +19.64 | 45.67 | +18.64 | 46.64 | +17.30 | 47.48 | +17.00 | 48.23 | +15.74 | 49.13 | +14.52
SSOD-AT(Random) | 44.10 | +21.23 | 45.30 | +20.02 | 46.31 | +19.28 | 46.87 | +17.53 | 47.56 | +17.11 | 48.50 | +16.01 | 49.71 | +15.10
SSOD-AT(Ours) | 44.20 | +21.33 | 45.54 | +20.26 | 46.75 | +19.72 | 47.67 | +18.33 | 48.14 | +17.69 | 49.21 | +16.72 | 49.90 | +15.29

(a)

(b)

Figure 2: Detection results of the different algorithms on the two remote-
sensing datasets. (a) DOTA. (b) DIOR

###  III-B Experimental Results and Analysis

####  III-B1 Compare With State-of-the-Art Methods

Table I and Table II show the experimental results on DOTA and DIOR
respectively. We also visualize the whole active learning procedure in Fig.
2(a) and Fig. 2(b) on DOTA and DIOR respectively. From the results, we can
observe that the proposed method outperforms the SOTA methods and achieves 1%
improvement at most cases in the whole active learning procedure. For SSOD-
AT(Random) with RoICM and random selected images, it consistently outperforms
the comparison methods at each labeling proportion, which indicates RoICM is
capable for detecting noisy RoIs in RSIs. Meanwhile, we should note that when
the Unbiased-Teacher and ILNet are trained with the proposed active learning
strategy, i.e., Unbiased-Teacher* and ILNet*, their performances always
achieve higher, which further indicates the effectiveness of our active
learning strategy. Hence, the proposed SSOD-AT is promising for SSOD in remote
sensing images.

TABLE III: Ablation Study of Proposed SSOD-AT on DOTA Dataset

Method | RoICM | Sampling Strategy | DOTA
---|---|---|---
Uncertainty | Diversity | 10.0% | 15.0% | 20.0%
Baseline | $\times$ | $\times$ | $\times$ | 44.53 | 47.36 | 49.79
SSOD-AT | ✓ | $\times$ | $\times$ | 45.43 | 48.75 | 51.04
✓ | ✓ | $\times$ | 45.34 | 49.17 | 51.87
✓ | $\times$ | ✓ | 45.69 | 48.50 | 51.46
✓ | ✓ | ✓ | 45.87 | 49.43 | 52.57

TABLE IV: Ablation Study of Proposed SSOD-AT on DIOR Dataset

Method | RoICM | Sampling Strategy | DIOR
---|---|---|---
Uncertainty | Diversity | 10.0% | 15.0% | 20.0%
Baseline | $\times$ | $\times$ | $\times$ | 43.25 | 45.65 | 47.20
SSOD-AT | ✓ | $\times$ | $\times$ | 44.10 | 46.31 | 47.56
✓ | ✓ | $\times$ | 44.15 | 46.60 | 47.88
✓ | $\times$ | ✓ | 44.28 | 46.37 | 47.82
✓ | ✓ | ✓ | 44.20 | 46.75 | 48.14

####  III-B2 Ablation Study

We show the variants of SSOD-AT with its important components, e.g. RoICM and
Sampling Strategy, on DOTA and DIOR datasets in Table III and IV. In both
tables, the first row represents a baseline teacher-student SSOD
method(Unbiased-Teacher). The second row adds our novel RoICM module into the
baseline with training samples randomly selected from datasets. The last three
rows shows the effectiveness of the uncertainty and diversity modules in our
sample strategy. It can be seen that the average accuracy of SSOD-AT is
improved with RoICM and the performance is degrading when uncertainty or
diversity strategy is removed. This demonstrates that each component is
essential for the proposed method.

####  III-B3 Visualization Analysis

Figure 3: Visualization of the images with top rank selection priority with
different active sampling strategies. The Prediction columns denotes the
pseudo-labels predicted by teacher network with 30%(DOTA) and 20%(DIOR)
labeled proportions, while the GT columns refers to the corresponding ground-
truths.

In Fig. 3, we visualize the examples selected by active sampling strategies
when 30% images are labeled in DOTA and 20% images are labeled in DIOR. It is
observable that the uncertainty selection usually selects images with objects
that are difficult to detect(e.g., small and occluded objects). Comparing
Prediction and GT, we can deserve that a large number of objects are missing
since the detector is highly uncertain about these images. However, the
samples selected by uncertainty are greatly susceptible to be category
imbalanced (e.g., a significant fraction of images containing vehicles), while
the addition of the diversity selection allows images containing more
categories.

##  IV Conclusion

We propose a novel method for semi-supervised object detection with active
learning for remote sensing images. The proposed method integrates the RoI
Comparison Module(RoICM) and Category Prototype to ensure the reliability of
pseudo-labels generated by the Teacher Detector and to effectively select the
most informative images for expert labeling. Our proposed method is
extensively evaluated on two remote sensing datasets, and it consistently
outperforms state-of-the-art methods.

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# New physics behind the new muon $g$-2 puzzle?

Luca Di Luzio  Dipartimento di Fisica e Astronomia ‘G. Galilei’, Università di
Padova, Italy  Istituto Nazionale di Fisica Nucleare, Sezione di Padova,
Padova, Italy Antonio Masiero  Dipartimento di Fisica e Astronomia ‘G.
Galilei’, Università di Padova, Italy  Istituto Nazionale di Fisica Nucleare,
Sezione di Padova, Padova, Italy Paride Paradisi  Dipartimento di Fisica e
Astronomia ‘G. Galilei’, Università di Padova, Italy  Istituto Nazionale di
Fisica Nucleare, Sezione di Padova, Padova, Italy Massimo Passera  Istituto
Nazionale di Fisica Nucleare, Sezione di Padova, Padova, Italy

###### Abstract

The recent measurement of the muon $g$-2 at Fermilab confirms the previous
Brookhaven result. The leading hadronic vacuum polarization (HVP) contribution
to the muon $g$-2 represents a crucial ingredient to establish if the Standard
Model prediction differs from the experimental value. A recent lattice QCD
result by the BMW collaboration shows a tension with the low-energy
$e^{+}e^{-}\to\text{hadrons}$ data which are currently used to determine the
HVP contribution. We refer to this tension as the new muon $g$-2 puzzle. In
this Letter we consider the possibility that new physics contaminates the
experimental extraction of the $e^{+}e^{-}\to\text{hadrons}$ cross-section.
This scenario could, in principle, solve the new muon $g$-2 puzzle. However,
we show that this solution is excluded by a number of experimental
constraints.

