# NanoNet: Real-Time Polyp Segmentation in Video Capsule Endoscopy and
Colonoscopy

Debesh Jha12, Nikhil Kumar Tomar1, Sharib Ali4, Michael A. Riegler1,
Håvard D. Johansen2, Dag Johansen2, Thomas de Lange5678, Pål Halvorsen13
1SimulaMet, Norway 2UiT The Arctic University of Norway, Norway 3Oslo
Metropolitan University, Norway 4Institute of Biomedical Engineering,
University of Oxford, Oxford, UK
5Department of Medical Research, Bærum Hospital, Norway 6Augere Medical AS,
Norway
7Medical Department, Sahlgrenska University Hospital-Mölndal Hospital, Sweden
8Department of Molecular and Clinical Medicine, Sahlgrenska Academy,
University of Gothenburg, Sweden

###### Abstract

Deep learning in gastrointestinal endoscopy can assist to improve clinical
performance and be helpful to assess lesions more accurately. To this extent,
semantic segmentation methods that can perform automated real-time delineation
of a region-of-interest, e.g., boundary identification of cancer or
precancerous lesions, can benefit both diagnosis and interventions. However,
accurate and real-time segmentation of endoscopic images is extremely
challenging due to its high operator dependence and high-definition image
quality. To utilize automated methods in clinical settings, it is crucial to
design lightweight models with low latency such that they can be integrated
with low-end endoscope hardware devices. In this work, we propose NanoNet, a
novel architecture for the segmentation of video capsule endoscopy and
colonoscopy images. Our proposed architecture allows real-time performance and
has higher segmentation accuracy compared to other more complex ones. We use
video capsule endoscopy and standard colonoscopy datasets with polyps, and a
dataset consisting of endoscopy biopsies and surgical instruments, to evaluate
the effectiveness of our approach. Our experiments demonstrate the increased
performance of our architecture in terms of a trade-off between model
complexity, speed, model parameters, and metric performances. Moreover, the
resulting models´ size is relatively tiny, with only nearly 36,000 parameters
compared to traditional deep learning approaches having millions of
parameters.

###### Index Terms:

Video capsule endoscopy, colonoscopy, deep learning, segmentation, tool
segmentation

##  I Introduction

Gastrointestinal (GI) endoscopy is a widely used technique to diagnose and
treat anomalies in the upper (esophagus, stomach, and duodenum) and the lower
(large bowel and anus) GI tract. Among the other GI tract organs, colorectal
cancer (CRC) has the highest cancer incidences and mortality rate [1]. There
are several CRC screening options. Theses are usually divided into two
categories, namely, invasive (visual examination-based test) and non-invasive
based tests (stool, blood, and radiological test). _Colonoscopy_ , the gold
standard for examining the large bowel (colon and rectum), is an invasive
examination used to detect, observe, and remove abnormalities (such as
polyps). It detects colorectal cancer with both high sensitivity and
specificity. _Sigmoidscopy_ is another invasive test. _Computed Tomography(CT)
Colonoscopy_ , _Fecal Occult Blood Test (FOBT) Fecal Immunochemical Test
(FIT)_ , and Video Capsule Endoscopy (VCE) are non-invasive tests. VCE is a
technology for capturing the video inside the GI tract. It has evolved as an
important tool for detecting small bowel diseases [2].

Deep Learning (DL) methods have made a significant breakthrough in several
medical domain such as lung cancer detection [3], diabetic retinopathy
progression [4], and obstructive hypertrophic cardiomyopathy detection [5]. It
has provided new opportunities to solve challenges such as bleeding, light
over/underexposure, smoke, and reflections [6]. However, DL normally needs a
large annotated dataset for the implementation of methods. It is difficult to
obtain a labeled medical dataset. First, it needs collaborations with the
hospitals. For data collection, the doctors require approval from various
authorities and patient consent. They need to set protocols for the
collection, and the collected data must be anonymized and cleaned with the
help of data engineers. Domain experts must label raw data, and after
labeling, the annotations must be done depending upon the need of the task.
The whole process requires an significant amount of expert time and is costly.
Additionally, it is an operator-dependent process. The quality of the data
labeling and annotation depends on the expertise of the clinicians. Therefore,
it is challenging to curate a larger dataset.

One way of solving the dataset issue is to create synthetic images using
Genearative Adversarial Network (GAN) [7]. However, generated synthetic images
may not always capture all the properties and characteristics of real
endoscopic images. Consequently, the model may only learn to predict the
properties from the synthetic images and may not perform well on a real
endoscopic dataset. Another solution could be domain adaptation from a similar
endoscopic dataset. However, we lack large publicly available labeled
endoscopic datasets. Thus, a viable and compelling approach to solve the
semantic segmentation task is to reuse ImageNet pre-trained encoders in the
segmentation model [8]. The predicted masks from the algorithm can provide
reliable information to the endoscopic model.

A lightweight Convolutional Neural Network (CNN) model can be essential for
the development of real-time and efficient semantic segmentation methods.
Usually, lightweight models are computationally efficient and require less
memory. A smaller number of parameters makes the network less redundant.
Lightweight CNN models are mainly being deployed in mobile applications [9]. A
lightweight model can play a crucial role from a system perspective with a
limited resource constraint for real-time prediction in clinics. Consequently,
we propose a novel architecture, NanoNet, optimized for faster inference and
high accuracy. An extremely lightweight model with very few trainable
parameters, faster inference, and higher performance would require less memory
footprint to be incorporated with any devices. Therefore, we put forward this
approach to address the challenges in endoscopy.

The main contributions of this work include the following:

  1. 1.

We proposed a novel architecture, named NanoNet, to segment video capsule
endoscopy and colonoscopy images in real-time with high accuracy. The proposed
architecture is very lightweight, and the model size is smaller, requiring
less computational cost.

  2. 2.

VCE datasets are difficult to obtain with pixel-wise annotations. In this
context, we have annotated 55 polyps from the “polyp” class of the Kvasir-
Capsule dataset with the help of an expert gastroenterologist. We have made
this dataset public and provided the benchmark.

  3. 3.

NanoNet achieves promising performance on the KvasirCapsule-SEG, Kvasir-SEG
[10], 2020 Medico automatic polyp segmentation challenge [11], 2020 EndoTect
challenge [12], and Kvasir-Instrument [13] datasets. All experiments conform
with state-of-the-art (SOTA) in terms of parameter uses (size), speed,
computation, and performance metrics.

  4. 4.

The model can be integrated with mobile and embedded devices because of fewer
parameters used in the network.

##  II Related work

###  II-A Semantic segmentation of endoscopic images

Semantic segmentation of endoscopic images has been a well-established topic
in medical image segmentation. Earlier work mostly relied on the handcrafted
descriptors for feature learning [14, 15]. The handcrafted features such as
color, shape, texture, and edges were extracted and fed to the Machine
Learning (ML) classifier, which separates lesions from the background.
However, the traditional ML methods based on handcrafted features suffer from
low performance [16]. The recent works on polyp segmentation using both video
capsule endoscopy and colonoscopy mostly relied on Deep Neural Network (DNN)
[17, 18, 19, 20, 21, 22, 23].

With the DNN methods, there is progress in the performance for segmenting
endoscopic images (for example, polyps). However, the network architectures
are often complex and requires high-end GPUs for training, and is
computationally expensive [24, 25, 22]. Additionally, real-time lesion
segmentation has often been ignored. Although there is some recent initiation
for the real-time detection of endoscopic images, they have mostly used
private datasets [26, 27, 28] for the experimentation. It is difficult to
compare the new methods on these datasets and extend the benchmark. Therefore,
there is a need for a benchmark on publicly available datasets to minimize the
research gap towards building a clinically relevant model.

###  II-B Lightweight model

There are few works in the literature that have proposed lightweight models
for image segmentation. Ni et al. [29] presented a novel bilinear attention
network-based approach with an adaptive receptive field for the segmentation
of surgical instruments. Wang et al. [30] proposed a lightweight encoder-
decoder network (LEDNet), an encoder-decoder network that uses ResNet50 in the
encoder block and attention pyramidal network in the decoder block. Beheshti
et al. [31] proposed SqueezeNet. The architecture of the SqueezeNet is
inspired by UNet [32]. The proposed model obtained a 12$\times$ reduction in
model size and showed efficient performance in multiplication accumulation
(mac) and memory uses.

