Ci a ion: Kubicek, J.; Va yso a, A.;
Ce ny, M.; Hanca o a, K.; Oczka, D.;
Augus ynek, M.; Penhake , M.;
P okop, O.; Scu ek, R. Pe o mance
and Robus ness o Regional Image
Segmen a ion D i en by Selec ed
E olu iona y and Gene ic
Algo i hms: S udy on MR A icula
Ca ilage Images. Senso s 2022,22,
6335. h ps://doi.o g/10.3390/
s22176335
Academic Edi o : Pawel S umillo
Recei ed: 6 July 2022
Accep ed: 18 Augus 2022
Published: 23 Augus 2022
Publishe ’s No e: MDPI s ays neu al
wi h ega d o ju isdic ional claims in
published maps and ins i u ional a il-
ia ions.
Copy igh : © 2022 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
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senso s
A icle
Pe o mance and Robus ness o Regional Image Segmen a ion
D i en by Selec ed E olu iona y and Gene ic Algo i hms:
S udy on MR A icula Ca ilage Images
Jan Kubicek 1,*, Alice Va yso a 1, Ma in Ce ny 1, K is yna Hanca o a 1, Da id Oczka 1,
Ma in Augus ynek 1, Ma ek Penhake 1, Ond ej P okop 2and Radomi Scu ek 3
1
Depa men o Cybe ne ics and Biomedical Enginee ing, VŠB—Technical Uni e si y o Os a a, 17.lis opadu
2172/15, Po uba, 708 00 Os a a, Czech Republic
2MEDIN, a.s., Vlacho icka 619, 592 31 No e Mes o na Mo a e, Czech Republic
3Depa men o Secu i y Se ices, Facul y o Sa e y Enginee ing, VŠB—Technical Uni e si y o Os a a, ul.
Lumi o a 3, 700 30 Os a a, Czech Republic
*Co espondence: [email p o ec ed]
Abs ac :
The analysis and segmen a ion o a icula ca ilage magne ic esonance (MR) images
belongs o one o he mos commonly ou ine asks in diagnos ics o he musculoskele al sys em o
he knee a ea. Con en ional egional segmen a ion me hods, which a e based ei he on he his og am
pa i ioning (e.g., O su me hod) o clus e ing me hods (e.g., K-means), ha e been equen ly used
o he ask o egional segmen a ion. Such me hods a e well known as as and well wo king in he
en i onmen , whe e ca ilage image ea u es a e eliably ecognizable. The well-known ac is ha
he pe o mance o hese me hods is p one o he image noise and a e ac s. In his con ex , egional
segmen a ion s a egies, d i en by ei he gene ic algo i hms o selec ed e olu iona y compu ing
s a egies, ha e he po en ial o o e come hese adi ional me hods such as O su h esholding o
K-means in he con ex o hei pe o mance. These op imiza ion s a egies consecu i ely gene a e
a py amid o a possible se o his og am h esholds, o which he quali y is e alua ed by using he
i ness unc ion based on Kapu ’s en opy maximiza ion o ind he mos op imal combina ion o
h esholds o a icula ca ilage segmen a ion. On he o he hand, such op imiza ion s a egies
a e o en compu a ionally demanding, which is a limi a ion o using such me hods o a s ack o
MR images. In his s udy, we publish a comp ehensi e analysis o he op imiza ion me hods based
on uzzy so segmen a ion, d i en by a i icial bee colony (ABC), pa icle swa m op imiza ion
(PSO), Da winian pa icle swa m op imiza ion (DPSO), and a gene ic algo i hm o an op imal
h esholding selec ion agains he ou ine segmen a ions O su and K-means o analysis and he
ea u es ex ac ion o a icula ca ilage om MR images. This s udy objec i ely analyzes he pe -
o mance o he segmen a ion s a egies upon a iable noise wi h dynamic in ensi ies o epo a
segmen a ion’s obus ness in a ious image condi ions o a a ious numbe o segmen a ion classes
(4, 7, and 10), ca ilage ea u es (a ea, pe ime e , and skele on) ex ac ion p eciseness agains he
ou ine segmen a ion s a egies, and las ly he compu ing ime, which ep esen s an impo an ac-
o o segmen a ion pe o mance. We use he same se ings on indi idual op imiza ion s a egies:
100 i e a ions and 50 popula ion. This s udy sugges s ha he combina ion o uzzy h esholding
wi h an ABC algo i hm gi es he bes pe o mance in he compa ison wi h o he me hods as om he
iew o he segmen a ion in luence o addi i e dynamic noise in luence, also o ca ilage ea u es
ex ac ion. On he o he hand, using gene ic algo i hms o ca ilage segmen a ion in some cases
does no gi e a good pe o mance. In mos cases, he analyzed op imiza ion s a egies signi ican ly
o e come he ou ine segmen a ion me hods excep o he compu ing ime, which is no mally lowe
o he ou ine algo i hms. We also publish s a is ical es s o signi icance, showing di e ences in he
pe o mance o indi idual op imiza ion s a egies agains O su and K-means me hod. Las ly, as a
pa o his s udy, we publish a so wa e en i onmen , in eg a ing all he me hods om his s udy.
Keywo ds:
medical image segmen a ion; a icula ca ilage; egional segmen a ion; ABC; PSO; DPSO;
O su h esholding; K-means clus e ing
Senso s 2022,22, 6335. h ps://doi.o g/10.3390/s22176335 h ps://www.mdpi.com/jou nal/senso s
Senso s 2022,22, 6335 2 o 38
1. In oduc ion
Medical image segmen a ion ep esen s one o he essen ial p ocedu es in medical
image analysis. The me hods, belonging in he a ea o medical image segmen a ion, play
an impo an ole o he image a ea decomposi ion wi h he ocus image unde s anding.
Such me hods enable wo impo an issues: (1) he ex ac ion o mo phological ea u es o
objec s o in e es and (2) he consequen ex ac ion o a ious ea u es o such objec s wi h
he aim o he quan i ica ion o biological issues [
1
–
7
]. In his con ex , we a e ou inely
ocused on ei he geome ical pa ame e s o image egions such as he a ea, pe ime e ,
diame e , o cu a u e pa ame e s o in ensi y pa ame e s, including a s a is ic es ima ion
o he in ensi y spec um o quan i y he image su ace [
8
,
9
]. No ha ing au oma ed
image segmen a ion me hods, clinical expe s would ha e o pe o m medical issues
segmen a ion manually by con ou ing objec s o in e es . Such a p ocedu e would be
su ely linked wi h subjec i e e o depended on he skills o he indi idual physician.
On he o he hand, manual con ou ing plays an impo an ole in he objec i iza ion o
au oma ed image segmen a ion me hods, whe e manual anno a ion no mally se es as
a gold s anda d o objec i ely e alua e he segmen a ion pe o mance based on selec ed
e alua ion pa ame e s such as he index o co ela ion, he mean squa ed e o (MSE), he
s uc u al simila i y index (SSIM), and many o he s [10–14].
Image segmen a ion includes a lo o me hods a ying in hei ma hema ical s a egy
and he aim o he segmen a ion. Edge de ec ion ep esen s one o he mos con en ional
me hods o he au oma ic con ou ing o objec s o in e es . He e, we ecognize mul iple
p inciples such as he maximum o he i s de i a i e, o he ze o c ossing de ec o s [
15
–
17
].
These me hods a e no mally linked wi h wo main limi a ions. Fi s ly, hey pe o m he
segmen a ion o he whole image ega dless o he ocus on a pa icula objec o in e es
and mainly hey a e p one o in ensi y a ia ions as he impac o image noise, which
may signi ican ly in luence he segmen a ion quali y. The e o e, hese me hods a e o en
comple ed wi h smoo hing il e s o imp o e he segmen a ion quali y. Mo e sophis ica ed
segmen a ion echniques ep esen ac i e shape models such as ac i e con ou s o le el
se me hods, which a e capable o ocusing on a pa icula objec o in e es and wi hin
a p ede ined numbe o i e a ions pe o m he g adual de ec ion o geome ical ea u es
o objec s o in e es . One o he majo limi a ions o hese me hods is i s compu ing ime,
because hey use a highe numbe o i e a ions, and he compu ing ime is also depended
on image esolu ion [18,19].