Introduction. The anomalous magnetic moment of the muon,
$a_{\mu}\\!\equiv\\!(g_{\mu}\\!-\\!2)/2$, has provided a persisting hint of
new physics (NP) for many years. The recent $a_{\mu}$ measurement by the Muon
$g$-2 collaboration at Fermilab Abi  _et al._ (2021); Albahri  _et al._
(2021a, b, c) has confirmed the earlier result by the E821 experiment at
Brookhaven Bennett  _et al._ (2006), yielding the average
$a_{\mu}^{\scriptscriptstyle\rm EXP}\\!=\\!116592061(41)\\!\times\\!10^{-11}$.
The comparison of this result with the Standard Model (SM) prediction
$a_{\mu}^{\scriptscriptstyle\rm SM}\\!=\\!116591810(43)\times 10^{-11}$ of the
Muon $g$-2 Theory Initiative Aoyama  _et al._ (2020) leads to an intriguing
$4.2\sigma$ discrepancy Abi  _et al._ (2021)

$\Delta a_{\mu}=a_{\mu}^{\scriptscriptstyle\rm
EXP}-a_{\mu}^{\scriptscriptstyle\rm SM}=251\,(59)\times 10^{-11}\,.$ (1)

The expected forthcoming results of the Fermilab experiment plan to reach a
sensitivity four-times better than the E821 one. Moreover, in a longer term,
also the E34 collaboration at J-PARC Abe  _et al._ (2019) aims at measuring
the muon $g$-2 through a new low-energy approach.

On the theory side, the only source of sizable uncertainties in
$a_{\mu}^{\scriptscriptstyle\rm SM}$ stems from the non-perturbative
contributions of the hadronic sector, which have been under close scrutiny for
several years. The SM prediction $a_{\mu}^{\scriptscriptstyle\rm SM}$ in Eq.
(1) has been derived using $(a_{\mu}^{\rm\scriptscriptstyle
HVP})^{\rm\scriptscriptstyle TI}_{e^{+}e^{-}}$, the leading hadronic vacuum
polarization (HVP) contribution to the muon $g$-2 based on low-energy
$e^{+}e^{-}\\!\to\\!{\rm hadrons}$ data obtained by the Muon $g$-2 Theory
Initiative Aoyama  _et al._ (2020) (see also Jegerlehner (2017); Davier  _et
al._ (2017); Keshavarzi  _et al._ (2018); Colangelo  _et al._ (2019);
Hoferichter  _et al._ (2019); Davier  _et al._ (2020); Keshavarzi  _et al._
(2020a); Hoid  _et al._ (2020)). Alternatively, the HVP contribution has been
computed using a first-principle lattice QCD approach Aoyama  _et al._ (2020)
(see also Borsanyi  _et al._ (2018); Blum  _et al._ (2018); Giusti  _et al._
(2019); Davies  _et al._ (2020); Gérardin  _et al._ (2019)). Recently, the BMW
lattice QCD collaboration (BMWc) computed the leading HVP contribution to the
muon $g$-2 with sub per-cent precision, finding a value,
$(a_{\mu}^{\rm\scriptscriptstyle HVP})_{\rm\scriptscriptstyle BMW}$, larger
than $(a_{\mu}^{\rm\scriptscriptstyle
HVP})_{e^{+}e^{-}}^{\rm\scriptscriptstyle TI}$ Borsanyi  _et al._ (2021). If
$(a_{\mu}^{\rm\scriptscriptstyle HVP})_{\rm\scriptscriptstyle BMW}$ is used to
obtain $a_{\mu}^{\scriptscriptstyle\rm SM}$ instead of
$(a_{\mu}^{\rm\scriptscriptstyle HVP})_{e^{+}e^{-}}^{\rm\scriptscriptstyle
TI}$, the discrepancy with the experimental result is reduced to $1.6\sigma$
only. The above results are respectively

$\displaystyle(a_{\mu}^{\rm\scriptscriptstyle
HVP})_{e^{+}e^{-}}^{\rm\scriptscriptstyle TI}=6931\,(40)\times 10^{-11}\,,$
(2) $\displaystyle(a_{\mu}^{\rm\scriptscriptstyle HVP})_{\rm\scriptscriptstyle
BMW}=7075\,(55)\times 10^{-11}\,.$ (3)

The present situation regarding the leading HVP contribution to the muon $g$-2
can be schematically represented as in Fig. 1,

Figure 1: The new muon $g$-2 puzzle: $4.2\sigma$ vs. $1.6\sigma$.

where $(a_{\mu}^{\rm\scriptscriptstyle HVP})_{\scriptscriptstyle\rm EXP}$ is
the value of the HVP contribution required to exactly match
$a_{\mu}^{\scriptscriptstyle\rm EXP}$ assuming no NP. Hereafter, the
difference between the discrepancies in Fig. 1 will be referred to as the new
muon $g$-2 puzzle.

Assuming that both $(a_{\mu}^{\rm\scriptscriptstyle
HVP})_{e^{+}e^{-}}^{\rm\scriptscriptstyle TI}$ and
$(a_{\mu}^{\rm\scriptscriptstyle HVP})_{\rm\scriptscriptstyle BMW}$ are
correct, we ask whether this puzzle can be solved thanks to NP effects which
would bring $(a_{\mu}^{\rm\scriptscriptstyle
HVP})_{e^{+}e^{-}}^{\rm\scriptscriptstyle TI}$ in agreement with
$(a_{\mu}^{\rm\scriptscriptstyle HVP})_{\rm\scriptscriptstyle BMW}$, _without_
spoiling the $1.6\sigma$ agreement of $(a_{\mu}^{\rm\scriptscriptstyle
HVP})_{\rm\scriptscriptstyle BMW}$ with $(a_{\mu}^{\rm\scriptscriptstyle
HVP})_{\scriptscriptstyle\rm EXP}$. Differently from what has been usually
done in the literature, here we do not assume a direct NP contribution to
$\Delta a_{\mu}$ (i.e. new states that couple directly to muons). If fact, by
itself this possibility could solve the longstanding discrepancy in Eq. (1),
but not the new muon $g$-2 puzzle. Instead, in order to solve the latter, we
invoke NP that modifies the $e^{+}e^{-}\\!\to\\!{\rm hadrons}$ cross-section
$\sigma_{\rm had}$.

An increase of $\sigma_{\rm had}$, due to an unforeseen missing contribution,
has been already proposed to enhance $(a_{\mu}^{\rm\scriptscriptstyle
HVP})_{e^{+}e^{-}}^{\rm\scriptscriptstyle TI}$ and solve $\Delta a_{\mu}$
Passera  _et al._ (2008); Crivellin  _et al._ (2020); Keshavarzi  _et al._
(2020b); Malaescu and Schott (2021); Colangelo  _et al._ (2021). However, the
required shift in $\sigma_{\rm had}$ is disfavoured by the electroweak fit if
it occurs at $\sqrt{s}\gtrsim 1$ GeV Keshavarzi  _et al._ (2020b). Hence, in
the following, we will consider NP modifications of $\sigma_{\rm had}$ below
the GeV scale. While Refs. Passera  _et al._ (2008); Crivellin  _et al._
(2020); Keshavarzi  _et al._ (2020b); Malaescu and Schott (2021); Colangelo
_et al._ (2021) did not specify the origin of the shift in $\sigma_{\rm had}$,
we here assume that it is due to NP. After classifying in a model-independent
way the general properties of such a NP, we investigate for the first time its
non-trivial impact on $e^{+}e^{-}$ and BMWc lattice results.

