This document discusses a method for detecting threefold symmetry in hexagonal images using finite Eisenstein fields. It begins with an introduction to symmetry detection and issues that arise when applying existing continuous techniques to digital images. It then describes finite Eisenstein fields, which are constructed as finite fields analogous to the complex integers. Elements of these fields correspond naturally to hexagons, allowing hexagonal images to be represented as functions over the fields. Polar coordinate transformations are introduced to represent field elements in exponential form, enabling the transfer of continuous symmetry detection methods to digital hexagonal images. In summary, the document proposes a novel approach for symmetry detection in hexagonal images based on the algebraic structure of finite Eisenstein fields.
A strictly face regular map is a k-valent plane graph on the sphere or the entire plane with faces of size a and b such that any a-gonal face is adjacent to exactly p a-gonal face and exactly q b-gonal faces. If only one of such rule is respected then we get a weak face-regular map.
We present here enumeration technique for the face regular maps that rely on polycycle and other techniques.
Un journal entièrement dédié aux clients du Marché reprenant toutes les informations et toute l'actualité du Marché de Gros Lyon-Corbas
Edition N°6 - Décembre 2011
A strictly face regular map is a k-valent plane graph on the sphere or the entire plane with faces of size a and b such that any a-gonal face is adjacent to exactly p a-gonal face and exactly q b-gonal faces. If only one of such rule is respected then we get a weak face-regular map.
We present here enumeration technique for the face regular maps that rely on polycycle and other techniques.
Un journal entièrement dédié aux clients du Marché reprenant toutes les informations et toute l'actualité du Marché de Gros Lyon-Corbas
Edition N°6 - Décembre 2011
U.S. Retail Banking: Prescriptions for Channel Integration and BeyondCognizant
To achieve the dual goals of satisfying tech-savvy customers and boosting the bottom line, banks must first lay the foundation for integrated channels and fulfillment processes. Here is how they can embark on this two-laned path.
El tratamiento de las enfermedades y las infraestructuras de sanidad ha ido modernizándose con el paso del tiempo. Algunos de los edificios antiguos conservan el aspecto exterior que tuvieron al ser construidos, desde el inmenso hospital General de San Carlos o el impresionante Hospital de Jornaleros hasta una modesta Casa de Curas.
Unos han seguido en uso sanitario y otros han sido reconvertidos pero siguen conservando el encanto y las soluciones técnicas de cuando fueron construidos: es muy interesante el sistema de escaleras interiores del actual Museo Reina Sofía, con las escaleras construidas con peldaños de muy poca altura (apenas unos centímetros), para permitir que se pudiesen subir y bajar las camillas con ruedas para los enfermos ya en el siglo XVII cuando aún ni se pensaba en la existencia de ascensores, y diseñados por los mejores arquitectos de la época, desde Sabatini a Antonio Palacios o Hermosilla.
Siyaset, Ekonomi ve Toplum Araştırmaları Vakfı'nın hazırladığı bu çalışma, Avrupalı yabancı savaşçılar vakasını üç katmanlı analitik bir düzlemde; kimlik-iddiaları, anlam oluşturma (motivasyon) ve radikalleşme araçları çerçevelerini kullanarak incelemektedir.
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
U.S. Retail Banking: Prescriptions for Channel Integration and BeyondCognizant
To achieve the dual goals of satisfying tech-savvy customers and boosting the bottom line, banks must first lay the foundation for integrated channels and fulfillment processes. Here is how they can embark on this two-laned path.
El tratamiento de las enfermedades y las infraestructuras de sanidad ha ido modernizándose con el paso del tiempo. Algunos de los edificios antiguos conservan el aspecto exterior que tuvieron al ser construidos, desde el inmenso hospital General de San Carlos o el impresionante Hospital de Jornaleros hasta una modesta Casa de Curas.
Unos han seguido en uso sanitario y otros han sido reconvertidos pero siguen conservando el encanto y las soluciones técnicas de cuando fueron construidos: es muy interesante el sistema de escaleras interiores del actual Museo Reina Sofía, con las escaleras construidas con peldaños de muy poca altura (apenas unos centímetros), para permitir que se pudiesen subir y bajar las camillas con ruedas para los enfermos ya en el siglo XVII cuando aún ni se pensaba en la existencia de ascensores, y diseñados por los mejores arquitectos de la época, desde Sabatini a Antonio Palacios o Hermosilla.
