Gualter License IC determination

7/26/2019 Gualter License IC determination

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MINISTRY OF EDUATION OF MOLDOVA

Free International University of Moldova

Falty of Infor!ati"s# En$ineerin$ and Desi$n

De%art!ent of Infor!ational Te"&nolo$ies and En$ineerin$

Accepted for defense Accepted for defense

Dean of the Faculty Head of the Department

Iuri Dubovehi, Dr, conf. univ.

___________________________ ___________________________

____________________!"#$ ____ _________________!"#$

LI'ENSE (RO)E'T

I!a$e 'o!%le*ity Deter!ination Syste!

Author

%odideal &heor'he, student 'r. ()*#

+roect -upervisor

eaceslav/. +eru, Dr. Hab., conf. univ.

&i+in, -./0

MINISTERUL EDUA1IEI AL RE(U2LIII MOLDOVA
http://vperju.ulim.md/http://vperju.ulim.md/


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Universitatea Li3er, Interna4ional, din MoldovaFaltatea Infor!ati, In$inerie i Di5ain

atedra Te&nolo$ii Infor!a ionale i In$inerie

Admis la susinere Admis la susinere

Deanul Fault0ii 1ef 2atedr0

Iuri Dubovehi, Dr, conf. univ.

___________________________ ___________________________

_____________________!"#$ ____ _________________ !"#$

TE6A DE LIEN17

Siste!l de Deter!inare a 'o!%le*it, ii I!a$inilor

34eutant

%odideal &heor'he, -tudentul 'rupei ()*!

2ondu0torul te5ei

+eru eaeslav, Dr. hab., conf. univ.

&i+in, -./0
http://var/www/apps/conversion/tmp/scratch_7/%5Chhttp://var/www/apps/conversion/tmp/scratch_7/%5Ch


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8 Rodideal 9&eor$&e# -./0


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1ef 2atedr0 6ehnolo'ii Informaionale 7i In'inerie

Iuri Dubovehi, Dr, 2onf. 8niv.

_________________________

___ ________________ !"...

S A R I N A

pentru te5a de lien0 a studentului 'rupei ()*!

%odideal &heor'he

Te!a: Siste!l de Deter!inare a 'o!%le*it, ii I!a$inilor aprobat0 prin ordinul nr. ________ din ___ _____________ !"...

on4intl notei e*%liative9 #. Anali5a al'oritmilor, metodelor 7i sistemelor e4istente de

determinare a comple4itatii ima'inelor: !. 3laborarea, reali5area 7i eretarea unui al'oritm

noude determinare a pi4elilor din ima'ini: ;. 3laborarea 7i eretarea sistemului de 'estioinare a

comple4itatii ima'inelor.

Lista !aterialli $ra%&i9 #. 2lasifiarea al'oritmilor, metodelor 7i sistemelor e4istente de

determinare: !. -trutura al'oritmului nou de determinare a comple4itatii: ;. -hema)blo a

softului elaborat de determinarea comple4itatii: *. %e5ultatele eret0rilor al'oritmului de

determinare a comple4itatii ima'inelor:


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ADNOTARE

Rodideal 9&eor$&e#Sistemul de Determinare a Complexit ii Imaginelor ,te5, de lien4,

la s%eialitatea 'al"latoare# &i+in,# -./0;

Aest proietul uprinde introduerea, trei apitole, onlu5ii u reomand0ri,

biblio'rafia din #> titluri. 3a este perfetat0 pe


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A2STRAT

Rodideal 9&eor$&e# =Image Complexity Determination System t&esis for s%eialty

'o!%ters# &isina# -./0;

6he thesis ontains the introdution, three hapters, onlusions and reommendations,

biblio'raphy of @@ titles. It onsists of @@ pa'es, inludin' @@ fi'ures and @ tables and @@

formule.

!e"#ords9al'oritmi,detetarea,reunoa7terea,sistem,multidimensional,marcareadetetion.

$ield of stud"of the thesis is information proessin'.

%oals and ob&etivesinlude researhin', multidimensional e4traction system accounts..

Novelt" and originalit"of this ?or is use of drivers to onnet ?ith ima'es.

'he theoretial signifianedevelopin' a soft?are that implements several al'orithms forIma'e (omple4ity Detection.

(ppliative value of the ?or is that this system an be ed'e detection in various

ompanies and areas.

Implementation results. 6he system developed has been onfi'ured and tested on

multiple omputers and multiple ima'es.


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'ontentsIntroduction.....................................................................................................................................B

#. 6H3 ACA/-I- EF 6H3 --63- ACD 36HED-.....................................................##

#.#. 6hG nGGd for dGfinition of im'G comlGJityKLM..........................................................##

#.!. (omlGJityK>M.................................................................................................................##

#.;. 3ntropy and mutual InformationK>M................................................................................#

#.M......................................................................................................!!

#.>. (omple4ity measuresK>M.................................................................................................!;

#.B. (omlGJity as m roGrtyK#;M..................................................................................!$

#.#". isul m comlGJity tGstin' mGthodsK#;M...............................................................!L

#.##. DGfinitions su''GstGd in visul sciGncGKLM..................................................................!>

#.#!. (omrisons ?ith GJGrimGntl GstimtGs of comlGJity KLM.....................................!B

#.#;. 3stimtin' im'G comlGJityK


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Introd"tion

In this paper, i introduce a ne? frame?or based on information theory and ima'e se'mentation

to study the comple4ity of an ima'e. Different authors have established a relationship bet?een

aesthetics and comple4ity. In #B!>, &.D. Nirhoff introduced the concept of the aesthetic

measure, deOned as the ratio bet?een order and comple4ity 6he comple4ity is rou'hly the

number of elements that the ima'e consists of and the order is a measure for the number of

re'ularities found in the ima'e 8sin' information theory, . Nense transformed NirhoffPs

measure into an informational measure9 redundance divided by statistical information. 6o

compute the comple4ity, he introduced the assumption that an input pattern can al?ays be

described as a t?o dimensional 'rid of discrete symbols from a pre)deOned repertoire. En the

other hand, he observed that order corresponds to the possibility of perceivin' lar'e structures A.

