[1]This document is the results of the research project funded by The Ocean
Cleanup.

[1] [1, 3]

Conceptualization of this study, Methodology, Software, Validation,
Investigation, Resources, Data Curation, Writing - Original Draft, Supervision

[2]

Software, Validation, Data Curation, Writing - Review and Editing, Project
administration

[2]

Writing - Review and Editing, Resources

[3] [1,3]

Supervision, Project administration, Writing - Review and Editing, Funding
acquisition, Resources

[cor1]Principal Author [cor2]corresponding author

# A Robotic Approach towards Quantifying Epipelagic Bound Plastic Using Deep
Visual Models

Gautam Tata<EMAIL_ADDRESS>California State University, Monterey Bay  Jay
Lowe<EMAIL_ADDRESS>Olivier Poirion<EMAIL_ADDRESS>The Jackson Laboratory,
Connecticut, United States of America  Sarah-Jeanne Royer<EMAIL_ADDRESS>Scripps Institution of Oceanography, San Diego, California, United States of
America  The Ocean Cleanup Foundation, Rotterdam, The Netherlands  Center for
Marine Debris Research, Hawaii Pacific University, Waimanalo 96795, Hawaii,
United States of America

###### Abstract

The quantification of positively buoyant marine plastic debris is critical to
understanding how plastic litter accumulates across the world’s oceans and is
also crucial to identifying hotspots for targeted cleanup efforts. Currently,
the most common method to quantify marine plastic is using manta trawls for
manual sampling. However, this method is cost-intensive and requires human
labor. This study removes the need for manual sampling by using an autonomous
method using neural networks and computer vision models, which trained on
images captured from various layers of the ocean column to perform real-time
plastic quantification. The best performing model has a Mean Average Precision
of 85% and an F1-Score of 0.89 while maintaining near real-time processing
speeds 2 ms/img.

###### keywords:

Marine Debris Deep Learning Computer Vision Climate Change

##  1 Introduction

Plastic pollution poses an imminent threat to the marine environment, food
safety [1], human health, eco-tourism, and contributes to climate change [2].
Global plastic production has exceeded 500 million tons of plastic, and
projections indicate that 30% of all produced plastic will end up discarded in
the oceans [3] [4]. Researchers have documented a five-fold increase in
plastic debris within the Central Pacific Gyre and have shown that plastic
pieces now outnumber the native plankton 6:1 in terms of abundance [5]. A
significant amount of marine plastic (about 80%) originates from land-based
sources [6]: Most commonly in the form of food containers, such as plastic
bags and bottles, and packaging materials. The other 20% stems from shipping
vessel discharges and discarded commercial fishing gear [6]. Studies have
shown that removing plastic from the oceans will exponentially benefit the
ecosystems. This includes the prevention of the movement of invasive species
between regions [7], the prevention of its degradation into micro-plastics
[8], and the decrease in emissions of greenhouse gases (thereby decelerating
climate change) [9]]. To understand the spatiotemporal distribution of
plastic, we require more accurate methods with reliable and low-cost
deployment strategies. Various in situ approaches to ocean plastic monitoring
have been proposed. These in situ methods include using SONAR/LIDAR to map
plastic debris [10], human counting via visual methods [11], and debris
sampling using fishing nets [12]. However, these methods are labor-intensive,
incur high financial costs, and do not cover large surface areas. Furthermore,
polymers such as polyethylene and polypropylene are affected by the growth of
a biofilm when submerged in water that will influence their sinking behaviors
[13]. Any polymer that has its density increased by biofilm to a certain point
sinks beyond surface sampling devices such as manta trawls. As a result, these
surface sampling limitations lead to quantity underestimations of floating
plastics. Creating an accurate marine plastic debris estimation requires
developing alternative methods to investigate the distribution of positively
buoyant plastic across the entire water column. Recently, several methods
using computer vision and modern deep learning technologies to quantify marine
plastic debris without physical removal have been suggested [14] [15]. The
Earth and Space Science journal illustrates a method using the two-stage
Faster R-CNN model to actively monitor and identify surface plastic as it
floats down a river [14]. This approach does not account for the sinking
polymer problem but shows that, on average, an automated method detects 34.6%
more plastic than human visual counting does. Remote sensing of plastic litter
provides a promising new and less labor-intensive tool for the quantification
and characterization of ocean plastic pollution [16]. A research team at the
University of Minnesota developed a computer vision model specialized for
marine plastic detection in deep-sea environments [15] which demonstrates that
quantification across the water column can be achieved. It also exemplifies
the relationship between object detection models and AUV’s to great success.
The AquaVision project [17] shows that object detection models can reach high
levels of precision utilizing open-source datasets and one-stage approaches
such as RetinaNet. Since AquaVision was trained on the TACO dataset. It also
indicates that a computer vision model trained on land-based images of plastic
can detect similar types of plastic in a marine environment. Unlike these
recently proposed algorithms that specialize in monitoring either floating
marine plastic [14], deep-sea specific environments [15], or models trained on
land-based plastic [17]. The Ocean Cleanup group has demonstrated that
computer vision models can detect floating plastic debris via cameras attached
to above-water vessels [16]. Their results show that macroplastics can be
successfully quantified for comparisons across methods. Our object detection
model (DeepPlastic) utilizes a training set composed exclusively of marine-
based plastic images and performs equally well across the entire water column
while producing significant results.

