Creating a Math Expression Filter to Extract
Concise Math Expression Images
Kuniko Yamada1 and Harumi Murakami2
1 Graduate School for Creative Cities, Osaka City University, Osaka, Japan
2 Graduate School of Informatics, Osaka Metropolitan University, Osaka, Japand18ud512@st.osaka-cu.ac.jp, harumi@omu.ac.jp

Keywords:

Math Expression Image, Concise Math Expression, Math Expression Filter, SVM, CNN.

Abstract:

Even though the web is an effective resource to search for math expressions, finding appropriate ones among
the obtained documents is time-consuming. Therefore, we propose a math expression filter that presents appropriate images for such searches. We call an appropriate math expression a concise math expression, suchx−x1
as fˆh (x) = Nh∑ni=1 K( h i ), written in a compact form whose content can be interpreted by the math expression itself. We determined the conditions satisfied by a concise math expression and developed classifiers
that discriminate the images of concise math expressions from web images using supervised machine learning methods based on these conditions. We performed two experiments: Experiment 1 used methods otherthan deep learning, and Experiment 2 used deep learning. A convolutional neural network (CNN) with transfer learning and fine tuning by VGG16 shows high performance with an obtained F-measure of 0.819. We
applied this filter to a task that presents math expression images by entering mathematical terms into a web
search engine as queries. All of the evaluation metrics outperformed the previous study, including F-measure,
MAP, and MRR.

1

INTRODUCTION

In recent years, the importance of mathematical information retrieval (MIR) has increased (Zanibbi andBlostein, 2012). Since mathematics is a tool for expressing various concepts in science fields, understanding math expressions is imperative in such fields.
When someone wants to clearly understand an expression, the web provides a natural searching resource. However, web search for math expressions
has its problems. After web documents are obtainedthat contain math expressions, finding the appropriate
math expressions in the located documents is timeconsuming. Such documents contain many various
types of math expression images. For example, suchvariables as “x” have no intrinsic meaning, and suchfragments as “log” and “≤” are not math expressions,and some long expressions simply show the process
for deriving a formula. Among expressions and fragments of expressions, we extracted math expressions,x−x1
including fˆh (x) = Nh∑ni=1 K( h i ), which are longenough for human interpretation without supplementary explanation, although shorter than expressions
from which a formula is derived. We call these concise math expressions. First, we collected candidate

expressions, analyzed their features, and determinedthe conditions of a concise math expression. We thenproduced a math expression filter by developing classifiers to perform binary classification and compared
the performance of each classifier with the others. Finally, we applied our filter to a task that presents mathexpression images on web searches and discussed theresults.

2

RELATED WORK

Various types of research have been pursued in MIR.
The first is math formula similarity searches, such as
adopting three different similarity measures (Ohashi
et al., 2016), using two different tree structures(Davila and Zanibbi, 2017), and focusing on the similarity of substructures (Zhong et al., 2020). Other
types of MIR include a math document classification
method based on text combined with the structures ofmath expressions (Suzuki and Fujii, 2017), extracting identifier-definiens pairs to improve performancein MIR tasks (Schubotz et al., 2017), and focusing
on partial equations within equations to analyze thefrequency distributions of math expressions in large
909

Yamada, K. and Murakami, H.
Creating a Math Expression Filter to Extract Concise Math Expression Images.
In Proceedings of the 15th International Conference on Agents and Artificial Intelligence (ICAART 2023) - Volume 3, pages 909-915
ISBN: 978-989-758-623-1; ISSN: 2184-433X
Copyright © 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