Siyaset, Ekonomi ve Toplum Araştırmaları Vakfı'nın hazırladığı bu çalışma, Avrupalı yabancı savaşçılar vakasını üç katmanlı analitik bir düzlemde; kimlik-iddiaları, anlam oluşturma (motivasyon) ve radikalleşme araçları çerçevelerini kullanarak incelemektedir.
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
Slide set presented for the Wireless Communication module at Jacobs University Bremen, Fall 2015.
Teacher: Dr. Stefano Severi, assistant: Andrei Stoica
Generalized sub band analysis and signal synthesisjournalBEEI
Currently, one of the main approaches used in analyzing properties and synthesis of signals in various classes is the subband methodology, which is carried out from the position of Fourier transform of signal samples (frequency representations) into subbands of the transform definition domain (transformants). In this case, the main tool, which is widely used for subband analysis (including wavelet analysis), is usage of bandpass filters (mainly those with finite impulse response or FIR filters). The present paper introduces the basics of building a theory forsubband analysis / signal synthesis for various classes, and using transformations based on any orthonormal basis with weight. This proposed approach is based on the concept of Euclidean signal norm square fraction in a given subband of the transformant definition domain. It is shown that the basis for mathematical apparatus of subband analysis is a new class of matrices, called subband ones. Some eigenvalue properties of these matrices are established, and the problem of optimal selection for additive signal components is formulated and solved.
ARRAY FACTOR OPTIMIZATION OF AN ACTIVE PLANAR PHASED ARRAY USING EVOLUTIONARY...jantjournal
Evolutionary algorithms (EAs) have the potential to handle complex, multi-dimensional optimization problems in the field of phased array. Out of different EAs, particle swarm optimization (PSO) is a popular choice. In a phased array, antenna element failure is a common phenomenon and this leads to degradation of the array factor (AF) pattern, primarily in terms of increased side lobe levels (SLLs), displacement of nulls and reduction in the null depths. The recovery of a degraded pattern using a cost and time-effective approach is on demand. In this context, an attempt made to obtain an optimized AF pattern after fault in a 49 elements quasi-circular aperture equilateral triangular grid active planar phased array using PSO. In the paper, multiple cases on recovery are discussed having a maximum 20% element failure. Each recovery is also further evaluated by different statistical analyses. A dedicated software tool was developed to carry out the work presented in this paper.
ARRAY FACTOR OPTIMIZATION OF AN ACTIVE PLANAR PHASED ARRAY USING EVOLUTIONARY...jantjournal
Evolutionary algorithms (EAs) have the potential to handle complex, multi-dimensional optimization
problems in the field of phased array. Out of different EAs, particle swarm optimization (PSO) is a popular
choice. In a phased array, antenna element failure is a common phenomenon and this leads to degradation
of the array factor (AF) pattern, primarily in terms of increased side lobe levels (SLLs), displacement of
nulls and reduction in the null depths. The recovery of a degraded pattern using a cost and time-effective
approach is on demand. In this context, an attempt made to obtain an optimized AF pattern after fault in a
49 elements quasi-circular aperture equilateral triangular grid active planar phased array using PSO. In
the paper, multiple cases on recovery are discussed having a maximum 20% element failure. Each recovery
is also further evaluated by different statistical analyses. A dedicated software tool was developed to carry
out the work presented in this paper.
ARRAY FACTOR OPTIMIZATION OF AN ACTIVE PLANAR PHASED ARRAY USING EVOLUTIONARY...jantjournal
Evolutionary algorithms (EAs) have the potential to handle complex, multi-dimensional optimization problems in the field of phased array. Out of different EAs, particle swarm optimization (PSO) is a popular choice. In a phased array, antenna element failure is a common phenomenon and this leads to degradation of the array factor (AF) pattern, primarily in terms of increased side lobe levels (SLLs), displacement of nulls and reduction in the null depths. The recovery of a degraded pattern using a cost and time-effective approach is on demand. In this context, an attempt made to obtain an optimized AF pattern after fault in a 49 elements quasi-circular aperture equilateral triangular grid active planar phased array using PSO. In the paper, multiple cases on recovery are discussed having a maximum 20% element failure. Each recovery is also further evaluated by different statistical analyses. A dedicated software tool was developed to carry out the work presented in this paper.
