# Conservation and breaking of pseudospin symmetry

T.-T. Sun<EMAIL_ADDRESS>School of Physics and Microelectronics,
Zhengzhou University, Zhengzhou 450001, China  Z. P. Li<EMAIL_ADDRESS>School of Physical Science and Technology, Southwest University, Chongqing
400715, China  P. Ring<EMAIL_ADDRESS>Physik-Department der Technischen
Universität München, D-85748 Garching, Germany

###### Abstract

It is well known that the pseudospin symmetry (PSS) is a relativistic
dynamical symmetry connected with the small component of the Dirac spinor. In
this paper, the conservation and breaking of PSS in nuclear single-particle
states are investigated in spherical Woods-Saxon potentials with the Green’s
function method which provides a novel way to precisely determine the
resonance parameters and properly describe the spacial density distributions
for all resonances of any widths, with which the process of symmetry
restoration and breaking are perfectly shown. In the PSS limit, i.e., when the
attractive scalar and repulsive vector potentials have the same magnitude but
opposite sign, PSS is exactly conserved with strictly the same energy and
width between the PS partners as well as identical density distributions of
the lower components. In the cases of finite-depth potentials, the PSS is
broken with obvious energy and width splittings and a phase shift between the
density distributions.

###### pacs:

25.70.Ef, 21.10.Pc, 21.60.Jz, 24.10.Eq, 21.10.Gv

Symmetries in the single-particle spectrum of atomic nuclei are of great
importance on nuclear structures and have been extensively studied in the
literature (see Refs. Ginocchio (2005); Liang et al. (2015) and references
therein). More than 50 years ago pseudospin symmetry (PSS) was found in atomic
nuclei, i.e., the two single-particle states with quantum numbers
($n,l,j=l+1/2$) and ($n-1,l+2,j=l+3/2$) are quasi-degeneracy and can be
redefined as the pseudospin (PS) doublets ($\tilde{n}=n$, $\tilde{l}=l+1$,
$j=\tilde{l}\pm 1/2$) Hecht and Adler (1969); Arima et al. (1969). The
pseudospin symmetry (PSS) is a relativistic dynamical symmetry, which has been
used to explain a number of phenomena in nuclear structures, such as
deformation Bohr et al. (1982), superdeformation Dudek et al. (1987); Bahri et
al. (1992), identical rotational bands Nazarewicz et al. (1990); Byrski et al.
(1990), magnetic moment Ginocchio (1999a), quantized alignment Stephens et al.
(1990) and so on. In addition, PSS is also of great concern in atomic and
molecular physics and has been discussed in some special atomic and molecular
potentials Jia et al. (2007); Zhang et al. (2008); Akcay and Sever (2013).

Since the recognition of PSS in the nuclear spectrum, comprehensive efforts
have been made to explore its origin until Ginocchio pointed out that PSS is a
relativistic symmetry in the Dirac Hamiltonian which is exactly conserved when
the scalar and vector potentials satisfying $\Sigma(r)\equiv S(r)+V(r)=0$
Ginocchio (1997). He also revealed that the pseudo-orbital angular momentum
$\tilde{l}$ is nothing but the orbital angular momentum of the lower component
of the Dirac wave function Ginocchio (1997) and there are certain similarities
in the relativistic single-nucleon wave functions of the corresponding
pseudospin doublets Ginocchio and Madland (1998). However, there is no bound
state in the PSS limit. Later, Meng et al. pointed out a more general
condition of $d\Sigma(r)/dr=0$, which can be approximately satisfied in exotic
nuclei with highly diffuse potentials Meng et al. (1998, 1999) and the onset
of the pseudospin symmetry to a competition between the pseudo-centrifugal
barrier (PCB) and the pseudospin-orbit (PSO) potential. Afterwards, PSS in
nuclear spectra have been studied extensively such as PSS in deformed nuclei
Lalazissis et al. (1998); Sugawara-Tanabe and Arima (1998); Sugawara-Tanabe et
al. (2000, 2002); Ginocchio et al. (2004); Sun et al. (2012); Guo et al.
(2014), SS in anti-nucleon spectra Zhou et al. (2003); Mishustin et al.
(2005); He et al. (2006); Liang et al. (2010); Lisboa et al. (2010), PSS and
SS in hypernuclei Song et al. (2009); Sun et al. (2017); Lu et al. (2017); Lu
and Sun (2019), perturbative interpretation of SS and PSS Alberto et al.
(2002); Lisboa et al. (2004, 2010); Liang et al. (2011); Li et al. (2011), and
PSS in supersymmetric quantum mechanics Leviatan (2004, 2009); Liang et al.
(2013); Shen et al. (2013).