Figure 1: Concept of real-time plastic detection via AUV’s equipped with
cameras and DeepTrash vision

In this study, we tested four state-of-the-art deep-learning architectures,
Faster-RCNN, Single Shot Multibox Detector, YOLOv4-Tiny and YOLOv5-S, then
reported their performances to infer marine plastic debris in real-time. The
main results will be described as follows: 1) the model’s precision and
accuracy to feasibly identify plastic debris, 2) insurability that this method
can successfully distinguish marine plastic debris from similar-looking non-
plastic objects, and 3) A generalized model capable of detecting marine
plastic in most oceanic environments. The results show that deep learning
models can identify plastic with significant accuracy while operating at a
rate that supports real-time applications such as autonomous underwater
vehicles (AUVs) for at-scale marine- plastic quantification and monitoring.

##  2 Related Work

Increasing demand for identifying and removing plastic from the world’s
waterways has led to a surge of research in computer vision and AUV solutions.
A team of researchers at the University of Minnesota robotics lab recently
experimented with AUV deployments for identifying deep ocean marine plastic
debris [15]. Another growing trend has been to utilize deep learning and
computer vision to identify floating marine plastic on the river automatically
and ocean surfaces [17]. Additionally, AUV’s have been used as a means for
environmental surveillance [18], mapping [19], and localization of marine
plastic debris [20]. Underwater vision technology has been pushed forward
thanks to work done by Ge et al. [20] with LIDAR technology to localize and
map marine-plastic debris on coastal beaches. Further research into
implementing LIDAR in conjunction with forward-facing SONAR image models
trained by deep convolutional neural networks was conducted by Howell et al.
[21], and Valdenegro-Toro et al. [10] which resulted in a model capable of
detecting underwater debris with 80% accuracy. Unfortunately, these methods
incur high expenses due to retrofitting sonar and an in-house water tank for
evaluation. The University of Minnesota robotics lab [15] annotated and
published a dataset of images collected by the Japan Agency for Marine-Earth
Science and Technology (JAMSTEC) [22]. JAMSTEC released the J-EDI (JAMSTEC
E-Library of Deep-Sea Images), which contains marine plastic debris dating
back to 1982 and provides data in the form of images and videos. The work
presented in this research paper has benefited from the University of
Minnesota team, which released close to 3000 annotated images from the JAMSTEC
J-EDI dataset. These datasets were used to train our convolutional neural
networks (CNNs) to identify features of plastic debris. Photography,
especially video-cameras, have found common application as environmental
monitoring systems [23] [24]. Underwater cameras provide a globally accessible
and low-cost quantification aid. Combining object detection models with
underwater cameras equipped on automobiles such as AUVs makes it possible to
observe and monitor sub-surface plastics in known hotspots worldwide [15]. By
mounting video cameras to AUV’s, buoys, and other submersibles, institutions
could feasibly quantify macro-plastics, which constitute 90% of the total
plastic mass in the oceans.

(a) Ocean

(b) Lake

Figure 2: Example images of marine plastic debris from the DeepTrash dataset
in different marine environments Figure 3: Methadology for Marine Plastic
Detection

##  3 Network Architecture

Four state-of-the-art object detection models were selected for this work.
Each architecture has different benefits and drawbacks, with the main trade-
off being speed for accuracy.

  * •

Faster RCNN Inception v2 Faster RCNN [25] is an improvement on R-CNN [26] that
introduces a Region Proposal Network to make the network trainable end to end.
The network uses the convolutional feature maps to produce region proposals,
which are fed to the fully connected (in our case softmax layers) for
detection. Faster R-CNN uses VGG-16 [27] for feature extraction while we use
the newer we use the Inception v2 [28] as feature extractor instead because of
it’s known capabilities to enhance object detection.

  * •

Single Shot Multibox Detector MobileNet v2 Single Shot MultiBox Detector (SSD)
[29] is another well-known detection model that performs object localization
and classification in a single forward pass of the network. This architecture
introduces additional Convolutional layers with the base network to improve
performance. We use a MobileNetv2 implementation [30] for faster performance
speeds.

  * •

YOLOv5-S [31] YOLOv5-S Unlike the official release of YOLOv4, YOLOv5 currently
exists in active development. There- fore, all YOLOv5 related code, and models
may be subject to modification or deletion without notice. YOLOv5-S has 7.5
million parameters, 140 layers and operates at a lightweight 7MB (14MB for
weights pre-trained on COCO). This architecture uses the Cross Stage Partial
Network (CSP) [32] as the processing backbone and was trained on MSCOCO to
extract rich/informative features from an input image. YOLOv5 also uses a
PANet [33] for the model-neck to generate feature pyramids and the
computational friendly LeakyReLU and Sigmoid activation function. The model
uses SGD as a default learning rate, but these tests were performed with the
ADAM adaptive learning rate enabled [34].

  * •

YOLOv4-Tiny [35]

Inference speeds on YOLOv4-Tiny can reach upwards of 400 frames/second when
using a 1080Ti GPU with accuracy, precision, and recall that meet the demands
of a production-ready robotics platform. YOLOv4- Tiny uses a CSPDarknet53-Tiny
neural network as opposed to the regular SPDarknet53 network. To simplify the
computation process, the YOLOv4-Tiny model uses the LeakyReLU as an activation
function.