The mos ex ensi e a ea o he medical image segmen a ion is egional image segmen-
a ion. These me hods no mally enable medical image decomposi ion in o a p ede ined
numbe o egions [
20
,
21
]. Such a egion is pe cei ed as a ini e numbe o image poin s
(pixels o oxels) ha mu ually sha e simila ea u es. This p ede e mines ha such e-
gionally o ien ed me hods a e able o well ecognize biological issues o in e es wi hin
indi idual egions. Rega ding he na u e o egional segmen a ion, we ecognize so called
nonin e p e ed me hods, which no mally u ilize con en ional segmen a ion s a egies and
only enable a decomposi ion o he image poin s in o indi idual egions wi hou in e -
p e ing he con en o indi idual egions [
22
–
24
]. Nowadays, he ecen ends in medical
image egional segmen a ion a e mainly ocused on so called in e p e ed me hods, enabling
he in e p e a ion o indi idual de ec ed objec s. Such me hods no mally use seman ic
segmen a ion [
25
,
26
]. Among he bene i s o such me hods, i is impo an o men ion ha
hese me hods equi e aining on huge da ase s, which may be a limi a ion in he con ex
o medical image a ailabili y [27–30].
In his pape , we p o ide a comp ehensi e insigh in he o m o a compa a i e analy-
sis o selec ed e olu iona y op imiza ion algo i hms and gene ic algo i hms pe o mances,
which a e used o he uning o con en ional segmen a ion s a egies o achie e a maximal
pe o mance unde a ious image condi ions and de e io a ion by image noise. In his
s udy, we compa e he pe o mance o op imized segmen a ion s a egies wi h he elemen s
Senso s 2022,22, 6335 3 o 38
o a i icial in elligence (e olu iona y and gene ic algo i hms) wi h con en ional s a egies
based on he ha d h esholding (O su me hod) and nonhie a chical clus e ing (K-means).
This analysis should objec i ely poin ou on he pe o mance and impac o mode n op i-
miza ion me hods wi h a i icial in elligence as om he iew o segmen a ion pe o mance
and obus ness unde addi i e noise and also compu ing ime, which is an impo an ac o
o each segmen a ion p ocedu e ega ding complex e ec i i y. This objec i iza ion analysis
in es iga es he obus ness and compu ing equi emen s o indi idual segmen a ion s a e-
gies p o ided unde he in luence o a ious ypes o addi i e de e minis ic noise wi h
dynamical in ensi y. Tha enables he s udy o he pe o mance o indi idual op imiza ion
echniques and i s de e mining pa ame e s along g adually de e io a ed condi ions and
shows he dynamical ea u es o obus ness o indi idual me hods. Besides he analysis o
pe o mance, we also publish he applica ion o egional segmen a ion o ca ilage ea u es
ex ac ion. As a pa o ou esea ch, we publish a es ing so wa e en i onmen , in eg a ing
indi idual me hods o egional segmen a ion wi h he possibili ies o selec indi idual
s ee ing segmen a ion pa ame e s. The so wa e applica ion enables he applica ion o a i-
ous de e minis ic noise gene a o s wi h he se ings o he noise pa ame e s, which con ol
noise in ensi y o simula e image deg ada ion o he es ing o indi idual segmen a ion
s a egies obus ness. This applica ion also enables he segmen a ion accu acy e alua ion
based on selec ed objec i iza ion pa ame e s, which a e also used in his s udy o e alua e
he segmen a ion pe o mance.
The o ganiza ion o he pape is ollowing. In Sec ion 2, we p o ide ecen no es and
ad ances in he a ea o medical image segmen a ion. In Sec ion 3, we in oduce indi idual
segmen a ion s a egies, guidelines o es ing o hese me hods, and da ase s o MR images
o a icula ca ilage used in his s udy. Sec ion 4is aimed on quan i a i e esul s, p o iding
a comp ehensi e insigh on he e ec i i y, obus ness, and limi a ions o segmen a ion
me hods. Sec ion 5is aimed on complex conclusions, discussion, and u u e ends o
his s udy.
2. Recen Wo k
In his sec ion, we ou line he con en ional p ocedu es, se ing o a spa ial image
domain decomposi ion, which is ou inely used as a undamen al ool o medical issues
iden i ica ion. The e a e a ious ma hema ical app oaches, which may be used o he
a o emen ioned image decomposi ion, including mainly echniques based on his og am
pa i ioning, edge de ec o s, con ou s acing, analysis pixel’s ela ionships, me hods o
a i icial in elligence, and o he s [30–38].
One o he mos popula echniques, and i is also he aim o his pape , is he egional
image segmen a ion bases on he his og am h esholding. These me hods no mally allow
o a his og am decomposi ion bases on ei he one, o mul iple h esholds, de ining indi-
idual image egions [
37
,
38
]. He e, one o he mos popula me hods is his o ically O su
segmen a ion, which de ines indi idual h esholds based on he minimiza ion o in a-class
in ensi y a iance and he maximiza ion o in e -class a iance [39–41].
O he popula a eas o egional medical image segmen a ion is clus e ing analysis.
He e, he mos popula me hods a e based on he nonhie a chical clus e ing such as K-
means o uzzy C-means (FCM) segmen a ion [
42
,
43
]. These me hods usually measu e
dis ance-based pa ame e s be ween indi idual pixels and clus e ( egion) cen oids o he
pixel’s classi ica ion [
44
,
45
]. Such me hods a e capable o pe o ming image decomposi ion
in o a ious isola ed classes based on hei ea u es as a le el o simila i y be ween he pixel
in ensi y and he cen oid [
46
–
50
]. O cou se, apa om he men ioned app oaches, he e
a e plen y o o he s, which a e no mally used o medical image segmen a ion, including
edge de ec o s, ou lining image boa de s, he me hods o consecu i e egions o ming,
such as egion g owing, o spli ing me hods, wa e shed o wa ele ans o ma ion, and
o he s [
51
–
53
]. O cou se, in ecen imes, one o he mos popula segmen a ion me hods
deals wi h a ious applica ions o machine and deep lea ning, enabling seman ic segmen a-
Senso s 2022,22, 6335 4 o 38
ion ins ead o he segmen a ion wi hou in e p e a ion as i is ypical in many con en ional
app oaches [54–68].
These ou ine app oaches, as we ou lined ea lie , a e usually easily implemen ed wi h
a easonable compu ing ime, on he o he hand hey no mally su e om ce ain limi a-
ions, which may ha e a signi ican in luence on hei e ec i i y and obus ness as well.
In he compa ison wi h con en ional segmen a ion me hods such as O su h esholding,
which is based on he ha d selec ion o indi idual h esholds in his og ams, we p oposed
schemes on h esholding s a egies u ilizing op imiza ion echniques o op imal h esholds
selec ion om a ious h esholds combina ions. This is supposed o be a mo e e ec i e
app oach, which be e e lec s a pixel’s dis ibu ion inside de ined egions. Fu he mo e,
we p o ide analysis o obus ness o each es ed me hod upon dynamic a ious noise
in luence o objec i ely show dynamic ea u es o pe o mance when an image domain is
g adually de e io a ed by addi i e noise. On he o he hand, we a e awa e ha e olu ion
s a egies and a gene ic algo i hm may ha e a signi ican in luence on he compu ing ime.
The e o e, we also publish ime complexi y analysis, showing hei complexi y. All he
op imized me hods we pu in a con as wi h he con en ional app oaches such as O su
h esholding and he K-means me hod o objec i ely poin ou di e ences in segmen a ion
pe o mance in he con ex bene i s and limi a ions o using e olu iona y and gene ic
algo i hms o a ious numbe s o h esholds and o he s ee ing op imiza ion pa ame e s
o hese op imiza ion echniques.
3. Ma e ials and Me hods
Recen ly, a ious implemen a ions o e olu iona y and gene ic op imiza ion s a egies
ha e inc eased in popula i y o sol ing a ious enginee ing p oblems in he a ea o
op imal se ings o s ee ing pa ame e s o a ious p ocedu es. The main aim o his pape
is o objec i ely poin ou on he e ec i i y and obus ness (a le el o s abili y in a ious
image en i onmen ) o selec ed h esholding-based egional segmen a ion s a egies, being
op imized wi h e olu iona y compu ing me hods (ABC and PSO wi h i s a ian s: FPSO
( uzzy pa icle swa m op imiza ion) and DPSO (Da winian pa icle swa m op imiza ion))
and gene ic algo i hms as om he iew o hei e ec i i y o segmen a ion, ea u es
ex ac ion, and also ime complexi y. On he o he hand, ABC algo i hm is used in he
combina ion wi h uzzy h esholding, which o ms indi idual segmen a ion egions based
on he membe ship unc ions o each egion, whe e pixels a e classi ied in o egions based
on he membe ship alues as we desc ibe u he . This is also an impo an issue o his
s udy as shows he impac o uzzy h esholding besides con en ional ha d h esholding
his og am pa i ioning.