In recent years, the study of single-particle resonant states has attracted
increasing attentions due to the essential roles in the exotic nuclei with
unusual $N/Z$ ratios. In exotic nuclei, the neutron or the proton Fermi
surface is very close to the continuum threshold and the valence nucleons can
be easily scattered to the single-particle resonant states in the continuum
due to the pairing correlations and the couplings between the bound states and
the continuum become very important Meng and Ring (1996); Dobaczewski et al.
(1996); Pöschl et al. (1997); Meng et al. (2006). Therefore, the study of the
PSS in the single-particle resonant states should also be of great interests
and importance. Until now, there are already some investigations of the PSS in
the single-particle resonant states. The PSS and SS in nucleon-nucleus and
nucleon-nucleon scattering have been investigated Ginocchio (1999b); Leeb and
Wilmsen (2000); Ginocchio (2002); Leeb and Sofianos (2004). In 2004, Zhang et
al. confirmed that the lower components of the Dirac wave functions for the
resonant PS doublets also have similarity properties Shi-Sheng et al. (2004).
Guo et al. investigated the dependence of the pseudospin breaking for the
resonant states on the shape of the mean-field potential in a Woods-Saxon form
Guo et al. (2005); Shi et al. (2020); Liu et al. (2022) as well as on the
ratio of neutron and proton numbers Guo and Fang (2006). In 2012, great
progress has been achieved by Lu et al. in Ref. Lu et al. (2012), where they
gave a rigorous justification of PSS in single-particle resonant states and
shown that the PSS in single-particle resonant states is also exactly
conserved under the same condition for the PSS in bound states, i.e.,
$\Sigma(r)=0$ or $d\Sigma(r)/dr=0$ Lu et al. (2012). However, the wave
functions of the PS partners in the PSS limit are still absent. And also their
research is mainly based on a radial square-well potential Lu et al. (2013).
Furthermore, a uniform description for the conservation and breaking of PSS,
i.e., from the PSS limit to cases with finite-depth potentials, is highly
expected.

In this letter, we will illustrate the exact conservation and breaking of PSS
in the nuclear single-particle states in spherical Woods-Saxon potentials.
Green’s function (GF) method Economou (2006); Tamura (1992); Foulis (2004);
Zhang et al. (2011); Sun (2016) is employed, which has been confirmed to be
one of the most efficient tools for studying the single-particle resonant
states with the following advantages: treating the single-particle bound and
resonant states on the same footing, precisely determining the energies and
widths for all resonances regardless of their widths, and describing properly
the spatial density distributions Sun et al. (2014, 2020); Chen et al. (2020);
Wang and Sun (2021). Besides, this method can describe the resonant states in
any potentials without any requirement on the potential shape.

In a relativistic description, nucleons are Dirac spinors moving in a mean-
field potential with an attractive scalar potential $S({\bm{r}})$ and a
repulsive vector potential $V({\bm{r}})$ Ring (1996). The Dirac equation for a
nucleon reads

$[\bm{\alpha}\cdot\bm{p}+V(\bm{r})+\beta(M+S(\bm{r}))]\psi_{n}(\bm{r})=\varepsilon_{n}\psi_{n}(\bm{r}),$
(1)

where $\bm{\alpha}$ and $\beta$ are the Dirac matrices and $M$ is the nucleon
mass. Based on the Dirac Hamiltonian $\hat{h}(\bm{r})$, a relativistic single-
particle Green’s function $\mathcal{G}(\bm{r},\bm{r^{\prime}};\varepsilon)$
can be constructed, which obeys

$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}^{\prime};\varepsilon)=\delta(\bm{r}-\bm{r}^{\prime}).$
(2)

With a complete set of eigenstates $\psi_{n}(\bm{r})$ and eigenvalues
$\varepsilon_{n}$, the Green’s function can be simply represented as

$\mathcal{G}(\bm{r},\bm{r}^{\prime};\varepsilon)=\sum_{n}\frac{\psi_{n}(\bm{r})\psi_{n}^{{\dagger}}(\bm{r}^{\prime})}{\varepsilon-\varepsilon_{n}},$
(3)

which is a $2\times 2$ matrix because of the upper and lower components of the
Dirac spinor $\psi_{n}(\bm{r})$. Equation (3) is fully equivalent to Eq. (2).