oles held that an aesthetic measure is closely related to ima'e comple4ity, and based his

measure of ima'e comple4ity on information theory +.achadoandA. (ardosoestablished that

anaesthetic visual measure depends on t?o factors9 processin' comple4ity and ima'e comple4ity

. 6hey consider that ima'es that are simultaneously visually comple4 and easy to process are the

ima'es that have a hi'her aesthetic value. From the above discussed ?ors, it appears that

comple4ity is at the core of aesthetics. Qith the 'uideline that under standin' comple4ity can

shedli'ht on a esthetics, ?e ?ill e4plore ima'e comple4ity from an information theoretic

perspective. Ima'e comple4ity has also been related to entropy of the ima'e intensity histo'ram.

Ho?ever, this measure does not tae into account the spatial distribution of pi4els, neither the

fact that a comple4ity measure must tae into account at ?hat level one ?ants to describe an

obect. For instance, a random seRuence reRuires a lon' description if all details are to be

described, but a very short one if a rou'h picture is reRuired . In ima'e processin', an ima'e is

se'mented by 'roupin' theima'ePspi4elsintounitsthatarehomo'eneousinrespect to one or more

characteristics, or features. -e'mentation of nontrivial ima'es is one of the most difOcult tass in

ima'e processin'. Ima'e se'mentation al'orithms are 'enerally based on one of t?o basic

properties of intensity values9 discontinuity and similarity. In the Orst cate'ory, the approach is to

partition the ima'e based on abrupt chan'es in intensity, such as ed'es in an ima'e. 6he principal

approaches in the second cate'ory are based in partitionin' an ima'e into re'ions that are similar

accordin' to a set of predeOned criteria. 6hresholdin', re'ion 'ro?in', and re'ionsplittin' and

mer'in' are e4amples of methods in this cate'ory . 6his paper is or'ani5ed as follo?s. In this

chapter, ?e present an al'orithm ?hich splits an ima'e in relatively homo'eneous re'ions usin'

abinary space partition SN-+Tora Ruad)tree. In ne4 chappter , comple4ity is deOned by usin' t?o


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measures ?hich tae into account the level at ?hich the ima'e is considered. Finally, in ne4t

chapter , ?e present our conclusions and future research.


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/; T>E ANALYSIS OF T>E SYSTEMS AND MET>ODS

/;/; T&? n??d for @ d?finition of i!@$? "o!l?BityC6hG concGt of im'G comlGJity is ?idGly usGd by comutGr sciGntists nd by Gn'inGGrs ?ho

dGsi'n nd construct informtion nGt?ors nd systGms for thG nlysis, rGco'nition,

rGconstruction, nd visuli5tion of im'Gs. 6hG concGt is lso usGd by nGurosciGntists, not only

thosG intGrGstGd in thG mGchnisms of obGct rGco'nition but lso thosG concGrnGd ?ith lGrnin'

nd mGmory. It is morG difficult, for GJmlG, to mGmori5G comlGJ im'G thn simlG onG.

Gt thGrG is no 'rGGd dGfinition for thG comlGJity of n im'G. DiffGrGnt dGfinitions hvG bGGn

offGrGd nd diffGrGnt l'orithms imlGmGntGd for Gstimtin' comlGJity. Urticulrly influGntil

hs bGGn Volmo'orovPs S#B$


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the meanin' of this Ruantity should be very close to certain measures of diffi)ult"concernin' the

obect or the system in Ruestion9 the difficulty in constructin' an obect, the difficulty in describ)

in' a system, the difficulty in reachin' a 'oal, the difficulty in performin' a tas. 6here are many

definitions of comple4ity correspondin' to the different ?ays of Ruantifyin' these difficulties.

A list of comple4ity measures provided by -eth /loyd is 'rouped under three Ruestions9 ho?

hard is it to describe, ho? hard is it to create, and ?hat is its de'ree of or'ani5ationZ In the first

'roup, entropy is ?idely applicable for indicatin' randomness. It also measures uncertainty,

i'norance, surprise, or information. In the second 'roup, the computational comple4ity Ruantifies

the amount of computational resources Susually time or spaceT needed to solve a problem.

Finally, in the third 'roup, mutual information e4presses the concept of comple4ity that

Ruantifies the de'ree of structure or correlation of a system or the amount of information shared

bet?een the parts of a system as a result of this or'ani5ational structure.6o our no?led'e, the only frame?or e4istin' until no? dealin' ?ith ima'e comple4ity is

defined in, ?hich deals ?ith comparin' the performance of A6% applications.

In this conte4t, ima'e comple4ity is defined as a measure of the inherent difficulty of findin' a

true tar'et in a 'iven ima'e. -uch a metric should predict the performance of a lar'e class of

A6%s on diverse ima'ery, ?ithout advanced no?led'e of the tar'ets. A split and mer'e

se'mentation al'orithm is first applied that partitions an ima'e into compact re'ions of uniform

'ray)level, no lar'er than the e4pected tar'et si5e. %ecursive thresholdin' determines the splits.

After the se'mentation procedure is applied, the tar'et similarity of each re'ion is estimated. 6he

distribution of this similarity is taen as a basis for comple4ity measurement. For instance, if

there are many re'ions ?ith tar'et similarity near the ma4imum the ima'e is relatively comple4.