##  4 Methodology

###  4.1 Dataset Construction

The dataset was curated by collecting videos of marine plastic from the field
in California (South Lake Tahoe, Bodega Bay, San Francisco Bay). The videos
vary significantly in quality, depth, and visibility to better represent the
harshness of marine environments.

After recording, manual identification of marine plastic captured in the still
images was performed, emphasizing choosing images containing complex object
detection scenarios such as illumination, noise and occlusion. Each image
would then get annotated to prepare them for object detection using the deep
learning models. This curation approach ensured that the dataset of images
would closely conform to real-world conditions. To further increase the
representation of marine plastics in different locations, images were also
sourced from datasets created by the Japan Agency for Marine-Earth Science and
Technology (JAMSTEC) [22]. Annotations were performed using the free tool
supervisely [36] to create the final dataset, which contains 3200 images.
Ocean environments provide a wide variety of visual challenges, so all plastic
instances get consolidated into a single classification labeled
“trash_plastic”. We called our final dataset, DeepTrash [37] and have open
sourced it to further help the research in this field.

###  4.2 Enhancements of Custom Dataset

The following procedures were implemented for the deep learning models to
detect marine plastic:

  1. a)

Dataset Formatting The input data constituted of images and annotation labels
for bounding boxes were converted into either a TFRecords (FasterRCNN and
SSD), PyTorch (YOLOv5-S) or a Darknet format (YOLOv4) to process each
respective model. The bounding boxes delimited each image’s regions of
interest based on 2D coordinates located in the respective annotation file.

  2. b)

Image Pre-processing To ensure that learning occurs on the same image
properties, auto orient was applied to strip images of their exchangeable
Image file format (EXIF) data [38] so that the models interpret images
regardless of image format. Finally, the input images get resized and bounding
boxes adjusted to 416x416 pixels.

  3. c)

Data Augmentation To mitigate the effects of the model generalizing towards
undesired features and to replicate underwater conditions such as variable
illumination, occlusion, and color–the dataset was further enhanced by
randomly changing the brightness and saturation of the images via PyTorch’s
built-in Transforms augmentation. These modified images were then added back
into the dataset, effectively tripling the size of our dataset.

###  4.3 Object Detection

We used four state-of-the-art neural network architectures FasterRCNN with
Inception v2, Single Shot Multibox Detector with MobileNet v2, YOLOv5-S and
YOLOv4, downloaded from their respective repositories . The following software
versions were used: Tensorflow1.5, PyTorch v1.8.1, Darknet, OpenCV version
3.2.0, and CUDA 11.2.

####  4.3.1 Fine Tuning Hyperparameters

This object detection model uses ADAM [34] as the adaptive learning rate,
which utilizes a decaying learning rate for a set number of epochs. The final
layer of the network uses Softmax and reflects the usage of a single class.

####  4.3.2 GPU Hardware

We tested two state of the art GPUs:NVIDIA P100 and NVIDIA Tesla V100®GPU
(version 460.32.03) were chosen due to their proven parallel computing
capability.

####  4.3.3 Training

After every 1000 epochs (iterations) of training, the model would be evaluated
on the validation dataset to calculate precision, recall, and mean average
precision (mAP). This means stopping training to check for the following:

     * •

When accuracy stops increasing, the model no longer needs additional training
to prevent overfitting.

     * •

Depending on performance, hyperparameters should receive adjustments to
optimize for evaluation metrics.

####  4.3.4 Evaluation Metrics

After the model has finished training, use the testing and validation datasets
containing images mutually exclusive from the training dataset as an input to
evaluate the network’s performance. The model draws a bounding box around
successfully detected objects with a confidence score of .50 or higher. The
number of true positive bounding boxes drawn around marine plastic debris and
true negatives provides the basis of evaluation. The following performance
metrics were utilized to produce results:

     * •

True positive and True negative values: True positive values represent an
outcome in which the models correctly predict a positive class, and
conversely, a true negative represents when the model correctly predicts the
negative class.

     * •

Precision and Recall – represents if the model suc- cessfully detected plastic
in an image.

$Recall=\frac{TP}{TP+FN}$ $Precision=\frac{TP}{TP+FP}$

     * •

Mean Average Precision – Evaluates how often the network can recognize plastic
in a group of images. After collecting the values for true and false
positives, generate a precision-recall curve using the Intersection over Union
(IoU) formula:

$\textit{IOU}=\frac{BBox_{predicted}\cap
BBox_{groundTruth}}{BBox_{predicted}\cup BBox_{groundTruth}}$

Where $BBox_{predicted}$ and $BBox_{groundTruth}$ Where BBoxpredicted and
BBoxgroundT ruth are the areas under the curve for predicted and ground truth
bounding boxes, respectively. A high threshold for confidence and IoU must be
set to ensure accuracy, with a correct detection represented by the threshold
being exceeded.

The mAP can then be obtained by integrating the precision-recall curve [39]:

[39]:

$mAP=\int_{0}^{1}p(x)dx$

     * •

F1-Score – Evaluates the balance between precision and recall values.

     * •

GPU Speed (ms/IMG) – Represents how fast the network can infer marine plastic
debris contained within an input image.

####  4.3.5 Visualizing results

For each processed image, the network populates arrays containing the
following data: Scores – Confidence scores for the predicted boxes and Number
of detections – The total number of detections made per image.