Crucial for our analysis is the dispersion relation used to determine
$(a_{\mu}^{\rm\scriptscriptstyle HVP})_{e^{+}e^{-}}^{\rm\scriptscriptstyle
TI}$ via $\sigma_{\rm had}$. This relation follows from the application of the
optical theorem to the photon HVP.

Figure 2:  Examples of NP contributions to $\sigma_{\rm had}$ via FSR (first
and second diagram) and via a NP tree-level mediator coupled both to hadrons
and electrons (third diagram).

It will be shown that scenarios in which NP couples only to hadrons are not
able to solve the new muon $g$-2 puzzle. Instead, if NP couples both to
hadrons and electrons (and hence it contributes to $\sigma_{\rm had}$ at tree
level), $(a_{\mu}^{\rm\scriptscriptstyle
HVP})_{e^{+}e^{-}}^{\rm\scriptscriptstyle TI}$ should be subtracted of NP in
the comparison with $(a_{\mu}^{\rm\scriptscriptstyle
HVP})_{\scriptscriptstyle\rm EXP}$, whereas $(a_{\mu}^{\rm\scriptscriptstyle
HVP})_{\rm\scriptscriptstyle BMW}$ should not. In fact, in this case, the
quantity that should enter the dispersion relation determining the HVP
contribution should not be the experimentally measured cross-section
$\sigma_{\rm had}$, but rather $\sigma_{\rm
had}-\Delta\sigma^{\rm\scriptscriptstyle NP}_{\rm had}$. Therefore, the
tension between $(a_{\mu}^{\rm\scriptscriptstyle
HVP})_{e^{+}e^{-}}^{\rm\scriptscriptstyle TI}$ and
$(a_{\mu}^{\rm\scriptscriptstyle HVP})_{\rm\scriptscriptstyle BMW}$ could be
solved by invoking a negative interference between the SM and NP, that is
$\Delta\sigma^{\rm\scriptscriptstyle NP}_{\rm had}<0$. As we will show below,
the above picture selects a very specific NP scenario which entails new light
particles with a mass scale $\lesssim 1$ GeV coupling to SM fermions through a
vector current.

Model-independent analysis. Hereafter we are going to examine the general
properties of NP models aiming at solving the new muon $g$-2 puzzle via a
modification of $\sigma_{\rm had}$. To this end, we introduce the dispersion
relation

$\displaystyle(a_{\mu}^{\rm\scriptscriptstyle HVP})_{e^{+}e^{-}}$
$\displaystyle=\frac{\alpha}{\pi^{2}}\int_{m^{2}_{\pi^{0}}}^{\infty}\frac{\text{d}s}{s}\,K(s)\,{\rm
Im}\,\Pi_{\rm had}(s)\,,$ (4)

where $K(s)$ is a positive-definite kernel function with $K(s)\approx
m^{2}_{\mu}/3s$ for $\sqrt{s}\gg m_{\mu}$. This equation defines the HVP
contribution to the muon $g$-2 in terms of the photon HVP, $\Pi_{\rm had}$,
which includes possible NP effects. If the possible NP entering the photon HVP
does not couple to electrons, i.e. it does not enter the hadronic cross-
section at tree level, then Eq. (4) can be written as

$\displaystyle(a_{\mu}^{\rm\scriptscriptstyle HVP})_{e^{+}e^{-}}$
$\displaystyle=\frac{1}{4\pi^{3}}\int_{m^{2}_{\pi^{0}}}^{\infty}\text{d}s\,K(s)\sigma_{\rm
had}(s)\,,$ (5)

where $\sigma_{\rm had}$ includes final-state radiation (FSR), whereas both
vacuum polarization and initial-state radiation (ISR) effects are subtracted.
In particular, vacuum polarization corrections can be simply accounted for by
multiplying the experimental cross-section by $|\alpha/\alpha(s)|^{2}$, while
the correction of ISR and ISR/FSR interference effects is addressed by each
experimental collaboration. In this Letter we will focus on the region where
$\sigma_{\rm had}$ is experimentally determined, i.e. $\sqrt{s}\gtrsim 0.3$
GeV, since this gives the by far dominant contribution to the dispersive
integral in Eq. (5).

In Fig. 2 we show a schematic classification of how NP can enter $\sigma_{\rm
had}$. The first two diagrams are representative of FSR effects, which also
unavoidably affect the photon HVP at the next-to-leading order (NLO). We can
safely neglect possible NP contaminations in ISR since the bounds on NP
couplings to electrons are very severe. The third diagram, where NP enters the
hadronic cross-section at tree level via a coupling both to hadrons and
electrons, also modifies the photon HVP at NLO. Crucially, however, its
dominant contribution to $a_{\mu}$ emerges via the tree-level shift of
$\sigma_{\rm had}$.

Hence, when invoking NP in $\sigma_{\rm had}$, there are two different
scenarios to be considered, depending on whether NP couples only to hadrons or
both to hadrons and electrons. In the following, we analyze these two
possibilities and their capability to solve the new muon $g$-2 puzzle.

1\. NP coupled only to hadrons. This scenario is schematically represented by
the first two diagrams of Fig. 2. As remarked above, real and virtual FSR must
be included in $\sigma_{\rm had}$. However, in order to establish the impact
of NP in FSR, it would be mandatory to perform dedicated experimental analyses
imposing the various selection cuts specific of each experimental setup, which
is beyond the scope of this Letter. Since the full photon FSR effect estimated
in scalar QED amounts only to $50\times 10^{-11}$ Aoyama  _et al._ (2020);
Schwinger (1989), a value well below the discrepancy between Eqs. (2)–(3), and
given that light NP couplings with the SM particles are tightly constrained,
the NP contributions in FSR cannot solve the new muon $g$-2 puzzle.

2\. NP coupled both to hadrons and electrons. If NP contributes to
$\sigma_{\rm had}$ at tree level (see third diagram in Fig. 2), then only the
subtracted cross-section $\sigma_{\rm had}-\Delta\sigma^{\rm\scriptscriptstyle
NP}_{\rm had}$ should be included in Eq. (5). We note that the latter can be
larger than $\sigma_{\rm had}$ if $\Delta\sigma^{\rm\scriptscriptstyle
NP}_{\rm had}<0$, thus requiring that the NP contribution is dominated by a
_negative_ interference with the SM. As $(a_{\mu}^{\rm\scriptscriptstyle
HVP})_{e^{+}e^{-}}^{\rm\scriptscriptstyle TI}$ has been computed using
$\sigma_{\rm had}$ rather than the subtracted cross-section $\sigma_{\rm
had}-\Delta\sigma^{\rm\scriptscriptstyle NP}_{\rm had}$, the theoretical
prediction of the HVP contribution is

$\displaystyle(a_{\mu}^{\rm\scriptscriptstyle
HVP})_{e^{+}e^{-}}^{\rm\scriptscriptstyle TI}+(a_{\mu}^{\rm\scriptscriptstyle
HVP})_{\rm\scriptscriptstyle NP}\,,$ (6)

where $(a_{\mu}^{\rm\scriptscriptstyle HVP})_{\rm\scriptscriptstyle NP}$
describes NP effects at LO, due to the tree-level exchange of the NP mediator
(see third diagram in Fig. 2), as well as at NLO. Instead,
$(a_{\mu}^{\rm\scriptscriptstyle HVP})_{\rm\scriptscriptstyle BMW}$ should be
shifted only by NLO NP effects. Remarkably, this scenario may allow to match
Eq. (6) with $(a_{\mu}^{\rm\scriptscriptstyle HVP})_{\rm\scriptscriptstyle
EXP}$, while keeping at the same time the agreement with the BMWc lattice
result.