6hree comple4ity measures are then 'iven. 6he first is the number of re'ions ?hose tar'et)

similarity e4ceeds a 'iven threshold, the second measures the distance from the body of the dis)

tribution to the most si'nificant outlier, and the third is the ?ei'hted avera'e of the distance to

all outliers.K>M

(i" /;/;/: In%t and ot%t distri3tions for t&e %artitionin$ of "&annel;


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I!@$? "o!l?BityC-0

[s mGntionGd bovG, in ordGr to GJtrct nd trc thG tr'Gt utomticlly it is nGcGssry for us to

mGsurG thG im'G comlGJity. (onsidGrin' thG liction nd rGsGrchGs in ctul liction

nd rGsGrch, ?G dGfinG thG im'G comlGJity s follo?s9 thG im'G comlGJity is mGsurG of

thG inhGrGnt difficulty of GJtrctin' nd trcin' tr'Gt.

NsGd on sclG, thG im'G comlGJity dGscrition cn bG clssifiGd into t?o ctG'oriGs9 onG is

bsGd on thG 'lobl chrctGristics nd thG othGr onG is bsGd on rG'ionl chrctGristics. NsGd

on for rticulr tr'Gt or not, thG im'G comlGJity dGscrition cn bG clssifiGd into t?o

ctG'oriGs9 onG is in connGction ?ith rticulr tr'Gt nd thG othGr onG is nothin' to do ?ith thG

tr'Gt. NsGd on diffGrGnt dGscritors, thG im'G comlGJity dGscrition cn bG clssifiGd into

thrGG ctG'oriGs9 6hGrG rG 'ry lGvGl bsGd, Gd'G bsGd nd shG bsGd.

[bout thG clssifiction of thG dGscrition mGthods of im'G comlGJity, sho?n in Fi'. #.!.#9

Fi$; /;-;/ T@Bono!y of i!@$? "o!l?Bity !?tri"s

6r'Gt

IndGGndGnt

6r'Gt

DGGndGnt

&ry)lGvGl &lobl %G'ionl %G'ionl

Yd'G &lobl %G'ionl %G'ionl

shG %G'ionl

NGcusG of diffGrGnt rGsGrch urosG, domGstic nd forGi'n scholrs hvG diffGrGnt focusGs on

im'G comlGJity. In this Gr, considGrin' thG vribility of tr'Gt in rGl)timG nd utomtic

tr'Gt GJtrction ?G dGscribG thG im'G comlGJity ccordin' to 'lobl chrctGristics nd tG

no ccount of thG sGcil tr'Gt.K!$M

I!@$? "o!l?Bity !?tri"sC-0

UGtGrs nd %ichrd hvG dGscribGd thG im'G comlGJity usin' thG 'lobl fGturGs. 6hGy

usGd 'ry lGvGl nd Gd'G chrctGristic to dGscribG thG im'G comlGJity, ?hich lost thG scG

distribution of 'ry lGvGl. 6hus this mGthod cnPt mGsurG thG im'G comlGJity ccurtGly.

In this Gr, considGrin' thG dt chrctGristics of thG im'G itsGlf nd thG dGmnds of

rcticl liction, ?G nly5G im'Gs from thG GrncG of 'ry lGvGl, thG GrncG of

tr'Gt nd thG rndomnGss of im'G tGJturG. NsGd on thG 'lobl fGturG bout thG GrncG of

'ry lGvGl, thG GrncG of tr'Gt nd thG rndomnGss of im'G tGJturG, this mGthod dGscrit


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thG im'G comlGJity usin' thG informtion Gntroy, thG Gd'G Gntroy nd tGJturG Gntroy. 6hG

GJGrimGnts sho? tht this mGthod could mGsurG im'G comlGJity RulittivGly \] 6hG

comlGJity 'ivGn by this mGthod is ccordin' ?ith thG difficulty of thG ?or to GJtrct nd trc

thG tr'Gt in tht im'G.

G?@r@n"? of $r@y l?v?l6hG morG informtion in thG im'G, thG morG comlGJ in thG im'G ?ill . 6hG GrncG

of 'ry lGvGl cn rGflGct thG 'ry lGvGl rich or not. 6hG informtion Gntroy is usGd to dGscribG

thG informtion continGd in 'ry lGvGl. 6hG formul of informtion Gntroy clcultion is s

follo?s9

S#.*.#T

In formul #, C is thG numbGr of 'ry lGvGls nd n is numbGr of thG iJGls in Gch 'ry

lGvGl. 6hG lr'Gr thG H, thG morG comlGJ thG im'G ?ill bG.

G?@r@n"? of ?d$?s

6hG Runtity nd thG comlGJity of thG tr'Gt cn bG GJrGssGd ?ith Gd'Gs. 6hus, ?G cn

tG dvnt'G of Gd'G Gntroy to chrctGri5G thG GrncG of thG tr'Gt. 6hG GrncG of

thG tr'Gt cn rGflGct thG im'GPs comlGJity. For GJmlG, GJtrctin' nd trcin' thG intGrGstin'

tr'Gt is RuitG difficult ?hGn thG numbGr of thG tr'Gts is lr'G. 6hG formul of Gd'G Gntroy

clcultion is s follo?s9

+^)S,g UTlo'!S,g,. S#.*.!T

In formul !, UGd'Gis thG numbGr of thG Gd'G oints in thG im'G. U is thG numbGr of im'G

iJGls. In this Gr, ?G usG thG (nny oGrtor to GJtrct Gd'Gs nd clcultG thG numbGr of thG

Gd'G oints.

R@ndo!n?ss of i!@$? t?Btr?s

Informtion Gntroy nd Gd'G Gntroy cnPt GJrGss thG scG distribution of 'ry lGvGl.