In he con as o he op imized me hods, we pu wo selec ed con en ional segmen-
a ion app oaches, which ha e been conside ed o a long ime as s anda ds o medical
image segmen a ion, as hey we e used in plen y o esea ch s udies, dealing wi h a -
ious objec de ec ions om medical images. The i s one is O su h esholding, which
is he implemen a ion o so-called ha d his og am h esholding, and he second me hod
is K-means, which de ines segmen ed egions based on he nonhie a chical clus e ing.
Implemen a ion o hese ou ine segmen a ion s a egies, which u ilize a ious p inciples
o image segmen a ion, poin ou on di e ences in pe o mance be ween non-hie a chical
clus e ing and h esholding o egional image segmen a ion. In a gene al way, we de ine
he op imiza ion p oblem o a se o h esholds:
T={T1,T2, . . . , Tn}(1)
In such con igu a ion, we sea ch o an op imal combina ion o indi idual h esholds
(
T1
,
T2
,
. . .
,
Tn
), de ining indi idual segmen ed egions, which he bes sa is ies op imiza-
ion c i e ia, which in e olu iona y and gene ic algo i hms is gi en by i ness unc ion,
which is desc ibed u he . As he de ini ion o i ness unc ion, we use a measu e o en opy
(Kapu en opy), which well de ines pixel’s dis ibu ions in segmen ed egions.
Senso s 2022,22, 6335 5 o 38
3.1. Segmen a ion Me hods
In his sec ion, we in oduce indi idual segmen a ion s a egies. He e, we desc ibe he
e olu iona y s a egies based on he ABC and PSO (and i s a ian s) and gene ic algo i hms
o his og am-based h esholding. In con as wi h hese s a egies, we pu he con en ional
me hods based on he ha d h esholding (O su h esholding) and K-means, which classi ies
pixels in o egions based on a simila i y (Euclidean dis ance) wi h egion’s cen oid. These
con en ional me hods do no con ain any op imiza ion elemen s, so i would be in e es ing
o compa e he pe o mance o op imiza ion s a egies wi h hese segmen a ion ou ines.
In o de o objec i ely epo he e ec i i y and obus ness o indi idual me hods, we
employ selec ed de e minis ic noise gene a o s wi h dynamic noise in ensi y (con olled by
hei s ee ing pa ame e s). Dynamical noise impac is mani es ed by g adual de e io a ion
and modi ica ion o pixel’s in ensi y dis ibu ion, which supposedly should ha e impac
on he segmen a ion obus ness as we p o ide modeling in he sec ion esul s. In o de
o p o ide such analysis, we employ Gaussian, sal and peppe , speckle, and Rician noise
gene a o s wi h dynamical ange o hei noise impac .
In o de o objec i ely measu e he noise impac o he segmen a ion pe o mance o
indi idual s udied me hods, we employ selec ed e alua ion pa ame e s, which a e ocused
on measu ing a le el o simila i y o di e ence be ween he na i e segmen a ion (wi h
a ze o le el o added noise) and espec i e noise le el. This app oach enables objec i e
pe o mance e alua ion o s udied segmen a ion s a egies. Fo his analysis we use he
ollowing objec i iza ion pa ame e s: s uc u al simila i y (SSIM), mean squa ed e o
(MSE), co ela ion coe icien (CORR), and signal noise a io (SNR). Figu e 1 ep esen s a
whole es ing en i onmen , which is he main aim o his pape , including applica ion o
noise gene a o s, segmen a ion s a egies, and pa ame e s o e alua ion.
3.1.1. O su Th esholding
O su me hod [
66
] is one o he sophis ica ed h esholding me hods, which is based on
he numbe o egions selec ed, O su me hod algo i hm de e mines he op imal h esholds
acco ding o he his og am. Image segmen a ion uses he classi ica ion o pixels in o
segmen a ion egions. The basis o he me hod is he s a is ical pa ame e o a iance,
which cha ac e izes he a iabili y o indi idual pixels in he image. The c i e ia o his
classi ica ion include minimizing he in a-class a iance o maximizing he in e -class
a iance. O su me hod uses his og am h esholding o de ine he numbe o egions. I
sea ches o he segmen a ion class wi h he smalles a iance, i.e., he op imally chosen
h eshold. This echnique is classi ied as a s a is ical me hod because i wo ks based on he
s a is ical pa ame e o a iance, which cha ac e izes he a iabili y o he dis ibu ion o
indi idual pixels in he image. The equa ion o calcula ion o he wi hin-class a iance a
any h eshold is:
σ2( )=ωbg(T)σ2
bg(T)+ω g(T)σ2
g(T), (2)
whe e
ωbg( )
and
ω g( )
a e he p obabili y o numbe o pixels o each class a h eshold
T and σ2is a iance o colo alues (pixels). The a iance is ep esen ed by equa ion:
σ2( )=∑(xi−x)2
N−1, (3)
whe e
xi
p esen s pixel alue in g oup o bg o g and
x
p esen s he mean pixel alue in
g oup o bg o g and N ep esen s he numbe o pixels.

Senso s 2022,22, 6335 6 o 38
Figu e 1. Complex gene al lowcha o es ing en i onmen o segmen a ion e alua ion.
Senso s 2022,22, 6335 7 o 38
ωbg and ω g a e calcula ed by:
ωbg =PBG(T)
Pall , (4)
ω g =PFG(T)
Pall , (5)
whe e P
BG
is coun o backg ound pixels a h eshold T, P
FG
is coun o o eg ound pixels
a h eshold T, and Pall is he o al coun o pixels in image.
O su me hod can be ex ended o a mul i egional segmen a ion scheme [
67
], whe e
mo e h esholds a e de ined. In his con igu a ion, he image his og am is di ided in o
equal a eas, and o each such a ea he own h eshold is de ined. Finally, he o iginal image
is segmen ed by using all hese h esholds. Supposing we ha e Limage in ensi ies in he
ange:
[0, 1, 2, . . . , L]
and he pa ame e ps ands o he numbe o h esholds. A wid h o
one a ea is de ined as he a io:
a=L
p(6)
Op imal h esholds (P) o indi idual a eas a e de ined as he maximiza ion o in e -
class a iance:
Pp=maxpσ2( )(7)
3.1.2. K-Means
K-means me hod [
68
] is classi ied as a non-hie a chical clus e ing me hod. K-means
me hod allows assigning indi idual pixels in o segmen a ion classes o which hey belong
based on dis ance. I uses nonhie a chical clus e ing o pixels assignmen and sea ches o
he minimum dis ance be ween he pixel and he selec ed cen e o g a i y (cen oid). A
gi en pixel is hen assigned o he egion o which i has he smalles dis ance. The mos
commonly used me ic is Euclidean, which measu es he dis ance o pixels in he ea u e
space acco ding o he ollowing equa ion:
D(→
x,→
y) = q∑n
i=1(xi−yi)2, (8)
whe e
→
x
,
→
y
a e he ea u e ec o s,
D(→
x,→
y)
is he esul ing dis ance, nis he dimension o
he space, xand ya e he pixel coo dina es.
Pixel xiis assigned o class yiacco ding o he ollowing ela ion:
yi=a gminj
xi−µj
(9)
The ollowing ec o ecalcula ion calcula es he new alues o he ec o s
µj
as he
mean alues om he pixels x
i
ha we e classi ied in o he class de e mined by he ec o
µj. The new alue o µjis calcula ed acco ding o equa ion:
µj=1
nj∑n
i=1,y1=j(xi), (10)
whe e n
j
deno es he numbe o pixels and x
i
classi ied in he second s ep in o he class
de e mined by he ec o
µj
. The ec o classi ica ion and ecalcula ion s eps a e epea ed
un il a leas one ec o x
i
is classi ied in o a di e en class han i was classi ied in he
p e ious s ep. The pixel in each class ha possesses he maximum equency is de e mined
as he cen oid. The disad an age o his me hod is ha by assigning objec s o each class,
i can only de e mine whe he he objec belongs o he clus e o no . The e o e, K-means
is classi ied as a ha d app oach echnique.
