Application of Bidimensional Empirical Mode Decomposition for
Particle Identification and Size Determination
DIANA RUBIO, NICOLAS SASSANO, MARCELA MORVIDONE, ROSA PIOTRKOWSKI
ITECA (UNSAM-CONICET), CEDEMA
Universidad Nacional de San Martin
25 de Mayo y Francia, 1650 San Martin, Buenos Aires
ARGENTINA
Abstract: - The analysis of surface texture appears in different disciplines of science and technology. Surface
texture is generally multiscale and can be separated into different spatial frequency or wavelength ranges
providing information on image characteristics such as shape, roughness, pseudoperiodic components and chaotic
components. Surface texture translates into image texture. Textures in images are complex visual patterns
composed of entities or subpatterns that have characteristic brightness, color, slope, size, etc. In this work, we
address the analysis of multimodal images and their decomposition using the bidimensional empirical mode
decomposition. This approach allows us to obtain component images from each original image, each of them
with a spatial frequency range. These analysis methods are currently used in images from various disciplines
such as biology (analysis of biological tissues), environmental and health sciences (particulate matter dispersed
in the atmosphere), materials sciences (texture on surfaces), earth sciences (SAR images). The main objective is
to present an algorithm that allows identifying, segmenting, and classifying structures and morphologies in each
image mode. The proposed numerical technique is applied to images from cytology analysis to study number of
particles present in fibroma (benign tumor) nuclei compared to the number in sarcoma (malignant tumor) nuclei
in order to investigate if there is a significant difference between them, sufficient to use this fact as part of a
diagnosis.
Key-Words: - Bidimensional Empirical Mode Decomposition, Texture Analysis, Particle Size, Particle
Asymmetry, Cytology Analysis, Fibrosarcoma.
Received: May 29, 2024. Revised: August 6, 2024. Accepted: September 7, 2024. Published: October 24, 2024.
1 Introduction
Properly characterizing the final structures has an
undoubted impact on applications. For example, it
helps to make better decisions in manufacturing
processes in the case of materials and to better
measure the consequences in the case of pollution.
Multiscale methods and algorithms such as
bidimensional wavelet transform and bidimensional
empirical mode decomposition, applied to surfaces
and images are constantly evolving. Their
optimization leads to representing the information in
a simpler, more accurate and precise way.
Empirical Mode Decomposition (EMD) was
pioneered by Huang et al. in 1989, for one dimension,
and it has proven to be a successful tool for analyzing
non-stationary and nonlinear time series. This
method finds application in diverse fields such as
acoustic emission studies and climate analysis, where
traditional linear methods may fall short. Time series
are decomposed into a finite number of zero-mean
components called Intrinsic Mode Functions (IMFs)
and a residue, where the number of IMFs depends on
the signal. Each IMF represents an oscillating
function that optimally captures the signal's
characteristics, allowing for the identification of
underlying physical processes. The Hilbert
Transform applied to each IMF enables the
determination of instantaneous frequency,
contributing to the overall Hilbert-Huang Transform
(HHT) procedure. These IMFs form a complete a
posteriori basis and are empirically derived from the
original series. Moreover, the residue, a monotonous
function, provides insight into the series' overall
trend. Beyond EMD, advancements like Ensemble
Empirical Mode Decomposition (EEMD) and
Variational Mode Decomposition (VEMD) have
further expanded its utility across various domains.
The decomposition strategies are varied and adapted
to different purposes. One of the authors, R.P., has
been actively engaged in diverse applications,
including the evaluation of material damage via
acoustic emission analysis, investigation of
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
COMPUTATIONAL SCIENCE AND SYSTEMS ENGINEERING
DOI: 10.37394/232026.2024.6.16
Diana Rubio, Nicolas Sassano,
Marcela Morvidone, Rosa Piotrkowski
E-ISSN: 2766-9823
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electrocardiograms (ECG) for ischemia detection and
examination of climate-related data [10]. In 2003,
Nunes et al. [11] extended EMD to BEMD for
analyzing image texture. Since then, BEMD and
bidimensional wavelet transform methods continue
evolving.
In [1], Clausel et al. introduced a two-dimensional
version of the synchrosqueezed wavelet transform,
which extends the concept of analytical signal to
images. Kim et al. [7] proposed the two-dimensional
statistical empirical modal decomposition (BSEMD)
algorithm, which employs an alternative smoothing
procedure for constructing 2D upper and lower
envelopes instead of interpolation. In [17], Xie et al.
presented optimization approaches based on
Delaunay triangulation of local maxima (minima)
and scale, using the initial modal image to effectively
capture multiscale features of input images. Ding et
al. [4] utilized a wavelet transform in combination
with a deep neural network to establish a one-to-one
correspondence of semantic structures in images,
thereby avoiding artificial style transitions around
these structures. In [13], Oyelade et al. addressed
image degradation issues and improved image
quality through a wavelet transform-based algorithm.