ARRAY FACTOR OPTIMIZATION OF AN ACTIVE PLANAR PHASED ARRAY USING EVOLUTIONARY...jantjournal
Evolutionary algorithms (EAs) have the potential to handle complex, multi-dimensional optimization problems in the field of phased array. Out of different EAs, particle swarm optimization (PSO) is a popular choice. In a phased array, antenna element failure is a common phenomenon and this leads to degradation of the array factor (AF) pattern, primarily in terms of increased side lobe levels (SLLs), displacement of nulls and reduction in the null depths. The recovery of a degraded pattern using a cost and time-effective approach is on demand. In this context, an attempt made to obtain an optimized AF pattern after fault in a 49 elements quasi-circular aperture equilateral triangular grid active planar phased array using PSO. In the paper, multiple cases on recovery are discussed having a maximum 20% element failure. Each recovery is also further evaluated by different statistical analyses. A dedicated software tool was developed to carry out the work presented in this paper.
ARRAY FACTOR OPTIMIZATION OF AN ACTIVE PLANAR PHASED ARRAY USING EVOLUTIONARY...jantjournal
Evolutionary algorithms (EAs) have the potential to handle complex, multi-dimensional optimization problems in the field of phased array. Out of different EAs, particle swarm optimization (PSO) is a popular choice. In a phased array, antenna element failure is a common phenomenon and this leads to degradation of the array factor (AF) pattern, primarily in terms of increased side lobe levels (SLLs), displacement of nulls and reduction in the null depths. The recovery of a degraded pattern using a cost and time-effective approach is on demand. In this context, an attempt made to obtain an optimized AF pattern after fault in a 49 elements quasi-circular aperture equilateral triangular grid active planar phased array using PSO. In the paper, multiple cases on recovery are discussed having a maximum 20% element failure. Each recovery is also further evaluated by different statistical analyses. A dedicated software tool was developed to carry out the work presented in this paper.
ARRAY FACTOR OPTIMIZATION OF AN ACTIVE PLANAR PHASED ARRAY USING EVOLUTIONARY...jantjournal
Evolutionary algorithms (EAs) have the potential to handle complex, multi-dimensional optimization problems in the field of phased array. Out of different EAs, particle swarm optimization (PSO) is a popular choice. In a phased array, antenna element failure is a common phenomenon and this leads to degradation of the array factor (AF) pattern, primarily in terms of increased side lobe levels (SLLs), displacement of nulls and reduction in the null depths. The recovery of a degraded pattern using a cost and time-effective approach is on demand. In this context, an attempt made to obtain an optimized AF pattern after fault in a 49 elements quasi-circular aperture equilateral triangular grid active planar phased array using PSO. In the paper, multiple cases on recovery are discussed having a maximum 20% element failure. Each recovery is also further evaluated by different statistical analyses. A dedicated software tool was developed to carry out the work presented in this paper.
ARRAY FACTOR OPTIMIZATION OF AN ACTIVE PLANAR PHASED ARRAY USING EVOLUTIONARY...jantjournal
Evolutionary algorithms (EAs) have the potential to handle complex, multi-dimensional optimization problems in the field of phased array. Out of different EAs, particle swarm optimization (PSO) is a popular choice. In a phased array, antenna element failure is a common phenomenon and this leads to degradation of the array factor (AF) pattern, primarily in terms of increased side lobe levels (SLLs), displacement of nulls and reduction in the null depths. The recovery of a degraded pattern using a cost and time-effective approach is on demand. In this context, an attempt made to obtain an optimized AF pattern after fault in a 49 elements quasi-circular aperture equilateral triangular grid active planar phased array using PSO. In the paper, multiple cases on recovery are discussed having a maximum 20% element failure. Each recovery is also further evaluated by different statistical analyses. A dedicated software tool was developed to carry out the work presented in this paper.