ICAART 2023 - 15th International Conference on Agents and Artificial Intelligencescientific datasets (Greiner-Petter et al., 2020). For
math image recognition, there are conversions suchas image to LaTeX (Peng et al., 2021; Wang andLiu, 2016) and image to markup (Deng et al., 2017).
Other work summarizes content by generating headlines with math equations (Yuan et al., 2020). However, none of these are intended for the web, andthe datasets are also specific. For the web, a formula search available from a browser is “ApproachZero” (Zhong, 2022), which allows users to search
for formulae in specific databases. For PDFs, there
is research on analyzing PDFs using OCR softwareand presenting math expression images in response
to a query (Yamada and Murakami, 2020) and research on detecting math formula regions as boundingboxes around formulae in PDFs using a CNN (Deyand Zanibbi, 2021). Our research aims to extract concise math expression images from images in HTML
documents on the Web by binary classification without directly analyzing the contents of the images. To
our knowledge, no similar studies were found.

3

EXPERIMENTS

After setting a dataset, we applied preprocessing including elimination of duplicate images, and determined the concise math expression conditions. We
then checked the correct images of the dataset basedon the conditions and conducted two experiments forevaluating the performance of the created classifiers.
Experiment 1 used machine learning methods otherthan deep learning. Experiment 2 used CNNs. Finally we compared all of the classifiers and selected
the best one.

3.1

DatasetTable 1: Dataset. These raw data include duplicate images
and errors in preprocessing. “Other than html” includes
PDFs, slides, Google Books and so on.
DatasetD0 trnD0 val
D0 tstTotal

Image
Total Correct
19,470442
15,427351
23,988929
58,885
1,722

Acquired webpage breakdown
Other than html Error Html
1,091
80 1,829
1,269
60 1.671
1,662
72 2,266
4,022
212 5,766

We use the same dataset as studied in a prior work
(Yamada et al., 2018). We randomly selected 100 keywords from the index of Bishop’s “Pattern Recognition and Machine Learning” (Bishop, 2006) and performed a web search using these keywords as queries
to obtain the top 100 web pages. We created a dataset

910

by extracting all the images from those pages. In Table 1, D0 trn is the keywords from 31 to 60 as the training dataset, D0 val is the dataset from keywords 1 to30 as validation, and D0 tst is the keywords from 61 to
100 as the testing dataset. The first author manuallyjudged images to determine whether they were related
to the keywords. When unclear cases surfaced, judgments were made in consultation with another person(the same person throughout all judgments). Keyword
examples are softmax function, SVM, kernel density
estimation method, Heaviside step function, Gaussian kernel, convex function, Probit function, Boltzmann distribution, functional derivative, and leastmean-squares algorithm.

3.2

Preprocessing

Because of the method used to create D0 trn and D0 val ,they included the same image registered with different
IDs. Therefore, we deleted the ones with overlapping
features. In Experiment 1, the basic features (file size,
width, and height) were used, so images with thesevalues overlapping were deleted. In Experiment 2,
the images were used directly, so the images with thesame features and the same appearance were deleted.
In addition, unnecessary icons such as buttons and logos were removed from the dataset for Experiment
1. We extracted the common strings from the image
names of the unnecessary icons in D0 trn , and images
with these strings in their image names were deleted
in advance.

3.3

Determining Concise MathExpression Conditions

After preprocessing, we obtained 314 of the original 442 keyword-related correct images in D0 trn (Table 1) and analyzed them to identify the conditionsof a concise math expression. As a result of a web
search using the above keywords, many of the correct
math expressions have proper names such as Gaussian kernel. Therefore, they are written in an organized form and are interpretable by the expressions
themselves. That means these images are considered
suitable candidates for concise math expression images. We examined the following by directly viewing the images: “Number of horizontal characters (including symbols),” “number of vertical characters (including symbols),” “number of lines,” “number of expressions1 ,” and “number of concatenations” (=, <,and so on). Because the fonts used in web math ex1 The number of expressions in () is 1. Nested expres-sions in an expression are not counted.