The following equation converts the normalized coordinates into image
coordinates for rendering bounding boxes on top of images:

$imgCoord_{k}=BoxScore_{i}^{j}\cdot Width$ (1)

where $k\in$ (left,right,top,bottom), $i$ is an index of boxes,
$j\in(0,1,2,3)$, and $Width$ is a width of the image. These image coordinates
were used to visualize the results of predicted bounding boxes in Figure
LABEL:fig_results.

(a) Detection near water surface

(b) Detection for partially buried debris.

Figure 4: Results generated by the model with bounding boxes and confidence scores rendered over marine plastic debris. Network | mAP |  F1-Score | Precision
---|---|---|---
YOLOv5s | 85.0 |  0.89 | 0.93
Tiny-YOLOv4 | 84.0 |  0.80 | 0.96
Faster R-CNN | 79.0 |  0.76 | 0.84
SSD | 76.0 |  0.71 | 0.83
Table 1: Detection metrics in mAP, F1, and Precision. Network | P100 | V100
---|---|---
YOLOv5s | 2.8 | 1.4
Tiny-YOLOv4 | 1.9 | 1.2
Faster R-CNN | 2.4 | 1.5
SSD | 2.5 | 2.1
Table 2: Performance metrics for Inference (ms/img).

##  5 Results

All results expressed in Table I were produced from the validation dataset
presented in the methodology section. Since the images used in the training
dataset were not isolated laboratory creations but instead real-world images
directly from the field, the general object detection has a more accurate
representation of marine plastic debris. This approach comes with a set of
trade-offs:

  * •

The model performs stronger in real-world deployments, and therefore, the
evaluation results in Table II do not significantly differ from near-real-time
measurements taken from the field.

  * •

Peak performance of the object detection model in a perfectly controlled
environment could not be measured, and the highest possible benchmark of a
single detection remains unknown.

  * •

These trade-offs indicate the results of this paper better approximate long-
term performance across a wider variety of marine environments–leading to a
more substantial evaluation of the object detection model’s performance in the
field.

###  5.1 Quantitative Results

The results captured in Table I demonstrate that near-real-time object
detection of marine plastic debris in the epipelagic layer of the ocean is
both feasible and close to real-world execution. The tested models demonstrate
high average precision, mAP, and F1 scores relative to their inference speed.
Repeated testing of the model produced a results variance of 2Usually,
evaluation results between models showcase a clear relationship between
models, such as trading off significant inference speed for increased
accuracy. However, the results presented in this paper showcase that both
YOLOv4-Tiny and YOLOv5- S produce high debris localization metrics when it
comes to identifying epipelagic plastic in near-real-time. YOLOv5-S provides a
significantly higher F1 score in exchange for a slight dip in inference
performance. Reducing the number of classes to 1, i.e.,” trash plastic,”
ensures an even distribution of class examples within the training dataset.
The singular nature of this object detection model may reduce the total number
of use cases the model can be utilized for–but guarantees strong performance
on use cases within the domain of the model. A single classification also
builds upon the pre-trained weights’ performance during transfer learning, as
it meant less skewing towards unrelated classifications.

###  5.2 Evaluation Results

####  5.2.1 Object Detection

The mAP values obtained from the object detection models on the validation
dataset have been expressed in Table I. Both models demonstrate high accuracy
in plastic localization. It also reveals that the YOLOv5-S model has a higher
mAP than the YOLOv4-Tiny, Faster RCNN and SSD models.

####  5.2.2 Inference Speed

These speeds were dictated by the GPU (NVIDIA P100 and V100 using a batch size
of 32) and included image preprocessing. The YOLOv4-S model provided the
highest inference speed-to-maP performance ratio for the provided dataset.

###  5.3 Qualitative Results

This study focused on determining the feasibility of detecting marine plastic
debris for near-real-time monitoring/quantification purposes. To that end, the
results in Table I demonstrate that general object detection models can fill
this much-needed role. Since a relatively high level of performance can be
maintained with such fast inference speeds–we believe that models such as the
one presented in this paper can be applied to AUVs and other tools for real-
world solutions. Equally important is that these solutions now have a near-
future timeline of implementation and have been proven to be low-cost.

##  6 Discussion

In this study, we built a computer vision model that detects marine plastic
debris with high precision, visualizes the detections with bounding boxes, and
operates near real-time speeds. These conditions match the requirements for
robotic platforms such as AUVs or buoys. As one of the first object detection
models specialized for the epipelagic layer, direct comparison results can not
be readily performed—however, relative performance comparisons.

