Complex network-based classification of radiographic images for COVID-19 diagnosis

Abstract

In this work, we present a network-based technique for chest X-ray image classification to help the diagnosis and prognosis of patients with COVID-19. From visual inspection, we perceive that healthy and COVID-19 chest radiographic images present different levels of geometric complexity. Therefore, we apply fractal dimension and quadtree as feature extractors to characterize such differences. Moreover, real-world datasets often present complex patterns, which are hardly handled by only the physical features of the data (such as similarity, distance, or distribution). This issue is addressed by complex networks, which are suitable tools for characterizing data patterns and capturing spatial, topological, and functional relationships in data. Specifically, we propose a new approach combining complexity measures and complex networks to provide a modified high-level classification technique to be applied to COVID-19 chest radiographic image classification. The computational results on the Kaggle COVID-19 Radiography Database show that the proposed method can obtain high classification accuracy on X-ray images, being competitive with state-of-the-art classification techniques. Lastly, a set of network measures is evaluated according to their potential in distinguishing the network classes, which resulted in the choice of communicability measure. We expect that the present work will make significant contributions to machine learning at the semantic level and to combat COVID-19.

Full text

RESEARCH ARTICLE
Complex network-based classification of
radiographic images for COVID-19 diagnosis
Weiguang Liu
1
, Rafael Delalibera RodriguesID
2
, Jianglong Yan
2
, Yu-tao Zhu
3
, Everson
Jose
´de Freitas Pereira
5
, Gen Li
4
, Qiusheng Zheng
4
, Liang ZhaoID
3,5
*
1School of Computer Science, Zhongyuan University of Technology, ZhengZhou, China, 2Institute of
Mathematics and Computer Science (ICMC), University of São Paulo (USP), São Carlos, Brazil, 3China
Branch of BRICS Institute of Future Networks, ShenZhen, China, 4Henan Key Laboratory on Public Opinion
Intelligent Analysis, Zhongyuan University of Technology, ZhengZhou, China, 5Department of Computing
and Mathematics, FFCLRP, University of São Paulo (USP), Ribeirão Preto, Brazil
*zhao@usp.br
Abstract
In this work, we present a network-based technique for chest X-ray image classification to
help the diagnosis and prognosis of patients with COVID-19. From visual inspection, we per-
ceive that healthy and COVID-19 chest radiographic images present different levels of geo-
metric complexity. Therefore, we apply fractal dimension and quadtree as feature extractors
to characterize such differences. Moreover, real-world datasets often present complex pat-
terns, which are hardly handled by only the physical features of the data (such as similarity,
distance, or distribution). This issue is addressed by complex networks, which are suitable
tools for characterizing data patterns and capturing spatial, topological, and functional rela-
tionships in data. Specifically, we propose a new approach combining complexity measures
and complex networks to provide a modified high-level classification technique to be applied
to COVID-19 chest radiographic image classification. The computational results on the Kag-
gle COVID-19 Radiography Database show that the proposed method can obtain high clas-
sification accuracy on X-ray images, being competitive with state-of-the-art classification
techniques. Lastly, a set of network measures is evaluated according to their potential in dis-
tinguishing the network classes, which resulted in the choice of communicability measure.
We expect that the present work will make significant contributions to machine learning at
the semantic level and to combat COVID-19.
Introduction
In late 2019, a viral respiratory disease outbreak emerged, named “coronavirus disease 2019”
(COVID-19), caused by a new type of coronavirus called SARS-CoV-2 (Severe Acute Respira-
tory Syndrome Coronavirus 2) [1–4]. It presents unique virological features, which boosts its
transmission efficiency, e.g., presenting a high viral load during the first week of symptoms,
which increases the pharyngeal virus shedding [3]. This feature drastically reduces the interval
between symptoms onset and the peak of infectivity and in conjunction with the high propor-
tion of mild illness facilitates undetected transmission [1,5], resulting in a quite high R-naught
(basic reproduction number) [6–8]. Thus, the high transmission efficiency of the virus and the
abundance of international travel rapidly turned the COVID-19 outbreak into a worldwide
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OPEN ACCESS
Citation: Liu W, Delalibera Rodrigues R, Yan J, Zhu
Y-t, de Freitas Pereira EJ, Li G, et al. (2023)
Complex network-based classification of
radiographic images for COVID-19 diagnosis. PLoS
ONE 18(9): e0290968. https://doi.org/10.1371/
journal.pone.0290968
Editor: Zhaoqing Pan, Nanjing University of
Information Science and Technology, CHINA
Received: June 5, 2022
Accepted: August 3, 2023
Published: September 1, 2023
Copyright: ©2023 Liu et al. This is an open access
article distributed under the terms of the Creative
Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in
any medium, provided the original author and
source are credited.
Data Availability Statement: The source code for
the method presented in this manuscript is
available on a GitHub repository at https://github.
com/lscc-usp/Modified-High-Level-Classification.
We have also used Zenodo to assign a DOI to the
repository: 10.5281/zenodo.7317559. The data
underlying the results presented in the study are
available from Kaggle COVID-19 Radiography
Database at https://www.kaggle.com/datasets/
tawsifurrahman/covid19-radiography-database/
versions/1. All other relevant data are within the
manuscript.

pandemic. Therefore, containment measures are very important to reduce COVID-19 spread-
ing, requiring quick and precise testing for timely patient identification.
The early diagnosis of COVID-19 patients is of utmost importance, not only for the patient’s
prognosis but also for optimizing hospital resources in response to the COVID-19 pandemic.
Efficiency is critical to avoid a crisis in the healthcare system and, consequently, an increase in
the number of deaths [9]. In a pandemic, the shortage of resources is practically unavoidable,
thus any available resource and tool that can help should be employed to combat the COVID-
19 public health emergency. In this context, medical imaging techniques play a crucial role in
accurately diagnosing and assessing chest involvement in COVID-19 patients. While Magnetic
Resonance Imaging (MRI) and Computed Tomography (CT) imaging techniques provide
higher resolution, they are costlier and less accessible compared to X-ray imaging. This is an
important issue for public health organizations and low-income people. Therefore, chest X-ray
images are a more encompassing technique in the context of COVID-19.
The field of COVID-19 chest X-ray image processing has seen significant advancements
with various studies focusing on image segmentation and classification. Most of these works
rely heavily on deep learning techniques, such as Convolutional Neural Networks (CNNs) and
transfer learning, to achieve high accuracy in detecting COVID-19 and other diseases. How-
ever, these deep learning techniques often demand substantial computational resources and
lack explicit interpretability in their learning results.
Moreover, traditional data classification algorithms rely only on physical characteristics
extracted from data, such as similarity, distance, or distribution, to determine the representa-
tion of data classes. These methods are called low-level classification and are particularly sus-
ceptible to errors when dealing with complex problems, e.g., distinct contexts, overlapping
classes, etc. Oppositely, human beings intuitively deal with complex scenarios, classifying
objects at an organizational and semantic level, taking into consideration pattern recognition.
The computational methods that take into consideration not only physical aspects of data but
also pattern formation are called high-level classification [10].