㻲㼞㼑㼝㼡㼑㼚㼏㼥

Creating a Math Expression Filter to Extract Concise Math Expression Images

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㻺㼡㼙㼎㼑㼞㻌㼛㼒㻌㼏㼔㼍㼞㼍㼏㼠㼑㼞㼟pression images are quite different and the layouts ofexpressions are also different, the number of horizontal characters is counted regardless of the size of the
font. For the number of vertical characters, an exponent, a fractional line, and others are set to 0.5. For
example, if an image is x2x−1, the number of hori+x−1
zontal characters is 6 and the number of vertical characters is 3.
The thresholds were determined in order from theeasiest one in Fig. 1. (1) From A, a relational expression is first adopted. A relational expression is one inwhich the left and right sides are connected by “=,”“<,” and so on2 . (2) From B, i = 1 or 2, i: numberof expressions. (3) From C, j = 1 or 2, j: number ofconcatenations. (4) From D, k = 1 or 2, k: number oflines. Then (5) from Fig. 2, the frequency between
4 and 35 is 2 or more, so 4 ≤ l ≤ 35, l: number ofhorizontal characters. (6) From Fig. 3, the frequencyof 6 or less is 2 or more, so 1 ≤ m ≤ 6, m: number ofvertical characters.
Therefore, a concise expression satisfies (1) a relational expression, (2) i = 1 or 2, i: number of expressions, (3) j = 1 or 2, j: number of concatenations, (4)k = 1 or 2, k: number of lines, (5) 4 ≤ l ≤ 35, l: number of horizontal characters, and (6) 1 ≤ m ≤ 6, m:
number of vertical characters.
We manually checked the correct images of the total dataset based on the above conditions. Since ourdataset is very imbalanced, we randomly undersampled the data with “*” in Table 2. In Experiment 2,
we adjusted the numbers of datasets (other than test㻲㼞㼑㼝㼡㼑㼚㼏㼥

Figure 1: Threshold determination. The third row of eachfigure represents the frequency. We determined the thresholds in order from the easiest one (from top to bottom in thetable). After determining a threshold, we searched for the
next one in the images that were already within the threshold.

Figure 2: Number of horizontal characters. This figure
shows the number of characters from 1 to 40 (total frequency 257). The remaining 41 to 91 are omitted because
they are sparsely scattered (total frequency 18).
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Figure 3: Number of vertical characters.The frequency of 4
to 35 characters in Fig. 2 is 251, and this figure shows theresults using those images.

data D2 tst ) for batch processing. Figure 4 shows examples of correct and incorrect images. Correct image (a) is an expression for the complementary error
function and correct image (b) is an expression for thet-distribution. Incorrect image (c) is the coordinatesof a point, and (d) is the output of the math software
for a cell decomposition of an annulus.

3.4

Evaluation Metric for Classifiers

In Experiment1 and 2, our dataset is very imbalanced,
so we use the F-measure (Eq. (1)) to compare a large
number of classifiers.
2 · precision · recall
F-measure =,(1)
precision + recall
where
precision =true positives
true positives + false positives

andrecall =true positives.
true positives + false negatives

2 “=” is 299, “inequality sign” is 14, “≈” is 4, “'” is2, “≡” is 2, “:=” is 2, “∼” is 2, and “” is 1 (including
multiple concatenations in a single image).

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ICAART 2023 - 15th International Conference on Agents and Artificial Intelligence

Table 2: Data for experiments. After preprocessing dataset in Table 1, we randomly undersampled the data with “*” in thetable (The criteria for “Correct” in Table 1 and in this Table are different ).

DatasetCorrect
Incorrect
Total

Experiment 1
D1 trn D1 val
D1 tst
2,410
1,868
4,336
2,562∗ 6,076 12,689
4,972
7,944 17,025

Figure 4: Examples of correct and incorrect images. Images labeled (a) and (b) are correct, while (c) and (d) areincorrect.