Senso s 2022,22, 6335 8 o 38
3.1.3. ABC E olu iona y Op imiza ion
ABC algo i hm o a i icial bee colony is an algo i hm ha belongs o he algo i hms
based on he swa ming beha io o animals. Speci ically, i is he beha io o bees looking
o ood. The p inciple o he algo i hm is ha i ies o p o ide he bes app oxima e
solu ion wi h low compu a ional equi emen s. The bees unc ion as a whole in a ce ain
way and he alloca ion o he di e en oles in he communi y is au oma ic. These a e he
employed bees (EB), he onlooke bees (OB), and he scou s (SB) whose ask is o imp o e
he o e all ood esou ces. The basic pa ame e s o ABC include he numbe o ood
sou ces, he limi , and he numbe o i e a ions. The limi de e mines a e how many
i e a ions a wo ke will abandon hei solu ion i hey ha e no been able o imp o e i .
The o alpopula ion(SN)consis so anequalnumbe o EBsandOBs,wi heachEBha ingone
empo a y solu ion R
i
adjacen o he solu ion X
i
. Pa ame e X
i
= {X
i
, 1, X
i
, 2,
. . .
,X
i
,p} ep esen s
he i- h solu ion in he swa m o bees, whe e p ep esen s he numbe o pa ame e s ha
a e op imized. In he i s s age o he algo i hm, hese wo solu ions a e compa ed using
he i ness unc ion and i he solu ion o he i ness unc ion R
i
is be e , i is kep as he
new solu ion om his pai . O he wise, no change occu s. This is done o each o he pai s
X
i
and R
i
. I is necessa y o apply a selec ion limi L
speci ying he maximum numbe o
a emp s in selec ing a solu ion Xiin case an op imal solu ion Ricanno be ound.
Howe e , i an op imal solu ion canno be ound e en a e exhaus ing L
, we conside
i as a bu n -ou solu ion. The second s age is essen ially an ex ension o EB. I wo ks wi h
OB in which he exis ing indi idual solu ions a e es ed om di e en pe spec i es. The
mo e op imal solu ion should ha e a highe alue o P
i
. A e selec ing he ood sou ce X
i
,
a neighbo ing ood sou ce R
i
is de e mined and hei i ness alues a e compa ed. The las
pa is he scou s sea ching o new ood sou ces ins ead o deple ed sou ces. The p ocess
o e alua ing solu ions is i e a i e, in mos cases 100 cycles a e applied. The ou pu o
his op imiza ion me hod is he se o all admissible solu ions, whe e in he inal s ep he
solu ion possessing he maximum alue o Piis selec ed.
3.1.4. PSO E olu iona y Op imiza ion
PSO, o pa icle swa m op imiza ion, is an e olu iona y op imiza ion compu ing
echnique inspi ed by he social beha io o bi ds and ish swa ms. This me hod uses a
popula ion o pa icles ha ly in an i egula mo ion h ough a gi en space a a ce ain
speed. The posi ion o each agen is gi en by he ec o x
i
, and i s mo emen co esponds
o he eloci y i. The pa icle eloci y is de e mined as ollows:
i( ) = i( −1) + c1∗ and1(pi−xi( −1)) + c2∗ and2(pg−xi( −1)), (11)
whe e c
1
and c
2
a e posi i e numbe s, and
1
and and
2
deno e andom numbe s om
he ange 0 o 1. Equa ion is composed o h ee pa s. The ine ia and a ac ion o he
bes - ound posi ion o a gi en pa icle p
i
, and we deno e he alue o he i ness unc ion
a his posi ion by p
bes
. This a ac ion is mul iplied by a andom weigh c
1∗
and
1
and
is called he memo y o he pa icle. The hi d pa o he equa ion is he a ac ion o he
bes - ound posi ion o he pa icle p
g
, and we deno e he co esponding i ness alue by
g
bes
. The a o emen ioned a ac ion is again mul iplied by a andom weigh c
2∗
and
2
and
is called sha ed in o ma ion, o also sha ed knowledge. Each indi idual emembe s hei
p e ious bes alue and he bes alue o hei neighbo s. The agen s he e o e use he
in o ma ion om he bes pa icle, and he e o e his algo i hm is mo e memo y e icien
han he gene ic algo i hm.
3.1.5. FPSO E olu iona y Op imiza ion
FPSO s ands o uzzy pa icle swa m op imiza ion. I is a modi ied PSO algo i hm
using uzzy logic heo y. The posi ion and eloci y o he pa icles in his algo i hm a e
de ined o ep esen he ela ionship be ween he uzzy and he a iables. Fuzzy logic
con olle wi h wo inpu s and one ou pu imp o es he pe o mance o PSO. The wo
Senso s 2022,22, 6335 9 o 38
inpu a iables ep esen cu en bes pe o mance e alua ion (CBPE) and cu en ine ia
weigh . The ou pu a iable is change in ine ia weigh . CBPE needs o be no malized
acco ding o he ollowing o mula:
NCBPE =CBPE −CBPEmin
CBPEmax −CBPEmin , (12)
whe e CBPE
min
is he ue minimum and CBPE
max
ep esen s he subop imal CBPE. CBPE
no maliza ion is used o make he algo i hm applicable o a wide ange o op imiza ion
p ocesses. A non-op imal CBPE is conside ed o be any solu ion wi h a CBPE g ea e han
o equal o CBPE
max
. These uzzy a iables a e de ined as uzzy se s wi h nine ules o a
uzzy sys em.
3.1.6. DPSO E olu iona y Op imiza ion
DPSO is a Da winian algo i hm ex ending he PSO algo i hm by na u al selec ion and
su i al o he i es o inc ease he abili y o escape om local op ima. Da winian pa icle
swa m op imiza ion (DPSO) allows many swa ms o es solu ions o exis a any poin in
ime. Each swa m wo ks like a egula PSO algo i hm, excep ha i uses na u al selec ion
(Da winian su i al o he i es ) o enhance he abili y o escape om a local op imum.
The p inciple o he solu ion sea ch is ha i i is heading owa ds a local op imum,
hen he sea ch o a solu ion in ha egion is e mina ed, and he sea ch o ano he a ea
begins. Each swa m is moni o ed a e e y s ep. I swa ms imp o e, hey a e ewa ded. The
ewa d is o ex end he li e ime o he pa icles o o p oduce o sp ing. I swa ms s agna e,
hey a e punished. The punishmen consis s o sho ening he li e ime o he swa m o
emo ing pa icles. A e he emo ing he pa icle, ins ead o being se o ze o, he coun e
is ese o a alue app oaching he h eshold numbe , acco ding o:
SCC(Nkill)=SCmax
C1−1
Nkill+1, (13)
whe e N
kill
ep esen s a numbe o dele ed pa icles om he swa m,
SCmax
C
ep esen s
ha he maximum numbe o swa ms mus no be exceeded. Whe eas he new swa m is
c ea ed wi h a p obabili y based on he equa ion:
p=
NS, (14)
whe e pis he p obabili y, is a andom numbe in he in e al 0 o 1, NS ep esen s he
numbe o swa ms. Each swa m is e alua ed using he i ness unc ion o all pa icles. In
his way, i is possible o analyze he o e all s a e o he swa ms sepa a ely and hus upda e
he neighbo hood and indi idual bes posi ions o each pa icle. New pa icles a e c ea ed
i a new global solu ion is ound. Con e sely, pa icle ex inc ion occu s i he swa m does
no ind a mo e sui able s a e in a de ined numbe o s eps.
3.1.7. Gene ic Algo i hms-Based Op imiza ion
Gene ic algo i hms a e based on na u al p ocesses wi h g adual elimina ion and
subsequen selec ion o he mos sui able solu ions. I is a combina ion be ween biology
and ma hema ics. Pa e ns om li ing na u e a e used, which ini ially wo k by chance
and g adually p oduce be e solu ions. These pa e ns a e hen applied using a ma h-
ema ical model o a a ie y o echnical applica ions, including image p ocessing using
segmen a ion echniques.