From the above-related work, we identify a need for a real-time polyp
segmentation method. A real-time polyp segmentation method can be achieved by
building a lightweight network architecture by designing an efficient network
with blocks that require fewer parameters. A lower number of network
parameters will reduce the network complexity, leading to real-time or faster
inference. In this respect, we propose NanoNet, which uses a lightweight pre-
trained network MobileNetV2 [33], and simple convolutional blocks such as
residual block and squeeze and excite block.

Figure 1: Overview of the proposed NanoNet architecture

##  III Network architecture

The architecture of NanoNet follows an encoder-decoder approach as shown in
Figure 1. As depicted in Figure 1, the network architecture uses a pre-trained
model as an encoder, followed by the three decoder blocks. Using pre-trained
ImageNet [34] models for transfer learning has become the best choice for many
CNN architectures [25, 8]. It helps the model converge much faster and
achieves high performance compared to the non-pre-trained model. The proposed
architecture uses a MobileNetV2 [33] model pre-trained on the ImageNet [34]
dataset as the encoder. The decoder is built using a modified version of the
residual block, which was initially introduced by He et al. [35]. The encoder
is used to capture the required contextual information from the input, whereas
the decoder is used to generate the final output by using the contextual
information extracted by the encoder.

###  III-A MobileNetV2

The MobileNetV2 [33] is an architecture that is primarily designed for mobile
and embedded devices. The architecture performed well on a variety of
different datasets while maintaining high accuracy, despite having fewer
parameters. The architecture of MobileNetV2 is based on the architecture of
MobileNetV1, which uses depth-wise separable convolutions as the main building
block. A depth-wise separable convolution consists of depth-wise convolution
followed by a point-wise convolution. The MobileNetV2 introduces two main
ideas: inverted residual block and linear bottleneck block [33].

The inverted residual block is based on the bottleneck residual block as
described in the [35], which consists of three standard convolutions, which
are $1\times 1$, $3\times 3$, and $1\times 1$. Every convolution layer is
followed by a Rectified Linear Unit (ReLU) non-linearity. In the first
$1\times 1$ standard convolution, the number of feature channels are reduced,
and in the last $1\times 1$ standard convolution, the number of feature
channels are expanded. After that, an element-wise addition with the identity
mapping is performed. The inverted residual block also has three convolution
layers: a $1\times 1$ standard convolution, a $3\times 3$ depth-wise
convolution, and a $1\times 1$ standard convolution. Every convolution has a
ReLU activation function. Here, the exact opposite of the bottleneck residual
block is performed. The first $1\times 1$ standard convolution expands the
number of feature channels, and the last $1\times 1$ standard convolution
reduces the number of feature channels. Due to this opposite functionality, it
is referred to as an inverted residual block. The linear bottleneck block is
the same as the inverted residual block, except the last $1\times 1$ standard
convolution has a linear activation before an element-wise addition is
performed with the identity mapping.

###  III-B Modified Residual Block

The original residual block uses two $3\times 3$ standard convolutions, where
the first convolution is followed by a batch-normalization and a ReLU
activation function. After that, the second convolution is followed only by a
batch-normalization. An element-wise addition is performed between the output
of the batch-normalization and the identity mapping, followed by another ReLU
activation function. An identity mapping consists of a $1\times 1$ standard
convolution and a batch-normalization over the original input.

We have modified the residual block for our network. The modified residual
block starts with a $1\times 1$ convolution followed by a $3\times 3$
convolution. In both of these convolutions, we reduce the number of filters by
$\frac{1}{4}$, which are then followed by the batch normalization and the ReLU
activation function. We have a $3\times 3$ convolution with batch
normalization. Now, we perform an element-wise addition with the identity
mapping. Finally, we apply a ReLU activation function followed by the squeeze
and excitation block. The squeeze and excitation block improves the quality of
feature maps by increasing their sensitivity towards essential features.

TABLE I: Publicly available endoscopic datasets used in our experiments Dataset |  No. of Images |  Imaging Type |  Availability
---|---|---|---
KvasirCapsule-SEG |  55 |  Video capsule endoscopy |  https://www.dropbox.com/sh/hr46vieykbmvmkk/AAAs_V8ECG0wq51Fpw3rYU_5a?dl=0
Kvasir-SEG [10] |  1000 |  Colonoscopy |  https://datasets.simula.no/kvasir-seg/
2020 Medico automatic polyp segmentation challenge [11] |  160⋄ |  Colonoscopy |  https://multimediaeval.github.io/editions/2020/tasks/medico/
Endotect Challenge Dataset [12] |  200⋄ |  Colonoscopy |  https://endotect.com/
Kvasir-Instrument [36] |  590 |  Colonoscopy |  https://datasets.simula.no/kvasir-instrument/
⋄test images |

###  III-C The NanoNet architecture

Figure 1 shows the block diagram of the NanoNet architecture. The NanoNet
architecture starts with a pre-trained MobileNetV2 as an encoder followed by a
decoder. There is a modified residual block between the encoder and the
decoder, which acts like a bridge that connects the encoder and the decoder.
In the first step, we feed the image data into the pre-trained encoder. The
pre-trained encoder starts with a standard convolution with 32 feature
channels, followed by the bottleneck layer with ReLU6 as the activation
function. All the convolution operations use a standard $3\times 3$ kernel
size. The entire encoder network progressively downsamples the feature maps by
using strided convolution and slowly increases the number of feature channels
alternatively.

The output from the pre-trained encoder passes through the modified residual
block, which is fed to the decoder. Every step in the decoder uses a bilinear
upsampling to increase the spatial dimension (height and width) of the input
feature maps. After that, it is concatenated with the appropriate feature maps
from the pre-trained encoder using the skip connections. These skip
connections pass information that may be lost sometimes between the layers and
are used to improve the quality of the feature maps. These concatenated
feature maps are passed through the modified residual block, which further
increases the generalization capacity of the decoder. After the feature maps
pass through all the three decoder block, the output of the last decoder block
is fed to a $1\times 1$ convolution with a number of classes as the feature
channels. This is followed by the sigmoid activation if it is a binary
segmentation task, else we use the softmax activation function.

We have demonstrated three different NanoNet architectures: NanoNet-A,
NanoNet-B, and NanoNet-C. Each architecture consists of different feature
channels in its decoder block. NanoNet-A consists of $32$, $64$ and $128$
feature channels. In NanoNet-B, the number of feature channels is reduced to
$32$, $64$, and $96$. In NanoNet-C, these feature channels are further reduced
to $16$, $24$, and $32$. The reduction in the number of feature channels leads
to less trainable parameters, which simplifies the model complexity leading to
a light-weight network.

##  IV Experimental setup

In this section, we will describe the dataset, evaluation metrics,
implementation details, and data augmentation techniques used.

Figure 2: Polyps and corresponding masks from KvasirCapsule-SEG

###  IV-A Datasets

To address the polyp segmentation problem from video capsule endoscopy images,
we have selected the polyp class from labelled images folder of the Kvasir-
Capsule dataset [37] and annotated it with the help of an expert
gastroenterologist. The Kvasir-Capsule is an open-access dataset that contains
13 classes of labelled anomalies and findings. It only includes 55 polyp
frames out of 44,228 medically verified video capsule frames present in the
Kvasir-Capsule. We have annotated the polyp class of Kvasir-Capsule and
generated corresponding ground truth masks. Examples of polyps and their
corresponding masks from KvasirCapsule-SEG can be found in Figure 2.
Furthermore, we also provide bounding box information to be used for video
capsule endoscopy detection and localization tasks. The Kvasir-Capsule can be
downloaded from here 111https://osf.io/dv2ag/ and KvasirCapsule-SEG can be
downloaded from here
222https://www.dropbox.com/sh/hr46vieykbmvmkk/AAAs_V8ECG0wq51Fpw3rYU_5a?dl=0.