Distinctively, high-level classification techniques require a singular fundamental data struc-
ture to support an enhanced representation of data classes: complex networks. This represen-
tation arises from the complex networks’ research, which deals with many real systems that
naturally behave as a network or benefit a lot from an abstract representation in the network
form, also known as Network Science [11]. Complex networks refer to large-scale graphs with
nontrivial connection patterns [11–16]. This type of network is a suitable tool for characteriz-
ing data patterns due to its ability to capture spatial, topological, and functional relationships
in data. The interconnections between nodes in a network naturally allow for the identification
of patterns, requiring only some suitably defined measures; thus producing an intrinsic high-
level classifier.
From this perspective, by analyzing healthy and COVID-19 lung X-ray images, it is possible
to see that both classes present visually distinguishable patterns, such as the formation of fila-
ments on COVID-19 images that spread through the X-ray image as an opacity texture. Such a
finding implies that the two classes of images present different geometrical complexity. For
this reason, we inspect two complexity measures to characterize these differences: fractal
dimensions [17,18] and quadtree [19].
Thus, in this work, we seek to identify automatically the diagnosis of COVID-19 through
the analysis of chest X-ray images, using a new approach that combines complexity measures
and complex networks to provide a modified high-level classification technique. In the pro-
posed scheme, the complexity measures are employed in the feature extraction phase, extract-
ing characteristics of the images that are relevant to the applied problem, in this specific case,
those that help to distinguish the normal and COVID-19 classes. In a high-level classifier, this
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Funding: This work is carried out at the Center for
Artificial Intelligence (C4AI-USP), with support
from the São Paulo Research Foundation
(FAPESP) and the IBM Corporation under FAPESP
grant number 2019/07665-4, granted to LZ. This
work is also supported in part by the Coordenac¸ão
de Aperfeic¸oamento de Pessoal de Nı
´vel Superior -
Brasil (CAPES) - Finance Code 001, granted to
RDR. The funders had no role in study design, data
collection and analysis, decision to publish, or
preparation of the manuscript.
Competing interests: The authors have declared
that no competing interests exist.

step provides the data for constructing the network representations for each class. This distinct
approach emphasizes pattern formation within the data rather than relying solely on physical
features, providing a straightforward and explainable approach to data classification. By
leveraging complexity measures and complex networks, our method offers a unique perspec-
tive on COVID-19 chest X-ray image classification, combining accuracy with explainability.
Our work is inspired by the original idea of high-level classification, proposed in [10,20]
and extended in [21,22]. However, our modified high-level technique is not hybrid, not
requiring an applied combination with low-level classifiers, in contrast to [10,20]. Besides
that, although the works [21,22] also eliminate the need for the low-level classification part,
these works demand the comparison of diverse network measures, requiring the optimization
of various hardly tunable parameters. For this reason, we introduce a modified high-level clas-
sification technique using only one network measure, the communicability measure [23],
which eliminates those parameter calibration tasks.
Finally, our work aims to analyze chest X-ray images to assist in the diagnosis and prognosis
of patients with COVID-19. We employ a pair of complexity measures in conjunction with a
modified high-level classification technique. The technique captures and explores the complex
topological properties of the networks built for each of the classes from the input data, per-
forming the classification of healthy and COVID-19 X-ray images according to the conformity
of the testing sample to the network structure of each class, where its insertion causes the least
variation of the network measure under consideration. The experimental results show that the
proposed method achieves high accuracy in the classification task, being competitive with
state-of-the-art techniques. The primary contributions of this paper are summarized as
follows:
1. In this paper, we propose a new high-level data classification technique capable of classify-
ing data samples according to the pattern formation of each class instead of physical fea-
tures, such as distance, similarity, or distributions. Moreover, we find that complex
networks are suitable solutions to characterize data patterns.
2. Our hypothesis is that the COVID-19 images, although present high variations, may share
some hidden patterns, therefore, the high-level technique for COVID-19 image classifica-
tion is a suitable choice. The high classification precision obtained by our method in the
simulations using artificial and real datasets confirms such a hypothesis.
3. State-of-the-art classification techniques, such as deep learning techniques, require massive
computing power and do not provide a logical explanation of the learning results. On the
contrary, our method, although requires large memory space for a large network, presents a
straightforward and explainable way for data classification and requires the tuning of only a
single parameter k, which serves to network construction from the original data.
4. Instead of using statistical measures, we apply fractal dimension and quadtree measures to
characterize complexity levels between different classes of images.
Related works
Hereupon, many studies have achieved great successes in identifying COVID-19 through
chest image processing for computer-aided medical diagnosis [24–26]. These can differ in sev-
eral aspects, such as the encompassed stage or stages of image processing, the contemplated
medical imaging techniques, the applied classification approaches, and the types of medical
images, among others. In this context, we present here a brief review of some of the relevant
works.
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In terms of image segmentation, in [27] authors propose a method applied to Computed
Tomography (CT) chest images that are based on multi-agent deep reinforcement learning to
sharpen the automatic masking process. The technique has been evaluated in a combined data-
set collection, and compared against various similar-purpose state-of-the-art methods, all
based on deep learning frameworks, achieving an accuracy of 97.1%. Other studies pursue the
same objective of segmentation for COVID-19 CT images using deep learning architectures,
like in [28] where authors use UNet++ as a feature extractor to achieve the purpose of segmen-
tation, and later, combine it with ResNet-50 used as a backbone. In [29] authors evaluate the
VGG-19 deep learning architecture applied to the segmentation of nodules in lung CT images.
The proposed methodology shows promising results in a segmentation task that potentially
shares relevant similarities with COVID-19, achieving a maximum of 97.83% accuracy with
the SVM-RBF classifier.
A distinct approach involving COVID-19 CT image segmentation is presented in [30],
where an extended segmentation-based fractal texture analysis composes the feature extraction
phase united with other techniques, such as discrete wavelet transform. Then, an entropy-
based genetic algorithm performs the feature selection, generating an optimal fused feature
space, which is evaluated with various traditional classifiers. The method obtains the best accu-
racy with the naive Bayes classifier, presenting 92.6% of accuracy for a dataset obtained from
radiopaedia. On the same dataset, the study in [31] employs two pre-trained deep learning
models, AlexNet and VGG-16, for COVID-19 classification, not in a segmentation-based
approach. Instead, they first apply a hybrid contrast enhancement technique and, later, fine-
tune the deep methods. The features from both models are extracted and fused, and the opti-
mal features are selected with an entropy-controlled firefly optimization method. Lastly, these
features are, then, provided to various traditional machine learning classifiers, where the SVM
achieved the best accuracy of 98%.
For chest X-ray images, [32] proposes a multi-class framework for detecting 15 types of dis-
eases, including COVID-19. It implements a Convolutional Neural Network (CNN) for deep
feature extraction, later these are fed, via transfer learning, to traditional machine learning clas-
sification methods for boosting the prediction results. When evaluating the technique, the
authors combined two datasets, where the NIH Chest X-ray Dataset provides the samples for
14 types of chest-related diseases and the COVID-19 Chest X-ray Image Dataset provides the
samples for COVID-19 X-rays. This two-way classification scheme saturates the learning accu-
racy on 87.8% with CNN and later improves to an accuracy of 99.8% with k-NN. In [33],
authors combine three different sources of chest X-ray images with synthetic data from a Gen-
erative Adversarial Network (GAN) to obtain a balanced 15 disease classes (including COVID-
19) dataset. Then, four independent deep learning models are evaluated concerning the classifi-
cation capability, with ResNet-152 providing the higher accuracy of 87%. The work presented
in [34] proposes a new Domain Extension Transfer Learning (DETL) approach to deal with the
limitations presented by COVID-19 chest X-ray images available to that date. The DETL has
been evaluated along with AlexNet, VGG-16, and ResNet-50 for a classification problem con-
taining four classes. The overall best accuracy of 90.13% is obtained with VGG-16.