ARRAY FACTOR OPTIMIZATION OF AN ACTIVE PLANAR PHASED ARRAY USING EVOLUTIONARY...jantjournal
Evolutionary algorithms (EAs) have the potential to handle complex, multi-dimensional optimization problems in the field of phased array. Out of different EAs, particle swarm optimization (PSO) is a popular choice. In a phased array, antenna element failure is a common phenomenon and this leads to degradation of the array factor (AF) pattern, primarily in terms of increased side lobe levels (SLLs), displacement of nulls and reduction in the null depths. The recovery of a degraded pattern using a cost and time-effective approach is on demand. In this context, an attempt made to obtain an optimized AF pattern after fault in a 49 elements quasi-circular aperture equilateral triangular grid active planar phased array using PSO. In the paper, multiple cases on recovery are discussed having a maximum 20% element failure. Each recovery is also further evaluated by different statistical analyses. A dedicated software tool was developed to carry out the work presented in this paper.
ARRAY FACTOR OPTIMIZATION OF AN ACTIVE PLANAR PHASED ARRAY USING EVOLUTIONARY...jantjournal
Evolutionary algorithms (EAs) have the potential to handle complex, multi-dimensional optimization problems in the field of phased array. Out of different EAs, particle swarm optimization (PSO) is a popular choice. In a phased array, antenna element failure is a common phenomenon and this leads to degradation of the array factor (AF) pattern, primarily in terms of increased side lobe levels (SLLs), displacement of nulls and reduction in the null depths. The recovery of a degraded pattern using a cost and time-effective approach is on demand. In this context, an attempt made to obtain an optimized AF pattern after fault in a 49 elements quasi-circular aperture equilateral triangular grid active planar phased array using PSO. In the paper, multiple cases on recovery are discussed having a maximum 20% element failure. Each recovery is also further evaluated by different statistical analyses. A dedicated software tool was developed to carry out the work presented in this paper.
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Alexander Karkishchenko - Threefold Symmetry Detection in Hexagonal Images Based on Finite Eisenstein Fields
1. 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
2. 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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 2 / 28
3. 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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 3 / 28
4. 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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 4 / 28
5. Introduction
Note the complications:
ell the re—lEworld im—ges —re digitalD not ™ontinuousF
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 5 / 28
6. Introduction
Note the complications:
ell the re—lEworld im—ges —re digitalD not ™ontinuousF
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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 5 / 28
7. Introduction
Note the complications:
ell the re—lEworld im—ges —re digitalD not ™ontinuousF
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
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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 5 / 28
8. Introduction
Note the complications:
ell the re—lEworld im—ges —re digitalD not ™ontinuousF
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
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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 5 / 28
9. 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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 6 / 28
10. 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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 7 / 28
11. 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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 8 / 28
12. Finite Fields of Eisenstein Integers
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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 9 / 28
13. Finite Fields of Eisenstein Integers
‡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, . . . .
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 10 / 28
14. Finite Fields of Eisenstein Integers
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).
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 11 / 28
15. Finite Fields of Eisenstein Integers
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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 12 / 28
16. Polar Decompositions of Eisenstein Fields
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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 13 / 28
17. Polar Decompositions of Eisenstein Fields
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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 14 / 28
18. 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) .
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 15 / 28
19. 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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 16 / 28
20. EXAMPLE
µ = 0.8695
µ = 0.0411
µ = 0.0046
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 17 / 28
21. 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 .
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 18 / 28
22. 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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 19 / 28
23. 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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 20 / 28
24. Measure of Symmetry
Example 1
New Years Eve Times Square Ball.
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 21 / 28
25. Measure of Symmetry
Location of centers of 3-fold symmetry.
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 22 / 28
26. Measure of Symmetry
Example 2
Another Times Square New Years Eve Ball Bloomberg photo.
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 23 / 28
27. Measure of Symmetry
Location of centers of 3-fold symmetry.
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 24 / 28
28. Thanks for your attention!
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 25 / 28
29. 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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 26 / 28
30. 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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 27 / 28
31. 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
A. Karkishchenko, V. Mnukhin (SFU, Taganrog) Threefold Symmetry Detection 28 / 28