Table I shows the detailed information about the open imaging dataset used in
our experiments. Each of the datasets presented in Table I also has the
corresponding ground truth. The link for each of the datasets is provided in
the table. The standard setting for the “Medico automatic polyp segmentation
challenge” and “Endotect challenge” is that they use the Kvasir-SEG for
training. The challenge organizers have provided unseen 160 images in the
“Medico automatic polyp segmentation challenge” and released 200 images in the
“Endotect challenge” to test the participant’s approaches. For the Kvasir-
instrument dataset, we experimented with the official split provided by the
organizers. The detail explanation of these datasets and the baseline results
can be found in [10, 13, 11, 12].

###  IV-B Evaluation metrics

For the evaluation of our model, we have chosen standard computer vision
metrics such as Dice Coefficient (DSC), mean Intersection over Union (mIoU),
Precision, Recall, Specificity, Accuracy, and Frame-per-second (FPS). More
explanation of these metrics can be found in [10, 13, 11, 12].

TABLE II: Performance evaluation of the proposed networks and recent SOTA methods on KvasirCapsule-SEG Method | Parameters | DSC | mIoU | Recall | Precision | F2 | Accuracy | FPS
---|---|---|---|---|---|---|---|---
ResUNet (GRSL’18) [38] | 8,227,393 | 0.9532 | 0.9137 | 0.9785 | 0.9325 | 0.9677 | 0.9386 | 17.96
ResUNet++ (ISM’19)[24] | 4,070,385 | 0.9499 | 0.9087 | 0.9762 | 0.9296 | 0.9648 | 0.9334 | 15.39
NanoNet-A (Ours) | 235,425 | 0.9493 | 0.9059 | 0.9693 | 0.9325 | 0.9609 | 0.9351 | 28.35
NanoNet-B (Ours) | 132,049 | 0.9474 | 0.9028 | 0.9682 | 0.9308 | 0.9593 | 0.9324 | 27.39
NanoNet-C (Ours) | 36,561 | 0.9465 | 0.9021 | 0.9754 | 0.9238 | 0.9629 | 0.9297 | 29.48
TABLE III: Performance evaluation of the proposed networks and recent SOTA methods on Kvasir-SEG[10] Method | Parameters | DSC | mIoU | Recall | Precision | F2 | Accuracy | FPS
---|---|---|---|---|---|---|---|---
ResUNet (GRSL’18) [38] | 8,227,393 | 0.7203 | 0.6106 | 0.7602 | 0.7624 | 0.7327 | 0.9251 | 17.72
ResUNet++ (ISM’19) [24] | 4,070,385 | 0.7310 | 0.6363 | 0.7925 | 0.7932 | 0.7478 | 0.9223 | 19.79
NanoNet-A (Ours) | 235,425 | 0.8227 | 0.7282 | 0.8588 | 0.8367 | 0.8354 | 0.9456 | 26.13
NanoNet-B (Ours) | 132,049 | 0.7860 | 0.6799 | 0.8392 | 0.8004 | 0.8067 | 0.9365 | 29.73
NanoNet-C (Ours) | 36,561 | 0.7494 | 0.6360 | 0.8081 | 0.7738 | 0.7719 | 0.9290 | 32.17
TABLE IV: Performance evaluation of the proposed networks and recent SOTA methods on the Medico 2020 dataset [11] Method | Parameters | DSC | mIoU | Recall | Precision | F2 | Accuracy | FPS
---|---|---|---|---|---|---|---|---
ResUNet (GRSL’18) [38] | 8,227,393 | 0.6846 | 0.5599 | 0.7235 | 0.7236 | 0.6961 | 0.9231 | 18.54
ResUNet++ (ISM’19) [24] | 4,070,385 | 0.6925 | 0.5849 | 0.8249 | 0.6840 | 0.7434 | 0.8995 | 19.47
NanoNet-A (Ours) | 235,425 | 0.7364 | 0.6319 | 0.8566 | 0.7310 | 0.7804 | 0.9166 | 28.07
NanoNet-B (Ours) | 132,049 | 0.7378 | 0.6247 | 0.8283 | 0.7373 | 0.7685 | 0.9223 | 29.04
NanoNet-C (Ours) | 36,651 | 0.7070 | 0.5866 | 0.8095 | 0.7089 | 0.7432 | 0.9148 | 32.66
TABLE V: Performance evaluation of the proposed networks and recent SOTA methods on the Endotect 2020 dataset [12] Method | Parameters | DSC | mIoU | Recall | Precision | F2 | Accuracy | FPS
---|---|---|---|---|---|---|---|---
ResUNet (GRSL’18) [39] | 8,227,393 | 0.6640 | 0.5408 | 0.7510 | 0.6841 | 0.6943 | 0.9075 | 26.55
ResUNet++ (ISM’19) [24] | 4,070,385 | 0.6940 | 0.5838 | 0.8797 | 0.6591 | 0.7597 | 0.8841 | 18.58
NanoNet-A (Ours) | 235,425 | 0.7508 | 0.6466 | 0.8238 | 0.7744 | 0.7773 | 0.9255 | 27.19
NanoNet-B (Ours) | 132,049 | 0.7362 | 0.6238 | 0.8109 | 0.7532 | 0.7646 | 0.9252 | 29.91
NanoNet-C (Ours) | 36,651 | 0.7001 | 0.5792 | 0.8000 | 0.7159 | 0.7380 | 0.9091 | 32.98
TABLE VI: Performance evaluation of the proposed networks and recent SOTA methods on Kvasir-Instrument [13] Method | Parameters | DSC | mIoU | Recall | Precision | F2 | Accuracy | FPS
---|---|---|---|---|---|---|---|---
UNet (Baseline) [39] | - | 0.9158 | 0.8578 | 0.9487 | 0.8998 | 0.9320 | 0.9864 | 20.46
DoubleUNet (Baseline) [25] | - | 0.9038 | 0.8430 | 0.9275 | 0.8966 | 0.9147 | 0.9838 | 10.00
ResUNet++ (ISM’19) [24] | 4,070,385 | 0.9140 | 0.8635 | 0.9103 | 0.9348 | 0.9140 | 0.9866 | 17.87
NanoNet-A (Ours) | 235,425 | 0.9251 | 0.8768 | 0.9142 | 0.9540 | 0.9251 | 0.9887 | 28.00
NanoNet-B (Ours) | 132,049 | 0.9284 | 0.8790 | 0.9205 | 0.9482 | 0.9284 | 0.9875 | 29.82
NanoNet-C (Ours) | 36,561 | 0.9139 | 0.8600 | 0.9037 | 0.9452 | 0.9139 | 0.9863 | 32.18

###  IV-C Implementation details

We have implemented the NanoNet using Keras333https://keras.io/ with
TensorFlow [40] as backend. The experiments were run on the Experimental
Infrastructure for Exploration of Exascale Computing (eX3), NVIDIA DGX-2
machine. The code implementation of NanoNet can be found
here444https://github.com/DebeshJha/NanoNet. As the model has very few low
trainable parameters, we have set a batch size of 16. We have resized the
dataset images to $256\times 256$ pixels for better utilization of the GPU,
and it also helps to reduce the training time. The model is trained on 200
epochs with the Nadam optimizer [41] and dice coefficient as the loss
function. The learning rate for the optimizer is set to 1e-4. We prefer to
choose a low learning rate to update the parameters slowly and carefully. The
learning rate is reduced by a factor of 0.1 when the validation loss does not
decrease in $10$ consecutive epochs. It helps to improve model performance.
Additionally, we have used an early stopping mechanism to prevent over-
fitting.

###  IV-D Data augmentation

We use data-augmentation on the training set to increase diversity and to
improve the generalization of our model. Data augmentation techniques such as
random cropping, random rotation, horizontal flipping, vertical flipping, grid
distortion, and many more are used. We have used an offline data augmentation
technique. The validation and testing set is not augmented and is directly
resized into $256\times 256$.

Figure 3: Qualitative results of NanoNet-A on five different datasets

##  V Result and Discussion

In this section, we provide the experimental results for the segmentation task
of the endoscopic image dataset. For the evaluation, we have used performance
metrics such as DSC and mIoU, and FPS as the main evaluation metrics. We also
calculate recall, precision, F2, and overall accuracy to support a complete
set of metrics. Table II, Table III, Table IV, Table V, and Table VI show the
results of the NanoNet model experiments using different parameters. The
results are compared with the recent SOTA computer vision methods.