Other notable approaches to COVID-19 chest X-ray image classification should also be
mentioned. In [35], authors address the problem by proposing a multi-scale BoDVW-based
(Bag of Deep Visual Words) feature extraction, where the raw feature map is extracted from
the 4th pooling layer of a VGG-16 pre-trained model, capturing detailed semantic relation-
ships with three distinct kernels. The proposal is applied as a multi-class classifier and evalu-
ated with four datasets in comparison to five other state-of-the-art methods. The results
present a significant improvement in performance with a top accuracy of 90.29%. When con-
sidering improvements in explainability, the study in [36] exemplifies the ongoing research
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efforts to enhance the accuracy and interpretability of deep learning models for chest X-ray
image analysis. By incorporating XAI (eXplainable Artificial Intelligence) techniques into a
single lightweight CNN, researchers have provided a four-class classification model that out-
performed the existing methods, presenting an accuracy of 95.94%, while providing medical
radiology experts with noticeable tools for aiding in the interpretation.
High-Level classification preliminaries
Here, the foundations for the Network-Based High-Level classification [10,20] are presented
with some of its properties and distinguishing features, which are essential to the proper con-
ceptual understanding of the proposed method.
Generally, a data classification problem is presented in its supervised form, when there is a
set of known labels (indicating the corresponding classes) for a subset of data. This set can be
given to an algorithm to be used as the training set for the learning process; named supervised
learning process in this scenario. With it, the algorithm can check the quality of the relations
inferred between the attributes, and thus the learning is done by iteratively adjusting the
parameters for those relations and minimizing the error. Ultimately, it achieves the best
parameters, establishing a certain model for the data, with which it is possible to predict the
proper class, mapping data samples to the expected class with a certain accuracy.
For the described inference process, the vast majority of data classification methods con-
sider only the “physical features” from the provided data, e.g., distance, density, similarity, or
distribution, to guide the learning process. Also, the act of learning can be understood as the
establishment of some sort of decision boundary capable of segmenting the n-dimensional
space of data into distinct regions that represent the expected classes. Therefore, in the context
of a high-level classification, these methods are referred to as “low-level classification”.
Thereby, the central concept for the high-level classification contrasts with the prior meth-
odology by focusing on the identification of intrinsic pattern formation from the provided
data, instead of relying only upon physical relations between features. This conceptual change
intends to allow the task of classification to be done by analyzing the data at an organizational
and semantic level, no matter how similar or dissimilar when measured by the physical fea-
tures of the data.
With this, it is expected that a high-level classification technique can explore non-trivial
problems and scenarios, where the feature space of data is presented with complex geometric
distributions (e.g., twisted shapes, overlapping, contexts, etc.). The process employed by
low-level classification of determining regions and boundaries in the n-dimensional space
on complex scenarios becomes an unnatural way of addressing and modeling these
problems.
In order to illustrate the salient features of the high-level pattern-based classification. We
present the following simulation results on a toy dataset. Fig 1 illustrates an artificial and very
simple scenario where the low-level classification techniques fail to infer the correct expected
class for a certain data sample (in red). There are two classes: The blue data conforms to the
circle pattern and the green data conforms to a triangular pattern. The training data, shown by
Fig 1(a), is provided to various classical and state-of-the-art machine learning algorithms (Fig
1(b)–1(i)). After the training phase, it is possible to observe the varied decision boundary con-
figuration strategies. However, when presenting the red testing instance, according to the tra-
ditional low-level learning mechanism of relying on purely physical relations, the testing
sample is inferred as belonging to the blue circle class, delimited by its respective boundary.
This occurs due to the high affinity of the testing sample and the blue class data elements
according to the “physical” guiding measures: distance, density, similarity, and distribution.
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However, when examining Fig 1(a), our mind intuitively perceives the red sample as
belonging to the green triangular class in a much more natural way, as it exhibits a high degree
of conformity with the pattern displayed by the green class. In accordance with it, Fig 1(j)
shows that the proposed high-level classification technique correctly identifies the red testing
instance into the proper green class. Since it constructs a distinct network for each of the clas-
ses, the testing sample is evaluated according to the conformity it presents to each class, being
attributed to the one that causes less perturbation. In other words, the sample is attributed
according to the pattern conformation to the class network. Thereby, the high-level classifica-
tion technique gives higher relevance to pattern formation in the inference process, surpassing
the guidance purely based on physical measurements.
The high-level classification approach proposed in [10,20] addresses this concept by repre-
senting the input data as networks, where a distinct network is constructed for each one of the
classes during the training phase. The prediction (testing phase) is performed with the inser-
tion of the test sample, being evaluated, into each of the networks and, then, analyzing the con-
formity of the element with each network, concerning the caused perturbation in it. Thus, the
test sample is assigned to that class where its insertion in the corresponding network causes
the least variation of the measures under consideration.
In both works, a hybrid approach is implemented for the prediction phase, unifying the
low-level classification, which can be implemented by any traditional classification technique,
Fig 1. A simple non-trivial comparative classification scenario. The red sample is located in a dense blue region, despite presenting high conformity
with the sparse green triangular pattern. (a) The input training data provided to all algorithms is presented in the figure. The red sample is provided
only for the testing phase. (b-i) Various traditional machine learning techniques. All failed to predict the red testing sample into the green class. (j) The
high-level classification technique is the only one to correctly assign the red testing sample to the green class.
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with the high-level approach, which explores the complex topological properties of the net-
work built from the input data. The difference between these two works resides in the strategy
to evaluate the network perturbance when inserting the testing element. In the work intro-
duced in [10], the high-level classification is performed using three network measures: assorta-
tivity, clustering coefficient, and average degree. While in [20], the network perturbation is
characterized by measures obtained from the dynamics of tourist walks, extracting the tran-
sient and the cycle lengths to describe network patterns.
Later, this concept is extended in [21,22], where the low-level classification component has
been eliminated and pure high-level classification techniques have been proposed. However,
in those works, several network measures are employed in conjunction, introducing a large set
of parameters representing the weight for each measure, which are hard to correctly deter-
mine. For this reason, we introduce a modified high-level classification technique using only
one network measure, the communicability measure [23], which largely reduces the parameter
calibration task while keeping great performance.
Materials and methods
In this section, we present the proposed method for classifying chest X-ray images step by step.
In brief, our method starts to apply feature extraction methods to both COVID-19 and normal
X-ray images. This allows each image to be represented in the n-dimensional space by its fea-
ture vector. Then, the training phase is started by constructing a distinct network for each class
(normal and COVID-19). Having those networks, a new unlabeled data sample can be evalu-
ated in the testing phase. Given this sample, already, as a feature vector, it is incorporated into
each network, and later, the level of caused perturbation is calculated with the communicabil-
ity network measure. Lastly, the testing sample is associated with that class in which the sample
caused the least variation in the measure.
