#  Convolutional Neural Network (CNN) to reduce construction loss in JPEG
compression

Suman Kunwar
<EMAIL_ADDRESS>

###### Abstract

In recent decades, digital image processing has gained enormous popularity.
Consequently, a number of data compression strategies have been put forth,
with the goal of minimizing the amount of information required to represent
images. Among them, JPEG compression is one of the most popular methods that
has been widely applied in multimedia and digital applications. The periodic
nature of DFT makes it impossible to meet the periodic condition of an image’s
opposing edges without producing severe artifacts, which lowers the image’s
perceptual visual quality. On the other hand, deep learning has recently
achieved outstanding results for applications like speech recognition, image
reduction, and natural language processing. Convolutional Neural Networks
(CNN) have received more attention than most other types of deep neural
networks. The use of convolution in feature extraction results in a less
redundant feature map and a smaller dataset, both of which are crucial for
image compression. In this work, an effective image compression method is
purposed using autoencoders. The study’s findings revealed a number of
important trends that suggested better reconstruction along with good
compression can be achieved using autoencoders.

###### Keywords:

JPEG Compression, Discrete Fourier Transform (DFT), Construction Loss,
Convolutional Neural Network (CNN), Autoencoder

##  I Introduction

In today’s digital era, human beings are surrounded by digital gadgets.
Photographs are now an integral part of a person’s daily life, and digital
images are widely used in a variety of applications. As digital imaging and
multimedia services advance, more and more people can share their data on the
Internet. The number of internet users is growing day by day rapidly [1],
resulting in increased data transfer, which necessitates efficient image
compression. In multimedia, JPEG is one of the most commonly used lossy
compression techniques [2]. There are several variations of JPEG: JPEG 2000,
JPEG XS, JPEG Systems, JPEG Pleno, and JPEG XL [3]. Based on the data provided
by Web Technology Survey, 74.3% of websites use the JPEG image format [4]. In
digital images, pixels have high correlations, and the removal of this
correlation will not affect the visual quality of the image [5, 6]. To achieve
the best quality with the smallest possible size, the low frequency values are
preserved as they define the content of the image, and the high frequency
values are truncated by a certain amount [7, 8].

With the help of DFT, images in the spatial domain can be converted into the
frequency domain, and certain frequencies can be ignored or modified to
produce a low-information image with adequate quality [9, 10, 11]. In
practical application, when we compute the DFT of an image, it is impossible
to meet the periodic condition that opposite borders of an image are alike,
and the image always shows strong discontinuities across the frame border. As
a result this affects the registration accuracy and success rate [12]. To
solve this problem, various approaches have been taken. Among them, raised-
cosine window [13], blackman window [14], and flap-top window [15] are the
most popular ones.
As CNN have advanced rapidly[16], the problem of removing image artifacts from
the decoded images has been re-examined. Several state-of-the-art deep
learning-based algorithms [17, 18, 19, 20, 12] have been developed with great
success using this approach. Approaches like using the Hann windows for
reducing edge-effects in patch based image segmentation with CNNs has shown
promising results and pointed out the further investigation with different
window functions, and with reducing the amount of context needed [21]. While
these solutions are promising, still some information is lost in the process.
To minimize the reconstruction loss, an autoencoder network is purposed. The
encoder reduces the dimensionality of an image into a number of feature maps.
The feature map contains densely packed color values of an image whose
original value can be closely reconstructed using the decoder associated with
it. To measure the quality of the reconstructed image Peak Signal to Noise
Ratio (PSNR), and Structural Similarity Index (SSMI) are used. It is also
compare with the popular image compression standard JPEG. Here, SSIM measures
the similarity in pixels and is used as a second image quality metric.
The remainder of this paper is organized as follows. Proposed method is
described in Section II. In Sections III and IV, training and experimental
evaluations are described in detail. Section V presents the conclusions of
this research.

##  II Methods

This section gives more information about the implementation, including deep
learning network architecture, and the loss function.

###  II-A Network Architecture

As autoencoder pipeline claims to provide efficient compression efficiency
[22, 23], an autoenoder network is constructed. Here, the encoder takes the
input to represent the data in the simplest possible way. This is done by
extracting the most prominent features and presenting them in a way that the
decoder can understand. Whereas, the decoder learns to read compressed code
representations and generate them based on those representations. The proposed
network architecture is depicted in Figure 1.