They reconstructed images using diffuse contrast
enhancement to highlight details while preserving
spectral information.
These methodologies are employed in
sophisticated applications of science and technology
dedicated to texture analysis and they represent a
dynamic and continually evolving research domain.
In [15], Shao et al. analyzed surface topography by
extracting encrypted surface components using a
two-dimensional empirical wavelet transform. In
[16], Veluppal et al. conducted entropy-based
multiscale two-dimensional texture analysis to
identify texture alterations induced by Alzheimer's
disease in brain MRI images. In [18], Yang et al.
employed BEMD and multifractal geometry to
analyze retinal images for detecting various degrees
of diabetic retinopathy. Dong et al. [5] utilized
multiscale models to estimate the effective thermal
conductivities of particulate compounds with
complex microstructures. In [6], Gogolewski
employed fractional spline wavelets to analyze the
surface texture of industrial parts across various
scales, facilitating the generation of surface profiles
with a variable distribution of irregularities. In [19]
Yu et al. proposed a computer vision-based method
for measuring drill edge wear, incorporating an
adaptive contrast enhancement algorithm based on
two-dimensional local mean decomposition, which
groups in similarities in each image mode.
In this work we address the analysis of
multimodal images and their decomposition using the
BEMD applied to cytology images with the objective
to analyze differences in the structure of the nuclei of
benign and malignant tumors.
2 Formulation and Methodology
Real-world images often exhibit significant
heterogeneity, representing diverse scales and
morphologies. Consequently, employing a
decomposition method to characterize the various
component images becomes both useful and
necessary. BEMD stands out for its versatility and
adaptability to such diverse data.
The algorithm scheme is sketched in Fig. 1. and
then summarized in Sub-Section 2.3.
Figure 1. Algorithm scheme.
BEMD enables the extraction of component
images, called Bidimensional Intrinsic Mode
Functions (BIMF) that contain image information
ranging from high frequency to low frequency, along
with a residue that is a non-zero mean remainder.
After the BEMD is performed on the image, one
proceeds to identify particles or structures within
these components and classify them based on their
size and morphology. Finally, statistical analysis
could be eventually conducted to further examine and
understand the diverse characteristics present.
The proposed approach was applied to images
from cytology studies, where a benign tumor
(neurofibroma) and a malignant tumor
(fibrosarcoma) were identified by one of the authors,
the veterinary N.S., and later confirmed by a
pathologist.
Quantifying the difference in number of particles
present in fibroma and sarcoma nuclei, could help to
obtain clues for further diagnosis and treatment
planning. One expects to have robust statistical
information in future work.
2.1 Multiscale Decomposition
The BEMD method is applied to images that undergo
a decomposition process analogous to its one-
dimensional counterpart. BEMD spatial
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decomposition enables the extraction of meaningful
information from the data in both dimensions
simultaneously. By decomposing the image into
BIMFs, BEMD unveils intricate patterns and
structures that might remain unseen in the original,
often complex image, serving as a powerful tool for
uncovering hidden spatial features and enhancing our
understanding of multidimensional data.
2.2 Particle identification and Size
Determination
Once the image (matrix) has been decomposed into
BIMFs, the particles, or structure elements, can be
identified by locating the local maximum values
within each of these matrices. Subsequently, for
every detected particle, its size is determined by
assessing the local minimum values along the
horizontal and vertical direction from each local
maximum. This process yields four values, denoted
as
and
, as illustrated in Fig.2. Note that,
since the particles are randomly oriented, these
values are of statistical nature.
Figure 2: Size determination proposed in the algorithm.
The particle diameter and asymmetry are defined
as
(1)
(2)
where
. (3)
2.3 The Algorithm
For a 2D image I, the proposed algorithm consists in
the following steps:
1) Initialize r = I (residue) and define h=I, m=0
(mode).
2) Identify all extrema of h.
3) Interpolate extrema that verify the criterion to
obtain an upper and lower envelope, namely Emax and
Emin, respectively.
4) Compute the envelope mean EM.
5) Subtract the mean EM from the image h = h− EM.
6) Repeat steps 2-5 until h can be considered as zero-
mean.
7) m=m+1, BIMFm = h, r =r − h , h= r.
8) Repeat steps 2-7 until the number of extreme
values in r is less than 2.