An overview of the method is illustrated in Fig 2. The text that follows will cover all the
phases in the figure, together with the foundations, motivations, and techniques that compose
our method.
Image feature extraction
When dealing with image processing, feature extraction is a common step, if not mandatory.
Since each image contains a huge amount of information, it is important to find metrics or
techniques to extract relevant characteristics of this image. Otherwise, dealing directly with the
image pixels would be a hard task to treat variations of images in the same class and pose the
classification problem in a high dimensional space that would drastically hamper the classifica-
tion accuracy due to the curse of dimensionality and high computational complexity.
Usually, it requires some feature engineering to find appropriate or optimal features
according to the specificities of the problem. We address this feature extraction process by ana-
lyzing two metrics that present promising characteristics to the applied problem, they are frac-
tal dimension [17,18] and quadtree [19].
Fractal dimension. Fractal dimension [17,18] can be used to describe the geometrical
complexity of the images. It allows the quantification of an image according to complexity
analysis, where patterns of visual information contained in an image can be abstracted as frac-
tal geometry, according to how the detail in a pattern changes with scale. Thus, the fractal
dimension provides a measure that acts as an index of complexity, where a larger value indi-
cates a higher complexity level, and, on the contrary, smaller values indicate lower complexity.
These characteristics motivated the use of this measure as a feature extraction method in our
work.
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For calculating the fractal dimension, we use the box-counting method [18], since it is
widely used in fractal analysis. The method is applied to binary images. Therefore, given a
gray-level chest X-ray image, we first generate several binary images simply using a series of
increasing threshold values. Some examples of the original images are shown in Fig 3 and their
corresponding resulting binary images can be seen in Figs 4and 5.
Then, for each binary image, we cover the image with a grid and then count how many
boxes of the grid are covering the pattern in the image. Then we repeat the process but use
smaller boxes. By shrinking the size of the boxes repeatedly, we can accurately capture the
structure of the pattern. The fractal dimension Dis the slope of the line when we plot the value
of log(N) against the value of log(r):
D¼logðNÞ
logðrÞð1Þ
where Nis the number of boxes that cover the pattern and ris the inverse of the box size.
Fig 2. Overview of the modified high-level classification technique. (a) Image feature extraction phase maps images to points in n-dimensional space.
(b) The training phase constructs the network for each of the classes; normal and COVID-19. (c) The testing phase incorporates the unlabeled sample to
be evaluated by its impact on a defined network measure, predicting its membership to the network with the least variation of the measures under
consideration.
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Finally, for each gray-level image, we get a vector of values, each of which is the box-count-
ing dimension of one of its binary images. Thereby, each vector acts as a feature vector that
maps an image to a point in the n-dimensional space, as represented in Fig 2(a).
Quadtree. Analogous to fractal dimension, quadtree [19] can also be used to describe the
geometrical complexity of the images. From a slightly different perspective, it allows the
Fig 3. Chest X-ray images of healthy and infected COVID-19 lungs. (a-d) Healthy chest X-ray images. (e-h) COVID-19 chest X-ray images. The
images are taken from the COVID-19 Radiography Database [37].
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Fig 4. Four normal X-ray images and their corresponding binary images. The original gray-level images are leftmost.
Each binary image, from left to right, is generated with threshold values 100, 110, 120, 130, 140, and 150, respectively.
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decomposition of the two-dimensional space, by partitioning it recursively into four new
quadrants when a non-homogeneous area is found. This decomposition strategy leads to a few
blocks of big size covering the vast homogeneous regions and many blocks of small size wrap-
ping the very heterogeneous regions, richer in detail.
Then, the resulting distribution of block sizes and their corresponding quantity (occur-
rence) can be used to describe and compare images according to how homogeneous or hetero-
geneous they are. This is sufficient motivation for evaluating this algorithm as a
complementary feature extraction method in our work.
Here, we apply the quadtree algorithm directly for each gray-level image and later get the
histogram for the distribution of block sizes and quantities that compose the values for the fea-
ture vector. Again, each vector can map an image to a point in the n-dimensional space; Fig 2
(a). Both feature extraction strategies can be employed separately or in conjunction, united in
a mixture feature vector.
The modified high-level classification technique
As the original high-level classification proposed in [10,20], the modified high-level classifica-
tion technique, introduced in this work, is also divided into two phases: the training phase and
the testing phase. However, the modifications we propose reduce the complications of parame-
ter calibration since the number of parameters is diminished. We implement two modifica-
tions in this respect. The first modification incorporates the strategy used in [22], where the
two parameters (kand r) required for the network construction phase are reduced to one, with
the radius (r) being determined according to the parameter k. The second improvement
reduces the parameters in the testing phase with the employment of the communicability mea-
sure [23], which shows more robust performance when analyzing network perturbation for
this specific applied problem.
In the training phase, we construct a distinct network for each class of image features,
where the feature vector of each image is a node and the connections between nodes are
Fig 5. Four COVID-19 X-ray images and their corresponding binary images. The original gray-level images are
leftmost. Each binary image, from left to right, is generated with threshold values 100, 110, 120, 130, 140, and 150,
respectively.
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formed by a technique that uses either k-Nearest Neighbors (k-NN) or Radius Neighbors
(RN). Where the k-NN is used for sparse regions and RN for dense regions. An illustration of
this phase is in Fig 2(b).
If a feature vector of an image has a small number of similar feature vectors of other images,
the corresponding node in the network falls in a sparse region. In this case, this node is con-
nected to its kmost similar nodes. On the other hand, if a feature vector has a large number of
similar ones, it falls in a dense region and is connected to all the nodes within a predefined
similarity radius. Thus, the network construction criterion is defined by the following equa-
tion:
NðxiÞ ¼
RNðxi;YxiÞ;if jRNðxi;YxiÞj >k
kNNðxi;YxiÞ;otherwise
8
<
:ð2Þ
where x
i
is a data sample (already mapped as a feature vector) and Yxidenotes the class label of
x
i
, indicating that only neighbors of the same class should be considered.
In addition, we define the rradius of Radius Neighbors by the following equation:
r¼medianfkNNdistðxi;YxiÞg ð3Þ
where kNN
dist
brings the distances from all x
i
that are members of the same class in the train-
ing set to its k-NNs also in the same class.
Thus, the parameter ris calculated according to the value of k, being the median of the dis-
tance values returned when k-NN is applied to all points as the origin. This leaves the new pro-
posed model with basically a single parameter, it is worth remembering that kis a natural
integer parameter.
After the network construction, the testing phase can be initiated. The purpose of this
phase is to predict to which class a certain unlabeled testing data sample should belong. The
steps involved in this phase are shown in Fig 2(c). In this phase, we classify the unlabeled data
samples one by one. Firstly, a new data sample is inserted (temporarily) into each of the two
networks constructed so far based on the same principle used during the network construction
phase, as represented by Eq 2.