3.5

Experiment 1: Machine Learning
Methods Other Than Deep
Learning

In order to create classifiers, we used Weka (Franket al., 2022), which has an easy-to-understand interface that allows us to combine many classifiers andeasily change hyperparameters. It also outputs various evaluation metrics, such as confusion matrices,making it easy to compare classifiers. We createdclassifiers to discriminate concise math expressions
using image features. The basic features were “filesize in bytes,” “number of file width pixels,” and“number of file height pixels.” Furthermore, “density(Eq. (2)),” “aspect ratio (Eq. (3)),” and “gray value(Eq. (4))” were used.
density =filesize,width · height

(2)

width
,
(3)
heightwhere width is the number of width pixels and heightis the number of height pixels.
aspect ratio =gray = |R − G| + |G − B| + |B − R|,(4)

where R is the red component of the image, G isgreen, and B is blue. Eq. (4) is used because somemath expression images may appear to be gray but
actually have a color component. If the gray value is912

D2 trn2,400
2,400∗
4,800

Experiment 2
D2 val bal D2 val imbal
1,900
1,900
1,900∗
8,180
3,800
10,080D2 tst4,462
18,67823,1400, then the image is gray. The “+gray value” in Table 3 is calculated again as a correct image when thepredicted value of the classifier created with featuresother than the gray value is correct and the gray value
is 0. We performed 10-fold cross-validation on every classifier and searched for hyperparameters. Fora multilayer perceptron, we considered the number ofhidden layers, and for a random subspace (Ho, 1998),
which is an ensemble learning method, we consideredseveral classifiers for use. For an SVM (Chang andLin, 2011), we used a nonlinear SVM with an RBF
kernel and searched for the values of c and γ.
In Table 3, methods B and J were the best, so we
conducted an evaluation experiment using test data
D1 tst in Table 2. As a result, both had an F-measureof 0.628 using SVM with c = 7000 and γ = 6. Comparing both, true positives and true negatives were almost the same number, but B had less false positives
and more false negatives, and J had exactly the opposite.

3.6

Experiment 2: CNN Methods

We created CNNs to conduct binary classification that
discriminated between concise math expression images and other images, using Keras (Chollet et al.,2022) with a TensorFlow backend. Previously, a large
amount of image data was required for deep learning, but recently, trained models for transfer learning
have been widely released and are generally available.
We used “data augmentation,” “transfer learning,” and“fine tuning” methods and examined the models provided by Keras. Data augmentation is a common
data increasing operation to avoid overfitting causedby having too few data. Random rotation, translation, equal volume deformation, zooming, and horizontal inversion of half of the image were performed.
The models of transfer learning were pre-trained by
the ImageNet. The number of epochs for fine tuning
was set at 100, but only NASNetMobile was stopped
early to avoid overfitting. Table 4 shows the resultsof the final test data using Table 2’s data. First, as abaseline, we constructed method A with four Conv2D
layers and four MaxPooling2D layers. The activation
function is the sigmoid function, and the loss functionCreating a Math Expression Filter to Extract Concise Math Expression Images

Table 3: Creating classifiers using machine learning method other than deep learning. Top results by each feature.

Method
A
B
C
D
E
F
G
H
I
J
K
L

Featurebasic features
basic features, density
basic features, aspect ratio
basic features, density, aspect ratio
basic features, gray value
basic features, density, gray value
basic features, aspect ratio, gray value
basic features, density, aspect ratio, gray value
basic features, +gray value
basic features, density, +gray value
basic features, aspect ratio, +gray value
basic features, density, aspect ratio, +gray value

is binary cross entropy. We examined eight different
patterns for each model, which were whether to use
D2 val bal (balanced data) or D2 val imbal (imbalanceddata) as validation data, whether to use data augmentation or not, and whether to use fine tuning or not.
As a result, method N was found to be the best.