For a spherical nucleus, the Green’s function can be expanded as

$\mathcal{G}({\bm{r}},{\bm{r}^{\prime}};\varepsilon)=\sum_{\kappa
m}Y_{jm}^{l}(\theta,\phi)\frac{\mathcal{G}_{\kappa}(r,r^{\prime};\varepsilon)}{rr^{\prime}}Y_{jm}^{l*}(\theta^{\prime},\phi^{\prime}),$
(4)

where $Y_{jm}^{l}(\theta,\phi)$ is the spin spherical harmonic,
$\mathcal{G}_{\kappa}(r,r^{\prime};\varepsilon)$ is the radial Green’s
function, and the quantum number $\kappa=(-1)^{j+l+1/2}(j+1/2)$. The Eq. (2)
can be reduced as

$\left[\varepsilon-\left(\begin{array}[]{cc}\Sigma(r)&-\frac{d}{dr}+\frac{\kappa}{r}\\\
\frac{d}{dr}+\frac{\kappa}{r}&\Delta(r)-2M\\\
\end{array}\right)\right]\mathcal{G}_{\kappa}(r,r^{\prime};\varepsilon)=\delta(r-r^{\prime})I,$
(5)

where $\Sigma(r)\equiv V(r)+S(r)$, $\Delta(r)=V(r)-S(r)$, and $I$ is a two-
dimensional unit matrix. A radial Green’s function
$\mathcal{G}_{\kappa}(r,r^{\prime};\varepsilon)$ could be constructed with
exact asymptotic behaviors of the Dirac wave functions for bound states and
continuum. For these details, please see Refs. Sun et al. (2014); Sun (2016)

To study the conservation and breaking of PSS in resonant states, radial
Woods-Saxon potentials are considered both for $\Sigma(r)$ and $\Delta(r)$,

$\Sigma(r)=\frac{C}{1+e^{(r-R)/a}},~{}~{}\Delta(r)=\frac{D}{1+e^{(r-R)/a}}.$
(6)

Here, the potential depths $C=-66$ MeV and $D=650~{}$MeV, the width $R=7$ fm,
and the diffusivity parameter $a=0.3$ fm are adopted.

On the single-particle complex energy plane, bound and resonant states
respectively distribute on the negative real energy axis and in the fourth
quadrant. The energy $\varepsilon_{n}$ is real for bound states while complex
for the resonant states and in latter case $\varepsilon_{n}=E-i\Gamma/2$ with
$E$ and $\Gamma$ respectively being the resonant energy and width. Meanwhile,
these eigenvalues are the poles of Green’s function as shown in Eq. (3). Thus,
a direct approach that by searching for the poles of Green’s function to
determine the single-particle energies $\varepsilon_{n}$ has been proposed
naturally Sun et al. (2020); Chen et al. (2020); Wang and Sun (2021). In
practice, one can do this by calculating the integral function of the Green’s
function $G_{\kappa}(\varepsilon)$ for each partial wave $\kappa$ at different
energies $\varepsilon$ Wang and Sun (2021)

$G_{\kappa}(\varepsilon)=\int
dr\left(|\mathcal{G}_{\kappa}^{(11)}(r,r;\varepsilon)|+|\mathcal{G}_{\kappa}^{(22)}(r,r;\varepsilon)|\right),$
(7)

where $|\mathcal{G}_{\kappa}^{(11)}(r,r;\varepsilon)|$ and
$|\mathcal{G}_{\kappa}^{(22)}(r,r;\varepsilon)|$ are the moduli of the Green’s
functions respectively for the “11” and “22” matrix elements. To search for
the bound and resonant states, Green’s functions in a wide energy range are
calculated by scanning the single-particle energy $\varepsilon$. For the bound
states, the energies $\varepsilon$ are taken along the negative real energy
axis. For the resonant states, the energies $\varepsilon$ are complex
$\varepsilon=\varepsilon_{r}+i\varepsilon_{i}$ which are scanned in the fourth
quadrant of the complex energy plane $\varepsilon$, both along the real and
imaginary energy axes. In Fig. 1, the resonant parameters of the state
$3d_{5/2}$ are exactly determined to be $E=2.2728~{}$MeV and
$\Gamma/2=1.9949~{}$MeV by searching for the poles of the Green’s functions in
the fourth quadrant of the complex energy plane $\varepsilon$, where a sharp
peak is observed at $\varepsilon_{r}=2.2728$ MeV and
$\varepsilon_{i}=-1.9949~{}$MeV. Calculations are done with an energy step of
$0.1$ keV for the integral functions $G_{\kappa}(\varepsilon)$ in a coordinate
space with size $R_{\mathrm{max}}=20~{}$fm and a step of $dr=0.05$ fm. This
approach has been certified to be highly effective for all resonant states
regardless of whether they are wide or narrow Chen et al. (2020); Wang and Sun
(2021).