Gene ic algo i hms use special p ocedu es o ind he op imal solu ion using selec ion,
c osso e , and mu a ion ope a ions. They s a wi h andom selec ion and sea ch o new,
be e solu ions; he mos op imal solu ion is hen selec ed om he esul s. All algo i hms
include a i ness unc ion ha p o ides in o ma ion abou he quali y o he solu ion. Each
app oach has speci ic pa ame e s ha mus be se . Examples a e he numbe o egions, he
Senso s 2022,22, 6335 16 o 38
Figu e 5.
Example o p o on-dense sequence o in es iga ion o a icula ca ilage wi h ea ly ca ilage
loss. The i s ow shows a sequence o h ee na i e images om he MR da ase and he second ow
ep esen s image RoIs, ocusing on ca ilage a ea, whe e he ed squa es poin ou MR signal change
in ca ilage s uc u e ha indica e he ea ly ca ilage loss.
3.3. De e minis ic A i icial Noise Gene a o s
In o de o p o ide he analysis o obus ness o indi idual segmen a ion echniques,
we employ a ious image noise gene a o s, which simula e g adual de e io a ion o spa ial
image a ea by using hei s ee ing pa ame e s, as we desc ibe u he . Fo ou analysis, we
use he ollowing noise gene a o s: Gaussian noise, speckle noise, sal and peppe noise,
and Rician noise.
3.3.1. Gaussian Noise
Gaussian noise ep esen s whi e s a is ical noise. This ype o noise is due o na u al
sou ces such as ambien empe a u e. The dis ibu ion o Gaussian noise is uni o m in he
image and a ec s all pixels wi h he same in ensi y. I is a no mal dis ibu ion o noise
dis ibu ion in he image. Gaussian noise can be de ined using he ollowing o mula:
G(x) = 1
σ√2π
e
(x−µ)2
2σ2, (24)
whe e x ep esen s he luminance o he noise,
σ2
is he a iance, and
µ
ep esen s
he mean.
3.3.2. Speckle Noise
Speckle noise is a common noise ha occu s in all cohe en imaging sys ems (lase s,
acous ic sys ems, ul asound). The cause o his noise is he in e e ence o a signal ha
has a di e en phase when e u ning om he a ge . This noise is displayed in he image
as da k pixels wi h a highe b igh ness alue. The inpu pa ame e is he speckle noise
a iance. Speckle noise can be desc ibed by he o mula ion:
J=I+n∗I, (25)
whe e Ip esen s he inpu image, Jis he noise dis ibu ion in he inpu image, and n
p esen s uni ied ze o mean alue o he noise in inpu image.

Senso s 2022,22, 6335 17 o 38
3.3.3. Sal and Peppe Noise
Sal and peppe ep esen s impulse noise. The image deg ada ion akes place a se e al
pixels in he image, wi h he pixel ca ying no in o ma ion abou he o iginal alue. The
pixel alues in he image a e eplaced by alues o 255 o 0. Thus, his noise is ep esen ed
in he image as whi e and black do s esembling sal and peppe . The inpu pa ame e o
se ing he noise is he densi y. This noise is mos o en no iceable du ing da a ansmission.
3.3.4. Rician Noise
Rician noise ep esen s he mos ypical noise in images aken by magne ic esonance.
Rician noise is based on Gaussian noise in ha he eal and imagina y pa s o he signal a e
co up ed by an unco ela ed ze o mean. The magni ude o Rician noise can be exp essed
using he ollowing o mula:
M=q(I+n1)2+n2
2, (26)
whe e M ep esen s he signal magni ude, I ep esen s he o iginal image wi h negligible
noise in ensi y, and
n1
and
n2
a e Gaussian noise a iables wi h ze o mean and equal
a iance
σ2
n
. He e, we can de ine he p obabili y densi y unc ion (PDF) o an image, which
is co up ed by Rician dis ibu ion by he ollowing way:
pM|I,σ2
n=M
σ2
n
exp−M2+I2
2σ2
nI0IM
σ2
nu(M)(27)
In his de ini ion,
I0(.)
depic s he 0 h o de o modi ied Bessel unc ion o he i s
kind, and he pa ame e u(.) s ands o Hea iside s ep unc ion [56].
3.4. Applica ion and Se ings o Noise Gene a o s
Fi s ly, we in oduce he se ings o noise de e minis ic gene a o s, which a e used in
his s udy. Each o he noise gene a o s is de e mined by i s s ee ing pa ame e s, which
de e mine he noise in ensi y. We use a g adual ascended sequence o he noise in ensi ies
o e ec i ely simula e he segmen a ion pe o mance deg ada ion upon inc easing he
le el o he noise-based image de e io a ion. In he Gaussian noise (G), we use a cons an
dispe sion (
σ2=
0.01) and a iable mean alue o he noise (
µ
), in he sal and peppe (SaP)
noise we p o ide es ing o a iable noise densi y (d) and o speckle (Sp) and Rician noise
(Ric), we con ol he noise in ensi y ia he pa ame e a iance (
σ2
). Fo he pu poses o
es ing, we use 20 noise le els o simula e he segmen a ion pe o mance (Table 2).
Table 2. De ini ion o noise gene a o s o segmen a ion pe o mance analysis.
Noise Gene a o
Numbe o Regions G: (σ2=
0.01
), (µ)SaP: (d)Sp: (σ2) Ric: (σ2)
4 0.01–0.2 0.17–0.33 0.01–0.2 0.02–0.4
7 0.01–0.2 0.17–0.33 0.01–0.2 0.02–0.4
10 0.01–0.2 0.17–0.33 0.01–0.2 0.02–0.4
We g adually applied he noise gene a o s wi h he ange o he noise in ensi y pa-
ame e s o a i icially simula e he noise impac on he pixel’s dis ibu ion. As we s a ed
ea lie , o each noise, we se 20 le els on he noise in ensi y. Tha means each na i e
MR image con ains in o al 21 images o es ing o segmen a ion algo i hms (1 na i e
image + 20 noise le els). Fo each image, we de ined a mul idimensional a ay, whe e
all hese noisy images a e s o ed. In he ollowing ou pu s: Figu es 6–9, we p o ide
he examples o g adual de e io a ion o indi idual noise gene a o s, which we used o
he es ing.
Senso s 2022,22, 6335 18 o 38
Figu e 6.
Applica ion o Gaussian noise on na i e MR ca ilage image wi h in ensi ies ( om le ):
na i e image, σ2=0.01, µ= {0.005, 0.01}.
Figu e 7.
Applica ion o Rician noise on na i e MR ca ilage image wi h in ensi ies ( om le ): na i e
image, σ2={0.2, 0.4}.
Figu e 8.
Applica ion o sal and peppe noise on na i e MR ca ilage image wi h in ensi ies ( om
le ): na i e image, d={0.2, 0.4}.
Senso s 2022,22, 6335 19 o 38
Figu e 9.
Applica ion o speckle noise on na i e MR ca ilage image wi h in ensi ies ( om le ):
na i e image, σ2={0.2, 0.4}.
3.5. E alua ion Pa ame e s
All he pe o mance cha ac e is ics a e cons uc ed by he way we use he segmen a-
ion o na i e MR images as a gold s anda d agains indi idual segmen a ion in indi idual
noise le els. This app oach enables objec i e measu emen o he noise in luence o each
noise le el. This inally shows dynamical ea u es o pe o mance wi hin con inuous deg a-
da ion by image noise wi h a iable in ensi y. The ollowing pa ame e s a e conside ed o
his s udy.
SSIM o s uc u al simila i y index [
59
] is a pa ame e ha allows us o objec i ely
exp ess he simila i y o wo images xand yusing a me ic. This pa ame e is de ined by
he ollowing o mula:
SSIM (x,y)=2µxµy+C12σxy +C2
µ2
x+µ2
y+C1σ2
xσ2
yC2, (28)
whe e C
i
=
(k,l)2
, whe e l ep esen s he dynamic ange o pixel alues, k<< 1 a e small
cons an s wi h alues usually 0.02,
µ
ep esen s he weigh ed a e age o he xand y
images, and
σ
ep esen s he co a iance o xand y. These componen s in he o mula
allow o compa e be ween xand yimages: b igh ness (l), con as (c), and ex u e (s). The
compa ison me hod ex ac s s uc u al in o ma ion om he scene. This pa ame e akes
alues om
−
1 o 1, wi h 1 ep esen ing he absolu e ma ch be ween he xand yimages.