Light new physics analysis. We now explore whether the second scenario
envisaged above can be quantitatively realized in an explicit NP model.
Motivated by the fact that the kernel function in Eq. (5) scales like $1/s$
and by the fact that modifications of $\sigma_{\rm had}$ above $\sim 1$ GeV
are disfavoured by electroweak precision tests Keshavarzi  _et al._ (2020b),
we focus on the sub-GeV energy range, where the dominant contribution to
$\sigma_{\rm had}$ arises from the $e^{+}e^{-}\to\pi^{+}\pi^{-}$ channel. In
fact, in the SM, this channel accounts for more than $70\%$ of the full
hadronic contribution to the muon $g$-2. Furthermore, the requirement of
having a sizeable negative interference with the SM amplitude narrows down the
general class of NP models. Indeed, the interference of scalar couplings with
the SM vector current is suppressed by the electron mass, while pseudoscalar
and axial couplings do not interfere. Hence, we focus on the tree-level
exchange of a light $Z^{\prime}$ boson with the following vector couplings to
electrons and first-generation quarks 111Assuming only the interactions in Eq.
(7), the gauge current associated to the $Z^{\prime}$ would be anomalous.
Additional heavy fermions charged under the electroweak group can be
introduced in order to make the model UV consistent.

$\mathcal{L}_{Z^{\prime}}\supset(g_{V}^{e}\,\overline{e}\gamma^{\mu}e+g_{V}^{q}\,\overline{q}\gamma^{\mu}q)Z^{\prime}_{\mu}\,,$
(7)

with $q=u,d$ and $m_{Z^{\prime}}\lesssim 1$ GeV.

The matrix element of the two-pion intermediate state can be expressed in
terms of the pion vector form factor

$\langle\pi^{\pm}(p^{\prime})|J_{\rm
em}^{\mu}(0)|\pi^{\pm}(p)\rangle=\pm(p^{\prime}+p)^{\mu}F^{V}_{\pi}(q^{2})\,,$
(8)

where $q=p^{\prime}-p$ and $J_{\rm
em}^{\mu}=\frac{2}{3}\overline{u}\gamma^{\mu}u-\frac{1}{3}\overline{d}\gamma^{\mu}d$
is the electromagnetic current. Exploiting the charge conjugation invariance,
in the iso-spin symmetric limit, we find that

$\langle\pi^{\pm}|J_{\rm
em}^{\mu}|\pi^{\pm}\rangle=\langle\pi^{\pm}|\overline{u}\gamma^{\mu}u|\pi^{\pm}\rangle=-\langle\pi^{\pm}|\overline{d}\gamma^{\mu}d|\pi^{\pm}\rangle\,.$
(9)

Hence, $F^{V}_{\pi}(q^{2})$ also describes the matrix element of the
$Z^{\prime}$ quark current
$J^{\mu}_{Z^{\prime}}=g^{u}_{V}\overline{u}\gamma^{\mu}u+g^{d}_{V}\overline{d}\gamma^{\mu}d$

$\displaystyle\\!\\!\\!\\!\langle\pi^{\pm}\\!(p^{\prime})|J_{Z^{\prime}}^{\mu}(0)|\pi^{\pm}(p)\rangle$
$\displaystyle\\!=\pm(p^{\prime}+p)^{\mu}F^{V}_{\pi}\\!(q^{2})(g^{u}_{V}\\!-\\!g^{d}_{V}).$
(10)

Note that if the $Z^{\prime}$ interactions are iso-spin symmetric, then
$g_{V}^{u}=g_{V}^{d}$ and the contribution to the $\pi^{+}\pi^{-}$ amplitude
vanishes. Defining $\sigma_{\pi\pi}^{\rm\scriptscriptstyle
SM+NP}=\sigma_{\pi\pi}^{\rm\scriptscriptstyle
SM}+\Delta\sigma_{\pi\pi}^{\rm\scriptscriptstyle NP}$, the tree-level exchange
of the $Z^{\prime}$ with width $\Gamma_{Z^{\prime}}$ leads to

$\displaystyle\\!\\!\\!\frac{\sigma_{\pi\pi}^{\rm\scriptscriptstyle
SM+NP}}{\sigma_{\pi\pi}^{\rm\scriptscriptstyle SM}}$
$\displaystyle=\left|1+\frac{g_{V}^{e}(g_{V}^{u}-g_{V}^{d})}{e^{2}}\frac{s}{s-m^{2}_{Z^{\prime}}+im_{Z^{\prime}}\Gamma_{Z^{\prime}}}\right|^{2},$
(11)

where the pion vector form factor cancels out in the ratio.

The dispersive contribution to the muon $g$-2 due to SM and NP can be obtained
by using $\sigma_{\rm had}-\Delta\sigma^{\rm\scriptscriptstyle NP}_{\rm had}$
in Eq. (5). Imposing that the current discrepancy $\Delta a_{\mu}$ is solved
by NP in the hadronic cross-section, we obtain

$\Delta a_{\mu}=\frac{1}{4\pi^{3}}\int_{s_{\rm
exp}}^{\infty}\text{d}s\,K(s)(-\Delta\sigma^{\rm\scriptscriptstyle NP}_{\rm
had}(s))\,,$ (12)

where the lower integration limit is $s_{\rm exp}\approx(0.3\
\text{GeV})^{2}$, that is, the integral is performed in the data-driven region
for the $\pi\pi$ channel. Approximating $\Delta\sigma^{\rm\scriptscriptstyle
NP}_{\rm had}\approx\Delta\sigma_{\pi\pi}^{\rm\scriptscriptstyle NP}$ 222We
expect that this approximation reproduces $\Delta a_{\mu}$ with
$\mathcal{O}(20)\%$ accuracy. Indeed, in the SM, the $\pi^{0}\gamma$ and
$n\pi$ channels (with $n>2$) amount to 17$\%$ of the $\pi\pi$ channel
contribution to $(a_{\mu}^{\rm\scriptscriptstyle
HVP})_{e^{+}e^{-}}^{\rm\scriptscriptstyle TI}$ (see e.g. Keshavarzi  _et al._
(2020a)). Moreover, NP is assumed to couple only to up and down quarks so that
the contribution of other heavy mesons is negligible., from Eq. (11) we find