The quantitative results in these tables show that NanoNet consistently
outperforms or performs nearly equal to its competitors in terms of
performance. The quantitative results also show that NanoNet can produce real-
time segmentation (i.e., produces at least close to 30 FPS for each dataset
present in the Tables). This is one of the major contributions of the work.
The other strength of the work lies in the parameter use. From Table II, we
can observe that the best performing NanoNet (i.e., NanoNet-A) uses nearly 35
times less parameters as ResUNet [38]. Similarly, NanoNet-C uses 225 times
less parameters as compared to that of ResUNet and also produces better DSC,
mIoU and FPS with the Kvasir-SEG.

The qualitative results are displayed in Figure 3. The first, second, and
third columns show the image, ground truth, and prediction masks,
respectively. Similarly, the name of the dataset is provided on the left side.
One example image for each dataset is shown. The qualitative results with
diversified classes of medical datasets show that NanoNet can produce accurate
segmentation results with different types of lesions (polyps) and therapeutic
tools. The example images and the prediction also show that NanoNet produces
good segmentation masks for large, medium, and small polyps (see Figure 3).
From the qualitative results, we can derive and conclude that NanoNet produces
good results with small-sized polyps but produces over-segmentation for the
large-sized lesions upon detail dissection. For future work, one could create
a specific dataset consisting of a set of small and large-sized polyps to
explore this further.

From both evaluation metrics and qualitative results, the improvement is
remarkable. Thus, the proposed NanoNet architecture is simple, compact, and
provides a robust solution for real-time applications, as it produces
satisfactory performance despite having fewer parameters.

##  VI Conclusion

In this paper, we proposed a novel lightweight architecture for real-time
video capsule endoscopy and colonoscopy image segmentation. The proposed
NanoNet architecture utilizes a pre-trained MobileNetV2 model and a modified
residual block. The depthwise separable convolution is the main building block
of the network and allows the model to achieve high performance with minuscule
trainable parameters. The experimental results on varied endoscopy datasets
demonstrate the strength of our model compared to state-of-the-art models with
respect to their speed and performance. The presented model has the potential
to enable easier roll out of deep learning models in clinical systems due to
fewer parameters, competitive accuracy, and low-latency. In addition, the
model does not require any sort of initialization, post-processing, or
temporal regularization, considered as another strength of this work. In the
future, we will design an encoder lighter than the currently used pre-trained
MobilNetV2. Moreover, we aspire to utilize the currently built segmentation
module in the clinic and study the efficacy of our designed model.

## Acknowledgment

The research is partially funded by the PRIVATON project (263248) and the
Autocap project (282315) from the Research Council of Norway (RCN). Our
experiments were performed on the Experimental Infrastructure for Exploration
of Exascale Computing (eX3) system, which is financially supported by RCN
under contract 270053.

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  *[GI]: Gastrointestinal
  *[CRC]: colorectal cancer
  *[VCE]: Video Capsule Endoscopy
  *[DL]: Deep Learning
  *[GAN]: Genearative Adversarial Network
  *[CNN]: Convolutional Neural Network
  *[SOTA]: state-of-the-art
  *[ML]: Machine Learning
  *[DNN]: Deep Neural Network
  *[ReLU]: Rectified Linear Unit
  *[DSC]: Dice Coefficient
  *[mIoU]: mean Intersection over Union
  *[FPS]: Frame-per-second


# Combinatorial Calabi flows on surfaces with boundary

Yanwen Luo, Xu Xu

###### Abstract

Motivated by Luo’s combinatorial Yamabe flow on closed surfaces [17] and Guo’s
combinatorial Yamabe flow on surfaces with boundary [14], we introduce
combinatorial Calabi flow on ideally triangulated surfaces with boundary,
aiming at finding hyperbolic metrics on surfaces with totally geodesic
boundaries of given lengths. Then we prove the long time existence and global
convergence of combinatorial Calabi flow on surfaces with boundary. We further
introduce fractional combinatorial Calabi flow on surfaces with boundary,
which unifies and generalizes the combinatorial Yamabe flow and the
combinatorial Calabi flow on surfaces with boundary. The long time existence
and global convergence of fractional combinatorial Calabi flow are also
proved. These combinatorial curvature flows provide effective algorithms to
construct hyperbolic surfaces with totally geodesic boundaries with prescribed
lengths.

MSC (2020): 52C26

Keywords:  Vertex scaling; Surfaces with boundary; Combinatorial Yamabe flow;
Combinatorial Calabi flow; Fractional combinatorial Calabi flow

##  1 Introduction

The resolution of the Poincaré Conjecture by Ricci flow marks the culmination
of the research of curvature flows on manifolds. It led to a flurry of study
on various types of geometric flows to deform Riemannian metrics on a
manifold. The purpose of this paper is to extend this study to surfaces with
boundary in the combinatorial setting. We introduce combinatorial Calabi flows
on hyperbolic surfaces with boundary equipped with ideal triangulations, which
deform a hyperbolic metric on a surface with boundary within the discrete
conformal class of the metric. We prove that the combinatorial Calabi flows
converge to hyperbolic surfaces with totally geodesic boundary, whose lengths
can be arbitrarily prescribed.

The notion of discrete conformal structures on triangulated surfaces is a
discrete analogue of their smooth counterparts, conformal structures on smooth
surfaces. One motivation to study discrete conformal structures is to compute
the conformal maps between planar regions in applications. Thurston [23]
rediscovered the circle packing theorem initiated by Koebe [16] and Andreev
[1, 2], and proposed circle packings as a natural discretization of conformal
maps. This idea was carried out by Rodin-Sullivan [21]. Since then, different
types of discrete conformal structures on closed surfaces have been
extensively studied in the last two decades, including tangential circle
packings, Thurston’s circle packings, inversive distance circle packings,
Luo’s vertex scaling, mixed types of discrete conformal structures, etc. See,
for instance, [4, 5, 10, 11, 15, 17, 18, 25, 27, 29] and others. To find
circle packings with prescribed combinatorial curvatures on closed
triangulated surfaces, Chow-Luo [5] introduced the combinatorial Ricci flow
for Thurston’s circle packing metrics on closed surfaces. Luo [17] further
introduced a new type of discrete conformal structure called vertex scaling
for piecewise flat metrics on closed surfaces, and the corresponding
combinatorial Yamabe flow deforming discrete metrics within a discrete
conformal class. This new notion of discrete conformal structure leads to
rigidity results of polyhedral surfaces with respect to discrete curvature [3]
and discrete uniformization theorems on closed polyhedral surfaces [12, 13,
22]. These works give rise to tons of new results on combinatorial curvature
flows and discrete conformal structures on closed surfaces, and lead to
various applications in surface matching, surface parametrization, manifold
spline and others. See [30] for a comprehensive survey on this topic.

In contrast to the case of closed surfaces, there are only sporadic results on
discrete conformal structures and discrete curvature flows on surfaces with
boundary. The first notion of discrete conformal structure, also called vertex
scaling, was introduced by Guo in [14] for hyperbolic metrics on ideally
triangulated surfaces with boundary. Guo [14] further introduced the
corresponding combinatorial Yamabe flow and studied its properties. See also
[28]. In this paper, we introduce combinatorial Calabi flows for Guo’s vertex
scaling and study their long time behavior.

###  1.1 Set up

Suppose $\Sigma$ is a compact surface with boundary $B$ consisting of $n$
connected components, which are topologically circles. An ideal triangulation
$\mathcal{T}$ of $\Sigma$ could be constructed using topological hexagons as
follows. Suppose we have a disjoint union of finite colored hexagons such that
three non-adjacent edges of each hexagon are colored red and the other three
edges colored black. Then we identify the red edges of the colored hexagons in
pairs by homeomorphisms. The quotient space of these colored hexagons is an
ideally triangulated compact surface with boundary. For simplicity, we define
a face in the ideal triangulation $\mathcal{T}$ as the image of one hexagon
under the quotient map. Similarly, an edge in $\mathcal{T}$ is the image of
one red edge in the hexagons under the quotient map, and a boundary component
is the quotient of a sequence of black edges glued together at their
endpoints. We will denote the connected components of the boundary $B$ as
$\\{1,2,\cdots,n\\}$, and the set of edges and faces as $E$ and $F$
respectively.