Figure 1: Proposed Network Architecture

The encoders are trained along with the decoders without labels. Eight layer
of convolution is used in the auto encoder network. Batch Normalization is
used to enable faster and more stable training of deep neural networks [24].
The benefit of using a layer is, it allows similar operations to be performed
simultaneously. With more convolution kernels, the number of parameters
increases linearly. As a result, the number of output channels also increases
linearly and helps to reduce the artifacts caused by the gaussian noise [25].
The computation time is also proportional to the size of the input channel and
to the number of kernels.

Sigmoid activation function is used in the output layer, whose output is bound
between 0 and 1 range, and can be prone to suffering from the vanishing
gradient problem [26]. To overcome this, Rectified Linear Unit (ReLU) is used
as an activation function to increase to speed up the application and for
better results [27]. Upsampling is also done to increase the spatial
dimensions of the feature maps[28]. Adam optimizer is used as optimizer which
uses momentum and adaptive gradient to compute adaptive learning rates for
each parameter[29].

###  II-B Loss Function

Loss functions are used to calculate predicted errors created across the
training samples. Here, loss is measured in terms of Mean Squared Error (MSE).
It calculates the mean of square error of the predicted value and the actual
value. It results are derivable and is possible to control the update rate.

$\begin{gathered}\sum_{i=1}^{D}(x_{i}-y_{i})^{2}\end{gathered}$ (1)

##  III Training

This section describes the training procedure and parameter settings used in
this research.

###  III-A Datasets

CIFAR-10 dataset is used here to train and test the model[30]. It consists of
60000 32x32 colour images and is divided into training and test datasets.
Simple random sampling is used here to represent the different sets of images
as it gives an equal probability of selecting a particular item [31]. Here,
samples are only taken from the training images, the dataset used to train the
algorithm. As the method aims to evaluate itself using both real and
representative data, no training were done on holdout tests [32].

###  III-B Procedure

The training dataset is run through the model in an incremental fashion. While
training the model, Adam is used as a optimizer and loss is calculated in
terms of Mean Square Error (MSE) as it has better generalization performances
than the Cross Entropy (CE) loss [33]. Loss values and accuracy scores while
training at different iterations are measured. For error correction,
regularization options [34] such as early stopping and dropout are carried
out.

##  IV Evaluation

This section compares the results that happened while training the model with
different settings. First, the model is tested with lower epochs and then
gradually increased.

Figure 2: Loss vs Model Loss after 15 epoches of training Figure 3: Loss vs
Model Accuracy after 15 epoches of training

Certain spikes can be seen in Figure 5. Here, the loss first decreases,
increases, and decreases at last. These spikes are often encountered when
training with high learning rates, high order loss functions or small batch
sizes [35]. The batch size is reduced and early stopping is added to the model
for to avoid overfitting [36]. A checkpoint is also added to save the best
value weights obtained while training. The maximum accuracy value has reached
up to 0.90104 within 200 epochs training.

Figure 4: Loss vs Model Loss after 200 epoches of training Figure 5: Loss vs
Model Accuracy after 200 epoches of training

From the comparison, the model shows high accuracy with regularization and has
higher loss with low training. With the increasing number of training
variation on loss and accuracy can be seen. The model shows low accuracy with
the addition of noise value to the model with normalization.

TABLE I: Comparison of proposed model with different settings Model (variations) | Loss | Accuracy | Epoches
---|---|---|---
Model | 0.162 | 80.236 | 15
Model | 0.092 | 85.352 | 100
Model | 0.089 | 82.307 | 200
Model with batch size 32 | 0.13 | 84.269 | 200
Model with normalization | 0.07 | 85.94 | 100
Model with normalization | 0.08 | 87.22 | 200
Model with normalization and noise | 0.111 | 56.74 | 30

Image quality can be compromised as a result of distortions during the
acquisition and processing of images. Different metrics have been used to
measure the quality of compressed results, including Mean Square Error (MSE),
Peak Signal to Noise Ratio (PSNR), and Structural Similarity Index (SSMI)
[37]. To measure the performance of the reconstructed images of the purposed
system, PSNR and SSIM are used.