He e, we conside ha x ep esen s he segmen a ion wi h ze o le el o addi i e noise and
y ep esen s he segmen a ion ou pu wi h espec i e le el o he noise.
Co ela ion coe icien ep esen s he linea co ela ion be ween wo images xand y.
The co ela ion coe icien is de ined as he a io o he co a iance o he a iables xand y
mul iplied by hei s anda d de ia ions. The Pea son pai wise co ela ion coe icien can
be exp essed using he ollowing equa ion:
=∑(xi−x)·(yi−y)
(n−1)sxsy, (29)
whe e s
x
and s
y
ep esen s anda d de ia ions and
x
and
y
ep esen he a i hme ic mean o
each o he a iables xand y. The co ela ion coe icien akes alues om
−
1 o +1, and he
close he absolu e alue o he co ela ion coe icien is o one, he close he ela ionship
be ween he a iables x,y. The highe he alue, he be e he segmen a ion pe o mance.
Senso s 2022,22, 6335 20 o 38
SNR o signal o noise a io is a pa ame e ha allows us o exp ess he a io o
use ul powe o useless powe o a signal (image). This pa ame e is de ined by he
ollowing equa ion:
SNR =10log10 ·
∑M
i=1∑N
j=1g2
i,j − 2
i,j
∑M
i=1∑N
j=1gi,j − i,j2(30)
SNR o signal o noise a io is a pai whe e
gi,j
ep esen s he o iginal (gold s an-
da d) segmen a ion (wi hou addi i e noise) and
i,j
is he segmen ed image wi h espec-
i e addi i e noise le el. The SNR quan i y is decibels (dB). SNR alues can be in e -
p e ed in he o m he highe SNR alues we achie e o espec i e segmen a ion, he
be e ag eemen wi h he gold s anda d we ha e, and be e segmen a ion pe o mance
we achie e.
MSE, o mean squa ed e o , is a pa ame e ha can be used o objec i ely e al-
ua e image quali y. This pa ame e exp esses he deg ee o mean squa ed e o be-
ween he o iginal image and he segmen ed image. This pa ame e is de ined by he
ollowing o mula:
MSE =1
MN
M
∑
i=1
N
∑
j=1xi,j −yi,j2, (31)
whe e M ep esen s he image size in he ho izon al di ec ion, N ep esen s he image size
in he e ical di ec ion,
xi,j
co esponds o a pixel in he segmen ed image a coo dina es i
and j, and
yi,j
co esponds o a pixel in he o iginal image a coo dina es i and j. Fo his
pa ame e , he lowe he alue, he g ea e he simila i y be ween he images. P ac ically,
we compu e he squa ed di e ences be ween he pixels, ha ing he same coo dina es in
he segmen a ion ma ixes. Consequen ly, hese di e ences a e summed up, and las ly, i s
mean alue is compu ed. By his way, we compu e he mean quad a ic di e ence be ween
he gold s anda d segmen a ion and espec i e noise le el segmen a ion.
The segmen a ion esul s (mul i egional segmen a ion) a e e alua ed ia labeling
ma ix, whe e each egion has a unique numbe . This numbe ep esen s an in e al o
in ensi y alues, which a e classi ied in o a speci ic egion. Thus, his can be in e p e ed as a
ans o ma ion o a se o in ensi y alues om he image spa ial a ea in o he egion index.
The e alua ion pa ame e s e lec pixel eclassi ica ion among indi idual such egions
by he in luence o addi i e noise. By adding addi i e noise wi h g adually inc easing
in ensi y, a espec i e pixel will ha e signi ican ly di e en in ensi y alue when compa ing
wi h he si ua ion wi hou addi i e noise. The e o e, such pixels may be eclassi ied in
a di e en egion. The e alua ion o pixel-wise pa ame e s such as SNR o MSE, e lec
he impac o change o pixels assignmen among indi idual egions. The main aim o
hese pa ame e s is e lec ing he impac be ween change o pixel’s assignmen in adjacen
egions ( his is only small change on noise impac ) and he shi be ween mo e egions,
whe e we can suppose a highe noise impac . This si ua ion i s ly e lec s he change o
pixel egion eassignmen , bu also he shi o pixel in ensi y by addi i e noise. Finally,
he e alua ion pa ame e s quan i y he impac o a pixel’s assignmen change and hus
objec i ely e alua e a obus ness o a espec i e pixel’s classi ica ion upon he image noise
wi h g adual in ensi y. The highe shi be ween egions is egis able, he bigge impac on
e alua ion pa ame e s is, which quan i y he pe o mance and obus ness o he egional
segmen a ion upon dynamic noise in ensi y.
4. Resul s
In his sec ion, we in oduce quan i a i e esul s o es ing analyzed h esholding-
based segmen a ion s a egies. He e, we p o ide se e al ypes o cha ac e is ics o p o ide
an objec i e iew o he segmen a ion pe o mance and limi a ions. We p o ide examples
o g aphical compa isons o he segmen a ion me hods, which show he in luence o he
a iable image noise o segmen a ion maps. Fo he gene a ion o he segmen a ion maps,
we use an a i icial colo coding. Whe e each single colo ep esen s one egion o he
Senso s 2022,22, 6335 21 o 38
segmen a ion model. To p o ide a complex iew on he segmen a ion pe o mance, we p o-
ide his es ing o a a iable numbe o segmen a ion classes because his pa ame e has
a subs an ial e ec on he segmen a ion pe o mance. One o he impo an pe o mance
ea u es is he ime complexi y, hus we p o ide ime equi emen s o indi idual segmen-
a ion me hods. This aspec is subs an ially impo an when pe o ming a simul aneous
segmen a ion o a s ack o MR images. In o de o show a s a is ical signi icance be ween
he ou ine me hods and op imized segmen a ion models, we p o ide he s a is ical es ing
o signi icance o p- alues o median es s. The las quan i a i e analysis deals wi h he ex-
ac ion o clinically impo an ca ilage ea u es including he a ea, pe ime e , and ca ilage
skele on. He e, we show di e ences o au oma ic segmen a ion and he gold s anda ds.
Las ly, we p o ide a p esen a ion o he so wa e en i onmen , which in eg a es indi idual
epo ed segmen a ion s a egies wi h he possibili y o selec ing s ee ing pa ame e s o
segmen a ion.
4.1. Quan i a i e Segmen a ion E alua ion
G adual noise dynamics has a subs an ial e ec on he pixel’s dis ibu ion as we
men ion in he p e ious examples. In ou s udy, we u ilize his ac o es he obus ness o
segmen a ion s a egies o jus i y he impac o op imiza ion elemen s o he pe o mance
o egional segmen a ion as we desc ibe u he .
The i s analysis, which we p o ide is aimed on he g aphical e alua ion o he
analyzed segmen a ion me hods unde g adual inc easing noise in luence. Based on such
esul s, we can subjec i ely obse e clea ly isible no able di e ences in indi idual me hods
in segmen a ion maps. As he example, we p o ide he compa ison (Figu es 10 and 11) o
all he me hods o sal and peppe noise wi h h ee le els o densi y: 0.1, 0.5, and 0.7 and
Rician noise wi h h ee se ings: σ2={0.1, 0.5, 0.7}.
The segmen a ion esul s a e in e p e ed in he o m o segmen a ion maps in he
colo spec um (Figu es 10 and 11). The in e p e a ion o hese colo maps is ha each
segmen a ion egion in he segmen a ion map is ep esen ed by a single alue. Thus, he
numbe o colo s co esponds wi h he numbe o egions. Each such egional model can
be in e p e ed as a ans o ma ion o he scale o in ensi y alues o he numbe o egions.
Fo ins ance, 8-bi images (256 in ensi y alues) a e ans o med in o ou in ensi ies
(segmen a ion model wi h ou egions). The main aim o his analysis is o objec i ely
epo how he dis ibu ion o a pixel’s assignmen in o indi idual egions a e modi ied
unde he in luence o addi i e noise agains he gold s anda d (segmen a ion wi hou
addi i e noise in luence).