Figure 1: (Color online) Single-particle resonant state $3d_{5/2}$ located in
the fourth quadrant of the complex energy plane
$\varepsilon=\varepsilon_{r}+i\varepsilon_{i}$ determined by searching for the
poles of the Green’s function $G_{\kappa}(\varepsilon)$. Figure 2: (Color
online) Density distributions $\rho_{\kappa}(r,\varepsilon)$ of the resonant
state $3d_{5/2}$ with the contributions from the upper and lower components of
the Dirac wave functions.

Besides, with the Green’s function method, the density distributions in the
coordinate space can also be examined by exploring
$\rho_{\kappa}(r,\varepsilon)$ defined at the energy $\varepsilon=E$,

$\displaystyle\rho_{\kappa}(r,\varepsilon)$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi
r^{2}}\frac{1}{\pi}{\mathrm{Im}}\left[\mathcal{G}_{\kappa}^{(11)}(r,r;E)+\mathcal{G}_{\kappa}^{(22)}(r,r;E)\right],$

where the terms $\mathcal{G}_{\kappa}^{(11)}(r,r;E)$ and
$\mathcal{G}_{\kappa}^{(22)}(r,r;E)$ are respectively related with the upper
and lower components of the Dirac wave functions (c.f. Eq. (3)). In Fig. 2,
the density distributions for the resonant state $3d_{5/2}$ are also plotted,
with the red and blue lines the contributions from the upper and lower
components. In order to better display the density distribution, here and
hereafter, we adjust the highest peak of $\rho_{\kappa}(r,\varepsilon)$ to be
$1.0$ fm${}^{-1}\cdot$MeV-1 and ensure relative sizes of the different
components remaining unchanged. Since $3d_{5/2}$ is a low lying narrow
resonant state, the density distribution of the lower component is reduced to
be zero very soon while very slight oscillation can be observed in the upper
component.

Figure 3: (Color online) The poles of Green’s functions on the complex energy
plane in the Woods-Saxon potentials with depth $C=0$ (solid symbols),
$C=-33~{}$MeV (half-filled symbols) and $C=-66~{}$MeV (empty symbols) for the
PS partners $d_{5/2}$ (square) and $g_{7/2}$ (diamond).

In the following, the PSS in resonant states will be studied with the Green’s
function method. In Fig. 3, we show the solutions in different potential
depths on the complex energy plane for the PSS doublets with pesudospin
angular momentum $\tilde{l}=3$, i.e., $d_{5/2}$ with $\kappa=-3$ and $g_{7/2}$
with $\kappa=4$. In the PSS limit, i.e., the potential depth $C=0$, all the
roots locate in the lower half-plane and there is no bound states. Three pairs
of resonant PS doublets with exact the same energy and width are obtained,
indicating the exact conservation of PSS in resonant states. Besides, one
single intruder state $1d_{5/2}$ appears near the continuum threshold. With
finite potential depths, one finds the breaking of the PSS with obvious energy
and width splitting between the PS partners. More in details, for most PS
partners, $g_{7/2}$ with pseudospin $\tilde{s}=+1/2$ owns lower energy and
smaller width compared with the PS partner $d_{5/2}$ with $\tilde{s}=-1/2$ due
to the higher PCB potential of $g_{7/2}$. One exception is the PS partner
$(3d_{5/2},2g_{7/2})$ obtained when $C=-66$ MeV, the energies of which are
$\varepsilon(3d_{5/2})=2.2728-i1.9949$ MeV and
$\varepsilon(2g_{7/2})=2.5422-i0.1019~{}$MeV, respectively. Meanwhile, PS
partners move down and some resonant PS partners evolve to be bound states.
For PS doublets with other values of $\tilde{l}$, similar behaviors concerning
the exact conservation and the breaking of the PSS could be observed.

Figure 4: (Color online). Evolutions of the density distributions
$\rho_{\kappa}(r,\varepsilon)$ for the PS doublets $3d_{5/2}$ and $2g_{7/2}$
from the PSS limit with $C=0~{}$MeV to the cases with finite-depth potentials
$C=-15,-33,-45$ MeV. Densities plotted in the left and right columns are
contributed by the upper and lower components of the Dirac wave functions,
respectively.

To study the conservation and breaking of PSS, the similarities of the lower
component of the Dirac wave functions for the PS doublets are also examined.
In Fig. 4, the spacial density distributions $\rho_{\kappa}(r,\varepsilon)$ of
the PS partners $3d_{5/2}$ and $2g_{7/2}$ in different potential depths $C$
are plotted. The left and right columns are respectively the contributions
from the upper and lower components of Dirac wave functions. In the PSS limit,
i.e., $C=0$, the density distributions for the PS partner are exactly the same
for the lower component while differ one node for the upper component, which
provides a direct evidence for the exact conservation of the PSS in resonant
states and also certifies that PSS is a symmetry of the Dirac Hamiltonian
related with the lower component of the Dirac spinor. In the case of the
finite-depth potentials, the lower components of the density distributions for
the PS partners are no longer exactly the same, but it still remains strong
similarity. Their difference is manifested as an obvious phase shift, which
increases with the deepening of the potential depths. For example, when the
potential depth $C=-45~{}$MeV, there is almost one $\pi$ phase shift between
the PS partners for the density distributions outside the potential in the
area of coordinate $r>7.5~{}$fm.