$\displaystyle\Delta\sigma^{\rm\scriptscriptstyle NP}_{\rm had}(s)$
$\displaystyle\approx\sigma_{\pi\pi}^{\rm\scriptscriptstyle
SM}(s)\times\frac{2\epsilon
s(s-m^{2}_{Z^{\prime}})+\epsilon^{2}s^{2}}{(s-m^{2}_{Z^{\prime}})^{2}+m_{Z^{\prime}}^{4}\gamma^{2}}\,,$
(13)

where we introduced the effective coupling $\epsilon\equiv
g_{V}^{e}(g_{V}^{u}-g_{V}^{d})/e^{2}$ and the adimensional width parameter
$\gamma\equiv\Gamma_{Z^{\prime}}/m_{Z^{\prime}}$. If both the $Z^{\prime}\to
ee$ and $Z^{\prime}\to\pi^{+}\pi^{-}$ channels are kinematically open, the
associated decay widths (normalized to $m_{Z^{\prime}}$) read, respectively

$\gamma_{ee}\approx\frac{(g_{V}^{e})^{2}}{12\pi}=2.7\times
10^{-10}\left(\frac{g_{V}^{e}}{10^{-4}}\right)^{2}\,,$ (14)

up to $(m_{e}/m_{Z^{\prime}})^{4}$ corrections, and

$\gamma_{\pi\pi}=\frac{(g_{V}^{u}-g_{V}^{d})^{2}}{48\pi}|F^{V}_{\pi}(m^{2}_{Z^{\prime}})|^{2}\left(1-\frac{4m^{2}_{\pi}}{m^{2}_{Z^{\prime}}}\right)^{3/2}\,,$
(15)

where $|F^{V}_{\pi}(m_{Z^{\prime}}^{2})|^{2}$ (normalized to
$F^{V}_{\pi}(0)=1$) can be enhanced up to a factor of 45 by the $\rho$
resonance Colangelo  _et al._ (2019). For $m_{Z^{\prime}}<2m_{\pi^{+}}\approx
0.28$ GeV, $\gamma=\gamma_{ee}$, whereas for $m_{Z^{\prime}}\in[0.3,1]$ GeV we
can approximate $\gamma\approx\gamma_{\pi\pi}$ since the $e^{+}e^{-}$ channel
can be safely neglected given the tight bounds on $g^{e}_{V}$. Note that
possible contributions to $\gamma$ stemming from non-SM final states (e.g.
decays into a dark sector) yield a positive-definite shift to $\sigma_{\rm
had}$, since they cannot interfere with the SM. Hence, they contribute with a
negative shift to $\Delta a_{\mu}$ (cf. Eq. (12)), thus worsening the
discrepancy.

Figure 3: $\sigma^{\rm\scriptscriptstyle SM}_{\rm had}$ and
$\Delta\sigma^{\rm\scriptscriptstyle NP}_{\rm had}$ for some benchmark values
of the $Z^{\prime}$ model parameters solving the $\Delta a_{\mu}$ discrepancy,
see Eq. (1).

Figure 4: $Z^{\prime}$ contribution to $\Delta a_{\mu}$ via a modification of
$\sigma_{\rm had}$ vs. $Z^{\prime}$ constraints.

The profile of $\Delta\sigma^{\rm\scriptscriptstyle NP}_{\rm had}$, together
with its SM counterpart (taken from Ref. (Keshavarzi  _et al._ , 2020a)), is
shown in Fig. 3 for some representative benchmark values of the $Z^{\prime}$
model parameters addressing the $\Delta a_{\mu}$ discrepancy. In particular,
we find that $Z^{\prime}$ masses below the $\rho$ resonance require
$\epsilon<0$, whereas $Z^{\prime}$ masses above it require $\epsilon>0$.
Moreover, in order to obtain $\Delta a_{\mu}>0$, the interference term has to
dominate over the pure resonant effect in Eq. (13). All in all, we find that
the parameters of the $Z^{\prime}$ model that are needed to explain $\Delta
a_{\mu}$ can be divided in two regions: $i)$ $m_{Z^{\prime}}\gtrsim 0.3$ GeV
which requires $|\epsilon|\approx 10^{-2}$ and $\gamma\gtrsim 10^{-3}$ and
$ii)$ $m_{Z^{\prime}}\lesssim 0.3$ GeV which requires $|\epsilon|\approx
10^{-2}$ and basically no relevant constraint on $\gamma$ (as evident from Eq.
(13)).

In the following, we are going to inspect whether the region of the parameter
space of the $Z^{\prime}$ model needed to explain $\Delta a_{\mu}$ is allowed
by experimental constraints. These can be divided for convenience in three
classes: 1\. semi-leptonic processes; 2. purely leptonic processes and 3.
purely hadronic, iso-spin violating observables.

1\. Semi-leptonic processes. The $e^{+}e^{-}\to q\overline{q}$ cross-section,
$\sigma_{qq}$, has been measured with a per-cent level accuracy at LEP II for
center of mass energies $\sqrt{s}\in[130,207]$ GeV Schael  _et al._ (2013).
The leading effect to this process induced by a $Z^{\prime}$ exchange is given
by

$\displaystyle\frac{\sigma_{qq}^{\rm\scriptscriptstyle
SM+NP}}{\sigma_{qq}^{\rm\scriptscriptstyle SM}}$ $\displaystyle\approx
1+2\frac{g_{V}^{e}g_{V}^{q}}{e^{2}Q_{q}}\,,$ (16)

where $Q_{q}$ denotes the quark charge. Requiring that the deviation from
unity in Eq. (16) is less than 1$\%$ Schael  _et al._ (2013) leads to
$|g_{V}^{e}g_{V}^{q}|\lesssim 4.6\cdot 10^{-4}|Q_{q}|$ which translates into
$\epsilon\lesssim 3.3\cdot 10^{-3}$. Moreover, the bound does not depend on
the $Z^{\prime}$ mass and it acts on a coupling combination that is similar to
the one entering $\Delta\sigma_{\pi\pi}^{\rm\scriptscriptstyle NP}$, but not
vanishing in the iso-spin symmetric limit $g_{V}^{u}=g_{V}^{d}$. Hence, this
bound can be considered to be conservative.

2\. Leptonic processes. The $Z^{\prime}$ coupling to electrons is also tightly
constrained. In particular, the non-observation at BaBar of the process
$e^{+}e^{-}\to\gamma Z^{\prime}$ followed by the decay $Z^{\prime}\to
e^{+}e^{-}$ yields $g^{e}_{V}\lesssim 2\cdot 10^{-4}$ Lees  _et al._ (2014) if
the ${Z^{\prime}}$ decays dominantly into electrons. Therefore, in our
framework, this bound applies only for $m_{Z^{\prime}}\lesssim 0.3$ GeV where
the $Z^{\prime}\to\pi^{+}\pi^{-}$ decay is not kinematically allowed.
Moreover, another important bound arises from the electron $g$-2, yielding
$|g_{V}^{e}|\lesssim 10^{-2}\,(m_{Z^{\prime}}/0.5$ GeV) for
$m_{Z^{\prime}}\gtrsim$ MeV.