Then, the same measure of each network after the insertion, G
after
(class
i
), i= 1, 2 is calculated
and compared to the measure of each network (each class) before the insertion, G
before
(class
i
),
i= 1, 2. Now, we have the impact of the insertion of the new sample to each class, given by,
DGðclassiÞ ¼ jjGbeforeðclassiÞ  GafterðclassiÞjj;i¼1;2:ð4Þ
Finally, the new sample is classified to class j, where
DGðclassjÞ ¼ minfDGðclassiÞg;i¼1;2:ð5Þ
In our method, we use the average communicability measure hMvii[23] as G
before,after
(class
i
),
which accounts not only for the shortest paths connecting two nodes but also the longer paths
with a lower contribution. The communicability Mvifrom node v
i
to all other nodes of the net-
work is described by,
Mvi¼1
ðN1ÞX
j2N
1
s!PvivjþX
k>s
1
k!Wvivj
!;i6¼ jð6Þ
where sis the length of the shortest path between v
i
and v
j
,Pvivjis the number of shortest paths
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between v
i
and v
j
and Wvivjis the number of paths connecting v
i
and v
j
of size k>s. The reason-
ing behind this choice is that the shortest paths are significantly affected by structural changes in
a network.
In other words, the new sample conforms to the pattern formed by network jif it doesn’t
generate a larger perturbation to the network j. Observe that the new sample can even stay far
from the elements of class jin the physical space.
Algorithms
Here, we provide the algorithms for the modified high-level classification technique. Algo-
rithm 1 describes the steps for the training phase, while Algorithm 2 refers to the testing
phase.
This experimental setup is common to all supervised learning algorithms. In the training
phase, the algorithm constructs a model from the provided labeled training data, and in the
testing phase, it predicts the label for the queried unlabeled samples. Traditional machine
learning approaches determine the classification model relying on physical characteristics of
the data space, by the demarcation of decision boundaries.
Therefore, the main distinction of our approach resides in the fact that the classification
model is now represented by a network formation process, resulting in a distinct network
representation for each of the classes.
Algorithm 1 Training phase: Modified high-level classification
Input: A given training dataset composed of nimages and the corre-
sponding label vector.
Output: A distinct network representing each of the classes.
1: Calculate the feature vector for each of the nimages, using frac-
tal dimension and quadtree
2: for each class
c
do
3: Calculate the k-NN data structure for all samples of class
c
4: r medianfkNNdistðxi;YxiÞg
5: G(class
c
) Add an unconnected node representing each data sample
of class
c
6: for x
i
2class
c
do
7: if jRNðxi;YxiÞj >kthen
8: G(class
c
) Connect x
i
representative node with fRNðxi;YxiÞg
9: else
10: G(class
c
) Connect x
i
representative node with fkNNðxi;YxiÞg
11: end if
12: end for
13: GbeforeðclasscÞ hMvii
14: end for
In Algorithm 1, the “for” in Line 2 controls the construction of a network for each training
class. In Lines 6 to 12, a data sample x
i
is really inserted in the corresponding network accord-
ing to the rules defined by Eqs 2and 3. Line 13 calculates the average communicability of each
network using Eq 6.
Algorithm 2 Testing phase: Modified high-level classification
Input: A testing instance x
t
.
Output: The predicted label of the testing instance: ^
Yxt.
1: Calculate the feature vector for the testing instance, using frac-
tal dimension and quadtree
2: for each class
c
do
3: if |RN(x
t
,class
c
)| > kthen
4: G(class
c
) Connect x
t
representative node with {RN(x
t
,class
c
)}
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5: else
6: G(class
c
) Connect x
t
representative node with {kNN(x
t
,
class
c
)}
7: end if
8: GafterðclasscÞ hMvii
9: ΔG(class
c
) ||G
before
(class
c
) − G
after
(class
c
)||
10: ^
Yxt minfDGðclasscÞg
11: G(class
c
) Remove x
t
representative node
12: end for
In Algorithm 2, Line 2 controls the insertion of a testing data sample x
t
in each class net-
work. Specifically, x
t
is inserted into the network class
c
by Lines 3 to 7, according to Eq 2. Line
8 calculates the average communicability measure after the insertion of x
t
. Line 9 checks the
variation of the average communicability measure before and after the insertion, according to
Eq 4. Thus, Line 10 determines the class to which x
t
belongs based on the pattern conforma-
tion criteria. Lastly, Line 11 discards the testing sample by removing the corresponding node
from both networks.
The computational complexity of the modified high-level classification technique can be
determined as follows. We first examine the computational cost for the training phase, where a
network is constructed to represent each class of the provided training dataset. This phase ini-
tially requires a distance matrix for the k-NN and Radius Neighbors, presenting a complexity
of O(n
2
) to be constructed, and O(1) to query the neighbors of a data sample. Next, the median
to determine the radius ris calculated in O(n). Thus, the network formation can be concluded
with O(n
2
), and the network measure can be calculated with O(M). Resulting in a total compu-
tational complexity of O(n
2
+M) for the entire training phase.
Concerning the testing phase, given that the class networks are already formed, the testing
instance requires only one query to the k−NN and Radius Neighbors distance matrix to find
its neighbors and connect to them, which can be done in O(1). Thus, the incorporation of the
testing sample into the networks requires only O(1), and, finally, a new calculation of the net-
work measure is required. Thereby, the entire testing phase demands a total computational
complexity of O(M).
In this paper, the communicability measure is applied. It consists of calculating the l-th
power of the adjacent matrix and, then, the eigenvalues of each class, where lis the shortest dis-
tance between two nodes. Generally, l�n, therefore, the complexity order of
OðMÞ ¼ OððmaxðnCiÞÞ3Þ, where maxðnCiÞis the number of data samples of the largest class.
Certainly, maxðnCiÞ<n. Obviously, the computational complexity can be reduced if we apply
other network measures with lower complexity orders.
Results
In this section, we present the computational results of chest X-ray image classification using
the proposed method. Initially, we define the database used in the simulations, and its compo-
sition is characterized in detail. Next, the two complexity measures (fractal dimension and
quadtree) are analyzed to check their capabilities as feature extractors, and, then, the mixture
of these feature extractors is evaluated. Later, a network measure analysis is conducted to verify
which one presents the greater potential to distinguish the networks. And, lastly, the classifica-
tion results are exemplified and compared to other state-of-the-art techniques.
Database
All radiographic images used in this paper have been obtained from a public data source: the
COVID-19 Radiography Database, Version 1 [37–39]. The entire database contains 219
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COVID-19 positive images and 1341 normal chest X-ray images. Each of the images contains
1024×1024 pixels with 8 bits of depth gray-scales.
For the simulations of this paper, we construct a balanced dataset by randomly selecting
150 healthy lung images and 150 COVID-19 images. Therefore, this dataset contains a total of
300 samples. During the classification experiments, the dataset is randomly split in the training
and testing set, with a ratio of 9: 1; resulting in 270 samples for the training, and 30 for testing.
For each algorithm under comparison, the classification results are averaged over 50 execu-
tions. We denote the two classes of this dataset as “normal” and “COVID-19”. Fig 3 shows
some examples of the normal and COVID-19 chest X-ray images, respectively.