In summary, the conservation and breaking of the PSS in nuclear single-
particle states are investigated within the relativistic framework by
exploring the poles of Green’s function in spherical Woods-Saxon potentials.
Great advantages of the Green’s function method that precisely determining the
energies and widths for all resonances regardless of their widths and
describing properly the spatial density distributions provide a good platform
to study restoration and breaking of PSS. In the PSS limit, i.e.,
$\Sigma(r)\equiv V(r)+S(r)=0$, the PSS in resonant states is confirmed to be
strictly conserved with the exact same energy and width between the PS
partners. Besides, we also find identical density distributions of the small
components for the first time, which provides a direct evidence that the PSS
is a relativistic dynamical symmetry connected with the small component of the
Dirac spinor. In the cases with finite-depth potentials, the PSS is broken
with obvious energy and width splitting for the PS partners as well as a phase
shift between the spatial density distributions of the small components.

Helpful discussions with Hao-Zhao Liang are acknowledged. This work was partly
supported by the National Natural Science Foundation of China (No. U2032141
and No. 12375126), the Open Project of Guangxi Key Laboratory of Nuclear
Physics and Nuclear Technology (No. NLK2022-02), the Central Government
Guidance Funds for Local Scientific and Technological Development, China (No.
Guike ZY22096024), the Fundamental Research Funds for the Central Universities
, and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
under Germanys Excellence Strategy EXC-2094-390783311, ORIGINS.

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# Parametric entropy based Cluster Centriod Initialization for $k$-means
clustering of various Image datasets

Faheem Hussayn1, Shahid M Shah 2  Communication Control & Learning Lab
($C^{2}L^{2}$)
Department of Electronics & Communication Engineering,
National Institute of Technology Srinagar21
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

One of the most employed yet simple algorithm for cluster analysis is the
$k$-means algorithm. $k$-means has successfully witnessed its use in
artificial intelligence, market segmentation, fraud detection, data mining,
psychology, etc., only to name a few. The $k$-means algorithm, however, does
not always yield the best quality results. Its performance heavily depends
upon the number of clusters supplied and the proper initialization of the
cluster centroids or seeds.
In this paper, we conduct an analysis of the performance of $k$-means on image
data by employing parametric entropies in an entropy based centroid
initialization method and propose the best fitting entropy measures for
general image datasets. We use several entropies like Taneja entropy, Kapur
entropy, Aczel Daroczy entropy, Sharma Mittal entropy. We observe that for
different datasets, different entropies provide better results than the
conventional methods. We have applied our proposed algorithm on these
datasets: Satellite, Toys, Fruits, Cars, Brain MRI, Covid X-Ray.

###### Index Terms:

††publicationid: pubid:

##  I Introduction

A subset of artificial intelligence, machine learning, supplies machines with
the ability to learn and make decisions without needing to be programmed
explicitly. If learning is accomplished without supplying a machine with a
labeled dataset, the machine needs to find the implicit data patterns without
external support, this is known as unsupervised machine learning. Many real-
life problems are actually modeled this way because, in real life, the pattern
of data is difficult to know in advance [1].
Recent times have witnessed an avalanche of data. The use of data mining
techniques has thus seen a tremendous increase and clustering has been one of
the most used unsupervised techniques. Clustering, sometimes referred to as
cluster analysis, is an unsupervised machine learning technique wherein we
tackle the problem of grouping or division of data points such that those data
points that fall within the same group are more related to each other and less
related to the data-points clustered into other groups [2]. It can also be
simply defined as the collection or grouping of objects on the basis of
similarity and dissimilarity between the objects. This technique finds its
applications in Wireless networks, System diagnostics, Search engines, Fraud
detection, Market Segmentation, Satellite imagery, pattern recognition, big
data analytics, and so on.
The simplest unsupervised learning algorithm that is usually employed in
solving clustering problems is the $k$-means algorithm. $k$-means, being one
of the most famous algorithms employed for clustering, has also witnessed its
use as part of other algorithms [3]. The $k$-means algorithm is iterative in
nature and aims to assign every data-point of a data-set to one of the $k$
clusters based on the features supplied. The $k$-means algorithm partitions
’$n$’ data points or observations into ‘$k$’ groups or clusters. $K$-means
algorithm has applications in various areas, like energy analytics [4], attack
detection [5], [6], [7]. Very recently $k$-means algorithm has also been used
in activity detection in smart grid-based systems [8]. However, the quality of
the solution and convergence speed of the $k$-means algorithm largely depends
on the number of clusters supplied and the position of the initial seed points
or cluster centroids. The traditional $k$-means algorithm initializes cluster
centroids randomly but this has obvious drawbacks. Various methods have been
devised for alternate centroid initialization, which will be discussed in the
next section. In this study, we will focus on an centroid initialization
method based on the maximization of entropy measure.

