Alexander Karkishchenko - Threefold Symmetry Detection in Hexagonal Images Based on Finite Eisenstein Fields

Threefold Symmetry Detection
in Hexagonal Images
Based on Finite Eisenstein Fields
elex—nder u—rkish™henkoD karkishalex@gmail.com
†—leriy wnukhinD mnukhin.valeriy@mail.ru
Southern Federal University, Taganrog Campus, Russia
esƒ„9PHIT " ‚ussi—D ‰ek—terin˜urgD epril U!WthD PHIT
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 1 / 28

Introduction
Introduction
Symmetry is — ™entr—l ™on™ept in m—ny n—tur—l —nd m—nEm—de
o˜je™tsF ƒymmetry —ppe—rs in n—ture —t —ll s™—lesD r—nging from
di—mond l—tti™es to pl—ntsD —nim—lsD pl—netsD —nd g—l—xiesF

Introduction
Symmetry detection is — fund—ment—l t—sk in m—™hine visionD
p—ttern ™l—ssi(™—tionD im—ge d—t—˜—se retriev—lD etsF ƒymmetry
group of —n o˜je™t is invariant to rotation, scale, translation and
intensity transformationsF st is — strong des™riptor for o˜je™t
re™ognitionD m—t™hing —nd segment—tionF
genters of local rotational symmetries —re import—nt fe—ture points
of —n im—geF

Introduction
sn im—ge —n—lysis it is quite ™ommon to pro™eed on the —ssumption
of ™ontinuity of im—gesF
Continuous 2D gray-level image is just —n nonneg—tive ˜ounded
fun™tion f(x, y) : R2 → RF
„hen pol—r ™oordin—tes ™—n ˜e used to ™he™k if — point (x0, y0) is
the ™enter of lo™—l rot—tion—l symmetryX
gommonlyD to verify periodi™ity in r0ϕEdom—inD the pourier
tr—nsform is usedF ƒu™h method is known —s the Fourier-Mellin
transform —ppro—™h —nd h—ve ˜een extensively used for symmetry
re™ognition in the l—st de™—desF

Introduction
Note the complications:
ell the re—lEworld im—ges —re digitalD not ™ontinuousF

Introduction
st seems th—t ™urrently no w—ys to introdu™e ™orre™tly pol—r
™oordin—tes in the whole digit—l pl—ne Z2 —re knownF es —
resultD for digit—l im—ges pourierEwellin tr—nsform —ppro—™h
le—ds to systematic errorsF

Introduction
qener—llyD lo™—l symmetry dete™tion —re— is —ssumed to ˜e
rel—tively sm—llD whi™h in™re—ses the risk of errors in pol—r
™oordin—tesF

Introduction
qener—llyD lo™—l symmetry dete™tion —re— is —ssumed to ˜e
rel—tively sm—llD whi™h in™re—ses the risk of errors in pol—r
™oordin—tesF
por digit—l im—ges one m—y t—lk only —˜out some measure of
symmetryD whi™h depends on rot—tions —nd s™—lingF

Introduction
„his r—ises —n issue of the development of methods initi—lly fo™used
on dis™rete im—ges —nd ˜—sed on tools of —lge˜r— —nd num˜er
theoryF
yne of su™h symmetry dete™tion methods is the su˜je™t of the t—lkF
st is —˜out threefold symmetry detection " —nd sin™e su™h
symmetry is inherent in hex—gonsD our method is optimal for
hexagonal imagesF

Introduction
st should ˜e noted th—t the threefold symmetry dete™tion is
™urrently ™l—imed in v—rious —re—s of ™ryst—llogr—phyD virologyD
—n—lysis of ele™tron mi™ros™ope im—gesD etsF
ƒeeD for ex—mpleD
rundtD ‚FD ƒ™h¤onD tFgFD r—nnem—nnD eFD t—nsen wF
Determination of symmetries and idealized cell parameters for
simulated structuresF tourn—l of epplied gryst—llogr—phyD QPD
RIQERIT @IWWWAF
ƒpekD enthony vF Structure validation in chemical
crystallographyF e™t— gryst—llogr—phi™—F hTSD IRVEISS @PHHWA
eyun ‰uD f—j—jD gF Automatic ultrastructure segmentation of
reconstructed CryoEM maps of icosahedral virusesF siii
„r—ns—™tions on sm—ge €ro™essingD IR@WAD IQPREIQQU@PHHSA
—nd so onF