For GJmlG, somG im'Gs hvG thG smG informtion Gntroy nd Gd'G Gntroy but thGy hvG

diffGrGnt im'G comlGJity. 6hGrG forG ?G should hvG somG othGr chrctGristic to GJrGss thG

scG distribution of 'ry lGvGl.

6GJturG fGturG is ind of non)sGctrl fGturGs, ?hich is ?y to mGsurG thG stil

distribution of 'ry lGvGl. 6GJturG nlysis of im'Gs ?s dGvGloGd in thG #BL"s. 6hGrG rG

mny dGscrition mGthods bout im'G tGJturG. YJtrctin' tGJturG fGturGs bsGd on 'ry lGvGl

co)occurrGncG mtriJ is clssic sttisticl nlysis mGthod. %GsGrch on 'ry lGvGl co)

occurrGncG mtriJ hs lon' history. ost of scholrs 'rGG tht this is vGry rGliblG mGthod$)L

.


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HGrG ?G minly usG thG 'ry lGvGl co)occurrGncG mtriJ to GJtrct thG tGJturG informtion >. 6hG

formul of 'ry lGvGl co)occurrGncG mtriJ comuttion is s follo?s9

In formul ;, on thG ri'ht sidG of thG GRution thG molGculr is thG numbGr of iJGl

coulGs. 6hosG iJGl coulGs hvG somG ind of stil rGltions ?hosG 'ry lGvGls rG '# nd '!

rGsGctivGly. En thG ri'ht sidG of thG GRution thG dGnomintor is thG totl numbGr of thG iJGl

coulGs S` indictG thG numbGr of thG follo?in' fctorT. In tht ?y, ?G 'Gt thG normli5Gd .

S#.*.;T

From thG 'ry lGvGl co)occurrGncG mtriJ, ?G cn 'Gt thG sGcond)ordGr momGnts,

contrst, corrGltion, Gntroy nd sGriGs of tGJturG dGscritions. (onsidGrin' thG rGl)timG nd

dimGnsionl consistGncy, ?G ust usG 'ry lGvGl co)occurrGncG mtriJ to clcultG thG tGJturG

Gntroy to dGscribG thG rndomnGss of thG im'G tGJturG. 6hG formul of tGJturG Gntroy

clcultion is s follo?s9

S#.*.*T

6hG 'rGtGr thG vluG - is, thG morG rndomnGss thG tGJturG distribution, GJtrctin' nd

trcin' thG intGrGstin' tr'Gt is morG difficult. In this csG, ?G considGr tht thG im'G is morG

comlGJ.

Fi$; /;H;/ To i!@$?s it& t&? s@!? infor!@tion ?ntroy @nd ?d$?s# 3t diff?r?nt in ?ntroy

From ?ht discussGd bovG, ?G usG thG informtion Gntroy, thG tGJturG Gntroy nd thG

Gd'G Gntroy to mGsurG thG im'G comlGJity.K!$M

/;-; Entro%y and !tal Infor!ationC

6he Shannon entrop" /S@T of a discrete random variable0 ?ith values in the set @ ^4i,*1*n is defined as


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S#.M

Met&od 3ased on Entro%y

&iven an ima'e ?ith Npi4els and an intensity histo'ram ?ith n(pi4els in bin i, ?e

define a discrete information channel ?here inputX represents the bins of the histo'ram, ?ith

probability distribution pt ^ C , output Ythe pi4el)to) pi4el ima'e partition, ?ith uniform

distribution 7q j ^ N , and conditional probability pt of the channel is the transition

probability from bin iof the histo'ram to pi4eljof the ima'e. 6his inormation !"annelcan be

represented by

S#.


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As ?e have seen in content #.*, mutual informationI


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Qi" /;J; a 2S( r .;./.# MIR H-;H 3 adtree r .;./.# MIR K;/. " adtree r .;./H# MIR H-;H

/;; T&? ?ntroi" !od?l of vis@l "o!l?BityC06o crGtG mthGmticl modGl of thG visul comlGJity bsGd on stil rmGtGrs ?G

hvG rGviG?Gd mny of thG locl nd 'lobl fGturGs from litGrturG. &lobl fGturGs rG suitGd to

dGrivG sin'lG vluGs from thG 'GnGrl roGrtiGs of n im'G. /ocl fGturGs rG nGGdGd to tG

into ccount clssicl vGrbl GJlntions for thG mGnin' of comlGJity9 mny vGrsus fG?,

curvGd ndor dGtilGd vGrsus linGr nd lnr, comlGJ tGJturGs vGrsus flt rGs. 8sin' locl

fGturGs lso hGls rGducin' mbi'uitiGs in rGsults.

Lo"al Featres E*tra"tion

Uoints of intGrGst my bG idGntifiGd by usin' locl oGrtors. QG chosG t?o ?Gll) no?n

locl fGturGs9 thG im'G Gd'Gs nd thG locl symmGtriGs comutGd by thG Dis)rt S"mmtr"

'r=nsform i=l momntsof body round its cGntGr of 'rvity. In thG im'G csG, thG iJGls insidG

circulr ?indo? rG considGrGd s oint mssGs, ?ith thGir mss GJrGssGd by thGir 'ry vluG g6

Gn ntroi" M?@sr? of 'o!l?Bity

QG rG no? intGrGstGd in 'lobl l'orithm tht cn outut sin'lG vluG for Gch

filtGrGd im'Gs, ?hilG rGsGrvin' its clss of comlGJity. QG dGcidGd to invGsti'tG thG

usGfulnGss for this ts of thG fu55y Gntroic distncG functions dGtilGd in . 6hGrG rG lGnty of

rGsons for considGrin' thGsG functions mon' mny othGrs usully GmloyGd in this ind of


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bsis, ?ithout ny no?lGd'G of our rGsGrchPs ims. Urivcy of thG subGcts ?s tGn crG of

ccordin' to thG Itlin l? on Grsonl dt: only initils, 'G nd 'GndGr ?GrG rGcordGd for

Gch subGct.