##  II Related Works

For the problem of proper centroid initialization, two groups of studies
exist. The first group of studies has focused on improving the existing random
initialization. When incorporating entropy, Steinbach et al. found that the
”bisecting $k$-means” method generally outperformed the classical $k$-means
and when not measuring entropy, it almost performed similarly [9]. However,
they did not include any time-related metrics for comparison. $k$-means++ is
another such modification of random centroid initialization. In this approach,
Arthur and Vassilvitskii [10] initialized centroids from the data points at
random while using the squared distance from the already initialized centroids
to weigh potential centroids. The effect was that in contrast to the random
initialization, this approach would ensure a maximal distance or ”spread” of
cluster centers or centroids.
The other group of studies has devised alternative methods to random centroid
initialization for $k$-means and are consistent to the method we will discuss.
A method for seed point selection that is recursive in nature was discussed by
Duda and Hart [11]. A MaxMin algorithm was devised by Higgs et al. [12] and
Snarey et al. [13] which was based on selecting a subset from original
database to be used for initial centroids in order to create initial clusters.
The bilinear program was introduced by Bradley et al. [14] which determined
initial points in such a way that the sum of distances of each data point
should be minimized to the nearest centroid. Su & Dy [15] came up with a
deterministic method for centroid initialization. The method is hierarchically
divisive in nature and is based on Principal Component Analysis (PCA). Cao et.
al came up with an effective method to initialize clusters based on cohesion
and coupling degree [16]. Bai et. al proposed an initialization method based
on distance and density metrics [17]. Using the concept of Voronoi circles and
their radii, Reddy et. al came up with an initialization technique [18].
Mahmud et al. came up with a method which is faster than traditional k-means.
In this method, the selection of initial points is carried out using a
weighted average score on the sorted data [19]. A density based approach was
developed by Gingles and Celebi [20] which was based on the hypothesis that
centroids of clusters would naturally occur near the areas of high data-point
density. Another density-based approach by Dalhatu et. al [21] has also been
developed. One of the recent works for initialization based on entropy and
with respect to image segmentation (clustering of pixels) was carried out by
Chowdhury et al. [22]. In their work, they employed the maximization of
Shannon’s Entropy to determine the optimal initial positions of the cluster
centroids. This yielded lesser computation time and number of iterations with
respect to image datasets. The given method, however, was only tested on
select images using Shannon’s entropy only. In spite of all these methods,
presently, there is no universally accepted method for centroid initialization
of $k$-means algorithm which is the prime reason for the pursuit of this
study.

##  III Main Contribution

The traditional $k$-means algorithm initializes centroids randomly and as
already discussed, the quality of clustering depends upon the location of the
initial centroids. We have employed the entropy maximization algorithm devised
by Chowdhury et al. [22] to initialize the centroids. However, in place of
Shannon’s Entropy, we test out different parametric entropy measures on
contrasting image datasets with different parameters to yield the best fitting
entropy for centroid initialization. The entropy based initialization works on
the entropy maximization principle. Exploiting the fact that for a multi
spectral image, the intensity values for each color band of a particular pixel
are mutually independent, we can easily calculate the probability of a pixel.
Let N denote the total number of pixels in a given image and $a$, $b$, and $c$
be the intensity values of the Red, Green, and Blue color bands respectively.
Also, let $n_{a}$, $n_{b}$ and $n_{c}$ be the number of intensity values for
$a$, $b$ and $c$, respectively. Then, using the concept of independent random
variables we can arrive at the equation:

$P(R=a,G=b,B=c)$ $P(R=a)*P(G=b)*P(B=c)$
$=\frac{n_{a}}{N}*\frac{n_{b}}{N}*\frac{n_{c}}{N},\forall 0\leq a,b,c\leq 255$

Using this probability, we can calculate the entropy measure for all the
intensities in an image.