A well-known fact in hyperbolic geometry is that for any three positive
numbers, there exists a unique hyperbolic righted-angled hexagon up to
isometry, where the lengths of three non-adjacent edges are given by the three
positive numbers [20]. If each edge of an ideal triangulation $\mathcal{T}$ of
a compact surface with boundary $\Sigma$ is assigned with a positive number,
then each face in $\mathcal{T}$ can be realized as a hyperbolic right-angled
hexagon. Gluing these geometric hexagons along the edges in pairs by
hyperbolic isometries, we can construct a hyperbolic surface with totally
geodesic boundary $\Sigma$ from the ideal triangulation $\mathcal{T}$.
Conversely, any ideally triangulated hyperbolic surface with totally geodesic
boundary $(\Sigma,\mathcal{T})$ produces a function
$l:E\rightarrow(0,+\infty)$ sending each edge to the length of the unique
geodesic orthogonal to boundary geodesics in its homotopy class. The function
$l:E\rightarrow(0,+\infty)$ is called a _discrete hyperbolic metric_ on
$(\Sigma,\mathcal{T})$. Moreover, the lengths of its boundary components
define a function $K:B\to(0+\infty)$, called the _generalized combinatorial
curvature_.

Motivated by Luo’s definition of vertex scaling of piecewise flat metrics on
triangulated closed surfaces [17], Guo [14] introduced the following
definition of discrete conformality for discrete hyperbolic metrics on ideally
triangulated surfaces with boundary, called vertex scaling as well.

######  Definition 1 (Guo [14]).

Suppose $(\Sigma,\mathcal{T})$ is an ideally triangulated surface with
boundary. Let $l$ and $\tilde{l}$ be two discrete hyperbolic metrics on
$(\Sigma,\mathcal{T})$. If there exists a function $w:B\rightarrow\mathbb{R}$
such that

$\cosh\frac{l_{ij}}{2}=e^{w_{i}+w_{j}}\cosh\frac{\tilde{l}_{ij}}{2},$ (1.1)

then the discrete hyperbolic metric $l$ is called vertex scaling of
$\tilde{l}$. The function $w:B\rightarrow\mathbb{R}$ is called a discrete
conformal factor.

For a fixed discrete hyperbolic metric $\tilde{l}$, a discrete conformal
factor $w:B\rightarrow\mathbb{R}$ is admissible if (1.1) defines a discrete
hyperbolic metric $l:E\rightarrow(0,+\infty)$. The space of admissible
discrete conformal factors of a discrete hyperbolic metric $\tilde{l}$ is
denoted by $\mathcal{W}=\mathcal{W}(\tilde{l})$. The discrete hyperbolic
metric $l$ in (1.1) will be referred as $w*\tilde{l}$ for simplicity.

Guo [14] further introduced the following combinatorial Yamabe flow

$\displaystyle\left\\{\begin{array}[]{ll}\frac{dw_{i}(t)}{dt}=K_{i}(t),&\hbox{
}\\\ w_{i}(0)=0,&\hbox{ }\end{array}\right.$ (1.2)

and proved its long time existence and global convergence to a cusped
hyperbolic surface. Motivated by Guo’s combinatorial Yamabe flow (1.2), the
second author [28] introduced the following combinatorial Yamabe flow

$\displaystyle\left\\{\begin{array}[]{ll}\frac{dw_{i}(t)}{dt}=K_{i}(t)-\overline{K}_{i},&\hbox{
}\\\ w(0)=w_{0},&\hbox{ }\end{array}\right.$ (1.3)

where $\overline{K}:B\rightarrow(0,+\infty)$ is a function defined on the
boundary components and $w_{0}\in\mathcal{W}$. The combinatorial Yamabe flow
(1.3) aims at finding hyperbolic metrics on $\Sigma$ with a prescribed
generalized combinatorial curvature $\overline{K}:B\rightarrow(0,+\infty)$,
equivalently, prescribed lengths of connected components of $B$. The long time
existence and global convergence of the combinatorial Yamabe flow (1.3) were
proved in [28].

###  1.2 Main Results

In this paper, we introduce the following combinatorial Calabi flow for Guo’s
vertex scaling of discrete hyperbolic metrics on ideally triangulated surfaces
with boundary.

######  Definition 2.

Assume $(\Sigma,\mathcal{T})$ is a surface with boundary with an ideal
triangulation. Let $\tilde{l}\in\mathbb{R}^{E}_{>0}$ be a discrete hyperbolic
metric defined on $(\Sigma,\mathcal{T})$ and $\bar{K}\in\mathbb{R}^{n}_{>0}$
be a function defined on the boundary components $B=\\{1,2,\cdots,n\\}$. The
combinatorial Calabi flow is defined to be

$\displaystyle\left\\{\begin{array}[]{ll}\frac{dw_{i}(t)}{dt}=\Delta(\bar{K}-K)_{i},&\hbox{
}\\\ w(0)=w_{0},&\hbox{ }\end{array}\right.$ (1.4)

where $K_{i}(t)$ is the generalized combinatorial curvature of the discrete
hyperbolic metric $l(t)=w(t)*\tilde{l}$, $\Delta=(\frac{\partial
K_{i}}{\partial w_{j}})_{n\times n}$ is the discrete Laplace operator defined
by the Jacobian of $K(t)$ with respect to the discrete conformal factor
$w(t)$, and $w_{0}\in\mathcal{W}=\mathcal{W}(\tilde{l})$ is an admissible
discrete conformal factor.

The combinatorial Calabi flow was first introduced by Ge in [6] (see also [7])
for Thurston’s Euclidean circle packing metrics on closed triangulated
surfaces and then further studied for different discrete metric structures on
closed surfaces and 3-dimensional manifolds. See, for instance, [8, 9, 24, 26,
31] and others. This is the first time that the combinatorial Calabi flow is
introduced on ideally triangulated surfaces with boundary.

We prove the following result on the long time existence and global
convergence of combinatorial Calabi flows (1.4) on ideally triangulated
hyperbolic surfaces with boundary.

######  Theorem 1.1.

Assume $(\Sigma,\mathcal{T},\tilde{l})$ is an ideally triangulated surface
with boundary with a discrete hyperbolic metric
$\tilde{l}\in\mathbb{R}^{E}_{>0}$ and $\bar{K}\in\mathbb{R}^{n}_{>0}$ is a
function defined on the boundary components $B=\\{1,2,\cdots,n\\}$. Then the
solution $w(t)$ to the combinatorial Calabi flow (1.4) exists for all time and
converges exponentially fast for any initial $w_{0}\in\mathcal{W}$.

In [14], Guo proved the following basic properties of the vertex scaling in
Definition 1.

######  Theorem 1.2 (Guo [14]).

Assume $(\Sigma,\mathcal{T},\tilde{l})$ is an ideally triangulated surface
with boundary with a discrete hyperbolic metric
$\tilde{l}\in\mathbb{R}^{E}_{>0}$. Then

  * (1)

the admissible space $\mathcal{W}$ is a convex polytope;

  * (2)

the discrete Laplace operator $\Delta=(\frac{\partial K_{i}}{\partial
w_{j}})_{n\times n}$ is symmetric and strictly negative definite on the
admissible space $\mathcal{W}$.

  * (3)

The generalized combinatorial curvature map
$K:\mathcal{W}\rightarrow\mathbb{R}^{n}_{>0}$ is a diffeomorphism.