DeepPlastic and object detection models geared towards plastic detection in
deep-sea and river plastic environments reveal DeepPlastics’ state-of-the-art
performances. The article mentioned above in Earth and Space Science [40]
describes a two-stage reference model, utilizing cameras positioned above
water, capable of detecting plastic floating on rivers. It utilized 1272
images in its training set and the Faster R-CNN architecture for its second
stage. Across multiple experiments, this model’s highest precision rate was
68% when employing image flipping and the ADAM adaptive learning rate.
DeepPlastic was trained on a dataset using image flipping, in addition to
other data augmentation techniques, and also uses the ADAM learning rate–but
achieves a precision rate of 93% when detecting marine plastic debris
submerged in the ocean via underwater cameras. The University of Minnesota’s
(UoM) computer vision model [15] specialized for marine plastic detection in
deep-sea environments utilized 5720 images in its training set and three
classes. The DeepTrash training dataset shares many of the same images as both
include samples from JAMSTEC [22]. The UoM model achieved an mAP of 82.3% for
its plastic images class using the YOLOv2 architecture and a high of 83.3%
when using Faster R-CNN. DeepPlastic achieves an mAP of 93% when using the
YOLOv4 architecture and input images of marine plastic debris on the same
dataset in similar conditions. AquaVision [17] was trained on three datasets
totaling 4400 images, including images of both land-based and marine-based
debris and four classes. AquaVision’s highest performance for the plastic
class was an average precision of 81.5% when using the one-stage RetinaNet
method. DeepPlastic performs at an mAP of 85% when using YOLOv5. The specific
training datasets used by the three models described above are either not
public or utilize datasets outside of the domain of DeepPlastic (i.e., the
dataset images are not underwater). Therefore comparing performances via
dataset is not an option for this study.

###  6.1 Points of Improvement

This model can efficiently monitor and quantify marine plastic. Improvements
can be made in the following areas:

####  6.1.1 Data Augmentation Improvements

While grayscale, saturation, and vertical/horizontal flipping have been proven
data augmentation techniques–emerging techniques such as AutoAugment [35]
could be explored to improve the model’s variability in the future once ready
for adaptation. Other methods such as shear and the cutout regularization
technique would be great to utilize after integration technologies improve.

####  6.1.2 Dataset Improvements

The data set used in this study is unique and one of the first of its kind.
For the data set, we see three main improvements that can be made to enhance
the deep learning model:

  * •

Adding more images from different locations

  * •

Using more images from variable water conditions

  * •

Finally, acquiring a more extensive set of underwater plastic images

As more plastic images from different locations and oceanic conditions become
available, they will increase marine plastic debris representation–providing a
more comprehensive dataset for model training. We believe this will improve
the mAP and overall robustness of the object detection model.

####  6.1.3 Camera Improvements

Readily available off-the-shelf cameras have come a long way but still suffer
from certain limitations. The first and most substantial limitation revolves
around most underwater cameras that will only work during the daytime. If we
want to continue the monitoring process during the nighttime, better night-
vision underwater sensors need to be developed. The second limitation stems
from the common H.265 video compression techniques [41] underwater cameras
utilize to induce encoding artifacts. This impedes real-time detection by
deteriorating the image quality. Developments in end-to-end deep learning
video compression techniques [41] could lead to solutions for this limitation
once ready for implementation.

##  7 Code and Dataset Availability

All code/dataset and instructions to build and utilize the DeepPlastic object
detection model can be found  online via Zenodo – DeepPlastic.

##  8 Conclusion

This work’s objective was to develop a deep learning vision model capable of
consistently identifying and quantifying marine plastic near real-time. To
attain this objective, a pair of general object detection models were
constructed using two state-of-the-art deep learning models built for
inference speed to measure which performed best.
This study concludes that a marine plastic debris detection system based on
the YOLOv5-S model would be fast, accurate, and robust enough to enable real-
time marine plastic debris detection. This study shows that effective object
detection models can be constructed using readily available, pre-enabled GPUs
for reasonable costs.
Furthermore, the dataset created for and utilized by this general detection
model demonstrates that massive, highly curated datasets can be used in
conjunction with samples relative to the domain of object detection and web
scraping to produce promising results.

This computer vision system enables multiple deployment methods to
detect/monitor marine plastic and allows researchers to quantify marine
plastic debris without physical removal.

##  9 Future Work

Improvement of the dataset would have the highest impact on performance but
collecting additional images would require human labor in fieldwork or
preprocessing. A technology capable of producing synthetic images containing
marine plastic debris in an ocean environment could provide an automated
solution to dataset creation. This could be accomplished with a two-stage
autoencoder [37]. Object detection models trained on identifying jellyfish (or
other objects similar to marine plastic debris) paired with a our object
detection model could lead to a decrease in false positives. Inference speed
could be improved through specialized GPU technology or tailoring models
towards specific higher power GPUs than used in this study. An end-to-end
video compression technique explicitly developed for near real-time object
detection could lead to a better ratio of true positives to true negatives and
an improved range on object detection. Tailoring this object detection model
for vision-equipped AUVs could result in automated identification and plastic
removal devices capable of scalable deployment across large bodies of water,
as shown in figure 1. Further optimizations could build in support for
stationary monitoring devices such as buoys as well. We hope that such a
system will facilitate scalable adoption by researchers and civilians to
detect and clean up marine plastic.

##  10 Acknowledgements

We gratefully acknowledge the help and support of Nikhil Deshmudre for his
efforts and help with the deployment of this computer vision system.

The authors would also like to thank Joseph Nelson, Co-Founder of
Roboflow.com, for providing us with Roboflow Pro free of charge, making it
easier to iterate on the deep learning models. Some of the images in this
dataset were sourced from the TrashCan dataset, where the researchers hand-
annotated and open-sourced over 5000 images from the JAMSTEC-JEDI dataset.

The authors would like to thank the researchers from the University of
Minnesota, Robotics LAB, and the Japan Agency for Marine-Earth Science and
Technology for open sourcing this data to contribute to the advancement of
science. Finally, we would like to thank Rae Rose Lowe for her support
throughout this process.

