Walter Zimmer, Jialong Wu, Xingcheng Zhou, Alois C. Knoll 11institutetext:
School of Computation, Information and Technology, Department of Computer
Engineering, Technical University of Munich (TUM), 85748 Garching, Germany,
11email<EMAIL_ADDRESS><EMAIL_ADDRESS>
https://www.ce.cit.tum.de/air

# Real-Time And Robust 3D Object Detection with Roadside LiDARs

Walter Zimmer${}^{\textrm{({\char 0\relax})}}$ *  — Jialong Wu *  — Xingcheng
Zhou *  —
Alois C. Knoll *  —

###### Abstract

This work aims to address the challenges in autonomous driving by focusing on
the 3D perception of the environment using roadside LiDARs. We design a 3D
object detection model that can detect traffic participants in roadside LiDARs
in real-time. Our model uses an existing 3D detector as a baseline and
improves its accuracy. To prove the effectiveness of our proposed modules, we
train and evaluate the model on three different vehicle and infrastructure
datasets. To show the domain adaptation ability of our detector, we train it
on an infrastructure dataset from China and perform transfer learning on a
different dataset recorded in Germany. We do several sets of experiments and
ablation studies for each module in the detector that show that our model
outperforms the baseline by a significant margin, while the inference speed is
at 45 Hz (22 ms). We make a significant contribution with our LiDAR-based 3D
detector that can be used for smart city applications to provide connected and
automated vehicles with a far-reaching view. Vehicles that are connected to
the roadside sensors can get information about other vehicles around the
corner to improve their path and maneuver planning and to increase road
traffic safety.

###### keywords:

Autonomous and Connected Vehicles, Traffic Perception, 3D Object Detection,
Roadside LiDAR, Point Clouds, Deep Learning, Transfer Learning,
Infrastructure, Testfield A9

##  1 Introduction

###  1.1 Problem Statement

High quality and balanced data is crucial to achieve high accuracy in deep
learning applications. We analyze and split the task into four challenges. The
first challenge is the lack of labeled frames of roadside LiDARs. Considering
the high labor cost of manual labeling for 3D bounding boxes in LiDAR point
clouds, we need to find a solution to deal with small datasets and use only
few labeled frames for training. Publicly available LiDAR datasets were
recorded and labeled from a driver perspective which makes is difficult to
apply these datasets for roadside LiDARs. The second research question this
work is dealing with lies in the area of domain adaptation. How can a neural
network that was trained in one operational design domain (ODD), e.g. on data
recorded by vehicle sensors like in the KITTI dataset [9, 8], be adapted to a
different domain (e.g. to data recorded by roadside sensors on a highway or a
rural area? Much research has been done in the area of domain adaptation and
transfer learning [29, 35] — training a model on a large dataset (source
domain) and fine-tuning it on a smaller dataset (target domain). Another
challenge is real-time 3D object detection on roadside LiDARs, i.e. to detect
objects with a high enough frame rate to prevent accidents. This highly
depends on the LiDAR type and the number of points per scan that need to be
processed. The final challenge this work is dealing with is a robust 3D
detection of all traffic participants. Detecting small and occluded objects in
different weather conditions and rare traffic scenarios is a highly important
research area to increase safety of automated vehicles.

Figure 1: Overview architecture of SE-ProPillars, a LiDAR-only single-stage
pillar-based 3D object detector. The detector is based on PointPillars [17],
with the following five additional extensions. 1) Shape-Aware Data
Augmentation [37], a training technique to improve the accuracy without adding
any additional costs in the inference time. 2) The Stacked Triple Attention
Mechanism [21] enhances the learned features from the raw point cloud using
the triple attention mechanism, including channel-wise, point-wise, and voxel-
vise attention. 3) The pillar feature net turns point-wise features into
pillar features and scatters the pillar features into a pseudo image. 4) We
propose the Attentive Hierarchical Middle Layers to perform 2D convolution
operations on the pseudo image. Hierarchical feature maps are concatenated
with an attentive addition operation. 5) The Multi-task detection head [36] is
used for the final prediction, that includes an IoU prediction to alleviate
the misalignment between the localization accuracy and classification
confidence. 6) Finally, a Self-Ensembling Training Architecture (a teacher &
student training framework) [37] is used as a training technique in the
training process that leads to a substantial increase in the precision without
affecting the inference time.

###  1.2 Objectives

The first objective is to create a large infrastructure dataset with
sufficient labeled point cloud frames. This dataset should be balanced in
terms of object classes and contain a high variety so that objects can be
detected in different scenarios and different environment conditions. Another
objective is to analyze whether transfer learning from a larger roadside LiDAR
dataset, such as the recently released IPS300+ dataset [30], can improve the
model performance. The first batch of the published IPS300+ dataset includes
1,246 labeled point cloud frames and contains on average 319.84 labels per
frame. In this work the aim is to design a single-stage 3D object detector
that can detect objects in real-time using roadside LiDARs. An example of an
intersection that is equipped with LiDARs is shown in Fig. 2. The goal is to
reach an inference rate of at least 25 Hz, which leads to a maximum inference
time of 40 ms per point cloud frame. In terms of accuracy the target is to
achieve at least 90% mean average precision (mAP) on the Car class within the
test set. In the end the designed model needs to reach a reasonable trade-off
between the inference time and the model performance. The final target is to
evaluate the model performance and speed on the first released batch of the
manually labeled A9-Dataset [2] and on publicly available infrastructure
datasets. The A9-Dataset was recorded on the Providentia++ Test Stretch in
Munich, Germany [16].

Figure 2: Left: Intersection where 3D Object Detection with roadside LiDARs is
performed. Right: Roadside LiDAR mounted above the Autobahn A9.

###  1.3 Contribution

The work provides a complete analysis and solution for the single LiDAR
detection task. The main contributions in this work can be summarized as
follows:

  * •

We propose a single-stage LiDAR-only detector based on PointPillars. We
introduce five extensions to improve PointPillars and evaluate the performance
of the designed model on KITTI’s validation set and the test set of IPS300+
and A9-Dataset to check its validity.

  * •

We design a point cloud registration algorithm for LiDARs installed on the
infrastructure to increase the point density.

  * •

We create a synthetic dataset, called proSynth, with 2k labeled LiDAR point
cloud frames using the CARLA simulator [5] and train our model on that.

  * •

Experiments show that our SE-ProPillars model outperforms the PointPillars
model by (+4.20%,+1.98%,+0.10%) 3D mAP on easy, moderate and hard difficulties
respectively, while the inference speed reaches 45 Hz (22 ms).

  * •

We train on the IPS300+ dataset, perform transfer learning on A9-Dataset, and
achieve a<EMAIL_ADDRESS>of 50.09% on the Car class of the A9 test set using
40 recall positions and a NMS IoU of 0.1.

##  2 Related Work

According to the form of feature representation, LiDAR-only 3D object
detectors can be divided into four different types, i.e. point-based, voxel-
based, hybrid and projection-based methods.

###  2.1 Point-based Methods

In point-based methods, features maintain the form of point-wise features,
either by a sampled subset or derived virtual points. PointRCNN [28] uses a
PointNet++ backbone [26] to extract point-wise features from the raw point
cloud, and performs foreground segmentation. Then for each foreground point,
it generates a 3D proposal followed by a point cloud ROI pooling and a
canonical transformation-based bounding box refinement process. Point-based
methods usually have to deal with a huge amount of point-wise features, which
leads to a relatively lower inference speed. To accelerate point-based
methods, 3DSSD [32] introduces a feature farthest-point-sampling (F-FPS),
which computes the feature distance for sampling, instead of Euclidean
distance in traditional distance farthest-point-sampling (D-FPS). The
inference speed of 3DSSD is competitive with voxel-based methods.

###  2.2 Voxel-based Methods

VoxelNet [39] divides the 3D space into equally spaced voxels and encodes
point-wise features into voxel-wise features. Then 3D convolutional middle
layers operate on these encoded voxel features. 3D convolution on sparse point
cloud space brings too much unnecessary computational cost. SECOND [31]
proposes to use a sparse convolutional middle extractor [10, 20], which
greatly speeds up the inference time. In PointPillars [17], the point cloud is
divided into pillars (vertical columns), which are special voxels without any
partition along the z-direction. The feature map of pillars can be treated as
a pseudo-image, and therefore the expensive 3D convolution is replaced by 2D
convolution. PointPillars achieves the fastest speed with the help of TensorRT
acceleration. SA-SSD [11] adds a detachable auxiliary network to the sparse
convolutional middle layers to predict a point-wise foreground segmentation
and a center estimation task, which can provide additional point-level
supervision. SA-SSD also proposes a part-sensitive warping (PS-Warp) operation
as an extra detection head. It can alleviate the misalignment between the
predicted bounding boxes and classification confidence maps, since they are
generated by two different convolutional layers in the detection head. CIA-SSD
[36] also notices the misalignment issue. It designs an IoU-aware confidence
rectification module, using an additional convolutional layer in the detection
head to make IoU predictions. The predicted IoU value is used to rectify the
classification score. By introducing only one additional convolutional layer,
CIA-SSD is more lightweight than SA-SSD. SE-SSD [37] proposes a self-
ensembling post-training framework, where a pre-trained teacher model produces
predictions that serve as soft targets in addition to the hard targets from
the label. These predictions are matched with student’s predictions by their
IoU and supervised by the consistency loss. Soft targets are closer to the
predictions from the student model and therefore can help the student model to
fine-tune its predictions. The Orientation-Aware Distance-IoU Loss is proposed
to replace the traditional smooth-$L_{1}$ loss of bounding box regression in
the post training, in order to provide a fresh supervisory signal. SE-SSD also
designs a shape-aware data augmentation module to improve the generalization
ability of the student model.