Figure 6: Original image (a) and the reconstructed image produced by (b) JPEG and the (c) purposed system TABLE II: Compression performance comparison | PSNR | SSIM
---|---|---
JPEG | 25.27 | 0.82
Purposed Model | 34.47 | 0.94

The results from Table II shows that the purposed system outperformed the
existing JPEG compression using DFT with a huge margin. Figure 6 presents few
sample images to compare the subjective quality of these methods.

##  V Conclusion

An autoencoder-based image compression technique is presented in the paper,
which uses a lower-dimensional representation to reconstruct the image with
less reconstruction loss by capturing the most significant elements. While it
outperforms JPEG compression using DFT, there are still other variations of
JPEG, that needs further research. Among others, the impact of noise as a
regularization method deserves further investigation. In our case, the
addition of noise resulted in moderate quality images.

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# On quotients of congruence subgroups of braid groups

Jessica Appel  Department of Mathematics, Indiana University, Bloomington,
Indiana 47305, USA<EMAIL_ADDRESS>,  Wade Bloomquist  School of Mathematics,
Georgia Institute of Technology, Atlanta, Georgia 30332, USA
<EMAIL_ADDRESS>,  Katie Gravel  Department of Mathematics,
Massachusetts Institute of Technology, Boston, Massachusetts, 02139, USA
<EMAIL_ADDRESS>and  Annie Holden  Department of Mathematics, University of
Notre Dame, Notre Dame, Indiana, 46556, USA<EMAIL_ADDRESS>
###### Abstract.

The integral Burau representation provides a map from the braid group into a
group of integral matrices. This allows for a definition of congruence
subgroups of the braid group as the preimage of the usual principal congruence
subgroups of integral matrices. We explore the structure these congruence
subgroups by examining some of the quotients that may arise in the series
induced by divisibility of levels. We build on the work of Stylianakis on
symmetric quotients of congruence subgroups, which itself generalizes the
quotient of the braid group by the pure braid group. We accomplish this by
utilizing results of Newman on integral matrices and explicitly finding
elements in the preimage of any transposition. Our generalization is made
possible by avoiding the use of a generating set for congruence subgroups. We
find further generalizations based on results of Brendle and Margalit as well
as Kordek and Margalit on the level four congruence subgroup. This gives
families of quotients which are not isomorphic to symmetric groups.

2020 Mathematics Classification: Primary 20F36. Secondary 20H05.
Key words and phrases: Braid group, Congruence subgroup.

##  1\. Introduction

In this brief note, we use elementary techniques to explore how known results
on factor groups in the series of congruence subgroups of braid groups can be
generalized. For each non-negative integer $\ell$, the level $\ell$ congruence
subgroup of the braid group, $B_{n}[\ell]$, is the kernel of the composition
of the integral Burau representation with the $\bmod$ $\ell$ reduction map.
The integral Burau representation has found uses in many branches of
mathematics. For the sake of brevity, we refer the reader to the introduction
of [3] where a list of citations covering applications in algebraic geometry,
dynamics, number theory, and topology can be found. Additional details on the
integral Burau representation can be found in Section $3$ of Margalit’s survey
of open problems and questions concerning mapping class groups [6]. Our
perspective will be based on viewing the integral Burau representation of the
braid group as an analog of the symplectic representation of the mapping class
group of a surface. This viewpoint was noted in Section $2.1$ of [3] for the
unreduced Burau representation and is stated here in Lemma 2.2 for the reduced
Burau representation, following the work of [4].