Based on such expe imen al esul s, i is no iceable ha an inc easing noise in ensi y
can signi ican ly impai he quali y o he segmen a ion esul s. Fo lowe noise le els he
segmen a ion esul s poin ou on a good pe o mance, o example he ABC algo i hm does
no exhibi mo e signi ican signs o he noise. On he o he hand, highe le els o noise
cause signi ican impai men o he segmen a ion model consis ency. In o de o objec i ely
jus i y his ac , we u he p o ide a obus es ing o hese segmen a ion me hods based
on he men ioned e alua ion pa ame e s, o objec i ely show he change o segmen a ion
pe o mance among indi idual me hods, and also how he numbe o egions in luence he
dynamic o segmen a ion pe o mance. To be e jus i y he es ing scheme, we p o ide
es ing on ou ine app oaches o O su h esholding and K-means clus e ing. He e, we
only se he numbe o he segmen a ion egions. Con a ily, in he e olu iona y s a egies,
including ABC, PSO, DPSO, and FPSO, we use a uni ied numbe o i e a ions, 100 (PSO
1
,
GA
1
and ABC
1
) and 500 (PSO
2
, GA
2
and ABC
2
), and a popula ion size o 50 (PSO
1
, GA
1
and ABC1) and 200 (PSO2, GA2and ABC2).

Senso s 2022,22, 6335 22 o 38
Figu e 10.
Compa ison o egional segmen a ion models wi h ou egions, which a e in e p e ed by
ou a ious single colo s o h ee le els o sal and peppe noise: om le na i e image and noise
densi y
d={0.1, 0.5, 0.7}
. All he e olu ion s a egies (ABC, PSO, DPSO, and FPSO) ha e he same
se ings: 100 i e a ions and 50 popula ions.
Senso s 2022,22, 6335 23 o 38
Figu e 11.
Compa ison o egional segmen a ion models wi h ou egions, which a e in e p e ed
by ou a ious single colo s o h ee le els o Rician noise: om le na i e image and noise
densi y
σ2={0.1, 0.5, 0.7}
. All he e olu ion s a egies (ABC, PSO, DPSO, and FPSO) ha e he same
se ings: 100 i e a ions and 50 popula ions.
He e, we p o ide he quan i a i e compa ison o indi idual op imiza ion echniques
o image h esholding-based egional segmen a ion agains selec ed con en ional seg-
men a ion app oaches, including O su ha d h esholding and K-means nonhie a chical
clus e ing o egional segmen a ion. We p o ide dynamical ea u e ex ac ions o hese
me hods, epo ing e ec i i y o each noise le el and obus ness in he o m o he end
o he e alua ion pa ame e s upon addi i e noise wi h dynamic in ensi y, measu ed by he
mean squa ed e o (MSE), he index o co ela ion (CORR), he s uc u al simila i y index
(SSIM), and he signal o noise a io (SNR). As he example, we p o ide hese cha ac e is ics
(Figu es 12–15) o he segmen a ion models wi h ou egions. The p o ided cha ac e is ics
a e cons uc ed o 1000 images, whe e he esul s o each le el o each noise a e a e aged.
Senso s 2022,22, 6335 24 o 38
Figu e 12.
Dynamical ea u es o Gaussian noise in luence o egional segmen a ion e ec i i y and
obus ness based on he MSE, SSIM, CORR, and SNR.
Figu e 13.
Dynamical ea u es o Rician noise in luence o egional segmen a ion e ec i i y and
obus ness based on he MSE, SSIM, CORR, and SNR.
Senso s 2022,22, 6335 25 o 38
Figu e 14.
Dynamical ea u es o sal and peppe noise in luence o egional segmen a ion e ec i i y
and obus ness based on he MSE, SSIM, CORR, and SNR.
Figu e 15.
Dynamical ea u es o speckle noise in luence o egional segmen a ion e ec i i y and
obus ness based on he MSE, SSIM, CORR, and SNR.
Judging by he expe imen al esul s, signi ican di e ences in e ec i i y among in-
di idual me hods a e no able. The ends o he pa ame e s o simila i y (SSIM, CORR,
and SNR) o O su and K-means exhibi signi ican ly lowe alues when compa ing wi h
he e olu iona y algo i hms. Tha indica es he no able wo se esul s o hese ou ine algo-
i hms in he compa ison wi h he op imiza ion echniques. The highe hese pa ame e s
a e, he be e he pe o mance o espec i e segmen a ion is achie ed. On he o he hand,
hese ou ine app oaches om he iew o MSE exhibi he mos apid inc easing end
when compa ing wi h op imiza ion echniques. This is also a sign o he much wo se
e ec i i y o O su and K-means agains he elemen s o a i icial in elligence.
Senso s 2022,22, 6335 32 o 38
Figu e 19.
Example o epo om he SW o egional segmen a ion es ing. This epo con ains he
segmen a ion esul s o all he in eg a ed segmen a ion echniques unde sal and peppe noise wi h
densi y 0.1.
4.3. Clinical Impo an Fea u es Ex ac ion o A icula Ca ilage
Based on he epo ed analysis o he segmen a ion pe o mance, mos ly he com-
bina ion o uzzy h esholding wi h he ABC e olu iona y algo i hms appea ed as he
bes segmen a ion s a egy, judging by epo ed objec i iza ion pa ame e s and mainly
p o ided s a is ical es s o signi icance. In his subsec ion, we would like o p o ide he las
analysis o selec ed ea u es ex ac ion o a icula ca ilage om MR images based on he
uzzy h esholding wi h he ABC algo i hm. The aim o his analysis is i s ly compu ing a
mul i egional segmen a ion model, allowing o a decomposi ion o he MR image in o a
ini e numbe (in his case i e) segmen a ion egions. Consequen ly, a egion, ep esen ing
he a icula ca ilage, is selec ed (Figu e 20) as he egion o in e es , while he es o
he segmen a ion egions a e supp essed om he segmen a ion model (Figu e 20). By
his selec ion scheme, we ob ain a bina y segmen a ion model, exclusi ely classi ying he
a icula ca ilage om he es o he issues in he MR images. Figu e 20 also p esen s
a mul i egional segmen a ion o a pa o a icula ca ilage ( emo al ca ilage) a ec ed
by os eoa h i is o I. g ade, which is no able by wo segmen ed lobes o he a icula
ca ilage, and be ween hem is a gap, whe e he ca ilage is missing. To objec ize he
quali y o he a icula ca ilage ex ac ion and he p eciseness o he epo ed ea u es,
we ex ac ed he same ea u es o he gold s anda d manual segmen a ion o a icula
ca ilage. Consequen ly, he ea u e di e ences a e compa ed o quan i y he segmen a ion
e ec i i y o a icula ca ilage de ec ion. No e ha we used he ollowing se ings o he
ABC algo i hm: 100 i e a ions and popula ion size 50. The ollowing ca ilage ea u es a e
conside ed o e alua ion:
•Ca ilage a ea
—a o al coun o he pixels, belonging o he model o a icula ca ilage.

Senso s 2022,22, 6335 33 o 38
•Ca ilage pe ime e
—a pe ime e o he ca ilage model. He e, we used Sobel edge
ope a o o he de ec ion o ca ilage bo de s, and consequen ly coun ed he bo de
pixels.
•Skele on o ca ilage— he de ec ion o ca ilage skele on and compu ing i s leng h.
Figu e 20.
Example o segmen a ion esul s o a icula ca ilage and i s ea u es ex ac ion based
on uzzy h esholding wi h ABC op imiza ion: (
a
) gold s anda ds by manual anno a ion, (
b
) bina y
segmen a ion, (
c
) na i e MR image wi h a ea o in e es indica ed by he g een squa e ( op) and
mul i egional segmen a ion wi h 4 egions (bo om), whe e yellow con ou s e lec wo lobes o
a icula ca ilage om egion o in e es , and (
d
) bina y ex ac ion o a icula ca ilage used wi h
he gold s anda d ( ed con ou ).
Based on he segmen a ion o m as bina y images, ep esen ing ex ac ed a icula
ca ilage and i s espec i e ea u es, we compu e desc ip i e s a is ics, poin ing ou on
indi idual dis ibu ion’s e o unc ions, which show pe cen ual di e ences o indi idual
ea u es, and a dis ibu ion o alues o he indi idual pa ame e s o segmen a ion pe o -
mance (SSIM and index o co ela ion). He e, he e alua ion pa ame e s we e compu ed
be ween he gold s anda d bina y image and he esul s o he uzzy h esholding wi h
he ABC algo i hm. Figu e 21 p o ides a g aphical ep esen a ion o he dis ibu ions o
di e ences o ca ilage ea u es and he dis ibu ion o alues o pe o mance pa ame e s
o uzzy so h esholding wi h ABC op imiza ion.