Fi$; /;J;/ B@!l?s of t?st i!@$?s# "l@ssifi?d 3y intitiv? "o!l?Bity: &i$& "o!l?Bity to !?di! "o!l?Bity !iddl?lo "o!l?Bity 3otto!

6hG GJGrimGnts ?GrG hGld in dim li'ht room to rGducG visul distrction, 'ivin' timG to

thG rticint for drnGss dttion. [ll thG usul Gr'onomic rGcutions, such s usin'

Rusi)soundroof room, ?GrG tGn, nd thG subGct ?s llo?Gd to choosG thGir o?n rGfGrrGd

osition nd visul n'lG. 6hG im'Gs ?GrG rGsGntGd full scrGGn. 6hG soft?rG usGd ?s homG)

mdG usin' thG multimGdi ro'rmmin' GnvironmGnt cromGdi DrGm?GvGr !""* on

n [lG cintosh comutGr ?ith 6F6 /(D monitor. 6hG chosGn im'Gs ?GrG comutGr

scns of intin's, dividGd in thrGG ctG'oriGs rGrGsGntin' diffGrGnt lGvGls of visul comlGJity,

bsGd on thG rGsGncG or bsGncG of cGrtin clssGs of fGturGs nd cuG oints. Fi'urG # sho?s

GJmlGs of intin' usGd in this study.

Ych im'G ?GrG rGsGntGd for fiJGd Griod of timG SB" sGcs.T, ?ith no tGmorl cluGs:

thG GJGrimGnt lso hd )ontrollddGsi'n in ordGr to minimi5G sidG GffGcts9 li'hts dimmGd nd

uniform, subGct lonG in soundroof room. 6hG subGct ?s lGrtGd to focus thGir ttGntion on

thG contGnts of thG dislyGd im'Gs. 6hG im'Gs usGd for thG GJGrimGnts ?GrG chosGn

ccordin' to thG intuitivG hyothGsis tht thG comlGJity of scGnG incrGsGs ?ith thG numbGr of

obGcts nd thGir rGltivG osition, nd ?ith its ovGrll structurG K>M. 6hG chosGn im'Gs ?GrG

intin's, dividGd in thrGG ctG'oriGs rGrGsGntin' diffGrGnt lGvGls of visul comlGJity. HGrG thG

GstimtG timG GrcGivGd by Gch subGct is rGortGd. QG considGr it s subGctivG mGsurGs of

comlGJity for thG thrGG ctG'oriGs of im'Gs introducGd bovG. In thG follo?in' it ?ill bG

dGnotGd s 6hG smlG mGn vluG nd thG vrincG of thG GrcGivGd timG S' u'T

rG rGortGd in 6blG #.


20/47

6hG roosGd normli5tion llo?s us bGttGr comrison ?ith thG rGsults obtinGd from

thG mthGmticl modGl, crriGd out in thG nGJt sGction. In this contGJt, " nd # hvG no strict

numGricl si'nificncG, but should bG intGrrGtGd morG liG subGctivG dG'rGGs of comlGJity,

?hich suits bGst ?ith our fu55y modGl.

%Gsults rG in 'rGGmGnt ?ith our modGl of timG GrcGtion9 comlGJ im'Gs SctG'ory IT

roducG shortGr timG Gstimtions thn im'Gs in ctG'ory II nd thG smG is truG for ctG'oriGs II

nd III.K$M

Fi$; /;J;- M?@n @nd Nor!@li5?d Ti!? sti!@tion

/;J; 'o!@rison of !?@sr?s @nd d@t@ v@lid@tionC0[s sho?n by comrin' thG GntriGs of 6blGs # nd !, our GJGrimGntl dt mtch thosGof thG mthGmticl modGl. In fct, im'Gs ?ith hi'h Gntroic comlGJity indGJ 'GnGrtG, on

vGr'G, shortGr Gstimtion of thG GrcGivGd timG. 6hGrGforG, ctG'ory I hs thG shortGst

Gvlutions nd ctG'ory III thG lon'Gst. 6hG stron' nti)corrGltion bGt?GGn thG Gntroic

mGsurG of comlGJity nd thG mGntl cloc is sho?n in Fi'. #.#


21/47

from thG GJGrimGntl dt nd thG comlGmGnts of thG Gntroic mGsurGs of comlGJity usin'

-GrmnsP @6 YvGn in thG ?orst csG, thG robbility of dt nd modGl sGRuGncGs bGin'

corrGltGd is morG thn ".B>. 6o confirm tht thG corrGltion is not duG to thG si5G of thG dt)sGt,

?G crriGd out mny nonrmGtric bootstr tGsts, usin' #",""" virtul sGts. In Gch tGst thG

diffGrGncG bGt?GGn thG mGn obtinGd from thG dt nd by thG bootstr mGthod ?s undGr

#"j*. [s ?G ?orGd mostly ?ith mGn vluGs, ?G lso usGd thG cnifG tGchniRuG, rG)

clcultin' thG rGsults s mny timGs s thG numbGr of im'Gs in our sGt, lGvin' out onG im'G

Gch timG: ll cnifG sGts hd thG smG distribution of vluGs, ?ith smll numGric diffGrGncGs.

K$M

/;0; I!a$e %artitionin$CIn this section, ?e present a 'reedy al'orithm ?hich partitions an ima'e in Ruasi)

homo'eneous re'ions. 6he optimal partitionin' al'orithm is C+)complete. 6o do this partition, a

natural approach could consider the above channel S#.