Analysis of fractal dimension as a feature extractor
From Fig 3, we can observe that the healthy lungs X-ray images present a clearly distinct pat-
tern when compared to the COVID-19 images. The COVID-19 lungs present the formation of
filaments that spread through the entire X-ray image as an opacity texture that impairs the
proper visualization of anatomical details. Therefore, since the fractal dimension is a geometri-
cal complexity measure we expect that it will capture this disparity in visual information pat-
terns. Thus, we extract the fractal dimension to compose feature vectors for characterizing the
two classes, where larger fractal dimension values are expected for those patterns that indicate
a higher complexity level.
In order to calculate the fractal dimension (box-counting dimension in this paper) for a
gray-level image, first, we generate a series of binary images from it imposing a series of
increasing threshold values, and then we calculate the box-counting dimension for each binary
image. These thresholds range from 100 until 150, with increments of 10, being a total of 6
thresholds, and consequently generating a feature vector with 6 fractal dimension values for
each image. Figs 4and 5show the binary images of the normal and COVID-19 gray-level
images, respectively, corresponding to those original images from Fig 3. The binarization also
shows the loss of the anatomical details for the COVID-19 lung X-ray images in comparison to
normal images, especially for lower threshold values.
Fig 6. Calculated fractal dimension values. (a) Fractal dimension values of binary images for the 8 gray-level images shown in Fig 3, where red curves
represent normal, and the purple represent COVID-19. (b) Mean value of the fractal dimensions for all images of the database.
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A box-counting dimension curve can be generated for each original gray-level image to
illustrate the behavior of fractal dimension values. Fig 6(a) shows the calculated box-counting
dimension curves for the 8 original gray-level images from Fig 3, while Fig 6(b) plots the mean
value for fractal dimensions of each class of images, considering the calculated measures for all
the 150 images of each class. From these figures, we see that the normal images and the
COVID-19 images have quite different complexity levels in terms of fractal dimension. Thus,
as we expected, the fractal dimension measure acts as a good feature extractor differentiating
the images concerning those distinct exhibited visual patterns.
Analysis of quadtree as a feature extractor
In a complementary analysis to the fractal dimension perspective, we can observe that the nor-
mal and COVID-19 lung X-ray images, illustrated in Fig 3, also show a very distinct spatial dis-
tribution of details. By a simple visual inspection, it is possible to notice that the opacity
texture present in COVID-19 images seems to cause a significant reduction in the heterogene-
ity of the lung X-ray image when compared to the normal images that are much richer in
detail. Then, the quadtree algorithm can also be applied as a feature extractor to quantify these
homogeneity differences, based on the values obtained from its histogram for the distribution
of block sizes and quantities determined by its partitioning.
To obtain the histogram of values for the block distribution, the quadtree algorithm is
directly applied to the gray-level image, with no need for binarization. Then, the histogram of
block distribution is analyzed for block sizes of 1 until 64, with increments of 2. The 7 resulting
values compose the feature vector for each inspected image.
The results for the quadtree division and the corresponding histogram of block distribu-
tions obtained from the same set of original images from Fig 3 are illustrated in Figs 7and 8
for the normal and COVID-19 class samples, respectively. From both figures, we can observe
that the lung X-ray images for normal class samples, Fig 7, always exhibit a higher number of
blocks with a small size and a lower number of blocks with a big size when compared to images
from COVID-19 class, Fig 8, which behave oppositely. Since more big blocks result in fewer
partitions, as a consequence, the COVID-19 images also present fewer partitions in total.
Fig 9 illustrates the average number of blocks, extracted by quadtree, according to the block
size for all the 150 images of each class in the dataset. Where the green bar represents values
for the normal class of images, and the red bar represents the values for the COVID-19 class. It
can be seen that in the normal category images when the blocks are relatively small, the num-
ber of blocks is large, that is, the details are richer and the edges are clearer. On the contrary, in
the COVID-19 category images, the smaller the blocks, the corresponding quantity is relatively
small; being scarce the occurrence of small blocks, since the detail level is not as rich as in the
normal category. Here, the feature vectors obtained from quadtree also show promising char-
acteristics to differentiate images from both classes, normal and COVID-19.
Mixture of feature extractors
Since both measures, fractal dimension, and quadtree, proved promising to characterize speci-
ficities for the elements of each class, we have analyzed those features according to the exhib-
ited parameters for the spatial distribution of each specific class in its n-dimensional feature
space. For a proper comparison, the measures are numerically normalized to the [0, 1] interval,
and the separability in the feature space is characterized by the calculation of the mean value of
the distances and the standard deviation of these distances for the elements of each class
independently.
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Fig 7. Quadtree analysis of four normal X-ray images. (a-d) Quadtree division of the 4 normal X-ray images shown in Fig 3. (e-
h) Quadtree block size distribution corresponds to each of the images on the left.
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Fig 8. Quadtree analysis of four COVID-19 X-ray images. (a-d) Quadtree division of the 4 COVID-19 X-ray images shown in
Fig 3. (e-h) Quadtree block size distribution corresponds to each of the images on the left.
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In addition, a mixture of those features is also evaluated by directly concatenating the fea-
ture vectors from both measures. Thus, the mixed feature vector is composed of those 6 values
originally from fractal dimension for the various binarized images concatenated with the 7 val-
ues from the histogram of block distribution of the quadtree, for block sizes of 1 until 64, with
increments of 2. Resulting in a mixed feature vector of 13 values for each image.
All the obtained values are presented in Table 1, where the evaluated feature vectors are
rows and the mean values and standard deviation are at the columns, always with the pair of
values by each corresponding class, normal and COVID-19, given respectively. From the table
values, we can observe that fractal dimension and quadtree measures behave quite similarly in
the capacity of separating the feature space, with a very similar absolute difference for mean
distance and standard deviation for the class differentiation. Thus, from the mixed features
perspective, the presented values evidence a more distinguishable absolute difference for the
mean distance between classes, while maintaining a similar absolute difference for the stan-
dard deviation.
These results support the use of the mixture of both measures as the input feature vector for
our modified high-level classification technique, especially for the network construction in the
training phase; Fig 2(b). As a result of the concatenation of features, the mixed feature vector
Fig 9. Histogram for the mean values of block sizes in quadtree division. The mean values are calculated for all
images of the database. The green bar refers to the normal class images, while the red bar refers to COVID-19 X-ray
images.
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Table 1. Comparison of the feature extractors.
Feature Mean distance Standard deviation
Normal COVID-19 Normal COVID-19
Fractal dimension 0.193 0.431 0.105 0.230
Quadtree 0.351 0.569 0.171 0.277
Mixed features 0.414 0.733 0.170 0.315
Comparisons are performed in vector space to analyze a particular feature extractor, where the numerical values are calculated from the average of the distances among
all vectors of a class, for all its images in the dataset.
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has an increase in the number of dimensions, which does not incur problems related to the
curse of dimensionality for our proposed technique since the feature vectors are only inputs
for the network construction, i.e. these features are mapped to a complex network (graph).