3\. Iso-spin breaking observables. The $Z^{\prime}$ contribution to
$\Delta\sigma^{\rm\scriptscriptstyle NP}_{\pi\pi}$ is proportional to the iso-
spin breaking combination $g_{V}^{u}-g_{V}^{d}$, which should be of
$\mathcal{O}(1)$ in order to explain $\Delta a_{\mu}$ in the
$m_{Z^{\prime}}\gtrsim 0.3$ GeV region. Therefore, it is natural to expect
sizeable effects on other iso-spin violating hadronic observables. A relevant
example is given by the charged vs. neutral pion mass squared difference,
$\Delta m^{2}=m^{2}_{\pi^{+}}-m^{2}_{\pi^{0}}$. Analogously to the QED case,
the quadratically divergent $Z^{\prime}$ loop leads to

$(\Delta
m^{2})_{Z^{\prime}}\sim\frac{(g_{V}^{u}-g_{V}^{d})^{2}}{(4\pi)^{2}}\Lambda^{2}_{\chi}\,,$
(17)

where $\Lambda_{\chi}\approx 1$ GeV is the cut-off scale of the chiral theory
and we chose $m_{Z^{\prime}}\ll\Lambda_{\chi}$. Instead, for
$m_{Z^{\prime}}\gg\Lambda_{\chi}$, the $Z^{\prime}$ contribution to $\Delta
m^{2}$ decouples as $\Lambda_{\chi}^{2}/m_{Z^{\prime}}^{2}\ll 1$. In practice,
the NP contribution from a light $Z^{\prime}$ is more reliably obtained by
rescaling the SM prediction from lattice QCD Frezzotti  _et al._ (2021) with
$(g^{u}_{V}-g^{d}_{V})^{2}/e^{2}$. Then, comparing the SM prediction of
$\Delta m^{2}$ with its experimental value, we find the $95\%$ C.L. bound
$\left|g_{V}^{u}-g_{V}^{d}\right|\lesssim 0.06$.

The interplay of the above constraints in the plane $-g_{V}^{e}$ vs.
$g_{V}^{u}-g_{V}^{d}$ is displayed in Fig. 4 for two representative scenarios
where $m_{Z^{\prime}}=0.1$ and 0.5 GeV. The directions of the arrows indicate
the excluded regions by the different experimental bounds. Instead, the red
band is the region favoured by the explanation of the muon $g$-2 discrepancy.
From Fig. 4 it is clear that, irrespectively of the $Z^{\prime}$ mass, there
are always at least two independent bounds preventing to solve the new muon
$g$-2 puzzle.

Conclusions. The recent lattice QCD result by the BMW collaboration shows a
tension with the low-energy $e^{+}e^{-}\to\text{hadrons}$ data currently used
to determine the HVP contribution to the muon $g$-2. In this Letter we
investigated the possibility to solve this tension, referred to as the _new
muon $g$-2 puzzle_, invoking NP in the hadronic cross-section. A possible way
to restore full consistency into the picture is to postulate a _negative_
shift in $\sigma_{\rm had}$ due to NP. We showed that this scenario requires
the presence of a light NP mediator that modifies the experimental cross-
section $\sigma_{\rm had}$. However, this non-trivial setup, where NP hides in
$e^{+}e^{-}\to\text{hadrons}$ data, is excluded by a number of experimental
constraints. Alternative confirmations of the $e^{+}e^{-}$ determinations of
the HVP contribution to the muon $g$-2, based on either additional lattice QCD
calculations or direct experimental measurements, as proposed by the MUonE
experiment Carloni Calame  _et al._ (2015); Abbiendi  _et al._ (2017, 2019)
333It is very unlikely that NP contributions will contaminate the MUonE’s
extraction of $a^{\rm\scriptscriptstyle HVP}_{\mu}$ Masiero  _et al._ (2020),
will hence be crucial to shed light on this intriguing puzzle.

###### Acknowledgements.

Acknowledgments. We thank M. Fael, A. Keshavarzi, L. Vecchi and G. Venanzoni
for helpful discussions. This work was supported by the European Union’s
Horizon 2020 research and innovation programme under the Marie Sklodowska-
Curie grant agreement No 860881-HIDDEN and by the research grant “The Dark
Universe: A Synergic Multi-messenger Approach” number 2017X7X85K under the
program PRIN 2017 funded by the Ministero dell’Istruzione, Università e della
Ricerca (MIUR).

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# A Graph-Constrained Changepoint Learning Approach for Automatic QRS-Complex
Detection

Atiyeh Fotoohinasab, Toby Hocking, and Fatemeh Afghah A. Fotoohinasab, T.
Hocking and F. Afghah are with the School of Informatics, Computing and Cyber
Systems at Northern Arizona University.

###### Abstract

This study presents a new viewpoint on ECG signal analysis by applying a
graph-based changepoint detection model to locate R-peak positions. This model
is based on a new graph learning algorithm to learn the constraint graph given
the labeled ECG data. The proposed learning algorithm starts with a simple
initial graph and iteratively edits the graph so that the final graph has the
maximum accuracy in R-peak detection. We evaluate the performance of the
algorithm on the MIT-BIH Arrhythmia Database. The evaluation results
demonstrate that the proposed method can obtain comparable results to other
state-of-the-art approaches. The proposed method achieves the overall
sensitivity of Sen = 99.64%, positive predictivity of PPR = 99.71%, and
detection error rate of DER = 0.19.

###### Index Terms:

Changepoint detection, ECG fiducial points detection, Constraint learning,
Graph learning.

##  I INTRODUCTION

The electrocardiogram (ECG) is a quasi-periodic biomedical signal, which
serves as the most commonly used non-invasive tool in the diagnosis of
cardiovascular diseases. One cardiac cycle in a typical ECG signal is
identified by arrangements of P, QRS-complex, and T waveforms as well as PQ
and ST segments. In most automated ECG analysis tools, correct R-wave
detection is of great importance as it is used for detecting other ECG
fiducial points and is served a significant criterion for the diagnosis of
many heart arrhythmias. Nevertheless, efficient R-peak detection is still a
challenging task due to the time-variant waveform morphology caused by noise
corruption or a specific cardiac condition.

There are various methods in the literature that have been proposed to detect
R-peak in ECG signal [1]. However, these methods mainly suffer from some
critical drawbacks that limit their implementations in practical applications.
First, the non-stationary morphology of the QRS complex can lead to the
misdetection of R-peak, especially in some determinant morphological patterns
concerning certain life-threatening heart arrhythmias. Considering this
limitation, it is apparent that incorrect identification of other waves can
occur subsequent to the incorrect detection of R-peaks. Second, the
performance of most R-peak detection algorithms highly depends on a
preprocessing step to remove the impact of different types of noise
contaminating the ECG signal. However, in real-time data processing and
ambulatory care settings, preprocessing-based algorithms are less effective.
Several studies have utilized deep learning techniques to detect the ECG
waveforms motivated by the high performance of deep learning methods in
various classification tasks [9]. However, the problem with deep learning-
based algorithms is that they need large scale datasets to train the algorithm
and often suffer from the imbalanced class problem [8].