22/47

&iven the error probability3eallo?ed in partitionin', Fanos ineRuality provides us ?ith

a lo?er bound for the 'ain of mutual information. 6ain' the eRuality, ?e obtain the minimum

value of I needed in the partitionin' al'orithm for a 'iven probability of error9

Imin


23/47

6o introduce our comple4ity frame?or, ?e ?ill reinterpret the previous partitionin'

approach from the point of vie? of the ma4imi5ation of the Xensen)-hannon diver'ence. 6his

perspective, althou'h eRuivalent to the ma4imi5ation of mutual information, is more appropriate

to deal ?ith ima'e comple4ity and has been introduced in the study of the DCA comple4ity .

First, ?e define a comple4ity measure, the Xensen) -hannon diver'ence, ?hich e4presses

the image )ompositional )omple*it"SI((T of an ima'e. 6his measure can be interpreted as the

spatial hetero'eneity of an ima'e from a 'iven partition. From, the Xensen)-hannon diver'ence

applied to an ima'e is 'iven by

S#.>.#T

?here+is the number of re'ions of the ima'e, @, is the random variable associated ?ith re'ionirepresentin' the intensity histo'ram of this re'ion, mis the number of pi4els of re'ion i, andN

is the total number of pi4els of the ima'e. Ebserve that for the information channel S#.$.*T, the

Xensen) -hannon diver'ence coincides ?ith the I. 6he compositional comple4ity S#.L.#T fulfils

the follo?in' properties9

It increases ?ith a finer partition.

It is null for a sin'le partition.

For a random ima'e and a coarse resolution it ?ould be close to ".

For a random ima'e and the finest resolution it ?ould be ma4imum and eRual to/


24/47

compression error and thus the number of re'ions is also related to the difficulty of compression.

K>M

Reslts

Qe use a uniform partition to test the compositional comple4ity on the ima'es in Fi'. #.#.#. 6he

results obtained are sho?n in Fi'. #.L.! for the number of partitions runnin' from ! 4 ! to the

number of pi4els in the respective ima'es. Qe observe that the relative orderin' of the

comple4ities depends on the resolution level Snumber of partitionsT. For instance, the earth rise

ima'e appears to be the most comple4 at resolution * 4 * ?hile the ?ild flo?ers appears as the

least one. Ho?ever, this behavior is reversed at hi'h resolution.

In Fi'ure #.L.# ?e can analy5e the behavior of the second proposed comple4ity measure. Qhile

the lines in the 'raph in Fi' #.L.! cross themselves, the ones in Fi'ure #.L.# eep a re'ular

orderin'. Ebserve their e4ponential 'ro?in' ?ithJI+that is due to the increasin' cost of the I

e4traction. It is important to note that forJI+ 2".< ?e obtain a 'ood Ruality?ith a fe? number

of re'ions. Qith respect to the number of re'ions, the most comple4 ima'e appears to be the

Haboon and the least one is theKarth rise6

It can also be sho?n SFi'ure #.L.;T that ?hile blurrin' an ima'e ?ill cause a loss of comple4ity,

increasin' the contrast causes the opposite effect. For instance, for a JI+ 2# and the luminance

channel L"B, the contrasted /ena ima'e of Fi'ure #.L.;.b Sr ^ B#.LT needs more re'ions than

the ori'inal /ena ima'e Sr ^ >B.*T and the blurred ima'e of Fi'ure #.L.;.a Sr ^ *>.;T needs less

re'ions.K>M

Fi$re /;;/ : Ratio of t&e n!3er of re$ions r it& res%e"t to MIR for t&e i!a$es of Fi$; / it& l!inan"e Y.K;


25/47

Fi$re/;;-: 'o!%ositional "o!%le*ity I'' over t&e n!3er of re$ions R of t&e %artitioned i!a$es of Fi$; / it&l!inan"e Y.K; T&e n!3er of %artitions $oes fro! - * - to t&e n!3er of %i*els N in t&e res%e"tive i!a$es;

/;; 'o!l?Bity as @ !@ ro?rtyC/

(omlGJity hs bGGn thG crto'rhGrsP obGct of intGrGst for mny yGrs, s it influGncGs

rGdbility nd GffGctivGnGss of crto'rhic roducts. (omlGJity rGsults from numbGr of

symbols on thG m, thGir divGrsity nd thG distncG bGt?GGn thGm SdGnsityT. (omlGJity my bG

considGr s intGrction bGt?GGn thGsG GlGmGnts rGltin' to t?o fundmGntl mPs sGcts )

syntctic nd sGmntic, hGncG it corrGsonds to t?o comlGJity sGcts ) visul nd intGllGctul

comlGJity ScYchrGn, #B>!T. 6hG intGllGctul comlGJity is minly dGtGrminGd by thGmount of rGsGntGd informtion, thG chrctGr of its rGsGnttion, rocGssin' lGvGl nd thG

clssifiction mGthod s ?Gll s numbGr of clssGs. YvGn if thG m 'rhics is roritGly

sGlGctGd nd obGcts rGsGntGd on thG m rG lG'iblG Gnou'h, thG usGr my hvG difficultiGs in

undGrstndin' its contGnt if thG mount of rGsGntGd informtion is too hi'h SHun', !""!T.