###  2.3 Hybrid Methods

Hybrid methods aim to take advantage of both point-based and voxel-based
methods. Point-based methods have a higher spatial resolution but involve
higher computational cost, while voxel-based methods can efficiently use CNN
layers for feature extraction but lose local point-wise information. Hybrid
methods try to strike a balance between them.

HVPR [23] is a single-stage detector. It has two feature encoder streams
extracting point-wise and voxel-wise features. Extracted features are
integrated together and scattered into a pseudo image as hybrid features. An
attentive convolutional middle module is performed on the hybrid feature map,
followed by a single-stage detection head. STD [33] is a two-stage detector
that uses PointNet to extract point-wise features. A point-based proposal
generation module with spherical anchors is designed to achieve high recall.
Then a PointsPool module voxelizes each proposal, followed by a VFE layer. In
the box refinement module, CNNs are applied on those voxels for final
prediction. PV-RCNN [27] uses the 3D sparse convolution for voxel feature
extraction. A Voxel Set Abstraction (Voxel-SA) module is added to each
convolutional layer to encode voxel features into a small set of key points,
which are sampled by farthest point sampling. Key point features are then re-
weighted by foreground segmentation score. Finally, they are used to enhance
the ROI grid points for refinement. H23D R-CNN [4] extracts point-wise
features from multi-view projection. It projects the point cloud to a bird’s
eye view under Cartesian coordinates and a perspective view under the
cylindrical coordinates, separately. BEV feature and point-voxel (PV) features
are concatenated together for proposal generation in BEV and fused together as
point-wise hollow-3D (H3D) features. Then voxelization is performed on the 3D
space, and point-wise H3D features are aggregated as voxel-wise H3D features
for the refinement process.

###  2.4 Projection-based Methods

RangeDet [7] is an anchor-free single-stage LiDAR-based 3D object detector
that is purely based on the range view representation. It is compact and has
no quantization errors. The inference speed is 12 Hz using a RTX 2080 TI GPU.
The runtime is not affected by the expansion of the detection distance,
whereas a BEV representation will slow down the inference time as the
detection range increases. RangeRCNN [18] is another 3D object detector that
uses a range image, point view and bird’s eye view (BEV). The anchor is
defined in the BEV to avoid scale variation and occlusion. In addition, a two-
stage regional convolutional neural network (RCNN) is used to improve the 3D
detection performance.

##  3 Data Generation

Figure 3: Left: Example of the intersection that is part of the A9 Test
Stretch that was modelled in the CARLA simulator. Right: Synthetic point cloud
frames with labeled ground truth vehicles extracted from CARLA simulator.

###  3.1 Real Data Generation

The first step is to record point cloud data from roadside LiDARs. These
recordings should cover a large variety of different scenarios (e.g. traffic
jam, overtaking, lane change, lane merge, tail-gate events, accidents, etc.).
In the next step (data selection), scenarios of high importance are selected
for labeling. The point clouds are then converted into the right format (.pcd)
and the ground is removed using the RANSAC algorithm. Dynamic objects are then
labeled using a custom 3D point cloud labeling tool called proAnno, that is
based on the open source 3D bounding box annotation tool 3D-BAT [40]. 456
LiDAR frames were manually labeled to form the first batch of the A9-Dataset.

###  3.2 Synthetic Data Generation

We created a synthetic dataset (proSynth) with 2000 point cloud frames using
the CARLA simulator and train our SE-ProPillars model on it. Figure 3 shows an
example of an intersection with generated traffic in the CARLA simulator. A
simulated LiDAR sensor represents a real Ouster OS1-64 (gen. 2) LiDAR sensor
with 64 channels and a range of 120 m. In the simulation a noise model is used
with a standard deviation of 0.1 to disturb each point along the vector of its
raycast. The LiDAR emits 1.31 million points per second and runs at 10 Hz
(131,072 points per frame). The extracted labels were stored in .json files
according to the OpenLABEL standard [3].

##  4 Approach

We design a real-time LiDAR-only 3D object detector (SE-ProPillars) that could
be applied to real-world scenarios. The architecture of the designed SE-
ProPillars model is shown in Fig. 1. We describe the each part of the detector
in the following sections.

###  4.1 Point Cloud Registration

We first design a point cloud registration algorithm for LiDARs installed on
the infrastructure to increase the point density and facilitate the detection
task. The two Ouster OS1-64 (gen. 2) LiDARs are mounted side by side (about 13
m apart from each other) on the gantry bridge. Both LiDARs are time
synchronized using the ROS time that itself is synchronized with an NTP time
server. One LiDAR is treated as the source ($L_{s}$) and the other one as
target ($L_{t}$). Both LiDAR sensors capture $N$ point cloud scans and each
scan is marked with a Unix timestamp: $A=\\{a_{i}\\}_{i=1,...,N}$ and
$B=\\{b_{i}\\}_{i=1,...,N}$. The goal is to put both point clouds into the
same coordinate system, by transforming the source point cloud into the target
point cloud’s coordinate system. To achieve this a set of correspondence pairs
$S=\\{s_{i}\\}_{i=1,...,n}$ and $T=\\{t_{i}\\}_{i=1,...,n}$ need to be found,
where $s_{i}$ corresponds to $t_{i}$. Then the rigid transformation $T$ that
includes the rotation $R$ and translation $t$ is estimated by minimizing the
root-mean-squared error (RMSE) between correspondences. Inspired by [15], we
provide an initial transformation matrix to help the registration algorithms
better overcome local optima. The initial transformation was acquired with a
Real-time Kinematic (RTK) GPS device. With this initial transformation the
continuous registration is less likely to be trapped in a local optimum. The
continuous registration is done by point-to-point ICP. The registration
process of two Ouster LiDARs, running at 10 Hz, takes 18.36 ms on an Intel
Core i7-9750H CPU. A RMSE of 0.52164 could be achieved using a voxel size of 2
m. Figure 4 shows the point cloud scans before and after registration.

Figure 4: Left: Two point cloud scans before registration. The displacement
error is marked in yellow. Right: Two point cloud scans after registration.

###  4.2 Voxelization

We divide the raw point cloud into vertical pillars before feeding them into a
neural network. These are special voxels that are not split along the vertical
axis. Pillars have several advantages over voxels. A pillar-based backbone is
faster than a voxel-based backbone due to fewer grid cells. Time consuming 3D
convolutional middle layers are also being eliminated and instead 2D
convolutions are being used. We also do not need to manually tune the bin size
along the z-direction hyperparameter. If a pillar contains more points than
specified in the threshold, then the points are being subsampled to the
threshold using farthest point sampling [6]. If a pillar contains fewer points
than the threshold, then it is padded with zeros to make the dimensions
consistent. Due to the sparsity issue most of the pillars are empty. We record
the coordinates of non-empty pillars according to the pillar’s center. Empty
pillars are not being considered during the feature extraction until all
pillars are being scattered back to a pseudo image for 2D convolution.

###  4.3 Stacked Triple Attention

The Stacked Triple Attention module is used for a more robust and
discriminative feature representation. Originally introduced in TANet [21] by
Liu et. al., this module enhances the learning of objects that are hard to
detect and better deals with noisy point clouds. The TA module extracts
features inside each pillar, using point-wise, channel-wise, and voxel-wise
attention. The attention mechanism in this module follows the Squeeze-and-
Excitation pattern [13].

Figure 5: Structure of the triple attention module [21].

The structure of the triple attention module is shown in Fig. 5. The input $V$
to the module is a $(P\times N\times C)$ tensor, where $P$ is the number of
non-empty pillars, $N$ is the maximum number of points in each pillar, and $C$
is the dimension of the input point-wise feature. In the upper branch point-
wise attention, following the Squeeze-and-Excitation pattern, we firstly
perform max pooling to aggregate point-wise features across the channel-wise
dimensions, and then we compute the point-wise attention score $S$ using two
fully connected layers. Similarly, the middle branch channel-wise attention
aggregates channel-wise features across their point-wise dimensions, to get
the channel-wise attention score $T$. Then $S$ and $T$ are combined by
element-wise multiplication, followed with a sigmoid function to get the
attention scale matrix $M$, $M=\sigma(S\times T)$. $M$ is then multiplied with
input $V$, to get the feature tensor $F1$. In the bottom branch voxel-wise
attention, the $C$-dim channel-wise feature in $F1$ is enlarged by the voxel
center (arithmetic mean of all points inside the pillar) to $C+3$-dim for
better voxel-awareness. Then the enlarged $F1$ is fed into two fully connected
layers. The two FC layers respectively compress the point-wise and the
channel-wise dimensions to 1, to get the voxel-wise attention score. Finally,
a sigmoid function generates the voxel-wise attention scale $Q$, multiplied
with the original $F1$ to generate the final output of the TA module $F2$.

To further exploit the multi-level feature attention, two triple attention
modules are stacked with a structure similar to the skip connections in ResNet
[12] (see Fig. 6). The first module takes the raw point cloud as input, while
the second one works on the extracted high dimensional features. For each TA
module the input is concatenated or summed to the output to fuse more feature
information. Each TA module is followed by a fully connected layer to increase
the feature dimension. Inside the TA modules, the attention mechanism only re-
weights the features, but does not increase their dimensions.