Introduction
e ™onsider—˜le —mount of rese—r™h in hex—gon—lly s—mpled im—ges
pro™essing @rs€A is t—king pl—™e now despite the f—™t th—t there —re
no h—rdw—re resour™es th—t ™urrently produ™e or displ—y hex—gon—l
im—gesF
Middleton, L., Sivaswamy, J. Hexagonal Image Processing:
A Practical Approach. Springer (2005), 259 pp.
por thisD software resampling is in useD when the origin—l d—t— is
s—mpled on — squ—re l—tti™e while the desired im—ge is to ˜e
s—mpled on — hex—gon—l l—tti™eF
ƒo the proposed —lgorithm ™—n ˜e used for regul—r squ—re im—ges
@in f—™tD th—t is how further ex—mples h—ve ˜een worked outAF
st is ˜—sed on the interpret—tion of hex—gon—l im—ges —s fun™tions
on 4nite Eisenstein elds4F

Finite Fields of Eisenstein Integers
sn num˜er theoryD Eisenstein integers —re ™omplex num˜ers of the
form z = a + bωD where a, b ∈ Z —nd ω = exp(2πi/3) ∈ C is —
primitive @nonEre—lA ™u˜e root of unityD so th—t ω2 + ω + 1 = 0F
xote th—t within the ™omplex pl—ne the iisenstein integers m—y ˜e
seen to ™onstitute — triangular latticeF

‡e will ™onstru™t now 4nite elds of Eisenstein integers4F
vet Zn = Z/nZ ˜e — residue ™l—ss ring modulo —n integer n ≥ 2F
Lemma 1
sf p = 12k + 5 is — primeD the polynomi—l x2 + x + 1 is irredu™i˜le
over ZpD ˜ut x2 + 1 = 0 is notF
Denition 2
vet p ≥ 5 ˜e — prime su™h th—t p ≡ 5 (mod 12)F „he (nite (eld
E(p)
def
== Zp[x]/(x2
+ x + 1) GF(p2
)
will ˜e ™—lled Eisenstein eldF
„husD iisenstein (elds h—ve p2 elementsD where
p = 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, . . . .

ilements of iisenstein (elds —re of the form z = a + bωD where
a, b ∈ Zp —nd ω denotes the ™l—ss of residues of xD so th—t
ω2 + ω + 1 = 0F „he produ™t of a + bω ∈ E(p) —nd c + dω ∈ E(p)
is given ˜y (a + bω)(c + dω) = (ac − bd) + (bc + ad − bd)ω.
The Eisenshtein eld E(5).

Hexagonal Images as Functions on Eisenstein Fields
ilements of —n iisenstein (eld E(p) —re in n—tur—l ™orresponden™e
with hex—gons in — (p × p)Edi—mondEsh—ped fr—gment of — regul—r
pl—ne tilingF
ren™eD fun™tions on iisenstein (elds m—y ˜e ™onsidered —s
hex—gon—l im—ges @of spe™i—l sizes3AF
‡e will use —n—logy ˜etween the ™omplex (eld C —nd E(p) to
represent elements of E(p) in —n 4exponential form4—nd to
introdu™e log-polar coordinates in hex—gon—l im—gesF

Polar Decompositions of Eisenstein Fields
vet C∗ ˜e the multipli™—tive group of ™omplex num˜ers —nd
R = R, + ˜e the —dditive group of re—lsF „he ™orresponden™e
0 = z = reiθ
= eln r+iθ
↔ (l, θ),
where l = ln r ∈ R —nd 0 ≤ θ 2π ,
˜etween nonEzero ™omplex num˜ers z —nd its logEpol—r ™oordin—tes
(l, θ) m—y ˜e ™onsidered —s —n isomorphism
C∗
R × (R/2πZ).
sn f—™tD any dire™t produ™t de™omposition of C∗ produ™es —
represent—tion of ™omplex num˜ersF
vet us tr—nsfer the previous ™onstru™tion to E(p)F

xote th—t the multipli™—tive group E∗(p) is ™y™li™D E∗(p) = g F
Lemma 3
por every p = 12k + 5D num˜ers m = 2(p − 1) = 8(3k + 1) —nd
n = (p + 1)/2 = 3(2k + 1) —re relatively primeD gcd(m, n) = 1F
xote th—t mn = p2 − 1 = |E∗(p)|F
Theorem 4
por every iisenstein (eld E(p)D its multipli™—tive group is
de™omposed into dire™t produ™t of ™y™li™ groups of orders m —nd nD
E∗
(p) = g Zm × Zn .
‡e will ™—ll it the polar decomposition of E(p)F
„he pol—r de™omposition ™—n ˜e used to tr—nsfer onto E(p) the
™on™ept of complex logarithmF