3.7

Experimental Results

APPLICATION

We applied the experimental results to the previous
study (Yamada et al., 2018). The task performs web
searches using a math term as a query and presents
the top ten math expressions related to the term with
its surrounding information. Score(ik ) is given to each
expression ik by Eq (5):score(ik ) = xline + xkey + xsvm + xbonus ,(5)where xline = 1 if ik is in a separate line, 0 otherwise;
xkey = 1 if ik has a keyword within a window size set
from -200 to +200 characters, 0 otherwise; xsvm = 1
if ik is discriminated as positive by the SVM, 0 otherwise; and xbonus = 1 if ik has a perfect score so far
and appears first in the same web document, 0 otherwise. The classifier is a nonlinear SVM with an RBF
kernel. The correct image is a math expression image
related to the keyword, which is different from thisF-measure
0.654
0.677
0.661
0.669
0.638
0.649
0.635
0.653
0.658
0.677
0.623
0.626

Rank
6
1
4
3
9
8
10
7
5
1
12
11

study. Evaluation metrics are the F-measure (Eq. (1))
where precision = nr and recall = cr , Mean Reciprocal
Rank (MRR) (Eq. (6)), and Mean Average Precision(MAP). MAP is the macro mean of the Average Precision(AP) (Eq. (7)):
1 n 1∑ r0 ,
n k=1
k

(6)

1
I(s)Prec(s),min(n, c) ∑
s

(7)

MRR =

Despite various performances in Experiment 1,
method N (Table 4) was the best throughout Experiment 1 and 2. The conditions were (1) use VGG16(Simonyan and Zisserman, 2014) as transfer learning,(2) use balanced data as validation data, (3) do not use
data augmentation, and (4) use fine tuning. Thus, thedesign of our math expression filter was completed.

4Classifier
SVM
SVM
SVM
SVM
MultilayerPerceptron
RandomSubSpaceMultilayerPerceptron
RandomSubSpace
SVM
SVM
SVM
SVM

AP =

where r is the number of correct images of top n, c
is the total number of correct images, rk0 is the rankof the correct images at the top of the k-th keyword,
I(s) is a flag indicating whether the image at s-th iscorrect, and Prec(s) is the precision at s-th.
We reranked them by replacing the xsvm values in
Eq. (5) with our filter’s values. Table 5 shows that ourfilter worked well.

5

DISCUSSION

Concerning our data, using data augmentation should
be avoided because none of the math web images aretilted or upside down. However, since the sizes of theimage margins vary, and some of the images are unclear after preprocessing (α channel removal and soon), data augmentation might be considered in thesecases. We plan to study data augmentation in suchcases in our future work. For the validation data when
dealing with imbalanced data, Table 4 does not show
which choice was better, balanced or imbalanced data.
For the application, since the original classifier discriminated whether an image was related to a given
keyword, the same image may be correct or incorrect,
depending on its context. The ability to identify correct images regardless of context led to improvement.
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ICAART 2023 - 15th International Conference on Agents and Artificial Intelligence

Table 4: Creating classifiers using CNNs. Top results by each model, where “+” is used and “−” is not used.

Method
M
N
O
P
Q
R
S
T
U
V
W
X
Y

Transfer learning model

Validation databaseline
VGG16
DenseNet169
VGG19
DenseNet201
DenseNet121
ResNet50
InceptionResNetV2
MobileNet
MobileNetV2
Xception
InceptionV3
NASNetMobile

D2 val imbal
D2 val bal
D2 val imbal
D2 val imbal
D2 val bal
D2 val bal
D2 val imbal
D2 val bal
D2 val imbal
D2 val bal
D2 val imbal
D2 val imbal
D2 val imbal

Table 5: Comparison of evaluation metrics@10.

Previous study
Using filter

6

F-measure
0.40
0.55

MRR
0.84
0.86

MAP
0.42
0.45

CONCLUSION

We developed a math expression filter to extract concise math expression images. We determined the conditions satisfied by a concise math expression anddeveloped classifiers that discriminate the images ofconcise math expression images from web images using supervised machine learning methods based onthese conditions. To investigate our filter’s performance, we applied it to a task that presents math expression images and obtained good results.

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