Based on Theorem 1.2, we can define the fractional discrete Laplace operator
$\Delta^{s}$ for Guo’s vertex scaling of discrete hyperbolic metrics on
ideally triangulated surfaces with boundary for any $s\in\mathbb{R}$ as
follows. Recall that if $A$ is a symmetric positive definite $n\times n$
matrix and $P\in O(n)$ is an orthogonal matrix with

$\displaystyle
A=P^{T}\cdot\text{diag}\\{\lambda_{1},\cdots,\lambda_{n}\\}\cdot P,$

where $\lambda_{1},\cdots,\lambda_{n}$ are positive eigenvalues of the matrix
$A$. Then $A^{s}$ is defined to be

$\displaystyle
A^{s}=P^{T}\cdot\text{diag}\\{\lambda_{1}^{s},\cdots,\lambda_{n}^{s}\\}\cdot
P.$

The $2s$-th order fractional discrete Laplace operator $\Delta^{s}$ for Guo’s
vertex scaling of discrete hyperbolic metrics on ideally triangulated surfaces
with boundary is defined to be

$\displaystyle\Delta^{s}=-(-\Delta)^{s},$ (1.5)

where $\Delta=(\frac{\partial K_{i}}{\partial w_{j}})_{n\times n}$ is the
standard discrete Laplace operator in Theorem 1.2. If $s=0$, the fractional
discrete Laplace operator $\Delta^{s}$ is reduced to the minus identity
operator. If $s=1$, the fractional discrete Laplace operator $\Delta^{s}$ is
reduced to the discrete Laplace operator $\Delta=(\frac{\partial
K_{i}}{\partial w_{j}})_{n\times n}$. By Theorem 1.2, the fractional discrete
Laplace operator $\Delta^{s}$ is strictly negative definite on the admissible
space $\mathcal{W}$ for any $s\in\mathbb{R}$.

Following Wu-Xu [24], we introduce the following fractional combinatorial
Calabi flow for Guo’s vertex scaling of discrete hyperbolic metrics on ideally
triangulated surfaces with boundary.

######  Definition 3.

Assume $(\Sigma,\mathcal{T})$ is an ideally triangulated surface with
boundary. Let $s\in\mathbb{R}$, $\tilde{l}\in\mathbb{R}^{E}_{>0}$ be a
discrete hyperbolic metric defined on $(\Sigma,\mathcal{T})$ and
$\bar{K}\in\mathbb{R}^{n}_{>0}$ be a function defined on the boundary
components $B=\\{1,2,\cdots,n\\}$. The fractional combinatorial Calabi flow is
defined to be

$\displaystyle\left\\{\begin{array}[]{ll}\frac{dw_{i}(t)}{dt}=\Delta^{s}(\bar{K}-K)_{i},&\hbox{
}\\\ w(0)=w_{0},&\hbox{ }\end{array}\right.$ (1.6)

where $K_{i}(t)$ is the generalized combinatorial curvature of the discrete
hyperbolic metric $w(t)*\tilde{l}$, $\Delta^{s}$ is the fractional discrete
Laplace operator of $w(t)*\tilde{l}$ defined by (1.5) and
$w_{0}\in\mathcal{W}=\mathcal{W}(\tilde{l})$ is an admissible discrete
conformal factor.

If $s=0$, the fractional combinatorial Calabi flow (1.6) is reduced to the
combinatorial Yamabe flow (1.3). If $s=1$, the fractional combinatorial Calabi
flow (1.6) is reduced to the combinatorial Calabi flow (1.4). The fractional
combinatorial Calabi flow (1.6) further covers the case of $s\neq 0,1$. Note
that the fractional combinatorial Calabi flow (1.6) is in general a non-local
combinatorial curvature flow. We prove the following result on long time
existence and global convergence of the fractional combinatorial Calabi flow
(1.6).

######  Theorem 1.3.

Assume $(\Sigma,\mathcal{T})$ is an ideally triangulated surface with
boundary. Let $\tilde{l}\in\mathbb{R}^{E}_{>0}$ be a discrete hyperbolic
metric defined on $(\Sigma,\mathcal{T})$ and $\bar{K}\in\mathbb{R}^{n}_{>0}$
be a function defined on the boundary components $B=\\{1,2,\cdots,n\\}$. Then
the solution $w(t)$ to the fractional combinatorial Calabi flow (1.6) exists
for all time and converges exponentially fast for any initial
$w_{0}\in\mathcal{W}$.

###  1.3 Organization of the paper

The paper is organized as follows. In Section 2, we study the basic properties
of the combinatorial Calabi flow (1.4) and prove Theorem 1.1. In Section 3, we
summarize the basic properties of the fractional combinatorial Calabi flow
(1.6) and prove Theorem 1.3.

Acknowledgements
The authors thank Professor Ren Guo for his constant encouragements and
supports and thank Professor Ze Zhou for valuable communications. The research
of the second author is supported by the Fundamental Research Funds for the
Central Universities under Grant no. 2042020kf0199.

##  2 Combinatorial Calabi flow on surfaces with boundary

In this section, we prove the main result of this paper about the convergence
of combinatorial Calabi flow on ideally triangulated surface with boundary.

###  2.1 Basic properties of combinatorial Calabi flow

######  Lemma 2.1.

The combinatorial Calabi flow (1.4) is a negative gradient flow of the
combinatorial Calabi energy defined by

$\mathcal{C}(w)=\frac{1}{2}\sum_{i=1}^{n}(K_{i}(w)-\bar{K}_{i})^{2}.$

Proof.  By direct calculations, we have

$\frac{\partial\mathcal{C}}{\partial w_{j}}=\sum_{i=1}^{n}\frac{\partial
K_{i}}{\partial
w_{j}}(K_{i}-\bar{K}_{i})=\Delta(K-\bar{K})_{j}=-\frac{dw_{j}}{dt},$

which implies that the combinatorial Calabi flow (1.4) is a negative gradient
flow of the combinatorial Calabi energy $\mathcal{C}(w(t))$. $\square$

As a corollary, we have the following result.

######  Corollary 2.2.

The combinatorial Calabi energy $\mathcal{C}(w)$ is decreasing along the
combinatorial Calabi flow (1.4).

Proof.  By direct calculations, we have

$\frac{d\mathcal{C}(w(t))}{dt}=\sum_{i=1}^{n}\frac{\partial\mathcal{C}}{\partial
w_{i}}\frac{dw_{i}}{dt}=-\sum_{i=1}^{n}(\Delta(K-\bar{K})_{i})^{2}\leq 0,$

the right side of which is strictly negative unless $K=\bar{K}$. $\square$

By Theorem 1.2, the following function

$\mathcal{E}(w)=-\int_{0}^{w}\sum_{i=1}^{n}(K_{i}-\bar{K}_{i})dw_{i}$ (2.1)

is a well-defined smooth convex function defined on the admissible space
$\mathcal{W}$. Guo [14] first proved that the energy function $\mathcal{E}(w)$
with $\bar{K}=0$ is a strictly convex energy function on the admissible space
$\mathcal{W}$. The global rigidity of Guo’s vertex scaling of discrete
hyperbolic metrics with respect to the lengths of the boundary components in
[14] follows from the convexity of this function.

The following monotonicity property of the energy function $\mathcal{E}(w)$
holds along the combinatorial Calabi flow (1.4).

######  Lemma 2.3.

The energy function $\mathcal{E}(w)$ defined by (2.1) is decreasing along the
combinatorial Calabi flow (1.4).

Proof.  By direct calculations, we have

$\frac{d\mathcal{E}}{dt}=\sum_{i=1}^{n}\frac{\partial\mathcal{E}}{\partial
w_{i}}\frac{dw_{i}}{dt}=\sum_{i=1}^{n}(K-\bar{K})_{i}\Delta(K-\bar{K})_{i}=(K-\bar{K})^{T}\Delta(K-\bar{K})\leq
0,$

where the last inequality follows from Guo’s Theorem 1.2 on the negative
definiteness of the discrete Laplace operator $\Delta$. $\square$

###  2.2 The long time behavior of combinatorial Calabi flow

As the combinatorial Calabi flow (1.4) is essentially a system of ordinary
differential equations, the solution to the combinatorial Calabi flow (1.4)
exists locally by the standard theory in dynamical system. The focus of this
subsection is on the long time behavior of the combinatorial Calabi flow
(1.4).

######  Lemma 2.4.

If the solution $w(t)$ to the combinatorial Calabi flow (1.4) exists for all
time and converges to $\bar{w}\in\mathcal{W}$, then $K(\bar{w})=\bar{K}$.