Historically, the study of quotients of congruence subgroups of braid groups
began with Arnol‘d [2]. He showed that the level $2$ congruence subgroup of
the braid group may be identified with the pure braid group. Alternatively
stated as:

######  Theorem 1.1 (Arnol‘d, 1968).

For each $n$,

$B_{n}/B_{n}[2]\cong S_{n}.$

As a corollary to constructing generators of congruence subgroups of prime
level Stylianakis, [8], proved the following extension:

######  Theorem 1.2 (Stylianakis, 2016).

For each $n$ and each odd prime $p$,

$B_{n}[p]/B_{n}[2p]\cong S_{n}.$

Without constructing a generating set, we provide an additional extension of
this result using results on integral matrices due to Newman [7] and providing
an explicit surjection from $B_{n}[\ell]$ onto $S_{n}$, namely:

######  Theorem 1.3 (Theorem 3.1).

For each $n$ and each odd $\ell$,

$B_{n}[\ell]/B_{n}[2\ell]\cong S_{n}.$

The first generalization of the above results to even levels, can be found in
Brendle and Margalit’s study of the level $4$ braid group [3]. They found the
following:

######  Theorem 1.4 (Brendle and Margalit, 2018).

For each $n$,

$B_{n}[2]/B_{n}[4]\cong(\mathbb{Z}/2\mathbb{Z})^{\binom{n}{2}}.$

Again using results on integral matrices and providing the surjection
explicitly, we extend as follows:

######  Theorem 1.5 (Theorem 3.2).

For each $n$ and each odd $\ell$,

$B_{n}[2\ell]/B_{n}[4\ell]\cong(\mathbb{Z}/2\mathbb{Z})^{\binom{n}{2}}.$

In exploring the cohomology of the level $4$ subgroups, Kordek and Margalit,
[5], proved the following

######  Theorem 1.6 (Kordek and Margalit, 2019).

For each $n$,

$B_{n}/B_{n}[4]\cong\mathcal{Z}_{n},$

where $\mathcal{Z}_{n}$ is a non-split extension of $S_{n}$ by
$(\mathbb{Z}/2\mathbb{Z})^{\binom{n}{2}}$.

We see that taken with this result our Theorems 3.1 and 3.2 imply the
following generalization:

######  Theorem 1.7 (Theorem 3.3).

For each $n$ and each odd $\ell$,

$B_{n}[\ell]/B_{n}[4\ell]\cong\mathcal{Z}_{n}.$

A natural direction for future work would be to look into levels which are
multiples of higher powers of $2$. The first step in this direction would be
understanding either $B_{n}/B_{n}[8]$ or $B_{n}[4]/B_{n}[8]$.

## Acknowledgments

The authors would like to thank Dan Margalit for his helpful conversations and
guidance. The first, third and and fourth authors were supported by the $2019$
Georgia Tech REU (funded in part by NSF Grant DMS-17455). The second author is
supported in part by NSF Grant DMS-17455.

##  2\. Preliminaries

We begin by recalling the definition of congruence subgroups of braid groups.
Our description of the integral Burau representation will be given explicitly
on generators due to the nature of the arguments we will use below. Although
this choice may seem to hide some of the rich topological content of this
representation, we will describe how that can be recovered in Lemma 2.2.

######  Definition 1.

The reduced integral Burau representation is a representation of the
$n-$strand braid group, $\rho_{-1}:B_{n}\rightarrow GL(n-1,\mathbb{Z})$
defined on generators as

$\displaystyle\rho_{-1}(\sigma_{1})=\left(\begin{array}[]{cc}1&0\\\
1&1\end{array}\right)\oplus\text{Id}_{n-3}$
$\displaystyle\rho_{-1}(\sigma_{n-1})=\text{Id}_{n-3}\oplus\left(\begin{array}[]{cc}1&-1\\\
0&1\end{array}\right)$

and for $1<i<n-1$

$\rho_{-1}(\sigma_{i})=\text{Id}_{i-2}\oplus\left(\begin{array}[]{ccc}1&-1&0\\\
0&1&0\\\ 0&1&1\\\ \end{array}\right)\oplus\text{Id}_{n-i-2}.$

Now recall for integral matrices it is possible to define a natural reduction
$\bmod$ $\ell$ map component wise, namely

$r_{\ell}:GL(n-1,\mathbb{Z})\rightarrow GL(n-1,\mathbb{Z}/\ell\mathbb{Z}).$

Then noting that each generator was sent to a integral matrix the level $\ell$
congruence subgroup of the braid group is the kernel of the composition of
these maps, meaning

$B_{n}[\ell]:=ker(r_{\ell}\circ\rho_{-1}).$

######  Remark 2.1.

This definition is equivalent to the definition found in [3], which uses the
unreduced Burau reprsentation. The equivalence can be seen as the unreduced
Burau representation factors into a direct sum of the reduced Burau
representation and a $1-$dimensional representation.