##  IV Methodology and Datasets

The algorithm for calculating the initial cluster centroids is given as
Algorithm 1. For the entropy calculation step, we use the following entropy
measures.
Shannon Entropy
Proposed by C.E Shannon [23], it is given as:

$H_{S}(P)=-\sum_{i=1}^{n}p_{i}\log p_{i}$ (1)

Kapur
Proposed by JN Kapur [24], this entropy is given by:

$H_{K}(P)=\frac{1}{1-\alpha}\log\left(\frac{\sum_{i=1}^{n}p_{i}^{\alpha+\beta-1}}{\sum_{i=1}^{n}p_{i}^{\beta}}\right)$
(2)

where $\alpha\neq 1,\alpha>0,\beta\geq 1$

Aczél Daróczy
Proposed by J Aczél, Z Daróczy [25], it can be calculated using:

$H_{AD}(P)=\frac{1}{\beta}\arctan\left(\frac{\sum_{i=1}^{n}p_{i}^{\alpha}\sin(\beta\log
p_{i})}{\sum_{i=1}^{n}p_{i}^{\alpha}\cos(\beta\log p_{i})}\right)$ (3)

where $\beta\neq 0,\alpha>0$

Havrda and Charvát
Proposed by J Havrda and F Charvát [26], this entropy is given by:

$H_{HC}(P)=\frac{1}{(2^{1-\alpha}-1)}\left[\sum_{i=1}^{n}p_{i}^{\alpha}-1\right]$
(4)

where $\alpha\neq 1,\alpha>0$

Taneja
Proposed by I.J Taneja [27], this entropy is calculated using the formula:

$H_{T}(P)=-\frac{2^{\alpha-1}}{\sin\beta}\sum_{i=1}^{n}p_{i}^{\alpha}\sin(\beta\log
p_{i})$ (5)

where $\alpha\neq k\pi,k=0,1,2,...,k>0$

Sharma Mittal
Proposed by Sharma and Mittal [28], this entropy is given by:

$H_{SM}(P)=\frac{1}{(2^{1-\alpha}-1)}\left[\left(\sum_{i=1}^{n}p_{i}^{\beta}\right)^{\frac{\alpha-1}{\beta-1}}-1\right]$
(6)

where $\alpha\neq 1,\alpha>,\beta\neq 1,\beta>0$
In this research, instead of using a few images to test the initialization, we
have used different real life data-sets containing similar images and averaged
out the results to evaluate time-related metrics for each entropy measure. The
data-sets were obtained from publicly available sources of Kaggle and some of
the images were manually curated from Google. A summary of the datasets used
is given in Table 1.
Since in this study we have focused on the initialization method of the
cluster centroids, we need a way to determine the optimal number of clusters
as it is also a factor on which the quality of $k$-means clustering depends as
already discussed. We used the classic ‘elbow method’ to achieve this. The
basic principle of the elbow method is that it plots the cost function (sum of
square error values) for different values of $k$. Clearly, as the number of
clusters increase, the SSE will reduce. A point will be reached where
increasing the number of clusters will not have a drastic effect on the cost
function. This we take as the optimal value of $k$. We can have the sum of
squared distances of all data-points to the cluster as the cost function,
where we call it ‘inertia’ or we can have the mean of squared distances of
each data point to its nearest cluster, where we call it ‘dispersion’.
Since there are multiple images in a particular dataset, using the fact that
the distribution of pixel intensities will be similar, we employed the elbow
method on any one of the images in a particular image dataset to determine the
optimal value for the number of clusters $k$ and conducted the analysis for
all the images.

TABLE I: Dataset Details Dataset | Image Count | Attributes | Optimal k | Source
---|---|---|---|---
Satellite | 25 | 3 | 3 | Kaggle
Toys | 50 | 3 | 4 | Google
Brain MRI | 30 | 2 | 3 | Kaggle
X-Ray | 25 | 2 | 2 | Kaggle
Fruits | 40 | 3 | 5 | Google
Cars | 50 | 3 | 3 | Kaggle

An example of clustering using this approach is shown in Figure 1 and the
corresponding comparison for the number of iterations utilized by $k$-means to
converge is given in Figure 2. The $th$ value for this image is set to 220.

Figure 1: Original Image  Figure 2: Scatter Plot Figure 3: Distortion based
Elbow Method Figure 4: Inertia based Elbow Method Figure 5: Optimal Clustering
(K=3) Figure 6: Iteration Comparison of Initializations for Car Image

It is clear from the above graph that the most appropriate initialization for
clustering this image using $k$-means algorithm is the Shannon and Taneja
Entropy. The random initialization denotes the default random initialization
of the $k$-means algorithm. The image shown in Figure 1: (a) is derived from
the car dataset. We cannot possibly show the clustering for each image used in
the experiment as we have employed image datasets and not single images.