Figure 6: Structure of the stacked triple attention module.

###  4.4 Pillar Feature Net

We choose PointPillars [17] as our baseline and improve its 3D detection
performance at the expense of inference time. The inference speed of
PointPillars is 42 Hz without the acceleration of TensorRT. Since there is a
trade-off between speed and accuracy, we can further boost the accuracy by
incorporating additional modules without sacrificing the inference speed too
much. The pillar feature net (PFN) shown in Fig. 1, introduced by Lang et. al,
takes pillars as input, extracts pillar features, and scatters pillars back to
a pseudo image for 2D convolution operations in the middle layers. The pillar
feature net acts as an additional feature extractor to the stacked triple
attention module. The point-wise pillar-organized features from the stacked TA
module with shape ($P\times N\times C$) are fed to a set of PFN layers. Each
PFN layer is simplified PointNet [25], which consists of a linear layer,
Batch-Norm [14], ReLU [22], and max pooling. The max-pooled features are
concatenated back to the ReLU’s output to keep the point-wise feature
dimension inside each pillar, until the last FPN layer. The last FPN layer
makes the final max pooling and outputs a ($P\times C$) feature as the pillar
feature. Pillar features are then scattered back to the original pillar
location, forming a ($C\times H\times W$) pseudo image, where $H$ and $W$ are
the height and width of the pillar grid. Here the location of empty pillars is
padded with zeros.

###  4.5 Attentive Hierarchical Middle Layers

We exchange the default backbone of PointPillars with an Attentive
Hierarchical Backbone to perform 2D convolution on the pseudo image from the
pillar feature net. Figure 7 depicts the structure of the attentive
hierarchical middle layers. In the first stage, the spatial resolution of the
pseudo image is gradually downsampled by three groups of convolutions. Each
group contains three convolutional layers, where the first one has a stride of
two for downsampling, and the two subsequent layers act only for feature
extraction. After downsampling, deconvolution operations are applied to
recover the spatial resolution. Deconvolutional layers (marked with an
asterix) recover the size of feature maps with stride 2 and element-wise add
them to upper branches. The remaining three deconvolutional layers make all
three branches have the same size (half of the original feature map). Then the
final three feature maps are combined by an attentive addition to fuse both,
spatial and semantic features. The attentive addition uses the plain attention
mechanism. All three feature maps are being passed through a convolutional
operation and are channel-wise concatenated as attention scores. The softmax
function generates the attention distribution and feature maps are multiplied
with the corresponding distribution weight. The element-wise addition in the
end gives the final attention output, a ($C\times H/2\times W/2$) feature map.

Figure 7: Left: Structure of the attentive hierarchical middle layers. Right:
Structure of the attentive addition operation.

###  4.6 Multi-task Head

The multi-task head outputs the final class (based on a confidence score), the
3D box position ($x,y,z$), dimensions ($l,w,h$), rotation ($\theta$) and the
direction of the detected object. The direction (front/back) is being
classified to solve the problem that the sine-error loss [31] cannot
distinguish flipped boxes. Four convolutional layers operate on the feature
map separately. Figure 1 shows the brief structure of the multi-task head in
the bottom right corner. One of the four heads is the IoU prediction head that
predicts an IoU between the ground truth bounding box and the predicted box.
It was introduced in CIA-SSD [36] to deal with the misalignment between the
predicted bounding boxes and corresponding classification confidence maps. The
misalignment is mainly because these two predictions are from different
convolutional layers. Based on this IoU prediction, we use the confidence
function (CF) to correct the confidence map and use the distance-variant IoU-
weighted NMS (DI-NMS) module post-process the predicted bounding boxes. The
distance-variant IoU-weighted NMS is designed to deal with long-distance
predictions, to better align far bounding boxes with ground truths, and to
reduce false-positive predictions. If the predicted box is close to the origin
of perspective, we give higher weights to those box predictions with high IoU.
If the predicted box is far, we give relatively uniform weights, to get a more
smooth final box.

###  4.7 Shape-Aware Data Augmentation

Data augmentation has proven to be an efficient way to better exploit the
training dataset and help the model to be more generalized. We use the shape-
aware data augmentation method proposed by SE-SSD [37] (see Fig. 8. This
module simplifies the handling of partial occlusions, sparsity and different
shapes of objects in the same class. The ground truth box is divided into six
pyramidal subsets. Then we independently augment each subset using three
operations, random dropout, random swap and random sparsifying.

Figure 8: Illustration of the shape-aware data augmentation module [37]. It
includes random dropout in the blue subset, random swap between green and
yellow subsets, and random sparsifying from the yellow to the red subset.

Some traditional augmentation methods are also applied before the shape-aware
augmentation, e.g. rotation, flipping, and scaling.

###  4.8 Self-Ensembling Training Framework

In addition, we introduce the self-ensembling training framework [37] to do a
post training: We first train the model shown in Fig. 9 but without the self-
ensembling module, and then we take the pre-trained model as a teacher model
to train the student model that has the same network structure.

Figure 9: Self-ensembling training architecture.

The self-ensembling training architecture is show in Fig. 9. The teacher model
is trained with the following parameters: An adapted learning rate for each
epoch was implemented with a maximum of 0.003 and a OneCycle learning rate
scheduler. The model is trained for 80 epochs with a batch size of 2 for pre-
training and for 60 epochs with a batch size of 4 for post-training to
accelerate the training process. Adam is used as optimizer. The predictions of
the teacher model can be used as soft targets. Combined with the hard targets
from the ground truth, we can provide more information to the student model.
The student model and the teacher model are initialized with the same pre-
trained parameters. During training, we firstly feed the raw point cloud to
the teacher model and get teacher predictions. Then we apply global
transformations to the teacher predictions as soft targets. For the hard
targets (ground truths), we apply the same global transformations and
additionally the shape-aware data augmentation. After that we feed the
augmented point cloud to the student model and get student predictions. The
orientation-aware distance-IoU (OD-IoU) loss is introduced in the hard
supervision, to better align the box centers and orientations between the
student predictions and hard targets. Comparing to the plain IoU loss [34],
OD-IoU loss also considers the distance and the orientation difference between
two boxes. Finally, we use an IoU-based matching to match student and teacher
predictions. We use the consistency loss over classification scores and
bounding box predictions to provide soft supervision to the student model. The
overall loss for training the student model consists of:

$\mathcal{L}_{student}=\mathcal{L}_{cls}^{s}+\omega_{1}\mathcal{L}_{OD-
IoU}^{s}+\omega_{2}\mathcal{L}_{dir}^{s}+\lambda\mathcal{L}_{iou}+\mu_{t}\mathcal{L}_{consist},$
(1)

where $\mathcal{L}_{cls}^{s}$ is the focal loss [19] for box classification,
$\mathcal{L}_{OD-IoU}^{s}$ is the OD-IoU loss for bounding box regression,
$\mathcal{L}_{dir}^{s}$ is the cross-entropy loss for direction
classification, $\mathcal{L}_{iou}$ is the smooth-$L_{1}$ loss for IoU
prediction in the detection head, $\mathcal{L}_{consist}$ is the consistency
loss, $\omega_{1}$, $\omega_{2}$, $\lambda$ and $\mu_{t}$ are weights of
losses.

During the post-training, the parameters of the teacher model are updated
based on the parameters of the student model using the exponential moving
average (EMA) strategy with a weight decay of 0.999.

##  5 Evaluation

To prove the effectiveness of our proposed modules in SE-ProPillars, we
perform some ablation studies in that we stepwise include some modules into
our training pipeline. We train and evaluate the model on the KITTI dataset
which is one of the most popular datasets in the autonomous driving domain.
Furthermore, we train and evaluate the model on two infrastructure datasets
with roadside LiDAR sensors: the IPS300+ dataset and the recently released A9
dataset. The evaluation was performed on a Nvidia GeForce RTX 3090 GPU.

###  5.1 Ablation Studies

By adding the Stacked Triple Attention module (S-TA module) the $mAP_{3D}$ can
be increased by 1.3% $mAP_{3D}$. This module adds 2 ms of inference time to
the baseline model and runs with 35 ms per frame (28.57 FPS). Considering only
the nearby area of the LiDAR point cloud (0-30 m), the $mAP_{3D}$ can even be
increased by 8.21% compared to 1.3% when taking the whole area of 70 m into
account. The second module (Attentive Hierarchical Backbone) increases the
$mAP_{3D}$ again by 2.26% compared to the baseline model.

Method | 3D mAP | BEV mAP | Time(ms)
---|---|---|---
| Easy | Moderate | Hard | Easy | Moderate | Hard |
ProPillars | 92.46 | 80.53 | 75.50 | 94.25 | 89.80 | 84.98 | 32.0
naive without CF | 89.64 | 79.59 | 74.43 | 93.88 | 89.29 | 86.29 | 19.4
naive | 89.71 | 79.95 | 75.03 | 93.85 | 89.55 | 86.75 | 20.1
naive+DI-NMS | 90.25 | 80.48 | 75.60 | 94.18 | 89.84 | 85.11 | 21.4
naive+DI-NMS+SA-DA | 92.38 | 80.93 | 76.05 | 96.30 | 89.85 | 87.21 | 22.0
SE-ProPillars (ours) | 93.10 | 81.26 | 78.17 | 96.44 | 89.88 | 87.21 | 22.0
Improvement | +0.64 | +0.33 | +2.12 | +0.14 | +0.03 | +0.00 | +0.00
Table 1: Ablation studies of the SE-ProPillars 3D object detector after the
pre and post training (last two rows). We report the 3D and BEV mAP of the Car
and Van category (similar type enabled) on the KITTI validation set under 0.7
IoU threshold with 40 recall positions. Post-training especially improves the
accuracy on the hard examples (+2.12 3D mAP).