9) Determine all local maxima, namely Max, in
BIMFmode. Each local maximum corresponds to
a particle in BIMFmode.
10) For each local maximum in Max
a. Find r1: the closest local minimum below, r3:
the closest local minimum above, r2: the
closest local minimum to the right, r4: the
closest local minimum to the left.
b. Define the particle size as
c. Define
and the
particle asymmetry as
.
11) Repeat 9)-10) for the significant modes.
Note that steps 1)-8) are based on Nunes et
al. [12] and its implementation is given in [14].
3 Application to Cytology Images
In veterinary medicine, cytology is valuable for
morphological cellular diagnosis, relying on
microscopic characteristics of cells and extracellular
components from various biological samples. It
complements clinical diagnosis, which considers the
patient holistically alongside all available methods.
The aim is to identify at least 80% of cells in
preparations, discerning quantities, proportions, and
relationships between cells. Thus, cytology can be
conclusive, suggestive, or inconclusive in diagnosis.
In recent years, there has been an increased demand
for cytology diagnosis, especially in skin lesions of
domestic canines and felines, due to skin
accessibility. Cytology involves microscopic
examination of cells from healthy tissues, lesions, or
body fluids [2,3,9].
The main objective of cytology is to differentiate
between inflammatory and non-inflammatory
processes and sometimes between benign and
malignant neoplasms within non-inflammatory
processes.
There are general and nuclear criteria of
malignancy to characterize non-inflammatory
processes. The general criteria for malignancy allow
the assertion of the presence of reactive cells,
meaning cells that have undergone morphological
changes. If most cells do not exhibit general criteria
of malignancy, the process can be classified as
benign. Nuclear criteria of malignancy allow the
assertion of the presence of a neoplasm. Neoplastic
processes are categorized based on cellular
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characteristics. In this sense, four main
groups are distinguished:
epithelial cell tumors; round cell tumors; naked
nuclei neoplasms; mesenchymal cell tumors.
In most malignant processes, a large proportion
of cells exhibit pleomorphism in shape, cell size, and
nuclear size when comparing cells within the same
population [8,9]. More than three nuclear criteria of
malignancy should be present in most of the cellular
population to classify the process as malignant since
changes in the cytoplasm are not as reliable as nuclear
ones.
Cytology analysis in general, and specifically
nuclear criteria of malignancy, is considered to carry
a degree of cytologist-dependent subjectivity [8,9].
To introduce a more quantifiable criterion, it is
suggested that certain nuclear features observed
under the microscope, captured in photographs, could
be identified as particles due to their intranuclear
heterogeneity. Consequently, malignant neoplasm
nuclei are expected to exhibit a higher particle count
compared to benign neoplasm nuclei, reflecting their
increased nuclear criteria of malignancy. To explore
this concept further, cytology preparations from
samples of a neurofibroma and a fibrosarcoma are
considered. The slides containing the samples were
stained with Giemsa. Initial microscopic evaluation
involved lower magnification objectives (4x and 10x)
to assess sample quality and staining. Subsequent
individual cell assessment and comparison were
performed using the 40x objective, while the 100x
immersion objective facilitated detailed observation
of organisms, inclusions, and certain cellular
characteristics.
Figure 3: Typical images from Neurofibroma nuclei
(left) and Fibrosarcoma nuclei (right) received at the
Guernica Veterinary Clinic in the South Zone of
Greater Buenos Aires, Argentina,
Figure 3 shows typical images from a cytology
study. The image on the left side of Fig.3 corresponds
to a 12-year-old male mixed-breed dog that presents
a subcutaneous neoformation on the neck. The
cytology diagnosis indicated a tumor of
mesenchymal cells. The diagnosis of benign
neurofibroma was confirmed by histopathological
study. On the other hand, the image on the right side
of Fig.3 corresponds to a 6-year-old female cat,
common European breed, presented with a
subcutaneous neoformation in the hip region. The
cytology diagnosis indicated a tumor of
mesenchymal cells with malignant characteristics.
The diagnosis of fibrosarcoma was confirmed by
histopathological study.
4 Results on Particles identification
In this study, we consider neurofibroma and
fibrosarcoma cytology samples for the application of
the method presented in section 2. A number of
representative photos of a neurofibroma at 100x and
of a fibrosarcoma at 100x were chosen. Fifteen nuclei
images were obtained from the samples of each type
of tumor with the aim of analyzing whether there is a
significant difference between the quantities of
particles for each type. Each of the nuclei images was
processed separately, decomposing it into 4
component BIMF images. Then, the particles are
detected in each of them, and the size measured as
described above.