We now provide a collection of lemmas.

######  Lemma 2.2 (Proposition $2.1$ [4]).

The reduced integral Burau representation is conjugate to the symplectic
representation

$\rho:B_{2g+b}\rightarrow\begin{cases}\mathrm{Sp}(2g,\mathbb{Z})&b=1\\\
(\mathrm{Sp}(2g+2,\mathbb{Z}))_{y_{g+1}}&b=2\end{cases}$

coming from the standard action on homology of the surface which double covers
the punctured disk, where $(\mathrm{Sp}(2g+2,\mathbb{Z}))_{y_{g+1}}$ is the
subgroup of $\mathrm{Sp}(2g+2,\mathbb{Z})$ which fixes the homology class
associated one of the two boundary components.

######  Lemma 2.3.

For each $m$ and for each $1\leq i\leq n$, we have $\sigma_{i}^{m}\in
B_{n}[m]$.

###### Proof.

This follows either from a direct computation of matrix multiplication using
the above definition, or as a direct application of Lemma 2.2. For a
discussion of the latter approach see relation “$R3$: Powers of Dehn Twists”
in [8]. ∎

######  Corollary 2.4.

For each $m$ and for each $1\leq i<j\leq n$ we have $A_{i,j}^{m}\in
B_{n}[2m]$, where

$A_{i,j}=\sigma_{j-1}\sigma_{j-2}...\sigma_{i+2}\sigma_{i+1}\sigma_{i}^{2}\sigma_{i+1}^{-1}\sigma_{i+2}^{-1}...\sigma_{j-2}^{-1}\sigma_{j-1}^{-1}$

are generators of $PB_{n}$.

###### Proof.

For any $i$ and $j$, we have that $A_{i,j}$ is a conjugate of
$\sigma_{i}^{2}$. By Lemma 2.3, $\sigma_{i}^{2m}\in B_{n}[2m]$. Then as
$r_{2m}\circ\rho_{-1}$ is a group homomorphism, the result follows. ∎

Made relevant in this context by Lemma 2.2, we have the following result on
integral symplectic matrices due to Newman.

######  Lemma 2.5 (Theorem $VII.22$ [7]).

Let $D=\frac{ab}{GCD(a,b)}$, then

$\mathrm{Sp}(2n,\mathbb{Z})[D]=\mathrm{Sp}(2n,\mathbb{Z})[a]\cap\mathrm{Sp}(2n,\mathbb{Z})[b]$

We now apply this to congruence subgroups of the braid group.

######  Lemma 2.6.

Let $m=\prod_{i}p_{i}^{\alpha_{i}}$. Then

$\bigcap_{i}B_{n}[p_{i}^{\alpha_{i}}]=B_{n}[m],$

and in particular, for $\ell$ odd we have

$B_{n}[2^{\alpha}\ell]=B_{n}[2^{\alpha}]\cap B_{n}[\ell].$

###### Proof.

The following is a generalization of Lemma $6.2$ in [8]. As any multiple of
$m$ is also a multiple of each $p_{i}^{\alpha_{i}}$ we have immediately from
the definition that $B_{n}[m]\leq\bigcap_{i}B_{n}[p_{i}^{\alpha_{i}}]$. The
reverse inclusion, $\bigcap_{i}B_{n}[p_{i}^{\alpha_{i}}]\leq B_{n}[m]$,
follows from Lemma 2.2 and repeated applications of Lemma 2.5.

∎

##  3\. Quotients of Congruence Subgroups

We may now prove our generalizations as stated in the introduction.

######  Theorem 3.1.

Let $\ell$ be odd, then

$B_{n}[\ell]/B_{n}[2\ell]\cong S_{n}.$

###### Proof.

The following is a variation of the proof of Theorem $6.1$ given in [8]. We
utilize our more general Lemma 2.6, and the use of a generating set for
$B_{n}[\ell]$ is avoided.