Proof.  By the proof of Corollary 2.2, we have

$\frac{d\mathcal{C}(w(t))}{dt}=-\sum_{i=1}^{n}(\Delta(K-\bar{K})_{i})^{2}\leq
0,$

which implies that the combinatorial Calabi energy $\mathcal{C}(w)$ is
decreasing along the combinatorial Calabi flow (1.4). Note that
$\mathcal{C}(w)\geq 0$ by definition, and
$\lim_{t\rightarrow+\infty}\mathcal{C}(w(t))$ exists. As a result, there
exists $\xi_{n}\in(n,n+1)$ such that

$\mathcal{C}(w(n+1))-\mathcal{C}(w(n))=\frac{d}{dt}\Big{|}_{t=\xi_{n}}\mathcal{C}(w(t))=-\sum_{i=1}^{n}\big{(}\Delta(K(w(t))-\bar{K})_{i}\big{)}^{2}|_{t=\xi_{n}}\rightarrow
0,\ \text{as}\ n\rightarrow+\infty.$

By the continuity of the generalized combinatorial curvature $K$ in the
discrete conformal factor $w$, we have

$\Delta(K(w)-\bar{K})|_{w=\bar{w}}=0$

by the assumption $\lim_{t\rightarrow+\infty}w(t)=\bar{w}$. By the strictly
negative definiteness of the discrete Laplace operator $\Delta$ in Theorem
1.2, we have $K(\bar{w})=\bar{K}.$ $\square$

######  Lemma 2.5.

Suppose $\bar{K}\in\mathbb{R}^{n}_{>0}$ is a function defined on the boundary
components $B=\\{1,2,\cdots,n\\}$. Then for any initial value
$w(0)=w_{0}\in\mathcal{W}$, the solution $w(t)$ to the combinatorial Calabi
flow (1.4) stays in a bounded subset of the admissible space $\mathcal{W}$.

Proof.  By Theorem 1.2, for any $\bar{K}\in\mathbb{R}^{n}_{>0}$, there exists
$\bar{w}\in\mathcal{W}$ such that $\bar{K}=K(\overline{w})$, which implies

$\nabla\mathcal{E}(\bar{w})=-(K(\bar{w})-\bar{K})=0.$

Note that $\mathcal{E}(w)$ is a strictly convex function on
$\mathcal{W}\subset\mathbb{R}^{n}$, we have

$\lim_{w\to\infty,w\in\mathcal{W}}\mathcal{E}(w)=+\infty.$ (2.2)

Since $\mathcal{E}(w)$ is decreasing along the combinatorial Calabi flow (1.4)
by Lemma 2.3, we have

$\mathcal{E}(w(t))\leq\mathcal{E}(w(0)).$ (2.3)

Combining (2.2) and (2.3), we prove that the solution $w(t)$ to the
combinatorial Calabi flow (1.4) stays in a bounded subset of $\mathbb{R}^{n}$
along the flow. $\square$

To prove the long time existence of the solution $w(t)$ to combinatorial
Calabi flow (1.4), we need to further prove that the solution $w(t)$ stays in
a compact subset of the admissible space $\mathcal{W}$. Recall the following
characterization of the admissible space $\mathcal{W}$ obtained by Guo [14].

######  Lemma 2.6 (Guo [14]).

The admissible space

$\mathcal{W}=\cap_{\\{ij\\}\in E}\mathcal{W}_{ij},$ (2.4)

where

$\mathcal{W}_{ij}=\\{w\in\mathbb{R}^{n}|w_{i}+w_{j}>-\ln\cosh\frac{\tilde{l}_{ij}}{2}\\}.$

The boundary of each $\mathcal{W}_{ij}$ in $\partial\mathcal{W}$ is a portion
of the hyperplane induced by $l_{ij}=0$, namely,

$\partial_{ij}\mathcal{W}=\\{w\in\partial\mathcal{W}|w\in\mathbb{R}^{n},w_{i}+w_{j}=-\ln\cosh\frac{\tilde{l}_{ij}}{2}\\}.$

Note that any point of $\partial\mathcal{W}$ lies on some
$\partial_{ij}\mathcal{W}$. We will prove that the solution $w(t)$ to the
combinatorial Calabi flow (1.4) will never reach the boundary
$\partial\mathcal{W}$. First, we have the following result.

######  Lemma 2.7 ([14, 28]).

For any $M>0$, there exists a positive constant
$\epsilon_{ij}=\epsilon_{ij}(M)$ such that if $w\in\mathcal{W}$ satisfies

$w_{i}+w_{j}<-\ln\cosh\frac{\tilde{l}_{ij}}{2}+\epsilon_{ij},$

then

$K_{i}>M,K_{j}>M.$

Proof.  For completeness, we sketch the proof here. For a hyperbolic right-
angled hexagon $\\{ijk\\}\in F$ adjacent to $i,j\in B$, we have

$\displaystyle\cosh\theta_{i}^{jk}=\frac{\cosh l_{ij}\cosh l_{ik}+\cosh
l_{jk}}{\sinh l_{ij}\sinh l_{ik}}>\frac{\cosh l_{ij}\cosh l_{ik}}{\sinh
l_{ij}\sinh l_{ik}}>\frac{\cosh l_{ij}}{\sinh l_{ij}},$ (2.5)

by the cosine law for hyperbolic right-angled hexagons, where
$\theta_{i}^{jk}$ is the length of the hyperbolic arc at $i\in B$ in the
hyperbolic right-angled hexagon $\\{ijk\\}\in F$. The formula (2.5) implies
that $\theta_{i}^{jk}\rightarrow+\infty$ uniformly as $l_{ij}\rightarrow
0^{+}$, which further implies that $K_{i}\rightarrow+\infty$ uniformly as
$w\rightarrow\partial_{ij}\mathcal{W}$. The same arguments apply to $K_{j}$.
$\square$ As an application of Lemma 2.7, we have the following result.

######  Lemma 2.8.

Assume $(\Sigma,\mathcal{T})$ is an ideally triangulated surface with
boundary. Let $\tilde{l}\in\mathbb{R}^{E}_{>0}$ be a discrete hyperbolic
metric defined on $(\Sigma,\mathcal{T})$ and $\bar{K}\in\mathbb{R}^{n}_{>0}$
be a function defined on the boundary components $B$. For any initial value
$w_{0}\in\mathcal{W}$, there exists a constant
$\epsilon=\epsilon(w_{0},\bar{K})>0$ such that the solution $w(t)$ to the
combinatorial Calabi flow (1.4) can never be in the region

$\mathcal{W}_{\epsilon}=\\{w\in\mathcal{W}|d(w,\partial\mathcal{W})<\epsilon\\},$

where $d$ is the standard Euclidean metric on $\mathbb{R}^{n}$.

Proof.  Set

$M=\max_{i}\\{|\bar{K}_{i}|+\sqrt{2\mathcal{C}(w_{0})}\\}.$

By Lemma 2.7, there exists $\epsilon_{ij}=\epsilon_{ij}(M)>0$ such that if

$w_{i}+w_{j}<-\ln\cosh\frac{\tilde{l}_{ij}}{2}+2\epsilon_{ij},$

then

$K_{i}(w)>M,K_{j}(w)>M.$

Set $\epsilon_{0}=\min_{\\{ij\\}\in E}\epsilon_{ij}>0$, then if
$w\in\mathcal{W}$ satisfies

$w_{i}+w_{j}<-\ln\cosh\frac{\tilde{l}_{ij}}{2}+2\epsilon_{0}$

for some edge $\\{ij\\}\in E$, then $K_{i}(w)>M,$ which further implies that

$|K_{i}(w)-\bar{K}_{i}|\geq|K_{i}(w)|-|\bar{K}_{i}|>M-|\bar{K}_{i}|\geq\sqrt{2\mathcal{C}(w_{0})}.$
(2.6)

We claim that the solution $w(t)$ to the combinatorial Calabi flow (1.4) can
never be in the region $\mathcal{W}_{\epsilon_{0}}$. Otherwise, there exists
some $t_{0}\in[0,+\infty)$ and an edge $\\{ij\\}\in E$ such that the solution
$w(t)$ to the combinatorial Calabi flow (1.4) satisfies
$w(t_{0})\in\mathcal{W}$ and

$w_{i}(t_{0})+w_{j}(t_{0})<-\ln\cosh\frac{\tilde{l}_{ij}}{2}+2\epsilon_{0},$

which implies that

$|K_{i}(w(t_{0}))-\bar{K}_{i}|>\sqrt{2\mathcal{C}(w_{0})}$ (2.7)

by (2.6). Note that $\mathcal{C}(w)$ is decreasing along the combinatorial
Calabi flow (1.4) by Corollary 2.2. Therefore, for any $t>0$, the solution
$w(t)$ to the combinatorial Calabi flow (1.4) satisfies

$|K_{i}(t)-\bar{K}_{i}|\leq\sqrt{2\mathcal{C}(w(t))}\leq\sqrt{2\mathcal{C}(w_{0})}$

for any $i\in B$, which contradicts to (2.7). Therefore, the solution $w(t)$
to the combinatorial Calabi flow (1.4) can never be in the region
$\mathcal{W}_{\epsilon_{0}}$. $\square$

######  Remark 1.

The result in Lemma 2.8 is independent of the existence of
$\bar{w}\in\mathcal{W}$ with $K(\bar{w})=\bar{K}$.