Figure 4: Neurofibroma nucleus, its four BIMFs and
quantities of particles.
In Fig. 4 and Fig. 5 it is shown a nucleus of
neurofibroma and a nucleus of fibrosarcoma,
respectively, with their four bidimensional mode
decomposition matrices and the number of particles
found in each mode. It is observed that BIMF1 shows
a more complex structure than the other image
Neurofibroma Nucleus
Mode 1
Mode 2
Mode 3
Mode 4
NEU
ROFI
BRO
MA
NUC
LEUS
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modes. Hence, it is expected that most of the smaller
particles will be found in mode 1.
Figure 5: Fibrosarcoma nucleus, its four BIMFs and
quantities of particles.
The number of particles identified in each
component image was counted and the results
obtained are plotted as star points in Fig. 6 and shown
in Table 1 for the 15 nuclei.
Figure 6: Number of particles found in each
bidimensional image mode matrix. Neurofibroma
(left) and Fibrosarcoma (right).
A tendency is observed where the number of
particles for fibrosarcoma nuclei exceeds that of
neurofibroma nuclei. The difference is particularly
evident in mode 1, where the higher number of
particles are found, since the smallest particles are
captured in this mode. Notice that points are
collapsed particularly at the higher modes meaning
that a similar number of particles are found in these
modes.
Table 1: Ordered number of particles found in each
Neurofibroma (left) and Fibrosarcoma (right) BIMF.
The total number of particles in each nucleus was
obtained by adding the partial amount from each
image mode. Fig. 7 shows the quantities of particles
ordered from smallest to largest, for the 15 nuclei of
the neurofibroma and the 15 nuclei of the
fibrosarcoma. The results obtained suggest a
potential difference in the cellular characteristics
between fibrosarcoma and neurofibroma nuclei. The
fact that the particle number of fibrosarcoma nuclei
exceeds that of neurofibroma nuclei implies that
there may be a higher level of nuclear complexity in
fibrosarcoma compared to neurofibroma.
Figure 7: Total number of particles found in
Neurofibroma nuclei and Fibrosarcoma nuclei.
It's worth noting that some neurofibroma nuclei
have a greater number of particles than certain
fibrosarcoma nuclei, indicating that the particle count
in a particular nucleus is not decisive. Therefore, it's
crucial not only to consider the quantity of cells for
analysis but also their representativeness in the
underlying biological process. Hence, the expertise
Nucleus Mode 1 Mode 2 Mode 3 Mode 4 Mode 1 Mode 2 Mode 3 Mode 4
1404 50 5 2 689 49 10 6
2472 36 9 4 779 31 5 0
3490 49 6 0 846 93 11 3
4490 49 6 0 935 65 10 3
5584 58 7 3 959 100 16 5
6602 57 11 31002 57 10 0
7679 47 7 3 1009 49 8 4
8679 47 7 3 1019 58 12 4
9754 66 12 31186 108 6 3
10 754 66 12 31190 78 13 4
11 770 54 12 31303 129 25 5
12 923 94 9 5 1389 84 9 4
13 923 94 9 5 1498 74 18 6
14 1088 91 12 41767 139 28 7
15 1088 91 12 41978 146 21 6
Neurofibroma
Fibrosarcoma
Fibrosarcoma Nucleus
Mode 1
Mode 2
Mode 3
Mode 4
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of a veterinarian specialized in cytology is still
necessary for achieving proper sampling.
The findings obtained through the proposed
algorithm are extremely valuable and could have
implications for diagnosis, prognosis and treatment
planning, since they can help distinguish between
these two types of tumors according to their
cytological characteristics. Additional analysis will
be performed on a larger number of samples to obtain
more robust and accurate statistics.
5 Conclusion
In this work we present a methodology for image
analysis based on mode decomposition. We have
described a simple and useful technique for particle
identification, together with a size and morphology
determination.
The proposed method was applied to images from
cytological studies, corresponding to a benign tumor
(neurofibroma) and a malignant tumor
(fibrosarcoma). The results are promising indicating
that this method can be a valuable tool for
veterinarians and could provide valuable insights into
the underlying biological differences between these
tumors. More studies will be conducted with a larger
number of samples to obtain more robust and
accurate statistical information on this particular
topic.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Marcela Morvidone, Diana Rubio and Rosa
Piotrkowski formulated the problem and wrote the
article.
Marcela Morvidone and Diana Rubio carried out the
numerical simulations.
Diana Rubio and Nicolas Sassano designed the
application of the technique to cytology images.
Nicolas Sassano provided the images and analyzed
the results.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0
/deed.en_US
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