The result of single-class pre-training is shown in Table 1 (second last row).
After applying the confidence function (CF), we call this model naive SE-
ProPillars (no training tricks). We also apply the distance-variant IoU-
weighted NMS (DI-NMS) and the shape-aware data augmentation (SA-DA) module in
the pre-training. After enabling the multi-task head but without applying the
CF, although the precision drops, the inference speed gets a large increase.
The runtime is reduced from 32 ms to 19.4 ms. The CF in the lightweight multi-
task head is trying to solve the misalignment problem between the predicted
bounding boxes and corresponding classification confidence maps. After
applying the confidence function, an increase is visible in all metrics,
except the BEV easy difficulty keeps at the same level. After adding the DI-
NMS module, the precision becomes competitive with ProPillars considering the
balance of precision and speed, although there is a relatively large gap in
the 3D easy mode. The shape-aware data augmentation leads to a further
increase in precision. After the pre-training our model outperforms the
baseline in almost all metrics, except a tiny $0.1\%$ gap in the 3D easy
difficulty. The speed of our model is much faster compared to the baseline.
There is a small latency in the runtime after enabling the shape-aware data
augmentation module, because as the precision increases, the computation of
the DI-NMS increases.

###  5.2 KITTI Dataset

We evaluate our SE-ProPillars detector on the KITTI validation set using the
conventional metric of KITTI, i.e. the BEV and 3D mAP result under 0.7 IoU
threshold with 40 recall positions. All three levels of difficulty are taken
into account. We follow the NMS parameters in [37], set the NMS IoU threshold
to 0.3, and the confidence score threshold to 0.3. We evaluate the performance
on both single-class detection — Car category, and multi-class detection —
Car, Pedestrian, and Cyclist. In the single-class detection, we enable the
similar type, which means we train on both Car and Van categories, but treat
them all as Cars. ProPillars [38] is used as our baseline.

We use the pre-trained model as the initial teacher model in the self-
ensembling training architecture. Results are shown in Table 2. In the post-
training, the consistency loss is enabled to provide supervision for the
student model, as well as the replacement from the traditional smooth-$L_{1}$
loss to the orientation-aware distance-IoU loss for the bounding box
regression. The SA-DA module is compulsory for providing more noisy samples to
the student model. After 60 epochs of post-training, all metrics gain further
improvement. Our model outperforms the baseline by $(0.64\%,0.73\%,2.67\%)$ 3D
mAP and $(2.19\%,0.08\%,2.23\%)$ BEV mAP on easy, moderate, and hard
difficulties respectively. At the same time, we have a much lower inference
time of 22 ms, compared to our baseline of 32 ms. The inference time includes
5.7 ms for data pre-processing, 16.3 ms for network forwarding and post-
processing (NMS). The hard disk used to load the model is relatively slow,
which leads to a higher data loading time. The self-ensembling architecture is
a training technique and therefore has no additional cost in the inference
time. Some qualitative results are shown in Fig. 10. We also compare our SE-
ProPillars model with other state-of-the-art detectors on the KITTI validation
set (see Table 2).

Method |  Year | 3D mAP (Car)
---|---|---
|  |  Easy |  Medium |  Hard
VoxelNet [28] | 2018 | 82.00 | 65.50 | 62.90
PointPillars [17] | 2019 | 87.75 | 78.39 | 75.18
F-PointPillars [24] | 2021 | 88.90 | 79.28 | 78.07
SE-ProPillars (ours) | 2022 | 93.10 | 81.26 | 78.17
Improvement |  | +4.20 | +1.98 | +0.10
Table 2: Comparison on the KITTI validation set. We show 3D multi-class
detection results of SE-ProPillars and report the 3D mAP of the Car category
under 0.7 IoU threshold with 40 recall positions. Figure 10: Left: BEV
visualization of single-class detection results on the KITTI validation set.
Ground truth bounding boxes are marked with GT, predicted bounding boxes with
P. In KITTI, some ground truth boxes are filtered out because they are smaller
than 25 pixels in height in the corresponding captured image, or completely
out of the image. Some of these filtered objects can still be detected by our
SE-ProPillars which shows the good performance for detecting small objects.
Right: 3D front view visualization of single-class detection results on the
KITTI validation set. In this example a larger vehicle is detected as a Car.

###  5.3 IPS300+ Roadside Dataset

Table 3 shows the evaluation results on the full unsplit (original) IPS300+
roadside test set as well as two modified test sets: A single-LiDAR on a split
IPS300+ test set and two registered LiDAR sensors on the split IPS300+ test
set. The IPS300+ roadside dataset contains full-surround 3D labels whereas the
KITTI dataset covers only the frontal view of the vehicle. That is why we
split the IPS300+ dataset into four sub areas that represent the four
different driving directions at the intersection where the dataset was
recorded.

| 3D<EMAIL_ADDRESS>| 3D<EMAIL_ADDRESS>| BEV<EMAIL_ADDRESS>| BEV<EMAIL_ADDRESS>
---|---|---|---|---
1 LiDAR, split-IPS300+ | 54.91 | 83.76 | 74.57 | 85.97
2 LiDARs, split-IPS300+ | 63.58 | 87.52 | 78.34 | 89.42
Full (unsplit) IPS300+ | 68.90 | 97.82 | 95.53 | 98.16
Improvement | +13.99 | +14.06 | +20.96 | +12.19

Table 3: 3D object detection results of the naive SE-ProPillars model with SA-
DA. We report the 3D and BEV mAP of the Car class on a single LiDAR and two
registered LiDARs on the split IPS300+ test set under 0.7 and 0.5 IoU
thresholds with 40 recall positions. Using the full unsplit (original) IPS300+
dataset, we noticed a performance improvement of +13.99 3D mAP at an IoU
threshold of 0.7. This shows the effect of splitting the IPS300+ point clouds
into four sub-areas. Objects at the split boundary get truncated and become
difficult to detect. Hence the 3D mAP values on the split datasets are lower.
Figure 11: Left: Visualization of detection results of a single LiDAR on the
split IPS300+ validation set. Ground truth bounding boxes are marked with GT
and predicted bounding boxes with P. Here only 18 out of 26 vehicles were
detected because of the sparse point cloud from a single LiDAR. Right:
Visualization of detection results of two registered IPS300+ LiDAR point cloud
scans of the IPS300+ validation set. Here 31 out of 32 vehicles could be
detected and only one vehicle was missed because of the far range.

###  5.4 A9-Dataset

We use the naive SE-ProPillars with SA-DA module trained on a single-LiDAR
split-IPS300+ as our model, fine-tune on the training set of the A9 dataset
for 100 epochs, and test on the testing set. We report our result using 3D and
BEV mAP under an IoU threshold of 0.5 and 0.25 like in [38]. The confidence
score is set to a threshold of 0.1, and we again decrease the NMS IoU
threshold to 0.2. This is exactly the same setting as the convention in the
nuScenes [1] dataset defines. Since the LiDAR frames of the first batch of the
A9 dataset are captured on the Highway A9, the distance between cars is
further and there are no parked cars. We also test the case of using 0.1 as
the NMS threshold. The result is shown in Table 4.

Metric | 3D<EMAIL_ADDRESS>| 3D<EMAIL_ADDRESS>| BEV<EMAIL_ADDRESS>| BEV<EMAIL_ADDRESS>
---|---|---|---|---
ProPillars | N/A | N/A | N/A | 47.56
Ours (NMS IoU=0.2) | 31.03 | 48.88 | 39.91 | 49.68
Ours (NMS IoU=0.1) | 30.13 | 50.09 | 40.21 | 51.53
Improvement | - | - | - | +3.97

Table 4: Results of naive SE-ProPillars with SA-DA. We report the 3D and BEV
mAP of Car on the A9 test set under 0.5 and 0.25 IoU threshold, with 40 recall
positions.

##  6 Conclusion

In this work we present our SE-ProPillars 3D object detector that is an
improved version of the PointPillars model. We replace the detection head from
[11] with a multi-task head for the final prediction, that includes an IoU
prediction to alleviate the misalignment between the localization accuracy and
classification confidence. Attentive hierarchical middle layers were proposed
to perform 2D convolution operations on the pseudo image. The attentive
addition operation concatenates the hierarchical feature maps. We add two
training techniques to our baseline: the shape-aware data augmentation module
and the self-ensembling training architecture to improve the accuracy without
adding additional costs in the inference time. We show the generalization
ability of our model and make it more robust to noise by applying multiple
data augmentation methods. Sufficient data collection is key to achieve a good
accuracy. That is why we generated a synthetic dataset proSynth with 2k
labeled LiDAR point cloud frames and trained our model on that. We do several
sets of experiments for each module to prove its accuracy and runtime
performance in different ablation studies. Finally, we evaluate our 3D
detector on different datasets that shows that our model significantly
ourperforms the baseline model.