Logarithms in Eisenstein Fields
Logarithms in Eisenstein Fields
por —ny primitive g de(ne the m—pping Exp g : Zm × Zn → E∗(p)X
Exp g(l, θ) = gnl+mθ
= z ∈ E∗
(p), where (l, θ) ∈ Zm × Zn .
„husD Exp g is —n isomorphism ˜etween the —dditive group of the
ring Zm × Zn —nd the multipli™—tive group of E(p)F
Denition 5
„he m—pping Exp g : Zm × Zn → E∗(p) is ™—lled the modular
exponent to base gD —nd its inverse Ln g : E∗(p) → Zm × Zn is the
modular logarithm to base gF
„he basic logarithmic identity follows immedi—telyX
Ln g(z1z2) = Ln g(z1) + Ln g(z2) .

Polar Forms of Hexagonal Images
Polar Forms of Hexagonal Images
‡e h—ve seen th—t fun™tions on E(p) m—y ˜e ™onsidered —s
hex—gon—l im—gesF
€ol—r de™ompositions of E(p) provide other w—yX
with a p × p-hexagonal image f(z) a square-sampled
image ψ of size m × n can be associated.
por thisD (x —ny primitive g ∈ E∗(p) —nd de(ne — fun™tion
ψ : Zm × Zn → R su™h th—t
ψ(Ln g(z)) = f(z), 0 = z ∈ E∗
(p) .
Denition 6
„he tr—nsform Pg[f] = ψ is ™—lled log-polar transform to base g of
—n hex—gon—l im—ge fD or just its polar transform P ˜rie)yF „he
squ—reEs—mpled im—ge ψ is the polar form of fF

EXAMPLE
µ = 0.8695
µ = 0.0411
µ = 0.0046

Threefold Symmetry in Hexagonal Images
Threefold Symmetry in Hexagonal Images
„he geometri™ interpret—tion of iisenstein (elds immedi—tely
implies the following de(nitionF
Denition 7
e hex—gon—l im—ge f is s—id to h—ve threefold central rotational
symmetry if —nd only if f(ωz) = f(z)F
vet ψ ˜e the pol—r form of — hex—gon—l p × pEim—ge fF st is —
(m × n)Em—trixD where n = 3(2k + 1)D m = 8(3k + 1) —nd
k = (p − 5)/12 ∈ ZF
Decompose ψ into three blocks ψ0, ψ1, ψ2 of equal size m × n/3.
Theorem 8
en im—ge f is threefold ™entr—l symmetri™ if —nd only if
ψ0 = ψ1 = ψ2 .

Measure of Symmetry
Measure of Symmetry
por re—lEworld im—ges f —pproxim—te equ—lities ψ0 ψ1 ψ2 holdF
„o measure symmetryD t—ke the norm—lized pol—r form m—trix
˜ψ = ψ/ max{ψ} —nd ev—lu—te
µ(f) = exp −αxβ
, where x = max
0≤ij≤2
˜ψi − ˜ψj
˜ψi + ˜ψj
.
rere α —nd β —re nonneg—tive re—lsD whose pre™ise v—lues ™ould
v—ry depending on the pr—™ti™—l pro˜lem to ˜e solvedF
„hen µ(f) is the measure of central threefold symmetry of fF

Measure of Symmetry
Example 9
€revious ex—mple demonstr—tes pol—r forms for squ—reEs—mpled
113 × 113Eim—ges of triskelionD ™ross —nd letter pF „o em˜ed them
into the (eld E(113) the st—nd—rd res—mpling —lgorithm w—s usedF
€ol—r forms were ev—lu—ted for g = 1 + 5ω ∈ E∗(113)F „he
symmetry me—sures —re
µ(f1) = 0.8695 , µ(f2) = 0.0411 , µ(f3) = 0.0046 ,
where the pro˜enius m—trix norm w—s used —nd α = 32D β = 4F
xote th—t it is —n e—sy m—tter to prove inv—ri—n™e of µ(f) under
™h—nge of the primitive element gF
tointly with —ny of the sliding window methodsD the introdu™ed
—ppro—™h ™—n ˜e used to dete™t ™enters of lo™—l threefold symmetry
in im—gesF

Measure of Symmetry
Example 1
New Years Eve Times Square Ball.