6hG visul comlGJity rGsults from stil divGrsity of visul m structurG nd dGGnds

on dG'rGG of GJtGnsivGnGss, 'GnGrli5tion nd thG dG'rGG of visul vriblG ordGr. 6hG visul

comlGJity cn bG rG'rdGd s thG oositG to thG rGdbility. Qin'Grt S#BL*T rovGd Gmiriclly

tht thG hi'h im'G dGnsity SovGrlodGd ?ith dGtilsT si'nificntly rGducGs thG stil structurG

Fi$re /;;: Lena i!a$e: a Ot of fo"s and 3 !ore "ontrasted t&an its ori$inal;


26/47

informtion GJtrction ccurcy. NGrtin S#B$LT dGscribGd rGdbility s thG bility to distin'uish

thG vriblGs from thG bc'round nd considGrGd tht it is ffGctGd by 'rhicl dGnsity,

divGrsity nd rGsolution connGctGd ?ith thG numbGr of symbols, thGir si5G nd roortions,

?hGrGby 'rhicl dGnsity ?s rG'rdGd s thG most imortnt fctor.K#;M

/;K; Vis@l !@ "o!l?Bity t?stin$ !?t&odsC/6hG ms comlGJity s n obGctivG fGturG cn bG studiGd GJclusivGly t thG visul lGvGl

sincG only t this lGvGl it is ossiblG to sGrtG subGctivG nd obGctivG lyGrs, hGncG Grform

ustifiGd comrison. In thG initil st'G of rGsGrch on thG visul msP comlGJity most of thG

?ors ?GrG concGrnGd ?ith thG thGmtic ms in rGsGct to ?hich it ?s ossiblG to usG

mGtrics tht llo?Gd Runtifyin' thGir comlGJity in simlG ?y. [ccordin' to cYchrGn

S#B>!T, numbGr of oly'ons, Gd'Gs nd nodGs on thG m lr'Gly rGflGcts its visul comlGJity.

ullGr S#BL$T liGd such comlGJity dGtGrminnt in his ?ors on chorolGt ms. 6hG

rGsults of his ?ors rtly rGflGctGd thG rGsult of rGvious studiGs on visul comlGJity crriGd

out by &ttrGll S#BL*T, ?ho notGd tht thG coGfficiGnts chrctGri5in' thG visul comlGJity

should bG insGrbly rGltGd to such m fGturGs s thG numbGr of oint si'nturGs or thG linG

lGn'th dGfinin' thGir boundriGs. 6hG mGnin' of mGsurblG nodGs, Gd'Gs nd lins bGt?GGn thG

GlGmGnts on thG m ?s dGGly studiGd by Y'GnhofGr SY'GnhofGr Gt l, #BB*T. 6hG linGs nd thG

nodGs ?GrG lso crucil for Ybi SYbi Gt l, #BB!T nd Il' S#BB"T in thG studiGs on im'Gs

comlGJity nd thG ossibility of thGir rGconstruction vi thG utomtic di'iti5tion rocGss.GrsGy S#BB"T roosGd thG clcultion mGthod Gstimtin' 'rhicl comlGJity similr to

cYchrGnsP S#B>!T utili5in' thG thGory of 'rhs nd bsGd on thG ?Gi'htGd numbGr of Gd'Gs

on thG m. In DiGt5GlPs ?ors S#B>;T, thG 'rh thGory ?s lso liGd. 6hG GJGrimGntl

studiGs of urry nd /iu S#BB*T should lso bG RuotGd. 6hGy too dvnt'G of 'Go'rhic

informtion systGms in ?hich dt is dislyGd in thG form of 'rhs, ?hich rGsGmblG ms. It

turnGd out tht thG 'rhicl m comlGJity should bG dGfinGd tin' into ccount its stil

vribility, nd not only simlG mGsurGs such s numbGr of linGs or numbGr of rticulr tyG ofsurfcG obGcts. Nsin' on thG forGmGntionGd ?ors c(rty nd -lisbury ScYchrGn,

#B>!T dGvGloGd mGsurG, ?hich llo?s dGtGrminin' thG comlGJity of contour ms. 6hG

similr indicGs tin' into ccounts thG stil distribution of m 'rhicl dGnsity ?GrG ?orGd

u by lyin' thG frctl dimGnsion SNurrou'h, cDonnGll, #BB>T nd thG mGthod of stil

utocorrGltion SNonhm)(rtGr, #BB*T.

Yntroy is nothGr vGry romisin' RuntittivG mGsurG, ?hich llo?s dGtGrminin' thG

'rhicl lod of thG nly5Gd m SHG Gt l, #BBLT. 6ht mGsurG hs dirGct connGction ?ith

thG m informtion contGnt nd is connGctGd ?ith thG ttGmts to chrctGri5G RuntittivGly


27/47

trnsmission of informtion throu'h thG communiction systGm. 6hG ?ors on thG mthGmticl

bc'round of trnsfGrrin' thG informtion by thG communiction systGm nd dGtGrminin' its

informtion contGnt ?ith thG usG of Gntroy ?GrG GrformGd by -hnnon nd QGvGr S#B*BT. [

sGrious dr?bc of -hnnon nd QGvGr mGthod, ?hich /i nd Hun' S!""!T ointGd out, is

thG lc of ossibility for considGrtion of stil distribution of obGcts. 6hGrGforG /iu nd

Hun' ostultGd tht comlGJity mGsurGs should lso tG this sGct into considGrtion nd

otGd for thG coGfficiGnts such s 6hiGssGn oly'on. 6hG most ctivG rGsGrchGr of Gntroy

mGsurGmGnt lictions in crto'rhicl rcticG ?s NorG S!"";T. 6in' dvnt'G of

usGful informtion concGt, hG sho?Gd ho? thG chn'Gs of symbols usGd on thG ms, thGir

ccurcy nd Gstimtion of disordGr cn ffGct thG GffGctivGs of m drftin' nd GrcGivin'

rocGss. Dt comrGssion tGchniRuG SdGrivGd from I6T is nothGr vGry intGrGstin' roch to

thG roblGm of controllin' m visul comlGJity S(ovGnGy, Hi'hfiGld, #BB


28/47

tht thG visul systGm mi'ht usG stil)frGRuGncy nlysis to rocGss visul im'Gs. 6his lGd to

thG ssumtion tht thG morG comlGJ n obGct, thG morG hi'h frGRuGnciGs rGRuirGd in its

sGctrum for rGco'nition. 6his roch is ?idGly usGd by Gn'inGGrs ?ho Gmloy stil)

frGRuGncy or ?vGlGt dGscrition of im'Gs. In thGsG tGrms comlGJity cn thGn bG dGfinGd s thG

numbGr of ctivG stil frGRuGnciGs or s thG numbGr of ctivG ?vGlGts.