##  7 Future Outlook

Our SE-ProPillars 3D object detector is a significant contribution within the
area of roadside LiDAR-based 3D perception and can be used for smart city
applications in the future. A far-reaching view can be provided to connected
and automated vehicles and detected vehicles around the corner can be
communicated to all traffic participants to improve road traffic safety. LiDAR
sensors can be used for anonymous people and vehicle counting to calculate the
traffic density in real-time. Traffic monitoring will improve traffic flow and
increase public safety. Anonymous 3D detections will be sent to connected
vehicles to improve path and maneuver planning. Detecting edge cases and rare
events like accidents or breakdowns is part of the future work to notify
emergency vehicles immediately. The used LiDAR sensors are part of an
Cooperative Intelligent Transport System (C-ITS), that can share dangers on
the road (e.g. vehicle breakdowns or falling objects) with all connected
vehicles in real-time. Using these LiDAR sensors in combination with smart
traffic lights will improve the traffic flow in the future by enabling a green
wave for traffic participants (e.g. cyclists). Connecting roadside LiDARs with
vehicles will lead to a safe and efficient driving and accidents will be
prevented before they happen. Connected and cooperative vehicles will be able
to make safer and more coordinated decisions and improve their path and
maneuver planning in real-time. Finally, an online and anonymized database of
traffic participants can be created and shared with connected traffic
participants to automatically react to potential incidents. This platform will
share all traffic data and stream live sensor data to improve public safety.

## Acknowledgements

This work was funded by the Federal Ministry of Transport and Digital
Infrastructure, Germany as part of the research project Providentia++ (Grant
Number: 01MM19008A). The authors would like to express their gratitude to the
funding agency and to the numerous students at TUM who have contributed to the
creation of the first batch of the A9-Dataset.

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# Prime subcomplexes

Ismael Akray  Mathematics Department, Faculty of Science
Soran University
44008, Soran, Erbil Kurdistan Region, Iraq<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In 1859, Ernst E. Kummer [7] introduced the concept prime ideal and after that
in 1983 this concept was generalized to prime submodules by R. McCasland [8].
In this article, by considering complexes as a generalization of modules, I
introduce the concept prime subcomplex and study some properties analogous to
that of prime submodules.

###### Key words and phrases:

complex; prime submodule; prime subcomplex

###### 2000 Mathematics Subject Classification:

16E05, 16P70, 11R44

##  1\. Introduction

Throughout this article all rings are assumed to be commutative with identity
and all modules will be unitary. The symbol $\sqrt{P}$ and $Ann(P)$ indicates
to the radical and the annihilator of $P$. E. E. Kummer [7] introduced the
notion prime ideal and then many authors discussed their properties like
Krull, Fitting and McCoy in [6, 5, 9]. A proper ideal $P$ of a ring $R$ is
said to be prime if $ab\in P$ for $a,b\in R$ implies $a\in P$ or $b\in P$. In
other words, an ideal $P$ is prime if and only if $R/P$ is an integral domain.
E. Noether [10] introduced primary ideal in 1921, where a proper ideal $P$ of
a ring $R$ is said to be primary if $ab\in P$ for $a,b\in R$ implies $a\in P$
or $b\in\sqrt{P}$. R. McCasland [8] in 1983 generalized this concept into
module theory and introduced prime submodules where a proper submodule $N$ of
an $R-$module $M$ is prime if $rm\in N$ for $r\in R$ and $m\in M$ implies
$m\in N$ or $r\in(N:_{R}M)=Ann_{R}(M/N)$. Equivalently, $N$ is prime submodule
if it is a primary submodule whose radical is identical with $(N:M)$, where a
proper submodule $K$ of $M$ is primary (see [11]) if $rm\in K$ for $r\in R$
and $m\in M$ implies $m\in K$ or $r\in\sqrt{(N:M)}$. In the theory of rings,
prime ideals plays an important role everywhere and so does prime submodules
in the theory of modules. In particular, primary submodules are the major
characters in Noetherian modules.

H. Cartan [4] introduced the notion $R-$complex in a siminar in 1959. An
$R-$complex $C=(C,d)_{i\in I}$ is a sequence

$\cdots\stackrel{{\scriptstyle
d_{n+2}}}{{\rightarrow}}C_{n+1}\stackrel{{\scriptstyle
d_{n+1}}}{{\rightarrow}}C_{n}\stackrel{{\scriptstyle
d_{n}}}{{\rightarrow}}C_{n-1}\stackrel{{\scriptstyle
d_{n-1}}}{{\rightarrow}}\cdots$

of $R-$modules $C_{n}$ and $R-$morphisms $d_{n}$ such that $d_{n}d_{n+1}=0$
for all $n\in I$ [12]. Also, a complex $S=(S_{i},d_{i})$ is called a
subcomplex of an $R-$complex $C=(C_{i},d_{i})$ briefly $S\leq C$ if
$S_{i}\subseteq C_{i}$ and $d_{i+1}(S_{i+1})\subseteq S_{i}$ for all $i\in I$.
Furthermore, if $S_{i}\neq C_{i}$ for some $i\in I$, then $S$ is called a
proper subcomplex of $C$. We denote the set of all indices $i\in I$ in which
$S_{i}\neq C_{i}$ by $P_{I}$. Define the set of zero divisors $Z_{R}(C)$
(briefly $Z(C)$) on an $R-$complex $C$ to be the intersection of all zero
divisors sets on the components of $C$, that is, $Z_{R}(C)=\cap_{i\in
I}Z_{R}(C_{i})=\\{r\in R:rm_{i}=0,\hskip 5.69046pt\text{for some}\hskip
5.69046ptm_{i}\in C_{i},\forall i\in I\\}$. Moreover, the annihilator of an
$R-$complex $C$ is $Ann_{R}(C)=\\{r\in R:rC_{i}=0,\hskip 5.69046pt\text{for
all}\hskip 5.69046pti\in I\\}$. So, the zero subcomplex of an $R-$complex $C$
is prime subcomplex if $Ann(C)=Z(C)$. Furthermore, we define the residual of a
subcomplex $S$ of a complex $C$ by $(S:C)=\cap_{i\in I}(S_{i}:C_{i})=\\{r\in
R:rC_{i}\subseteq S_{i},\forall i\in I\\}$. Every $R-$module $M$ can be
considered as a complex with $C_{0}=M$ and $C_{i}=0$ for all $(i\neq 0)\in I$
and all morphisms be zero. This motivate us to think about how can we
introduce or generalize many notions in ring theory and module theory into
$R-$complexes.

The purpose of this paper is to introduce and investigate the notion of
primeness in a more wide category, the category of $R-$complexes. A proper
subcomplex $S$ of an $R-$complex $C$ is said to be a prime subcomplex if
$S_{i}$ is a prime submodule of $C_{i}$ for all $i\in P_{I}$. In other sense,
A proper subcomplex $S$ of a complex $C$ is prime if for each $i\in I$, either
$\frac{C_{i}}{S_{i}}$ is identically zero or a torsion-free
$\frac{R}{P_{i}}$-module where $P_{i}=(S_{i}:C_{i})$. In such case, we call
$S$ a $P_{i}-$prime subcomplex. A proper subcomplex $S$ of an $R-$complex $C$
is primary if $S_{i}$ is primary for all $i\in P_{I}$, that is, for all $r\in
R$, $i\in P_{I}$ and $m\in C_{i}$ with $rm\in S_{i}$ implies $m\in S_{i}$ or
$r\in\sqrt{(S_{i}:C_{i})}$. Equivalently, the zero-divisors of
$\frac{C_{i}}{S_{i}}$ are nilpotent $\forall i\in P_{I}$. In this case we say
$S$ is $P_{i}-$primary where $P_{i}=\sqrt{(S_{i}:C_{i})}$. It is clear that a
proper subcomplex $S$ of an $R-$complex $C$ is prime if and only if the chain
map $f=<f_{i}>:\frac{C}{S}\rightarrow\frac{C}{S}$ defined by
$f_{i}(m+S_{i})=rm+S_{i}$, for $m\in C_{i}$ is either monic or nilpotent for
all nonzero $r\in R$.

In section two, we establish the most fundamental theorems of prime submodules
to prime subcomplexes. We first point out that the notion of prime subcomplex
have many equivalents (Theorem 2.9). We show that a subcomplex $S$ of an
$R-$complex $C$ is prime if $(S_{i}:C_{i})$ is a maximal ideal of $R$ for all
$i\in P_{I}$ (Theorem 2.10). Also, for any faithfully flat $R-$complex $F$, a
subcomplex $S$ of a complex $C$ is prime if and only if $F\otimes S$ is a
prime subcomplex of $F\otimes C$ (Theorem 2.17). Finally, the prime avoidance
theorem was generalized to prime subcomplexes (Theorem 2.20).

##  2\. Prime subcomplexes

In this section we introduce prime subcomplex and investigate some properties
and results on such concept. We begin with their definition.

######  Definition 2.1.

A proper subcomplex $S$ of an $R-$complex $C$ is said to be a prime subcomplex
if $S_{i}$ is a prime submodule of $C_{i}$ for all $i\in P_{I}$.

Let $\mathbb{N}$ be the set of natural numbers and
$\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$. It is well-known that for any non-
nilpotent element $a\in R$, the localization of a ring $R$ at the
multiplicative subset $S=\\{a^{n}:n\in\mathbb{N}_{0}\\}$ is
$R_{(a)}=\\{\frac{r}{a^{n}}:r\in
R,n\in\mathbb{N}_{0}\\}=R[\frac{1}{a}]=R[a,\frac{1}{a}]$, which is also called
the Laurent polynomial ring. This localized ring has a principal prime ideal
$<\frac{1}{a}>=\\{\frac{r}{a^{n}}:r\in R,n\in\mathbb{N},a\nmid r\\}$ and a
primary ideal $<\frac{1}{a^{2}}>$. We denote such ideals by $P_{a}(R)$ and
$(\frac{1}{a})P_{a}(R)$ respectively. Now, we give an example on prime and
primary subcomplexes in $\breve{C}$ech complexes [3, 5.1.6 Example, pp. 84].