Measure of Symmetry
Location of centers of 3-fold symmetry.

Measure of Symmetry
Example 2
Another Times Square New Years Eve Ball Bloomberg photo.

Measure of Symmetry
Location of centers of 3-fold symmetry.

Thanks for your attention!

References
References
„hrunD ƒF —nd ‡eg˜reitD fF ƒh—pe from symmetryF €ro™F sntF
gonferen™e on gomputer †ision @sgg†AD vFPD IVPREIVQI@PHHSAF
ghertokD wFD uellerD ‰FX ƒpe™tr—l ƒymmetry en—lysisF siii
„r—nsF on €—ttern en—lysis —nd w—™hine sntelligen™eF QP@UAD
IPPU!IPQV @PHIHA
herrodeD ƒFD qhor˜elD pFX ƒh—pe —n—lysis —nd symmetry
dete™tion in gr—yElevel o˜je™ts using the —n—lyti™—l
pourierEwellin represent—tionF ƒign—l €ro™essingF VR@IAD PSQW
@PHHRA
u—rkish™henkoD eFxFD wnukhinD †FfFX ƒymmetry re™ognition in
the frequen™y dom—in @in ‚ussi—nAF snX Wth sntern—tion—l
gonferen™e on sntelligent snform—tion €ro™essingD ppF RPT!RPWF
„y‚…ƒ €ressD wos™ow @PHIPA

References
g—mpello de ƒouz—D ‚FwFD de yliveir—D rFwF —nd ƒilv—D hF „he
Z „r—nsform over pinite pieldsF —rˆiv preprint ISHPFHQQUI
pu˜lished online pe˜ru—ry IID PHISF
g—mpello de ƒouz—D ‚FwF —nd p—rrellD ‚FqF pinite (eld
„r—nsforms —nd symmetry groupsF his™rete x—them—ti™sD STD
IIIEIIT@IWVSA
f—ndeir—D tF —nd g—mpello de ƒouz—D ‚FwF xew „rigonometri™
„r—nsforms yver €rime pinite pields for sm—ge pilteringF inX †s
sntern—tion—l „ele™ommuni™—tions ƒymposium @s„ƒPHHTAD
port—lez—EgeD fr—zilF TPVETQQ@PHHTA
f—kerD rFqF gomplex q—ussi—n sntegers for q—ussi—n qr—phi™sF
egw ƒigpl—n xoti™es PVD II @xovem˜er IWWQAD PPEPUF
wnukhin †FfF „r—nsform—tions of higit—l sm—ges on gomplex
his™rete „oriF €—ttern ‚e™ognition —nd sm—ge en—lysisX
edv—n™es in w—them—ti™—l „heory —nd eppli™—tionsD PHIRD volF
PRD noF RD ppF SSPESTHF

References
wnukhin †FfF higit—l im—ges on — ™omplex dis™rete torus @sn
‚ussi—nAF w—™hine ve—rning —nd h—t— en—lysisD I@SAD
SRHESRV@PHIQAF
u—rkish™henkoD eFxFD wnukhinD †FfFX eppli™—tions of modul—r
log—rithms on ™omplex dis™rete tori in pro˜lems of digit—l im—ge
pro™essing @sn ‚ussi—nAF fulletin of the ‚ostov ƒt—te …niversity
of ‚—ilw—y „r—nsportF QD IRUEISQ@PHIQAF
wnukhin †FfF pourierEwellin tr—nsform on — ™omplex dis™rete
torusF €ro™F IIth sntF gonfF 4€—ttern ‚e™ognition —nd sm—ge
en—lysisX xew snform—tion „e™hnologies4@€‚seEIIEPHIQAD
ƒeptem˜er PQEPVD PHIQD ƒ—m—r—D PHIQD ppF IHPEIHSF
rerD sF qeometri™ „r—nsforms on the rex—gon—l qridF siii
„r—ns—™tions on sm—ge €ro™essingF R@WAD IPIQEIPPP@IWWSAF
ˆi—ngji—n reD ‡enjing ti—D x—mho rurD i—ng ‡u —nd
tinwoong uimF sm—ge „r—nsl—tion —nd ‚ot—tion on rex—gon—l
ƒtru™tureF snX „he Tth siii snternF gonfF on gomputer —nd
snformF „e™hnology @gs„9HTAD ƒeoulD IRI@PHHTAF

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Alexander Karkishchenko - Threefold Symmetry Detection in Hexagonal Images Based on Finite Eisenstein Fields