Based on he esul s o he quan i a i e analysis o di e ence unc ion o he ex ac ed
ea u es, we did no achie e signi ican di e ences be ween he gold s anda d images and
uzzy so h esholding wi h he ABC algo i hm. Mos ly he dis ibu ions o di e ence
unc ion a e kep unde 6% o di e ence. Based on his analysis, we p o ide he desc ip i e
cha ac e is ics (Table 13), which epo s he median and s anda d de ia ion o each pa am-
e e . Based on hese esul s, he bes esul in median di e ence is achie ed o he ea u e
o skele on leng h (2.42%); con a ily, he wo s median di e ence is achie ed o he a ea
(4.12%). F om he iew o measu ing a iabili y (s anda d de ia ion) o he di e ence
unc ion, he lowes di e ence is achie ed o he skele on (1.38%) in he con as wi h he
pa ame e a ea, whe e he di e ence was he wo s (2.44%). The second s udied aspec is
he pe o mance pa ame e s: he index o co ela ion and he SSIM. He e, we achie ed a
highe median o co ela ion (0.94), whe e he median o he SSIM was 0.89. Fu he mo e,
om he iew o s anda d de ia ion, ep esen ing he concen a ion o alues is be e han
he index o co ela ion (0.017), while in SSIM we achie ed 0.028.
Senso s 2022,22, 6335 34 o 38
Figu e 21.
E alua ion o pe cen age di e ence dis ibu ions o uzzy so h esholding wi h ABC
algo i hm (agains manual segmen a ion) o ca ilage ea u es: a ea, pe ime e , and skele on and
dis ibu ions o pe o mance pa ame e s: index o co ela ion and SSIM.
Table 13.
Desc ip i e s a is ic o di e ence unc ions o ex ac ed ea u es, whe e he bes esul s a e
indica ed as g een and he wo s as ed.
Ca ilage Fea u es Median Di (%) S anda d De ia ion Di (%)
A ea 4.12 2.44
Pe ime e 3.51 1.85
Skele on 2.42 1.38
5. Discussion and Conclusions
Based on he p o ided esul s, signi ican di e ences among indi idual me hods
can be obse ed. In such compa isons, ou ine me hods show signi ican ly wo se esul s
when compa ing wi h he e olu iona y algo i hms. Along hese cha ac e is ics, we also
p o ide he a e age alues o all he noise le els o p o ide a global iew on all he
s udied me hods. By his s a is ical compa ison, in mos cases he ABC algo i hm seems
o be he mos e ec i e. On he o he hand, he use o he gene ic algo i hm o medical
image segmen a ion s a egies does no gi e sa is ac o y esul s. Fu he mo e, his s a egy
is enough ime demanding. We also s udied he ime complexi y o all he s udied
me hods o h ee di e en numbe o segmen a ion egions. He e, we can conclude ha
he inc easing numbe o egions inc eases he ime complexi y. These compa isons also
b ing no able di e ences among ou ine me hods and op imiza ion s a egies. The ou ine
me hods a e less ime demanding in he con as wi h he op imiza ion s a egies, which is
p edic able because he e olu ion s a egies usually ep esen complex p ocedu es. The
in e es ing no able ac om his s udy is he compa ison be ween he ha d h esholding-
based app oaches wi h PSO and i s a ian s and so h esholding wi h he ABC algo i hm.
Mos ly, so h esholding o e came he concep o ha d h esholding. In his iew, he so
h esholding appea s as mo e e icien . On he o he hand, he ha d h esholding s a egies
in his s udy a e less ime demanding. I is impo an o men ion ha he quan i a i e
Senso s 2022,22, 6335 35 o 38
cha ac e is ics o he segmen a ion pe o mance a e ep esen ed by he end cha ac e is ics
(Figu es 14–17) o he e alua ion pa ame e s, including he index o co ela ion, SSIM,
MSE, and SNR. Ideally, hese cha ac e is ics would be ep esen ed by a mono onous end
clea ly de ining a p og ess o he segmen a ion pe o mance upon dynamic noise. The
eal esul s some imes show ce ain a ia ions in he o m o local oscilla ions whe e he
complex ends do no ha e o be always mono onous. This may be caused by he ac ha
upon a ious noise in ensi ies, indi idual segmen a ion egions a e di e en ly a ec ed by
addi i e noise, which con ibu es o he o al e ec i i y. This phenomenon is connec ed
wi h he ac ha he noise gene a o s wo k on he p inciple o andom de ini ion o noise.
Al hough his s udy e eals a complex iew o selec ed aspec s o he employmen o
e olu iona y compu ing me hods and gene ic algo i hms o medical image segmen a ion,
he e a e s ill open issues o u he esea ch in his a ea. In he u u e esea ch, i would
be wo h s udying in de ail a ious se ings o popula ion size and he numbe o i e a ions
in he con ex o hei impac on he segmen a ion accu acy. The u he impo an aspec is
he de ini ion o c i e ia o he e alua ion o he mos sui able h eshold se ings. He e,
we use Kapu en opy as he i ness unc ion. Ne e heless, o he al e na i es may be
plausible. Fo ins ance, using a local s a is ic o a iabili y o pixel’s dis ibu ion inside o
egions appea s o be a easonable al e na i e. Based on hese open issues we would like o
build u u e esea ch in his a ea.
Au ho Con ibu ions:
Concep ualiza ion, J.K., O.P. and A.V.; me hodology, M.P. and M.C.; so wa e,
R.S. and D.O.; alida ion, K.H., M.P. and M.A.; o mal analysis, K.H. and R.S.; in es iga ion, K.H. and
D.O.; esou ces, M.P. and M.A.; da a cu a ion, M.C. and O.P.; w i ing—o iginal d a p epa a ion, J.K.
and A.V.; w i ing— e iew and edi ing, M.P.; isualiza ion, M.C. and M.A.; supe ision, R.S.; p ojec
adminis a ion, M.P. and M.C.; unding acquisi ion, M.P. All au ho s ha e ead and ag eed o he
published e sion o he manusc ip .
Funding:
This pape was suppo ed by p ojec No. CZ.02.1.01/0.0/0.0/17 049/0008441, Inno a-
i e The apeu ic Me hods o Musculoskele al Sys em in Acciden Su ge y wi hin he Ope a ional
P og amme Resea ch, De elopmen and Educa ion inanced by he Eu opean Union and by he
s a e budge o he Czech Republic. The wo k and he con ibu ions we e suppo ed by he p ojec
SV4502261/SP2022/98 ‘Biomedical Enginee ing sys ems XVIII’. P ojec Resea ch and de elopmen
o he new MEDIMONITOR sys em o in elligen p ognosis o ecas ing—CZ: Výzkum a ý oj
no ého sys ému MEDIMONITOR p o in eligen níp ognózo ání ý oje diagnóz-OPPIK–MPO:
CZ.01.1.02/0.0/0.0/20_321/0024858. This publica ion has been p oduced wi h he suppo o he
In eg a ed In as uc u e Ope a ional P og am o he p ojec : C ea ion o a Digi al Biobank o sup-
po he sys emic public esea ch in as uc u e, ITMS: 313011AFG4, co- inanced by he Eu opean
Regional De elopmen Fund. This s udy was suppo ed by he esea ch p ojec The Czech Science
Founda ion (TACR) No. TL02000313 De elopmen o in elligen neu o- ehabili a ion sys em o
pa ien s wi h acqui ed b ain damage in ea ly s ages o ea men .
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen :
No applicable. We used he MR images om publicly a ailable
da abase The Os eoa h i is Ini ia i e.
Da a A ailabili y S a emen :
Da a a e used om publicly open clinical da abase The Os eoa h i is
Ini ia i e. The published SW applica ion o a icula ca ilage segmen a ion based on he analyzed
segmen a ion s a egies wi h op imiza ion algo i hms can be ound ia he link: h ps://www.d opbox.
com/sh/m hnili z1 ccz/AAATYhbioFPHX9x3cd6gVyUNa?dl=0 (accessed on 20 June 2022).
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Senso s 2022,22, 6335 36 o 38
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