In [4], we introduced a new class of graph-based statistical models based on
optimal changepoint detection models, named graph-constrained changepoint
detection (GCCD) model to locate R-peak in the ECG signal. This work was the
first study performed to apply a changepoint detection model for extracting
ECG fiducial points. The GCCD model is constrained to a graph under which
prior biological knowledge of the signal is taken into account in order to
accomplish the segmentation task. However, the performance of the previous
GCCD model depends on the choice of the constraint graph, which is manually
defined by an expert with prior knowledge. To tackle this issue, in this
paper, we propose a new graph-based changepoint detection model that learns
the structure of the constraint graph in the labeled ECG data. It is worth
mentioning that, like the previous model, the proposed method does not require
any preprocessing step as it leverages the sparsity of changepoints to
denoising the signal as well as detecting abrupt changes.

The rest of the paper is organized as follows. In the next section, we
describe the proposed Graph-Constrained Changepoint Detection model and its
application in the localization of R-peak. Section III provides a description
of the dataset used in this study and a discussion of the results as well as a
comparison between the performance of the proposed algorithm and other state-
of-the-art algorithms. Finally, IV summarizes this research work and its
contributions.

##  II Methodology

In [4], we developed an initial GCCD model based on a manually defined
constraint graph using regarding the labeled ECG data. This GCCD model
extracted the R-peaks in an ECG signal by representing the periodic non-
stationary ECG signal as a piecewise-stationary time series with constant mean
values per segment [4]. To automate the ECG segmentation task, we propose a
new model, which takes a raw ECG signal and an initial graph structure as its
inputs and yields the onset/offset and the mean of desired segments regarding
the structure of the learned constraint graph. The model architecture is
composed of two steps, including training and detection. The training phase is
utilized for optimizing the structure of the constraint graph. It takes the
raw ECG signal and an initial graph structure as the inputs and tries to find
an optimum graph structure that can minimize the label errors over the
training set. Then, the changepoint detection model extracts the R-peaks in
the raw ECG record subject to the trained graph from the previous step. In the
following sections, we describe the details of various parts of the proposed
model.

###  II-A Graph-Constrained Changepoint Detection Model

ECG waves can be considered as abrupt up or down changes over time per cardiac
cycle. Thus, we employ the optimal changepoint detection model introduced in
[6] to localize R-peak positions in the ECG signal. In this framework, the
prior biological knowledge about the expected sequence of changes are
specified in a graph as the model constraints. Then, a functional pruning
dynamic programming algorithm is used to compute the globally optimal model
(mean, changes, hidden states) in fast log-linear $O(N\log N)$ time.

The constraint graph can be defined with a directed graph $G=(V,E)$, where a
vertex set $V\in\\{1,\dots,|V|\\}$ represents the hidden states/segments (not
necessarily a waveform), and the edge set $E\in\\{1,\dots,|E|\\}$ represents
the expected changes between the states/segments. Each edge $e\in E$ encodes
the following associated data based on the prior knowledge about the expected
sequences of changes:

  * •

The source $\underline{v}_{e}\in V$ and target $\overline{v}_{e}\in V$
vertices/states for a changepoint $e$ from $\underline{v}_{e}$ to
$\overline{v}_{e}$.

  * •

A non-negative penalty constant $\lambda_{e}\in\mathbb{R}_{+}$ which is the
cost of changepoint $e$.

  * •

A constraint function $g_{e}:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$
which defines the possible mean values before and after each changepoint $e$.
If $m_{i}$ is the mean before the changepoint and $m_{i+1}$ is the mean after
the changepoint, then the constraint is $g_{e}(m_{i},m_{i+1})\leq 0$. These
functions can be used to constrain the direction (up or down) and/or the
magnitude of the change (greater/less than a certain amount).

Given the input signal $Y=\\{y_{1},\dots,y_{n}\\}$ and the directed graph
$G=(V,E)$, the problem of finding changepoints $\mathbf{c}$, segment means
$\mathbf{m}$, and hidden states $\mathbf{s}$ can be mathematically stated as
follows;

$\displaystyle\operatorname*{minimize}_{\begin{subarray}{c}\mathbf{m}\in\mathbb{R}^{N},\,\mathbf{s}\in
V^{N}\\\ \mathbf{c}\in\\{0,1,\dots,|E|\\}^{N-1}\\\ \end{subarray}}\ \
\sum_{i=1}^{N}\ell(m_{i},z_{i})+\sum_{i=1}^{N-1}\lambda_{c_{i}}$ (1) s. t
$\displaystyle\ \text{no change: }c_{i}=0\Rightarrow
m_{i}=m_{i+1}~{}\&~{}s_{i}=s_{i+1}\,$ (2) $\displaystyle\ \text{change:
}c_{i}\neq 0\Rightarrow g_{c_{i}}(m_{i},m_{i+1})\leq 0~{}~{}\&$
$\displaystyle(s_{i},s_{i+1})=(\underline{v}_{c_{i}},\overline{v}_{c_{i}}).$
(3)

where the segment means $\mathbf{m}$, the hidden states $\mathbf{s}$, and the
changepoints $\mathbf{c}$ are the optimization variables. Also, the
changepoints $c_{i}$ can be any of the pre-defined edges
($c_{i}\in\\{1,\dots,|E|\\}$), or $c_{i}=0$ which indicates no change (and has
no cost, $\lambda_{0}=0$). The above objective function (1) consists of a
data-fitting term $\ell$ and a model complexity term $\lambda_{c_{i}}$. $\ell$
represents the negative log-likelihood of each data point, and
$\lambda_{c_{i}}$ is a non-negative penalty on each changepoint. The
constraint function $g_{e}$ also encodes the expected up/down change and the
least amplitude gap between the mean of two states. The constraint (2)
enforces the model to keep its current state $s_{i}=s_{i+1}$ with no change in
mean $m_{i}=m_{i+1}$ when there is no change $c_{i}=0$. On the other hand, the
constraint (3) forces a change in the mean implied by the constraint function
$g_{c_{i}}(m_{i},m_{i+1})\leq 0$, and a change in the state
$(s_{i},s_{i+1})=(\underline{v}_{c_{i}},\overline{v}_{c_{i}})$ when there is a
change $c_{i}\neq 0$.

###  II-B Constraint Graph Learning

The constraint graph $G=(V,E)$ in the optimization problem of (1) can be
designed manually or in a learning framework. In [4], we constructed the
constraint graph, including the graph topology and edge information manually
using some pre-specified categories for each waveform [12]. However, the
manual definition of the constraint graph can be inefficient for ECG signal
analysis considering the various morphological patterns for each waveform.
Besides, the performance of the model depends on the expert knowledge while
encoding the prior knowledge into the constraint graph. Therefore, in this
work, we propose a graph learning algorithm for learning the constraint graph
using the R-peak labels provided by a gold standard.