Algorithm 1 Entropy Maximization Initialization

Image (dataset) and number of clusters ($K$)

Initial Centroid List

  1. 1.

Input the number of cluster $K$ and $th$ (threshold for centroid spacing).

  2. 2.

Initialize the number of seeds $n_{cen}=1$

  3. 3.

Calculate needed entropy for each pixel in the image.

  4. 4.

Sort the pixels in descending order of entropy values.

  5. 5.

Take the first pixel from the sorted list and include in centroid list.

  6. 6.

Take the next pixel from the list and calculate its euclidean distance with
all the pixels in SE.

  7. 7.

if $min_{ED}>TH$, include this pixel in centroid list and perform
$n_{cen}=n_{cen}+1$. Otherwise goto step 6.

  8. 8.

if $n_{cen}=K$, stop. Otherwise goto step 6.

The $th$ or threshold value is a critical variable that essentially dictates
the distance between the clusters. If it is not properly initialised, the
distribution of centroids will not be appropriate. For example, if the value
of $th$ is too large, the computation cost will rise and so will be the time
taken for convergence. On the contrary, a value that is too small can cause
the centroids to be very near causing the algorithm to converge prematurely.
So, the choice of threshold value is important and and should be decided by
considering the ”spread” of the data.

##  V Results and Discussions

In our experiment, we used several images from contrasting image data-sets to
analyse the performance of multiple entropy measures on the cluster
initialisation. The results were evaluated using the metrics: number of
iterations of K-means (NIK), which essentially is the number centroid
movements it takes to converge KMeans, Computation Time ($C_{T}$), which is
the time to convergence, and Initialization Time ($I_{T}$). We club the
Computation and Initialization time and call it Total Time. Since the
$k$-means algorithm always converges and we are only modifying the
initialization approach using different entropies, the sum of squared errors
(SSE) metric may have similar results so our focus would be more on time-
related metrics. The comparative description of the results is given in Table
II. The initialization method using Shannon entropy is the original entropy
maximization method devised by Chowdhury et al. [22]. [22]

TABLE II: Comparison of Initialization for Image Datasets Dataset | Initialization | Avg. NIK | Total Time | SSE
---|---|---|---|---
Satellite | Random | 4.76 | 2.148 | 3751.72
| Shannon | 4.04 | 1.84 | 3751.5
Toys | Random | 4.07 | 1.4335 | 1493.13
| Shannon | 4.21 | 1.819 | 1492.3
| Taneja | 3.11 | 1.079 | 1492.3
Fruits | Random | 5.2 | 1.822 | 1564.1
| Shannon | 6.3 | 1.938 | 1564.6
| Taneja | 3.1 | 0.469 | 1563.6
Cars | Random | 3.9 | 0.9242 | 1331.91
| Shannon | 4.39 | 1.469 | 1332.02
| Taneja | 2.01 | 0.401 | 1330.32
Brain MRI | Random | 4.91 | 0.0203 | 1364.66
| Shannon | 6.285 | 0.065 | 1364.5
| Kapur | 4.285 | 0.036 | 1363.42
Covid X-Ray | Random | 4.89 | 0.05 | 1379
| Shannon | 3.69 | 0.079 | 1378.34
| Kapur | 2.24 | 0.038 | 1377.43

From our study we concluded that there was no single entropy that was
appropriate for the cluster initialization of every kind of image dataset. We
get the insight that for certain datasets, certain entropy measures worked
better. We summarize our results with the following insights:

  * •

For the datasets with natural intensity levels and a higher dynamic range such
as the images of cars, robots, toys, vegetables, fruits, etc. Taneja Entropy
was the most appropriate.

  * •

For the datasets with wide range of details like the satellite imagery,
Shannon Entropy was the most appropriate for cluster initialization.

  * •

For the datasets with similar saturation and less dynamic ranges, like the
medical datasets of X-Ray and MRI Images, Kapur’s Entropy was the most
appropriate.

##  VI Conclusion and Future Scope

In our study, we have extended the entropy based initialization method for
image clustering using $k$-means by employing parametric entropy measures and
demonstrated its effectiveness on contrasting image datasets. To generalize
the results found in this study and establish facts based upon them would be
irresponsible at this point due to the small size and lesser number of the
datasets used. However, our findings do point out that further research should
be pursued, and that the further exploration of the parametric entropies
discussed in the previous sections would prove beneficial in other entropy
based avenues of computing. In future, we would try to hypothesize or assert
why certain entropy measures work better with certain kinds of data and
include more entropy measures for testing. We would also study the effect of
using the generalized entropy measures in place of Shannon’s entropy with
other research problems, such as cluster validation, metric evaluation, and so
on. .

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