######  Example 2.2.

The $\breve{C}$ech complex of the ring of integers $\mathbb{Z}$ as a module
over itself with respect to integers $3$, $5$ and $7$ is

$C^{\bullet}:0\to\mathbb{Z}\stackrel{{\scriptstyle
d^{0}}}{{\to}}\mathbb{Z}_{(3)}\oplus\mathbb{Z}_{(5)}\oplus\mathbb{Z}_{(7)}\stackrel{{\scriptstyle
d^{1}}}{{\to}}\mathbb{Z}_{(35)}\oplus\mathbb{Z}_{(21)}\oplus\mathbb{Z}_{(15)}\stackrel{{\scriptstyle
d^{2}}}{{\to}}\mathbb{Z}_{(105)}\to 0$

where

$d^{0}(n)=(\frac{n}{1},\frac{n}{1},\frac{n}{1})$

$d^{1}((\frac{n_{1}}{3^{m_{1}}},\frac{n_{2}}{5^{m_{2}}},\frac{n_{3}}{7^{m_{3}}}))=(\frac{5^{m_{3}}n_{3}}{35^{m_{3}}}-\frac{7^{m_{2}}n_{2}}{35^{m_{2}}},\frac{3^{m_{3}}n_{3}}{21^{m_{3}}}-\frac{7^{m_{1}}n_{1}}{21^{m_{1}}},\frac{3^{m_{2}}n_{2}}{15^{m_{2}}}-\frac{5^{m_{1}}n_{1}}{15^{m_{1}}})$

$d^{2}((\frac{n_{1}}{35^{m_{1}}},\frac{n_{2}}{21^{m_{2}}},\frac{n_{3}}{15^{m_{3}}}))=\frac{7^{m_{3}}n_{3}}{105^{m_{3}}}-\frac{5^{m_{2}}n_{2}}{105^{m_{2}}}+\frac{3^{m_{1}}n_{1}}{105^{m_{1}}}$

Then the subcomplex

$S:0\to
0\stackrel{{\scriptstyle\bar{d^{0}}}}{{\to}}P_{3}\oplus\mathbb{Z}_{(5)}\oplus\mathbb{Z}_{(7)}\stackrel{{\scriptstyle\bar{d^{1}}}}{{\to}}\mathbb{Z}_{(35)}\oplus\mathbb{Z}_{(21)}\oplus\mathbb{Z}_{(15)}\stackrel{{\scriptstyle\bar{d^{2}}}}{{\to}}\mathbb{Z}_{(105)}\to
0$

is a prime subcomplex of $C^{\bullet}$ while the subcomplex

$T:0\to
0\stackrel{{\scriptstyle\bar{d^{0}}}}{{\to}}(\frac{1}{3})P_{3}\oplus\mathbb{Z}_{(5)}\oplus\mathbb{Z}_{(7)}\stackrel{{\scriptstyle\bar{d^{1}}}}{{\to}}\mathbb{Z}_{(35)}\oplus\mathbb{Z}_{(21)}\oplus\mathbb{Z}_{(15)}\stackrel{{\scriptstyle\bar{d^{2}}}}{{\to}}\mathbb{Z}_{(105)}\to
0$

is a primary subcomplex of $C^{\bullet}$ not a prime.

A proper subcomplex $S=(S_{i},d_{i})$ of a complex $C=(C_{i},d_{i})$ is said
to be a direct summand of $C$ if either $S_{i}=C_{i}$ or $S_{i}$ is a direct
summand of the $R-$module $C_{i}$ and denoted by $S\leq_{\oplus}C$. The
torsion subcomplex $S=T(C)$ of an $R-$-complex $C$ is a subcomplex
$S=\\{S_{i}\\}_{i\in I}$, where $S_{i}=T(C_{i})$ is the torsion submodule of
$C_{i}$, for all $i\in I$. We say that a complex $C$ is torsion if $C=T(C)$
and torsion-free if $T(C)=0$ is a zero complex.

######  Proposition 2.3.

Every direct summand subcomplex $S$ of a torsion-free complex $C$ is prime.

###### Proof.

Assume $S=(S_{i},d_{i})$ is a direct summand of a torsion-free complex
$C=(C_{i},d_{i})$. Hence there exists a direct summand subcomplex $T$ such
that $C=S\oplus T$, that means $C_{i}=S_{i}\oplus T_{i}$ for all $i$. Suppose
$rm\in S_{j}$ for all $j\in P_{I}$, $r\in R$ and $m\in C_{j}=S_{j}\oplus
T_{j}$. So, $m=a+b$ for $a\in S_{j}$, $b\in T_{j}$ and $rm-ra=rb\in S_{j}$ but
$rb\in T_{j}$, hence $rb=0$ and torsion-freeness of $M_{j}$ gives us either
$r=0$ or $b=0$. If $r=0$, then $rM_{j}=0\subseteq S_{j}$, while $b=0$ implies
that $m=a+0\in S_{j}$. Therefore $S$ is a prime subcomplex of $C$. ∎

A subcomplex $S$ of a complex $C$ is pure if $rC_{j}\cap S_{j}=rS_{j}$, for
$r\in R$ and $j\in P_{I}$. Now, we give a connection between primeness and
pureness of subcomplexes in torsion-free complexes.

######  Proposition 2.4.

A proper subcomplex $S$ of a torsion-free $R-$complex $C$ is prime with
$(S_{i}:C_{i})=0$, for all $i\in P_{I}$ if and only if $S$ is pure subcomplex
of $C$.

###### Proof.

Let $S$ be a prime subcomplex of $C$ with $(S_{i}:C_{i})=0$, for all $i\in
P_{I}$. Take $r\in R$ and $m\in C_{j}$ for $j\in P_{I}$. Then primeness of
$S_{j}$ implies $r\in(S_{j}:C_{j})=0$ or $m\in S_{j}$. In both cases we will
have $rS_{j}=S_{j}\cap rC_{j}$ and $S_{j}$ is pure submodule of $C_{j}$. So
$S$ is pure subcomplex of $C$. Now, suppose $S$ is a pure subcomplex of $C$
and $rm\in S_{j}$ for $r\in R$ and $m\in C_{j}$ for all $j\in P_{I}$. From the
identity $rC_{j}\cap S_{j}=rS_{j}$, we obtain $rm\in rS_{j}$, so there exists
$n\in S_{j}$ with $rm=rn$ and that $r(m-n)=0$. Torsion-freeness of $C$ implies
either $r=0$ or $m=n$, that is, $rC_{j}\subseteq S_{j}$ or $m\in S_{j}$. Thus
$S_{j}$ is a prime submodule of $C_{j}$ and that $S$ is prime subcomplex of
$C$. ∎

######  Proposition 2.5.

Suppose that $R$ is an integral domain and $C$ is an $R-$complex with
$T(C)\neq C$. Then $T(C)$ is a prime subcomplex of $C$.

###### Proof.

Suppose $rm\in T(C_{j})$ for nonzero $r\in R$ and $m\in C_{j}$ for $j\in I$
such that $T(C_{j})\neq C_{j}$. As $T(C_{j})$ is a torsion submodule of
$C_{j}$ and $R$ is an integral domain, there exists a nonzero $s$ in $R$ such
that $srm=0$ with $sr\neq 0$. Hence $m\in T(C_{j})$ and the result follows. ∎

Let $f:R\rightarrow S$ be a ring homomorphism and $C$ be an $S-$complex. Then
one can consider $C$ as an $R-$complex and the inverse image of each prime
subcomplex of an $S-$complex $C$ is a prime subcomplex of $C$ as $R-$complex.
For a primary subcomplex $S$ of $C$, $S$ is prime if and only if all proper
ideals $(S_{i}:C_{i})$ are prime $\forall i\in P_{I}$. Hence we have the
following result.

######  Proposition 2.6.

Suppose $S$ is $P_{i}-$primary subcomplex of $C$ containing a $P_{i}-$prime
subcomplex of $C$. Then $S$ is a $P_{i}-$prime.

###### Proof.

Consider a $P_{i}=(T_{i}:C_{i})-$prime subcomplex $T$ of $C$ with $T\subset S$
and $S$ a $P_{i}=\sqrt{(S_{i}:C_{i})}-$primary subcomplex. To prove $S$ is
$P_{i}-$prime, we have to prove $(S_{i}:C_{i})$ is prime foe all $i\in P_{I}$.
For this purpose, let $abm\in S_{i}$, for all $m\in C_{i}$, $a,b\in R$ and
$i\in P_{I}$. The primaryness of $S$ gives us $a\in\sqrt{(S_{i}:C_{i})}$ or
$b\in(S_{i}:C_{i})$. If $a\in\sqrt{(S_{i}:C_{i})}=(T_{i}:C_{i})$, then
$aC_{i}\subseteq T_{i}\subseteq S_{i}$ and so $a\in(S_{i}:C_{i})$. ∎

Let $P$ be a prime ideal of $R$ and $S$ a subcomplex of $C$. The localization
of $S$ at $P$ is $S_{P}=((S_{i})_{P},\bar{d}_{i})$ and the saturation of $S$
at $P$ is the subcomplex $\c{S}_{P}(S)=(f^{-1}((S_{i})_{P}),\bar{d}_{i})$,
where $f=<f_{i}>$ is the natural chain map defined by
$f_{i}:S_{i}\rightarrow(S_{i})_{P}$, $f_{i}(x)=\frac{x}{1}$, for $x\in S_{i}$.