The goal of constraint graph learning is to automatically discover the desired
graph topology property (i.e., identifying the desired $V$ and $E$ sets in
graph $G$) and the information of edges from data. In this study, the edge
information consists of the expected up/down change in the segment means, the
least amplitude gap between the mean of two states, and a non-negative penalty
corresponded to the edge transition. In order to learn the constraint graph,
the proposed graph learning algorithm starts with an initial graph, including
an initial graph topology and an initial edge information set; and then,
iteratively optimizes the graph parameters to find a graph that maximizes the
accuracy regarding the given labels. Figure 1a shows the simple initial graph
used for the optimization process. It should be noted that the initial edge
information is chosen based on the overall results obtained from the manual
definition of the constraint graph.

The algorithm considers 10 editing candidates per edge to heuristically
determine the graph candidate set in each iteration. Graph editing candidates
set in the proposed algorithm consists of three types of adding a node, two
types of deleting a node, one type of adding two nodes, and increasing or
decreasing the penalty and gap corresponding to an edge. We believe all
morphological patterns of the ECG waveforms can be constructed using these
editing candidates. Figure 1 exemplifies some of the applied graph editing
candidates related to the edge $(V_{i},V_{j})$ with an up change.

Figure 1: a: The initial constraint graph with two nodes labeled as $B$ and
$R$, representing the baseline and the R-peak segments, respectively, in a
cycle. b-d: Some of the applied graph editing candidates related to the edge
$(V_{i},V_{j})$ with an up change.

##  III Experimental Studies

###  III-A Dataset

We applied the publicly available MIT-BIH arrhythmia (MIT-BIH-AR) database to
evaluate the proposed model [7, 5]. The MIT-BIH-AR database contains 48 ECG
recordings taken from 47 subjects, and each recording is sampled at 360 Hz for
30 min with 200 samples resolution over a 10 mV range. Each recording consists
of two ECG leads, including leads V1, V2, V4, V5, and the modified lead II
(MLII). Only the MLII and V5 are used to evaluate the performance of the
algorithm. The database is annotated with both RR intervals and heartbeat
class information by two or more expert cardiologists independently. We employ
the provided annotations for R-wave positions in order to train the model and
evaluate its performance. The training and testing sets are also generated by
randomly dividing the ECG cycles per records with an approximate ratio of 3:1.

###  III-B Experimental Results

Figure 2 demonstrates the performance of the proposed model in the
localization of R-peak for subject 107 from the MIT-BIH-AR dataset. This
figure illustrates how the proposed algorithm iteratively edits the simple
initial graph to find a graph with maximum accuracy in detecting R-peaks
regarding the labels in the training set. The red part of the graph in each
iteration shows the modified part of the graph regarding the constraint graph
in the previous iteration. It is worth noting that the performance of the
proposed algorithm depends on the initial graph structure. Therefore, the
models for 5 records out of all records are trained based on a more complex
initial graph than the initial graph in 1a. We evaluate the performance of the
proposed algorithm using the sensitivity (Sen), positive predictivity rate
(PPR), and detection error rate (DER), which are defined as below:

Figure 2: The demonstration of constraint graph optimization using the
proposed graph learning algorithm for the subject 107 from the MIT-BIH-AR
dataset. a-e top: Extracted R-peak positions given the learned constraint
graph in each learning iteration. The red and orange coverage bands show the
used labels in the training procedure, including both training and validation
sets. a-e bottom: The learned constraint graph in each learning iteration.
Below each edge $e$, we show the penalty $\lambda_{e}$, and above, we show the
type of change (i.e., up/down) and the gap, which is the minimum magnitude of
change. The red part of the graph in each iteration shows the modified part of
the graph regarding the constraint graph in the previous iteration. f: Testing
the final learned constraint graph in a new window of data. The orange
coverage band shows the labels provided by the MIT-BIH-AR dataset.  Figure 3:
Label errors in detection of R-peak positions regarding the learned constraint
graph per 5 learning iterations for subject 107 in the MIT-BIH-AR dataset.

$\displaystyle\textit{Sen}(\%)=\frac{TP}{TP+FN}\times{100}$ (4)
$\displaystyle\textit{PPR}(\%)=\frac{TP}{TP+FP}\times{100}$ (5)
$\displaystyle\textit{DER}(\%)=\frac{FN+FP}{TP+FN}\times{100}$ (6)

where TP is true positives, FP is false positives, FN is false negatives, and
TN is true negatives. We used $k$-fold cross-validation approach to train and
test the proposed model with a $k$ size of 5. Indeed, we divided the dataset
into k= 5 folds. Then, for each fold of the 5 folds, one fold is used for
evaluating the model, and the remaining 4 folds are used to train the model.
In the end, the final result was averaged over all 5 folds. Figure 3
illustrates the training progress over 5 iterations for record 107 from the
MIT-BIH-AR dataset, where the Y axis shows the sum of false negative and false
positive error rates.

Table I represents the R-peak detection success for the proposed algorithm
against other state-of-the-art methods. As shown in the table, the proposed
algorithm achieves remarkable results, Sen = %99.64, PPR = %99.71, and DER =
0.19, in R-peak detection. We note that here our remarkable results were
obtained regardless of a preprocessing step, as opposed to other methods in
the literature. We noticed that the records with Sen and PPR values lower than
99% contain multiple different morphological patterns. In these records, more
than one optimum graph path is required in the constraint graph in order to
detect R-peaks. The performance of the proposed model for such cases can be
improved by learning a multi-path constraint graph, which is considered as
future work. Besides, learning a multi-path constraint graph can be applied
for detecting all ECG waveforms as considering morphological patterns of each
waveform leads to having cycles with completely various morphological
patterns.

TABLE I: Comparison of performance of several R-peak detection methods using
the MIT-BIH-AR database

Method | Sen (%) | PPR (%) | DER (%)
---|---|---|---
Park et al. [10] | 99.93 | 99.91 | 0.163
Farashi [3] | 99.75 | 99.85 | 0.40
Sharma and Sunkaria [11] | 99.50 | 99.56 | 0.93
Castells-Rufas and Carrabina [2] | 99.43 | 99.67 | 0.88
Proposed Model | 99.64 | 99.71 | 0.19

##  IV Conclusion

In this paper, the task of R-peak detection in the ECG signal is carried out
based on a new viewpoint using a changepoint detection model. The proposed
algorithm uses a changepoint detection algorithm constrained to a graph in
which prior biological knowledge of the expected changepoints per cardiac
cycle is encoded. We propose an algorithm for learning the constraint graph in
labeled data. The proposed learning algorithm starts from an initial graph and
optimizes the structure of the constraint graph as well as the edges
information to find a model that can maximize the detection accuracy. The
results provided in this paper demonstrates that changepoint detection models
constrained to a graph are promising approaches for detecting ECG waveforms.
The proposed learning algorithm can be advanced to learn a multi-path
constraint graph in order to detect all ECG waveforms and find the overall ECG
morphological patterns per cardiac cycle.

## References

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