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Stability in the homology of configuration spaces of high dimensional
manifolds

by

Muhammad Yameen

###### Abstract

Consider the configuration spaces of manifold (closed or open). An influential
theorem of McDuff and Segal shows that the (co)homology of unordered
configuration spaces of open manifold is independent of number of
configuration points in a range of degree called the stable range. Church
prove that the homological stability is also present in the closed background
manifold. The Church’s stable range is only depend on the number of
configuration points. In this paper, we prove that if the cohomology of
manifold (not necessarily orientable) is concentrated in even degree than the
stable range for odd degree homology groups of configuration spaces is also
depend on the dimension of background manifold. Moreover, we prove that if the
second cohomology group of manifold is nontrivial then our stable range for
even degree homology groups is optimal, and this stable range is depend only
on the number of configuration points.

> Key Words: Configuration spaces, homological stability, optimal stable
> range, even cohomology manifold
>
>
> Mathematics Subject Classification: Primary 55R80, Secondary 55P62.

##  1 Introduction

For any manifold $M$, let

$F_{k}(M):=\\{(x_{1},\ldots,x_{k})\in M^{k}|x_{i}\neq x_{j}\,for\,i\neq j\\}$

be the configuration space of $k$ distinct ordered points in $M$ with induced
topology. The symmetric group $S_{k}$ acts on $F_{k}(M)$ by permuting the
coordinates. The quotient

$C_{k}(M):=F_{k}(M)/S_{k}$

is the unordered configuration space with quotient topology. The geometry and
topology of the configuration spaces have attracted an important attention,
over the years. Configuration spaces is an active area of research of geometry
with important applications in several areas including mathematical physics
and computer science.

An important aspect of the study of configuration spaces of manifold is the
phenomenon of homological stability. Arnold proved that integral cohomology
groups of configuration spaces of Euclidean plane are stabilized:

$H^{i}(C_{2i-2}(\mathbb{R}^{2});\mathbb{Z})\cong
H^{i}(C_{2i-1}(\mathbb{R}^{2});\mathbb{Z})\cong
H^{i}(C_{2i}(\mathbb{R}^{2});\mathbb{Z})\cong\ldots$

The isomorphisms (for $k$ large depending on $i$)

$H^{i}(C_{k}(M);\mathbb{Z})\cong H^{i}(C_{k+1}(M);\mathbb{Z})\cong
H^{i}(C_{k+2}(M);\mathbb{Z})\cong\ldots$

were generalized for open manifolds by D. McDuff [10] and G. Segal [12]. The
corresponding theorem is false for closed manifolds [11]; for example

$H_{1}(C_{k}(S^{2});\mathbb{Z})=\mathbb{Z}_{2k-2}.$

Using representation stability [4], Church [3] proved that

$H^{i}(C_{k}(M);\mathbb{Q})\cong H^{i}(C_{k+1}(M);\mathbb{Q})\cong
H^{i}(C_{k+2}(M);\mathbb{Q})\cong\ldots$

for $k>i$ and $M$ a connected oriented manifold of finite type. This result
was extended by Randal-Williams [14].

######  Theorem 1.

Let $M$ be a connected manifold of dimension bigger than 2. The cap product
with unit in $H^{0}(M;\mathbb{Q})$ induces a map

$H_{i}(C_{k+1}(M);\mathbb{Q})\rightarrow H_{i}(C_{k}(M);\mathbb{Q})$

that is an isomorphism for $i\leq k.$

We will study the stability properties of the rational (co)homology of
unordered configuration spaces of connected even cohomology manifolds of
higher dimension. We prove that if the cohomology of high dimensional manifold
is concentrate in even degree then the stable range for odd degree homology
groups is also depend on the dimension of manifold. We will restrict our
attention to the case of even dimensional manifolds, since there is a simple
closed form available for odd dimensional manifolds [2] (see last section for
further commentary). A manifold $M$ has even cohomology if all its odd
cohomology groups are trivial. Without a special mention, the (co)homology
will have coefficients in $\mathbb{Q}.$

######  Theorem 2.

Let $M$ be a connected even cohomology manifold of even dimension $d\geq 6.$
The cap product with unit in $H^{0}(M;\mathbb{Q})$ induces a map

$H_{i}(C_{k+1}(M);\mathbb{Q})\rightarrow H_{i}(C_{k}(M);\mathbb{Q})$

that is
$\bullet$ an isomorphism for $i\leq 2k$ when $i$ is even, and
$\bullet$ an isomorphism for $i\leq 2k+d-5$ when $i$ is odd.

There is no optimal stable range present in literature for configuration space
of high dimensional manifolds. We prove that if the second cohomology group of
manifold is non-trivial then our stable range for even degree homology group
is optimal.

######  Theorem 3.

Let $M$ be a connected even cohomology manifold of even dimension bigger than
5. If $M$ is not closed and $H^{2}(M;\mathbb{Q})$ is not trivial then the
stable range for even degree homology group in Theorem 2 is optimal.

######  Remark 1.

_Our stable range can be further improved if the low-degree Betti numbers of
manifold vanish._

##  2 Lie algebra homology

The proof of Church is depend on the representation stability and Totaro’s
spectral sequence [13]. However, we use the Chevalley–Eilenberg complex (for
more details see [7][9][6] [5][8]).