Network measures analysis
Revisiting the proposed high-level classification technique steps, after the training phase, the
two classes, normal and COVID-19, are already represented by their own distinct network,
having mapped the points in the feature space to their corresponding networks through the
combination of Radius Neighbors and k-NN techniques; see Eq 2. And, as discussed in previ-
ous sections, the technique should, then, be able to perform the prediction of new elements
through the steps illustrated in Fig 2(c). First, extract features of the input image, resulting in
the mixed feature vector, and with it, incorporate the unlabeled data sample separately into
both networks representing the classes; normal and COVID-19. With that, the prediction can
be executed by analyzing the impact that the incorporation of the sample caused on a specified
network measure before and after its insertion.
In this scenario, many network measures can potentially address this differentiation to sup-
port the evaluation task in the proposed framework, simply, any measure that is robust enough
to capture the intrinsic network structure perturbance between the before and after incorpo-
ration of the unlabeled sample. From another perspective, the network measure aptitude can
be evaluated according to its capability to directly distinguish the networks representing the
normal and COVID-19 classes.
With this in mind, a set of network measures is calculated and analyzed for the dataset to
compare which best distinguishes the network structure for the normal and COVID-19 net-
works. The obtained values can be observed in Table 2, where the rows represent the normal
and COVID-19 classes and each column refers to a specific network measure. Thus, the calcu-
lated measures are Average Degree, Average Clustering Coefficient, Transitivity, Global Effi-
ciency, and Communicability.
Specifically for the network measure analysis experiment, the normal and COVID-19 net-
works are constructed from all 300 data samples, representing the 150 available elements in
each class of the dataset. To clarify, since the primary objective of the experiment is to assess
the discriminative power of network measures within each representative network, there is no
need for a training-test split configuration in the experimental design. The feature vectors pro-
vided to the network construction phase are obtained from the mixture of fractal dimension
and quadtree for each original image. Additionally, all the resulting values in Table 2 are
obtained for networks constructed with the parameter k= 5 of the k-NN technique, observing
that the rparameter for Radius Neighbors is calculated from the Eq 3.
Observing Table 2 values, the communicability measure values stand out as those that easily
distinguish both networks numerically, being other values with a much minor absolute
numerical difference. This corroborates with the characteristics presented in [23] for the
communicability measure, which gives evidence that it is capable of describing both the global
Table 2. Network measures analysis.
Average Degree Avg. Clustering Coefficient Transitivity Global Efficiency Communicability
Normal 6.382 0.382 0.366 0.319 37.981
COVID-19 4.940 0.436 0.404 0.258 0.198
Various network measures are calculated to compare the networks constructed for each class of the problem; normal and COVID-19.
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and local network scales simultaneously. Furthermore, in [23], the communicability measure
shows promising features for evaluating the structure-dynamic relationship of networks,
which can have common properties to our tasks of evaluating the measure’s impact on the net-
work before and after unlabeled sample incorporation.
Classification results
Having constructed the networks to represent both classes, normal and COVID-19, in the
training phase with a training-test configuration of ratio 9: 1, and defined the communicability
measure as a suitable network measure to assess the impact of the incorporation of new sam-
ples, here, we first provide an illustrative demonstration of the prediction process by the pro-
posed method for just 6 examples randomly picked from the testing dataset of 30 samples.
These examples are illustrated in Fig 10. The red nodes represent the network for the normal
class, the blue nodes represent COVID-19, and the black nodes are the testing samples. The
parameters for the training phase are k= 5 for the k-NN technique and r= 0.1703 for Radius
Neighbors, observing that this parameter is obtained directly from the Eq 3.
The prediction is performed through the analysis of the conformity of the testing data sam-
ple with each of the two networks, using the communicability measure [23] and comparing its
Fig 10. Six classification prediction examples. Six examples of classification prediction are randomly selected from the 30 testing dataset. In each of
the six subfigures, the red network is formed from training samples of the normal class, and the blue network is formed from training samples of the
COVID-19 class. For illustrating the classification process, in each of the subfigures, the black node is the testing node, which is briefly inserted into
both networks in order to be evaluated.
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values before and after the insertion of the sample. Thus, the new element is associated with
that class it conforms better (or provokes less perturbation), which is represented by the lowest
value of ΔG.
Table 3 reports the numerical results and the corresponding predicted class for all six exam-
ples in Fig 10. The communicability values for the generated networks are 44.8541 and 1.9568
for the normal and COVID-19 classes, respectively. Note that the communicability value prior
to each sample evaluation is always the same since the classification occurs during the testing
phase, where the networks representing each class are already constructed and the testing sam-
ples are not incorporated into the model (network) after each prediction.
For the testing data sample shown in Fig 10(a), the impacts for the insertions are calculated
as ΔG(class
normal
) = 0.0148 and ΔG(class
COVID−19
) = 0.0065, therefore, since this sample causes
less impact on the COVID-19 network, it is classified as belonging to this class. Regarding the
second example shown in Fig 10(b),ΔG(class
normal
) = 0.0126 and ΔG(class
COVID−19
) = 0.0132,
then, the sample is classified into the normal class. For Fig 10(c),ΔG(class
normal
) = 0.0117 and
ΔG(class
COVID−19
) = 0.0042, resulting in its prediction to the COVID-19 class. The Fig 10(d)
refers to ΔG(class
normal
) = 0.0114 and ΔG(class
COVID−19
) = 0.0035, and the corresponding sam-
ple is also predicted to the COVID-19 class. For the sample in Fig 10(e),ΔG(class
normal
) =
0.0083 and ΔG(class
COVID−19
) = 0.0128, being classified into the normal class. Lastly, Fig 10(f)
with ΔG(class
normal
) = 0.0146 and ΔG(class
COVID−19
) = 0.0079 has its sample associated with
the COVID-19 class.
Next, we perform a series of simulations to measure the performance of our modified high-
level classification technique in comparison with several state-of-the-art techniques. All the
simulations are averaged over 50 executions with the training and testing set randomly split in
a ratio of 9: 1 for the 150 normal and 150 COVID-19 X-ray images of the dataset. The obtained
results are shown in Table 4.
It is worth observing that the training phase of the proposed method is accomplished with
the construction of one independent network to represent each of the classes. During this
phase, the algorithm only has access to the data from the training samples. There is no interac-
tion between training and testing sets in the network construction phase. In other words, there
is no mixing between training and testing sets, and all the testing samples are unseen by the
constructed classifier. Once the training phase is completed, the “base” network that represents
a class never changes. During the testing phase, an unlabeled sample is temporarily incorpo-
rated into both networks just for the purpose of calculating the impact in the network measure
(communicability measure). With that, the sample is predicted as pertaining to that network it
causes less perturbation characterized by the least variation in the network measure or, in a
Table 3. Evaluating the six classification prediction examples.
Sample Normal COVID-19 Predicted Class
Before After ΔG(class
normal
) Before After ΔG(class
COVID−19
)
(a) 44.8541 44.1975 0.0148 1.9568 1.9698 0.0065 COVID-19
(b) 44.2862 0.0126 1.9308 0.0132 Normal
(c) 44.3288 0.0117 1.9484 0.0042 COVID-19
(d) 44.3399 0.0114 1.9498 0.0035 COVID-19
(e) 44.4804 0.0083 1.9316 0.0128 Normal
(f) 44.1979 0.0146 1.9413 0.0079 COVID-19
Evaluation of samples from Fig 10(a)–10(f) according to the communicability measure calculated before and after the training sample insertion, for each of the classes.