Let $\tau:B_{n}\rightarrow S_{n}$ be the natural surjective homomorphism which
sends each braid generator $\sigma_{i}$ to the transposition $(i,i+1)$. We
have that $\ker(\tau)=PB_{n}=B_{n}[2]$, as proved by Arnol‘d [2]. Let
$\tau_{\ell}$ be the restriction of $\tau$ to the subgroup $B_{n}[\ell],$

$\displaystyle\tau_{\ell}:=\tau|_{B_{n}[\ell]}:B_{n}[\ell]\rightarrow S_{n}.$

As $\ell$ is odd, we have that

$\displaystyle\tau(\sigma_{i}^{\ell})=(i,i+1)^{\ell}=(i,i+1),$

meaning $\tau_{\ell}$ is surjective utilizing Lemma 2.3.

Finally we have

$\displaystyle\ker(\tau_{\ell})=\ker(\tau)\cap B_{n}[\ell]\cong B_{n}[2]\cap
B_{n}[\ell]\cong B_{n}[2\ell],$

where the last isomorphism is a consequence of Lemma 2.6, and our result
follows from the first isomorphism theorem. ∎

######  Theorem 3.2.

Let $\ell$ be odd, then

$B_{n}[2\ell]/B_{n}[4\ell]\cong(\mathbb{Z}/2\mathbb{Z})^{\binom{n}{2}}.$

###### Proof.

Let

$\displaystyle\phi:PB_{n}\rightarrow
H_{1}(PB_{n};\mathbb{Z}/2\mathbb{Z})\cong(\mathbb{Z}/2\mathbb{Z})^{\binom{n}{2}}$

be the $\bmod$ $2$ abelianization map of the pure braid group. An explicit
description of $\phi$ is given by viewing
$(\mathbb{Z}/2\mathbb{Z})^{\binom{n}{2}}$ as the free abelian group on the
image of the $\binom{n}{2}$ generators of $PB_{n}$ which were denoted
$A_{i,j}$ in Corollary 2.4. The kernel of $\phi$ is known to be $B_{n}[4]$
(the main theorem of [3]). Now let $\phi_{\ell}$ be the restriction of $\phi$
to the subgroup $B_{n}[2\ell]\leq B_{n}[2]=PB_{n}$. From Corollary 2.4, we
have that $A_{i,j}^{\ell}\in B_{n}[2\ell]$. Now as $\ell$ is odd we have

$\displaystyle\phi(A_{i,j}^{\ell})=\phi(A_{i,j})^{\ell}=\phi(A_{i,j}),$

and so $\phi_{\ell}$ is a surjection. Additionally utilizing Lemma 2.6 we have

$\displaystyle ker(\phi_{\ell})=\ker(\phi)\cap B_{n}[2\ell]=B_{n}[4]\cap
B_{n}[2]\cap B_{n}[\ell]=B_{n}[4]\cap B_{n}[\ell]=B_{n}[4\ell],$

where we have also used that $B_{n}[4]\leq B_{n}[2]$ and so $B_{n}[4]\cap
B_{n}[2]=B_{n}[4].$ Then our desired result follows from the first isomorphism
theorem. ∎

######  Theorem 3.3.

Let $\ell$ be odd, then we have

$B_{n}[\ell]/B_{n}[4\ell]\cong B_{n}/B_{n}[4].$

###### Proof.

The third isomorphism theorem gives us that

$(B_{n}[\ell]/B_{n}[4\ell])/(B_{n}[2\ell]/B_{n}[4\ell])\cong
B_{n}[\ell]/B_{n}[2\ell],$

meaning the two rows of the following commutative diagram are exact:

${1}$${B_{n}[2]/B_{n}[4]}$${B_{n}/B_{n}[4]}$${B_{n}/B_{n}[2]}$${1}$${1}$${B_{n}[2\ell]/B_{n}[4\ell]}$${B_{n}[\ell]/B_{n}[4\ell]}$${B_{n}[\ell]/B_{n}[2\ell]}$${1}$

where the vertical arrows are $\ell^{\text{th}}$ power maps used to construct
isomorphisms in Theorem 3.1 and Theorem 3.2. Then our result follows from an
application of the five lemma.

∎

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