######  Proposition 2.7.

Suppose $S$ is a proper subcomplex of an $R-$complex $C$ such that
$(S_{i}:C_{i})=P_{i}$ is a prime ideal of $R$ for all $i\in P_{I}$ and set
$P=<P_{i}>$. Then the saturation $\c{S}_{P}(S)=f^{-1}(S_{P})$ of $S$ is a
$P_{i}-$prime, where

$(S_{P})_{i}=\begin{cases}{S_{i}}_{P_{i}},&\mbox{if }P_{i}\neq R\\\
(C_{i})_{P_{i}},&\mbox{otherwise}.\end{cases}$

and $f$ is the chain map $f_{i}:C_{i}\rightarrow(C_{i})_{P_{i}}$,
$f_{i}(m)=\frac{m}{1}$.

###### Proof.

Let $rm\in f^{-1}({S_{i}}_{P_{i}})$, for $r\in R$, $m\in C_{i}$ and $i\in
P_{I}$. Then $\frac{rm}{1}\in{S_{i}}_{P_{i}}$ and so $trm\in S_{i}$ for
$t\notin P$. If $r\notin P_{i}$, then
$f_{i}(m)=\frac{m}{1}=\frac{trm}{tr}\in{S_{i}}_{P_{i}}$ and $m\in
f^{-1}_{i}((S_{i})_{P_{i}})$. Therefore, $\c{S}_{P}(S)$ is a $P_{i}-$prime
subcomplex of $C$. ∎

A free complex of finite rank is a complex whose terms are free $R-$modules of
finite rank.

######  Theorem 2.8.

Let $S$ be a subcomplex of a free complex $F=(F_{i},d_{i})$ of finite rank.
Then $S$ is a $P_{i}-$prime if and only if $S=\c{S}_{P}(T)$, for some
subcomplex $T=(N_{i}+P_{i}F_{i},d_{i})$ of $F$, where $N=f^{-1}(A)$ for a
direct summand $A$ of $F_{P}=((F_{i})_{P_{i}},\bar{d}_{i})$ and the canonical
chain map $f:F\rightarrow F_{P_{i}}$, $f_{i}(x)=\frac{x}{1}$, $\forall x\in
F_{i}$.

###### Proof.

Clearly $(N_{i}+P_{i}F_{i}:F_{i})=P_{i}$ and so by Proposition 2.7,
$\c{S}_{P}(T)$ is a $P_{i}-$prime subcomplex of $F$. Now, let $S$ be a
$P_{i}-$prime subcomplex of $F$. So, $P_{i}F_{i}\subset S_{i}$ and we have the
composition of chain maps $F\stackrel{{\scriptstyle
f}}{{\to}}F_{P}\stackrel{{\scriptstyle g}}{{\to}}\dfrac{F_{P}}{PF_{P}}$.
Choose a basis $\\{e_{1},\cdots e_{v_{i}}\\}$ for $g_{i}({S_{i}}_{P_{i}})$ and
a set of elements
$m_{1},\cdots,m_{v_{i}}\in{S_{i}}_{P_{i}}\subseteq(F_{i})_{P_{i}}$ such that
$g_{j}(m_{i})=e_{i}$ for $i=1,\cdots,v_{j}$ and $j\in I$. Then one can check
that $(S_{i})_{P_{i}}=(m_{1},\cdots,m_{v_{i}})+P_{i}{F_{i}}_{P_{i}}$. Since
$P_{i}(F_{i})_{P_{i}}\subseteq(S_{i})_{P_{i}}$, we have an exact sequence
$0\to P_{i}(F_{i})_{P_{i}}\to S_{i}\to\frac{S_{i}}{P_{i}(F_{i})_{P_{i}}}\to
0$. Thus Nakayama lemma ensure that $<m_{1},\cdots,m_{v_{i}}>$ is a direct
summand of ${F_{i}}_{P_{i}}$. Put $N_{i}=f^{-1}_{i}(<m_{1},\cdots,m_{v_{i}}>)$
and $N=(N_{i},d_{i})$. So, we have $S_{i}\subset
N_{i}+F_{i}\subset\c{S}_{P}(N+P_{i}F)$ and $\c{S}_{P}(C+P_{i}F)$ is a
$P_{i}-$prime subcomplex of $F$. By [1, Corollary 3], there exists a bijection
correspondence between the $P-$prime subcomplexes of $F$ and those of
$F_{P_{i}}$. Since $S_{P}=C_{P}+P_{i}F_{P}$, we get that
$S=\c{S}_{P}(C+P_{i}F)$.

∎

From [2, Page 169, Ex. 12], every primary ideal in a von Neumann regular ring
is a prime. So we have every primary subcomplex of a complex over von Neumann
regular ring is prime.

######  Theorem 2.9.

Suppose $S$ is a proper subcomplex of an $R-$complex $C$ and
$(S_{i}:C_{i})=P_{i}$ for $i\in P_{I}$. Then the following assertions are
equivalent:

  1. (1)

$S$ is a prime subcomplex of $C$;

  2. (2)

every component $\frac{C_{i}}{S_{i}}$ of the quotient complex $\frac{C}{S}$ is
a torsion-free $\frac{R}{P_{i}}-$module;

  3. (3)

for each subcomplex $T$ of $C$ and a family of ideals $\\{J_{i}\\}_{i\in I}$
of $R$ with $J_{i}T_{i}\subseteq S_{i}$, $T_{i}\subseteq S_{i}$ or
$J_{i}\subseteq(S_{i}:C_{i})$, $\forall i\in P_{I}$;

  4. (4)

$(S_{i}:r)=S_{i}$ for all $r\in R-P_{i}$ and $i\in P_{I}$;

  5. (5)

$(S_{i}:J)=S_{i}$, $\forall i\in P_{I}$ and any ideal $J$ of $R$;

  6. (6)

$(S_{i}:m)=P_{i}$, for all $m\in C_{i}-S_{i}$ and $i\in P_{I}$;

  7. (7)

$(S_{i}:N_{i})=P_{i}$, for all subcomplex $N=(N_{i},d_{i})$ of $C$ with
$S\subset T$ and $i\in P_{I}$;

  8. (8)

$Ass(\frac{C_{i}}{S_{i}})=\\{P_{i}\\}$, $\forall i\in P_{I}$, where
$Ass(\frac{C_{i}}{S_{i}})$ indicates the set of associative primes of
$\frac{C_{i}}{S_{i}}$;

  9. (9)

$Z_{R}(\frac{C_{i}}{S_{i}})=P_{i}$, $\forall i\in P_{I}$;

  10. (10)

every nonzero subcomplex $\frac{T}{S}$ of $\frac{C}{S}$ has the same
annihilator, that is, $(T:C)=(S:C)$;

  11. (11)

for any family of ideals $J=<J_{i}>$ of $R$ with $(S_{i}:C_{i})\subsetneq
J_{i}$ and all subcomplex $T$ of $C$ with $S_{i}\subsetneq T_{i}$,
$J_{i}T_{i}\nsubseteq S_{i}$, $\forall i\in P_{I}$.

###### Proof.

$(1\rightarrow 10):$ It suffices to show for any
$m+S_{i}\in\frac{C_{i}}{S_{i}}$ with $m\in C_{i}-S_{i}$,
$Ann(m+S_{i})=(S_{i}:C_{i})$. Taking $r\in(S_{i}:<m+S_{i}>)$ gives $rm\in
S_{i}$ and we obtain from part (1) that $r\in(S_{i}:C_{i})$.

$(10\rightarrow 1):$ Suppose $rm\in S_{i}$, for $i\in P_{I}$, $m\in C_{i}\
S_{i}$ and $r\in R$. Part (10) gives $r\in(S_{i}:<m+S_{i}>)=(S_{i}:C_{i})$.

$(11\rightarrow 3):$ Suppose $J_{i}T_{i}\subseteq S_{i}$ for any subcomplex
$T$ of $C$ and a family of ideals $\\{J_{i}\\}_{i\in I}$. Set
$A_{i}=S_{i}+T_{i}$ and $B_{i}=J_{i}+(S_{i}:C_{i})$. Then $T_{i}\nsubseteq
S_{i}$ if and only if $A_{i}\nsubseteq S_{i}$ and
$J_{i}\nsubseteq(S_{i}:C_{i})$ if and only if $B_{i}\nsubseteq(S_{i}:C_{i})$.
Hence $B_{i}A_{i}\subseteq S_{i}$ and we conclude from part (10) that
$T_{i}\subseteq S_{i}$ or $J_{i}\subseteq(S_{i}:C_{i})$.

The remains are clear. ∎

######  Theorem 2.10.

A subcomplex $S$ of an $R-$complex $C$ with $P_{i}=(S_{i}:C_{i})$ a maximal
ideal of $R$ for all $i\in P_{I}$ is a prime subcomplex of $C$.

###### Proof.

The hypothesis gives us that $\frac{C_{i}}{S_{i}}$ is a vector space over the
field $\frac{R}{P_{i}}$ and so a torsion-free $\frac{R}{P_{i}}-$module. The
rest comes from Theorem 2.9. ∎

######  Corollary 2.11.

For each maximal ideal $m$ of $R$ with $mC\neq C$, $mC$ is a prime subcomplex
of $C$.