Let us introduce some notations. The symmetric algebra $Sym(W^{*})$ is the
tensor product of a polynomial algebra and an exterior algebra:

$Sym(W^{*})=\bigoplus_{k\geq 0}Sym^{k}(W^{*})=Poly(W^{even})\bigotimes
Ext(W^{odd}),$

where $Sym^{k}$ is generated by the monomials of length $k.$ The $n$-th
suspension of the graded vector space $W$ is the graded vector space $W[n]$
with $W[n]_{i}=W_{i-n},$ and the element of $W[n]$ corresponding to $a\in W$
is denoted $s^{n}a.$ We write $H_{-*}(M;\mathbb{Q})$ for the graded vector
space whose degree $-i$ part is the $i$-th homology group of $M.$

Now consider two graded vector spaces

$U^{*}=H^{-*}_{c}(M;\mathbb{Q}^{w})[d],\quad
V^{*}=H_{c}^{-*}(M;\mathbb{Q})[2d-1]:$

where

$U^{*}=\bigoplus_{i=0}^{d}U^{i},\quad V^{*}=\bigoplus_{j=d-1}^{2d-1}V^{j}.$

We choose bases in $U^{i}$ and $V^{j}$ as

$U^{i}=\mathbb{Q}\langle u_{i,1},u_{i,2},\ldots\rangle,\quad
V^{j}=\mathbb{Q}\langle v_{j,1},v_{j,2},\ldots\rangle$

(the degree of an element is marked by the first lower index; $x_{i}^{l}$
stands for the product $x_{i}\wedge x_{i}\wedge\ldots\wedge x_{i}$ of
$l$-factors). Always we take $U^{0}=\mathbb{Q}\langle u_{0}\rangle$. Now
consider the graded algebra

$\Theta^{*,*}_{k}(M)=\bigoplus_{i\geq
0}\bigoplus_{\omega=0}^{\left\lfloor\frac{k}{2}\right\rfloor}\Theta^{i,\omega}_{k}(M)=\bigoplus_{\omega=0}^{\left\lfloor\frac{k}{2}\right\rfloor}\,(Sym^{k-2\omega}(U^{*})\otimes
Sym^{\omega}(V^{*}))$

where the total degree $i$ is given by the grading of $U^{*}$ and $V^{*}$. We
called $\omega$ is a weight grading. The differential
$\partial:Sym^{2}(U^{*})\rightarrow V^{*}$ is defined as a coderivation by the
equation

$\partial(s^{d}a\wedge s^{d}b)=(-1)^{(d-1)|b|}s^{2d-1}(a\cup b),$

where

$\cup\,:H^{-*}_{c}(M;\mathbb{Q}^{w})^{\otimes 2}\rightarrow
H^{-*}_{c}(M;\mathbb{Q})$

(here $H^{-*}_{c}$ denotes compactly supported cohomology of $M$). The degree
of $\partial$ is $-1.$ The differential $\partial$ extends over
$\Theta^{*,*}_{k}(M)$ by co-Leibniz rule. By definition the elements in
$U^{*}$ have length 1 and weight 0 and the elements in $V^{*}$ have length 2
and weight 1. By definition of differential, we have

$\partial:\Theta^{*,*}_{k}(M)\longrightarrow\Theta^{*-1,*+1}_{k}(M).$

######  Theorem 4.

If $d$ is even, $H_{*}(C_{k}(M);\mathbb{Q})$ is isomorphic to the homology of
the complex

$(\Theta^{*,*}_{k}(M),\partial).$

##  3 Proofs of the main results

In this section, we give the proofs of main theorems.

Proof of Theorem 2. Let $d$ be even and $M$ is even cohomology manifold.

Case I. Assume $M$ is is not closed. We have decomposition of algebras:

$\Theta^{*,*}_{k+1}(M)\cong(\langle
u_{0}\rangle\otimes\Theta^{*,*}_{k}(M))\oplus\Theta^{*,*}_{k+1}(M)/\langle
u_{0}\rangle.$

The subspace $\Theta^{*,*}_{k+1}(M)/\langle u_{0}\rangle$ is
$\partial-$invariant:

$\partial(\Theta^{*,*}_{k+1}(M)/\langle
u_{0}\rangle)\subseteq\Theta^{*,*}_{k+1}(M)/\langle u_{0}\rangle.$

The degrees of elements of $U^{*}$ are less than $d.$ Also, the degree of
elements of $V^{*}$ are greater than and equal to $d-1.$ The differential
$\partial$ has degree $-1.$ Therefore, $\partial(\langle u_{0}\rangle\otimes
U^{*})=0.$ This implies that the subspace $\langle
u_{0}\rangle\otimes\Theta^{*,*}_{k}(M)$ is also $\partial-$invariant:

$\partial(\langle u_{0}\rangle\otimes\Theta^{*,*}_{k}(M))\subseteq\langle
u_{0}\rangle\otimes\Theta^{*,*}_{k}(M).$

As a consequence, we get the decomposition of complexes:

$(\Theta^{*,*}_{k+1}(M),\partial)\cong(\langle
u_{0}\rangle\otimes\Theta^{*,*}_{k}(M),\partial)\oplus(\Theta^{*,*}_{k+1}(M)/\langle
u_{0}\rangle,\partial).$

Moreover, we have an isomorphism :

$H_{*}(C_{k}(M);\mathbb{Q})\cong(\langle
u_{0}\rangle\otimes\Theta^{*,*}_{k}(M),\partial)$