As a direct corollary of Lemma 2.5 and Lemma 2.8, we have the following result
on the solution to the combinatorial Calabi flow (1.4), which implies the long
time existence of the solution to the combinatorial Calabi flow (1.4).

######  Corollary 2.9.

Suppose $(\Sigma,\mathcal{T},\tilde{l})$ is an ideally triangulated compact
surface with boundary with a discrete hyperbolic metric
$\tilde{l}\in\mathbb{R}^{E}_{>0}$ and $\bar{K}\in\mathbb{R}^{n}_{>0}$ is a
function defined on $B$. Then for any initial value $w_{0}\in\mathcal{W}$, the
solution $w(t)$ to the combinatorial Calabi flow (1.4) stays in a compact
subset $\Omega$ of the admissible space $\mathcal{W}$. As a result, the
solution $w(t)$ to the combinatorial Calabi flow (1.4) exists for all time.

Proof of Theorem 1.1: By Corollory 2.9, the solution $w(t)$ to the
combinatorial Calabi flow (1.4) stays in a compact subset $\Omega$ of the
admissible space $\mathcal{W}$. Note that the discrete Laplace operator
$\Delta$ is strictly negative by Theorem 1.2. By the continuity of the
eigenvalue $\lambda_{\Delta}$ of $\Delta$, there exists a uniform positive
constant $\lambda_{0}$ such that

$\lambda_{\Delta}(w)\leq-\sqrt{\lambda_{0}},$

when $w\in\Omega\subset\subset\mathcal{W}$. Therefore, along the combinatorial
Calabi flow (1.4), we have

$\frac{d\mathcal{C}(w(t))}{dt}=-(K-\bar{K})^{T}\Delta^{2}(K-\bar{K})\leq-\lambda_{0}\mathcal{C}(w(t)),$

which implies

$\mathcal{C}(w(t))\leq e^{-\lambda_{0}}\mathcal{C}(w_{0}).$

This completes the proof of the exponential convergence of the solution $w(t)$
to the combinatorial Calabi flow (1.4) to $\bar{w}$ with
$K(\overline{w})=\bar{K}$ by Theorem 1.2. $\square$

######  Remark 2.

One can also use the Lyapunov stability theorem ([19] Chapter 5) to prove the
exponential convergence of the solution $w(t)$ to the combinatorial Calabi
flow (1.4) to $\bar{w}$.

##  3 Fractional Combinatorial Calabi flow on surfaces with boundary

Based on the property that the fractional Laplace operator $\Delta^{s}$ is
strictly negative definite on the admissible space $\mathcal{W}$ for any
$s\in\mathbb{R}$, the fractional combinatorial Calabi flow (1.6) has many
properties similar to that of the combinatorial Calabi flow (1.4). As the
proofs for these properties are almost the same as that of the combinatorial
Calabi flow (1.4), we will only list these properties and omit the details of
the proofs.

######  Lemma 3.1.

The combinatorial Calabi energy $\mathcal{C}(w)$ is decreasing along the
fractional combinatorial Calabi flow (1.6).

######  Lemma 3.2.

The energy function $\mathcal{E}(w)$ defined by (2.1) is decreasing along the
fractional combinatorial Calabi flow (1.6).

######  Remark 3.

Different from the combinatorial Yamabe flow (1.3) and the combinatorial
Calabi flow (1.4), which corresponds to $s=0$ and $s=1$ for the fractional
combinatorial Calabi flow (1.6) respectively, the fractional combinatorial
Calabi flow (1.6) is generically not a gradient flow. As the fractional
Laplace operator $\Delta^{s}$ is generically a non-local operator, the
fractional combinatorial Calabi flow (1.6) is generically a non-local
combinatorial curvature flow defined on the ideally triangulated surface with
boundary $(\Sigma,\mathcal{T})$.

We also have the following long time behavior of the solution $w(t)$ to the
fractional combinatorial Calabi flow (1.6).

######  Lemma 3.3.

If the solution $w(t)$ to the fractional combinatorial Calabi flow (1.6)
exists for all time and converges to $\bar{w}\in\mathcal{W}$, then
$K(\bar{w})=\bar{K}$.

######  Lemma 3.4.

Suppose $\bar{K}\in\mathbb{R}^{n}_{>0}$ is a function defined on $B$. Then for
any $s\in\mathbb{R}$ and any initial value $w(0)=w_{0}\in\mathcal{W}$, the
solution $w(t)$ to the fractional combinatorial Calabi flow (1.6) stays in a
bounded subset of the admissible space $\mathcal{W}$.

######  Lemma 3.5.

Assume $(\Sigma,\mathcal{T})$ is an ideally triangulated surface with
boundary. Let $\tilde{l}\in\mathbb{R}^{E}_{>0}$ be a discrete hyperbolic
metric defined on $(\Sigma,\mathcal{T})$ and $\bar{K}\in\mathbb{R}^{n}_{>0}$
be a function defined on the boundary components $B$. Then for any
$s\in\mathbb{R}$ and any initial value $w_{0}\in\mathcal{W}$, there exists a
constant $\epsilon=\epsilon(w_{0},\bar{K})>0$ such that the solution $w(t)$ to
the fractional combinatorial Calabi flow (1.6) can never be in the region

$\mathcal{W}_{\epsilon}=\\{w\in\mathcal{W}|d(w,\partial\mathcal{W})<\epsilon\\}.$

As a direct corollary of Lemma 3.4 and Lemma 3.5, we have the following result
on the solution $w(t)$ to the fractional combinatorial Calabi flow (1.6),
which implies the long time existence of the solution to the fractional
combinatorial Calabi flow (1.6).

######  Corollary 3.6.

Suppose $(\Sigma,\mathcal{T})$ is an ideally triangulated compact surface with
boundary with a discrete hyperbolic metric $\tilde{l}\in\mathbb{R}^{E}_{>0}$
and $\bar{K}\in\mathbb{R}^{n}_{>0}$ is a function defined on $B$. Then for any
$s\in\mathbb{R}$ and any initial value $w_{0}\in\mathcal{W}$, the solution
$w(t)$ to the fractional combinatorial Calabi flow (1.6) stays in a compact
subset $\Omega$ of the admissible space $\mathcal{W}$. As a result, for any
$s\in\mathbb{R}$ and any initial value $w_{0}\in\mathcal{W}$, the solution
$w(t)$ to the fractional combinatorial Calabi flow (1.6) exists for all time.

The proof of Theorem 1.3 is paralleling to that of Theorem 1.1 with the
discrete Laplace operator $\Delta$ replaced by the fractional discrete Laplace
operator $\Delta^{s}$, which is also strictly negative definite on
$\mathcal{W}$ for any $s\in\mathbb{R}$. We omit the details of the proof here.

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(Yanwen Luo) Department of Mathematics, Rutgers University, New Brunswick NJ,
08817

E-mail<EMAIL_ADDRESS>

(Xu Xu) School of Mathematics and Statistics, Wuhan University, Wuhan 430072,
P.R. China

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