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complementary interpretation, to that network in which the sample best conforms to the pat-
terns expressed in the network topology. Lastly, the algorithm discards the testing sample by
removing the corresponding node from both networks before proceeding to the next sample,
preventing data leakage at the model level.
As shown in Table 4, our algorithm presents a competitive performance in terms of classifi-
cation accuracy compared to several traditional classification methods for COVID-19 identifi-
cation. The proposed technique achieves an average accuracy of 97.0% and an average
F1-Score of 0.953, which exceeds all other techniques, except for the Deep Learning (ResNet-
50) [40] that achieved 98.0% and 0.972, respectively.
Undoubtedly, the last decade’s impressive breakthroughs in image classification tasks have
been due to the evolution of deep learning architectures, thus the ResNet-50 superior result is
totally comprehensible and expected. However, it requires massive computing power [41] and
diverse computational strategies to optimize its training convergence, e.g., transfer learning
and fine-tuning of the hyperparameters. On the contrary, our technique, although requires
large memory space for a big network, shows a simpler setup, involving the tuning of only a
single parameter k, resulting in a much more manageable and explainable technique, in which
is possible to visualize the class structure represented directly as a network.
Compared to all other surpassed techniques, the greater average accuracy of our technique
is due to the fact that it creates a distinct network representation for each class of the problem,
which intrinsically captures patterns, and other organizational and semantic level information
since it is a high-level classification technique [10]. In contrast, all these surpassed techniques
are robust, well-known, and consolidated, but rely only on physical characteristics, such as
similarity, distance, or distribution to define data classes.
Conclusion
In this work, we present a new network-based high-level classification technique. From the
simulation on the artificial dataset, we see clearly that the network approach can capture data
patterns even if the data sample falls within another class measured by the physical feature (the
similarity or distance feature in this case). On the other hand, classical and state-of-the-art
classification techniques cannot recover this data sample from another class. This is because
those techniques only consider the physical features of the input data. Such a salient feature
Table 4. Classification accuracy comparison.
Classification Technique Accuracy F1-Score
AdaBoost 93.9% 0.917
Decision Tree 93.4% 0.908
Deep Learning (ResNet-50) 98.0% 0.972
Logistic Regression 96.8% 0.934
Multilayer Perceptron 89.1% 0.893
Naive Bayes 89.6% 0.907
Random Forest 93.7% 0.902
SVM 94.7% 0.926
Modified High-Level Classification Technique 97.0% 0.953
For all the experiments, each of the classes (150 normal and 150 COVID-19 images) is randomly split into two
subgroups: the training group (135 images) and the testing group (15 images). The results are averaged over 50
executions with randomly selected training and testing samples each.
https://doi.org/10.1371/journal.pone.0290968.t004
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implies that the proposed network approach may provide an elegant solution for invariant pat-
tern recognition problems, such as face recognition, where large variations among data sam-
ples appear. The simulations on the COVID-19 image dataset show that the proposed
network-based technique achieves a similar level of classification precision as the deep learning
technique. However, deep learning techniques, in general, do not have an explicit explanation
of the classification decisions. On the other hand, the proposed network-based technique gives
a straightforward reason why a data sample is classified into a determined class. Specifically,
the proposed technique classifies a data sample by checking whether it conforms to the pattern
formation of each class. Here, a network is constructed for the training samples of each class,
and the pattern of each class is characterized by network measure(s). The extraction of patterns
from X-ray images through complex network construction and the subsequent analysis of the
impact on network measures when a new testing sample is briefly incorporated shows that pat-
tern identification is an efficient and robust way for class prediction. This leads to new possi-
bilities not only for classification tasks but also empowers new perspectives for the analysis of
the structure of the data related to an applied problem.
Although the proposed technique also employs the concept of “network” as a key data
structure to perform the classification, there are considerable distinctions in the purpose of
these network representations when compared to deep learning architectures. As a funda-
mental difference, deep learning architectures usually employ networks with fixed topology
(fixed number of nodes, fixed layers, all-to-all connections, etc.), while our technique con-
structs its networks according to the presented training data, resulting in a complex network
without restriction on the topology. Therefore, our technique constructs an independent
network to store a representation for each of the classes, since it is expected that each class
presents inherent patterns and relationships. Furthermore, given that deep learning archi-
tectures use fixed networks as an information processing paradigm, at least part (some lay-
ers) of these “deep” networks require the adjustment of a large number of weights during
the training phase. These weight adjustments represent changes in edge values connecting
the “nodes” and, ultimately, will represent the learning and adaptation to the presented data.
In contrast, the networks constructed by the modified high-level classification technique
directly represent the data of a certain class, and the data patterns are characterized by com-
plex network measures.
Furthermore, even though the proposed technique presents salient features in semantic
data classification, it requires large memory space to store the large datasets and the corre-
sponding constructed networks for each class. It may also take a long time to calculate net-
work measures to characterize data patterns on large-scale networks. In future works, a new
approach to pattern identification will be tested to enable local sensing for the references in
the vicinity of the testing node instead of the entire network, to significantly reduce the
computational cost with the network growth. In addition, the technique will be extended to
include algorithms for coarsening and uncoarsening phases [42,43] for the constructed net-
works to deal with large-scale problems. Therefore, when addressing the original problem
with multiscale analysis, it’s expected to better control computational costs while maintain-
ing a compatible accuracy. Additionally, we will also investigate the possibility of subdivid-
ing each data class into more than one network, enabling the identification of subpatterns
embedded in each class and the discovery of hierarchical relations. Then, new network con-
struction techniques will be explored to evaluate advancements in the structural network
representation and its implications. Last but not least, the network representation and
proper measures will be studied to assess and predict different levels of severity for each
COVID-19 patient, which is critical for the prognosis of patients and important for optimiz-
ing hospital resources’ availability.
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Acknowledgments
This work is carried out at the Center for Artificial Intelligence (C4AI-USP), with support
from the São Paulo Research Foundation (FAPESP) and the IBM Corporation under FAPESP
grant number 2019/07665-4. This work is also supported in part by the Coordenac¸ão de Aper-
feic¸oamento de Pessoal de Nı
´vel Superior—Brasil (CAPES)—Finance Code 001.
Author Contributions
Conceptualization: Weiguang Liu, Yu-tao Zhu, Qiusheng Zheng, Liang Zhao.
Formal analysis: Rafael Delalibera Rodrigues, Jianglong Yan, Everson Jose
´de Freitas Pereira,
Gen Li.
Methodology: Weiguang Liu, Yu-tao Zhu, Qiusheng Zheng, Liang Zhao.
Software: Rafael Delalibera Rodrigues, Jianglong Yan, Everson Jose
´de Freitas Pereira, Gen Li.
Writing – original draft: Rafael Delalibera Rodrigues, Jianglong Yan, Everson Jose
´de Freitas
Pereira, Liang Zhao.
Writing – review & editing: Weiguang Liu, Rafael Delalibera Rodrigues, Jianglong Yan, Yu-
tao Zhu, Everson Jose
´de Freitas Pereira, Gen Li, Qiusheng Zheng, Liang Zhao.
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