Hence, we get an isomorphism :

$H_{*}(C_{k+1}(M);\mathbb{Q})\cong H_{*}(C_{k}(M);\mathbb{Q})\oplus
H_{*}(\Theta^{*,*}_{k+1}(M)/\langle u_{0}\rangle,\partial).$

Moreover, for each non-zero $u\in\Theta^{*,*}_{k+1}(M)/\langle u_{0}\rangle,$
we have

$\mbox{deg(u)}\geq\begin{cases}2k+2,&\mbox{ if deg(u) is even}\\\
2k+d-3,&\mbox{ if deg(u) is odd}.\\\ \end{cases}$

As a consequence, we conclude that the map

$H_{i}(C_{k+1}(M);\mathbb{Q})\rightarrow H_{i}(C_{k}(M);\mathbb{Q})$

is an isomorphism for $i\leq 2k$ when $i$ is even, and an isomorphism for
$i\leq 2k+d-5$ when $i$ is odd.

Case II. Assume $M$ is closed. The homology of $C_{k}(M)$ is stabilized in the
stable range of $H_{*}(C_{k}(M-\\{pt\\});\mathbb{Q})$ (see Theorem 3 of [14]).
This completes the proof. $\hfill\square$

Proof of Theorem 3. Let $d\geq 6$ be even and $M$ is even cohomology manifold
(but not closed). Let $\mbox{dim}H^{2}(M;\mathbb{Q})=n.$ Then the subspace
$U^{2}$ of $U^{*}$ is

$U^{2}=\langle u_{2,1},\ldots,u_{2,n}\rangle.$

The degree of elements of $V^{*}$ are greater than 4. The differential has bi-
degree $(-1,1).$ Therefore, $\partial(u_{2,i}\otimes u_{2,j})=0.$ Each
$u_{2,i_{1}}\ldots
u_{2,i_{k+1}}\in\text{I}^{k+1}\subseteq\Theta^{2k+2,0}/\langle u_{0}\rangle$
gives a cohomology class, where $\text{I}^{k+1}=\langle u_{2,i_{1}}\ldots
u_{2,i_{k+1}}|\,i_{j}\in\\{1,\ldots n\\}\rangle.$ The carnality of the bases
elements of $\text{I}^{k+1}$ is $\binom{(k+1)n-1}{n-1}.$ From the proof of
Theorem 1, we have decomposition

$H_{*}(C_{k+1}(M);\mathbb{Q})\cong H_{*}(C_{k}(M);\mathbb{Q})\oplus
H_{*}(\Theta^{*,*}_{k+1}(M)/\langle u_{0}\rangle,\partial).$

This implies that for each $i\geq 0,$ we have

$\mbox{dim}H_{i}(C_{k+1}(M);\mathbb{Q})\geq\mbox{dim}H_{i}(C_{k}(M);\mathbb{Q}).$

In particular, for $i=2k+2$ we have

$\mbox{dim}H_{2k+2}(C_{k+1}(M);\mathbb{Q})\geq\mbox{dim}H_{2k+2}(C_{k}(M);\mathbb{Q})+\binom{(k+1)n-1}{n-1}.$

This implies that for $n\neq 0$ we have

$\mbox{dim}H_{2k+2}(C_{k+1}(M);\mathbb{Q})>\mbox{dim}H_{2k+2}(C_{k}(M);\mathbb{Q}).$

As a consequence, we conclude that the homology groups
$H_{2k+2}(C_{k+1}(M);\mathbb{Q})$ and $H_{2k+2}(C_{k}(M)$ are not isomorphic.
$\hfill\square$

##  4 Odd dimensional case

We comment briefly on the case of odd dimensional manifolds, which is
encompassed implicitly by the results of [2]. We have an isomorphism [2]:

$H_{*}(C_{k}(M);\mathbb{Q})\cong Sym^{k}(H_{*}(M;\mathbb{Q})).$

We can write the decomposition of algebras:

$Sym^{k}(U^{*})\cong(\langle u_{0}\rangle\otimes Sym^{k-1}(U^{*}))\oplus
Sym^{k}(U^{*})/\langle u_{0}\rangle,$

here we take $U^{*}\cong H_{*}(M;\mathbb{Q})$ and $u_{0}$ is a corresponding
element of $H_{0}(M;\mathbb{Q}).$ Moreover, we have an isomorphism :

$H_{*}(C_{k-1}(M);\mathbb{Q})\cong\langle u_{0}\rangle\otimes
Sym^{k-1}(U^{*}).$

As a consequence, we get

$H_{*}(C_{k}(M);\mathbb{Q})\cong H_{*}(C_{k-1}(M);\mathbb{Q})\oplus
Sym^{k}(U^{*})/\langle u_{0}\rangle.$

This implies that for each $i\geq 0,$ the map

$k\rightarrow\emph{dim}H_{i}(C_{k}(M);\mathbb{Q})$

is eventually constant. If $M$ has even cohomology than the odd degree
homology groups of $C_{k}(M)$ are vanish.

Acknowledgement. The author gratefully acknowledge the support from the ASSMS,
GC university Lahore. This research is partially supported by Higher Education
Commission of Pakistan.

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Abdus Salam School of Mathematical Sciences,
GC University Lahore, Pakistan.
E-mail<EMAIL_ADDRESS>