######  Corollary 2.12.

Let $\\{m_{i}\\}_{i\in I}$ be a family of maximal ideals of $R$. A proper
subcomplex $S$ is $m_{i}-$prime if and only if $m_{i}C_{i}\subseteq S_{i}$. In
particular, if $S$ is an $m_{i}-$prime subcomplex of $C$, then each proper
subcomplex $T$ of $C$ containing $S$ is prime.

A proper subcomplex $S$ of an $R-$complex $C$ is said to be a maximal
subcomplex if there is no proper subcomplex contains $S$ properly, or
equivalently, if $S_{i}$ is a maximal submodule of $C_{i}$ for all $i\in
P_{I}$. In commutative ring with identity, every maximal ideal is prime. Now,
we prove this for subcomplexes.

######  Proposition 2.13.

Suppose $S$ is a maximal subcomplex of a complex $C$. Then $S$ is a prime
subcomplex of $C$ and $(S_{i}:C_{i})$ is a maximal ideal of $R$.

###### Proof.

For $i\in I$, $S_{i}$ is maximal submodule of $C_{i}$ if and only if
$\frac{C_{i}}{S_{i}}$ is a simple $R-$module. Thus $\frac{C_{i}}{S_{i}}$ a
cyclic $R-$module say $\frac{C_{i}}{S_{i}}=Rx_{i}$, $x_{i}\in C_{i}$ and
$Ann(Rx_{i})=Ann(\frac{C_{i}}{S_{i}})=(S_{i}:C_{i})$ is a maximal ideal of $R$
by [1, Proposition 3]. Hence $S$ is a prime subcomplex of $C$ due to
Proposition 2.10 ∎

A finitely generated complex is a complex whose terms are finitely generated.
So we have the following.

######  Corollary 2.14.

If $C$ is a finitely generated $R-$complex, then every proper subcomplex of
$C$ is contained in a prime subcomplex.

An $R-$complex $F=\\{F_{i},d_{i}\\}_{i\in I}$ is called flat if all its
component are flat over $R$.

######  Lemma 2.15.

Let $C$ and $F$ be $R-$complexes and $S$ be a subcomplex of $C$. Then
$F_{i}\otimes(S_{i}:r)\subseteq(F_{i}\otimes S_{i}:r)$, for any $r\in R$. The
equality holds if $F$ is a flat complex.

###### Proof.

Let $f\otimes c\in F_{i}\otimes(S_{i}:r)$ for $f\in F_{i}$, $rc\in S_{i}$ and
$i\in I$. Then $r(f\otimes c)=f\otimes rc\in F_{i}\otimes S_{i}$. Now, suppose
$F$ is flat. Then clearly
$F_{i}\otimes\frac{C_{i}}{S_{i}}\cong\frac{F_{i}\otimes C_{i}}{F_{i}\otimes
S_{i}}$. Also, exactness of

$0\to F_{i}\otimes(S_{i}:r)\to F_{i}\otimes C_{i}\stackrel{{\scriptstyle
i\otimes f_{r}}}{{\to}}\frac{F_{i}\otimes C_{i}}{F_{i}\otimes S_{i}}$

gives that $Im(i\otimes f_{r})=F_{i}\otimes(S_{i}:r)=(F_{i}\otimes S_{i}:r)$,
$\forall i\in I$. ∎

######  Theorem 2.16.

Let $S$ be a prime subcomplex of $R-$complex $C$ and $F$ be flat $R-$complex
with $F\otimes S\neq F\otimes C$. Then $F\otimes S$ is a prime subcomplex of
$F\otimes C$.

###### Proof.

We prove this theorem by the equivalence (4) of prime subcomplexes in Theorem
2.9. For this purpose, let $r\in R-(S_{i}:C_{i})$ for $i\in P_{I}$. Then the
primeness of $S$ and Theorem 2.9 (4) gives us $(S_{i}:r)=S_{i}$ and we obtain
from Lemma 2.15 that $(F_{i}\otimes
S_{i}:r)=F_{i}\otimes(S_{i}:r)=F_{i}\otimes S_{i}$. Therefore by Theorem 2.9
again, we conclude that $F\otimes S$ is a prime subcomplex of $F\otimes C$. ∎

The converse of Theorem 2.16 is true in the case where $F$ is faithfully flat,
that is, all components of $F$ are faithfully flat.

######  Theorem 2.17.

Let $S$ be a subcomplex of $R-$complex $C$ and $F$ be a faithfully flat
complex over $R$. Then $S$ is a prime subcomplex of $C$ if and only if
$F\otimes S$ is a prime subcomplex of $F\otimes C$.

###### Proof.

Since the exactness of the sequence

$0\to F_{i}\otimes S_{i}\to F_{i}\otimes C_{i}\to 0$

implies the exactness of of the sequence

$0\to S_{i}\to C_{i}\to 0,$

so we can assume that $F_{i}\otimes S_{i}\neq F_{i}\otimes C_{i}$, $\forall
i\in P_{I}$ and Theorem 2.16 tells us that $F\otimes S$ is a prime subcomplex
of $F\otimes C$. Now, let $F\otimes S$ be a prime subcomplex of $F\otimes C$
and so $S$ is a proper subcomplex of $C$. Thus for any $r\in R-(S_{i}:C_{i})$,
$\forall i\in P_{I}$, we have $r\in R-(F_{i}\otimes S_{i}:F_{i}\otimes C_{i})$
and Lemma 2.15 implies that

$F_{i}\otimes(S_{i}:r)=(F_{i}\otimes S_{i}:r)=F_{i}\otimes S_{i}$

As $F$ is faithfully flat complex, we have $(S_{i}:r)=S_{i}$, $\forall i\in
P_{I}$. Hence by Theorem 2.9 (4) again, $S$ is a prime subcomplex of $C$ as
desired. ∎

######  Corollary 2.18.

  1. (1)

Let $T$ be a multiplicative subset of $R$ and $S=(S_{i},d_{i})$ be a proper
subcomplex of an $R-$complex $C=(C_{i},d_{i})$ with $T\cap(S_{i}:C_{i})=\phi$,
$\forall i\in P_{I}$. Then $T^{-1}S=(T^{-1}S_{i},T^{-1}d_{i})$ is a prime
subcomplex of $T^{-1}C=(T^{-1}C_{i},T^{-1}d_{i})$ if and only if $S$ is a
prime subcomplex of $C$.

  2. (2)

Let $T=\\{T_{i}:i\in I\\}$ be a family of multiplicative subsets of $R$ and
$S=(S_{i},d_{i})$ be a proper subcomplex of $C=(C_{i},d_{i})$ such that
$(S_{i}:C_{i})\cap T_{i}=\phi$, $\forall i\in P_{I}$. Then
$T^{-1}S=(T_{i}^{-1}S_{i},T^{-1}_{i}d_{i})$ is a prime subcomplex of
$T^{-1}C=(T_{i}^{-1}C_{i},T_{i}^{-1}d_{i})$ if and only if $S$ is a prime
subcomplex of $C$.

###### Proof.

The proof come from Theorem 2.17 and the fact that $T^{-1}S=T^{-1}R\otimes S$.
∎

######  Corollary 2.19.

Let $S=(S_{i},d_{i})$ be a prime subcomplex of an $R-$complex
$C=(C_{i},d_{i})$ and $x$ be an indeterminate. Then
$S[x]=(S_{i}[x],d^{*}_{i})$ is a prime subcomplex of
$C[x]=(C_{i}[x],d^{*}_{i})$, where
$d_{i}^{*}(\sum_{i=1}^{n}m_{i}x^{i})=\sum_{i=1}^{n}d_{i}(m_{i})x^{i}$, for
$m_{i}\in S_{i}$.

###### Proof.

The proof follows from the fact that $S[x]=S\otimes R[x]$ and Theorem 2.17. ∎

We now generalize the prime avoidance theorem to prime subcomplexes.

######  Theorem 2.20.

Let $T_{1},\cdots,T_{n}$ be subcomplexes of an $R-$complex $C$ and let $S$ be
a prime subcomplex of $C$ with $T_{1}\cap T_{2}\cap\cdots\cap T_{n}\subseteq
S$. Then there exists an $i\in\\{1,\cdots,n\\}$ such that either
$T_{i}\subseteq S$ or $(T_{i}:C)\subseteq(S:C)$.

###### Proof.

Assume by contrary that $T_{1}\nsubseteq S$ and
$((T_{i})_{j}:C_{j})\nsubseteq(S_{j}:C_{j})$ for some $j\in P_{I}$. So, there
exist $m\in(T_{1})_{j}-S_{j}$ and $r_{i}\in((T_{i})_{j}:C_{j})-(S_{j}:C_{j})$
for all $i=1,\cdots,n$. Thus $r_{i}m\in(T_{1})_{j}\cap(T_{i})_{j}$ for
$i\in\\{2,\cdots,n\\}$, so $r_{1}r_{2}\cdots
r_{n}m\in(T_{1})_{j}\cap(T_{2})_{j}\cap\cdots\cap(T_{n})_{j}=(T_{1}\cap\cdots\cap
T_{n})_{j}\subseteq S_{j}$. The primeness of $S$ gives us $m\in S_{j}$ or
$r_{2}r_{3}\cdots r_{n}\in(S_{j}:C_{j})$ which contradicts our assumption. ∎

At the end of the paper I want to mention that there are a lot of open
questions on such subject. For example, one can do research on how to
generalize the supplemented modules and its generalizations into the category
of $R-$complexes.

The author declare that he has no conflict of interest.

This manuscript does not include any data.

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