[ similr, but morG formli5Gd dGfinition of im'G comlGJity cn bG found in CsnGn

Gt l S#BB;T. 6hGsG rGsGrchGrs su''Gst Gstimtin' thG comlGJity of im'Gs s thG roduct of thG

sRurGd mGdin of thG distribution of stil frGRuGnciGs nd thG im'G rG. 6hG incrGsin'

numbGr of rGltivG frGRuGnciGs, or hrmonics, in thG sGctrum rGsults in lr'Gr mGdins. Qith n

incrGsin' numbGr of non)Griodic linGs nd stroGs in thG obGct, thG numbGr of hi'h hrmonics

lso incrGsGs. In this csG thG mGdin of such n im'G sGctrum lso incrGsGs nd lGds to

hi'hGr GstimtG of im'G comlGJity, rovidGd thG rG is thG smG. [ccordin' to this dGfinition,Griodic ttGrns ?ith lr'Gr rG rG morG comlGJ. For sinusoidl 'rtin's, thG im'G

comlGJity, thus dGfinGd, is roortionl to thG numbGr of brs in thG 'rtin'. 6hus CsnGn Gt

lPs S#BB;T dGfinition incorortGs trditionl concGts of im'G comlGJity introducGd GrliGr in

thG ninGtGGnth cGntury.KLM

/;//; 'o!@risons it& ?B?ri!?nt@l ?sti!@t?s of "o!l?Bity CIt is not clGr ?hGthGr rticulr dGfinition of comlGJity cn bG liGd only to

sGcil clss of im'Gs or cn bG 'GnGrli5Gd to ll tyGs of visul obGcts, s fG? studiGs rGno?n ?hGrG GJGrimGntl Gstimtion of im'G comlGJity is comrGd to modGlin'. For

GJmlG, in CsnGn Gt l S#BB;T thG GfficiGncy coGfficiGnts of dGtGction ?GrG comrGd ?ith

comlGJity for only four filtGrGd im'Gs on noisy bc'round.

6o study comlGJity, [ttnGvG S#B obsGrvGrs ?GrG sGd to mG rtin's of thG comlGJity of thG

im'Gs usin' sGvGn)ctG'ory sclG. 6hG rGsults sho?Gd tht mtriJ 'rin nd curvGdnGss did

not hvG ny imct on comlGJity sclin': symmGtricl shGs ?GrG, in 'GnGrl, GstimtGd s

morG comlGJ if thGy hd thG smG numbGr of indGGndGnt turns, but thG most si'nificnt

vriblG ?s thG numbGr of turns, ?hich ccountGd for most of thG vribility of sclin'.


29/47

(onsGRuGntly, thG uthor concludGd tht im'G comlGJity is dGtGrminGd GssGntilly by thG

numbGr of turns in thG im'G.

In numbGr of rGvious n ttGmt hs bGGn mdG to corrGltG thG hysicl chrctGristics

of im'Gs, thG roGrtiGs of thG humn visul systGm, nd subGctivG GstimtGs of comlGJity. It

?s sho?n tht ?Gll)no?n stimuli rGrGsGntGd s blc nd ?hitG im'Gs hvG thGir o?n

miniml si5Gs for rGco'nition of ll dGtils. 6hGsG GJGrimGntlly obtinGd miniml si5Gs rG in

'rGGmGnt ?ith thGir thGorGticl GstimtGs clcultGd s thG rGRuirGd numbGr of smlin'

GlGmGnts ?hich rG hGJ'onl clustGrs of sGvGn conGs. It ?s sho?n tht thG miniml si5Gs rG

lr'Gr ?hGn thG stimulus is subGctivGly morG comlGJ.

In summry, sGvGrl studiGs hvG ttGmtGd to GstimtG visul comlGJity in GJGrimGnts,

but only limitGd comuttionl mGsurGs hvG bGGn liGd to thGsG rGsults. [ stil)frGRuGncy

roch ?s su''GstGd by CsnGn Gt l but ?s not tGstGd for n GJtGndGd sGt of im'Gs.DGsitG critics of thG liction of stil) frGRuGncy nlysis to vision, mny rGsGrchGrs usG

this roch. odGls oftGn usG filtGrin' of thG im'G ?ith DiffGrGncG of &ussins SDE&T or

&bor)liG tchGs simultin' rGcGtivG fiGlds t thG first lGvGl of rocGssin'. In our currGnt ?or

?G lso ly FouriGr nlysis to comutG thGorGticl GstimtGs of comlGJity.KLM

/;/-; Esti!@tin$ i!@$? "o!l?BityCJ&ivGn 'GnGrl comlGJity mGsurG (SJT for n im'G >onG cn try to GstimtG similritiGs

bGt?GGn im'Gs. [ nivG ssumtion ?ould bG tht thG diffGrGncG (SJoT (SJiT tGlls thGsimilrity bGt?GGn im'Gs J"nd>E68nfortuntGly such 'GnGrl comlGJity mGsurG doGs not

GJist. 6hG closGst thin' tht GJists is thG Volmo'orov comlGJity or l'orithmic Gntroy VSJT of

thG im'G Sor ny strin'T J. Volmo'orov comlGJity is not comutblG, ho?GvGr.

YvGn if thG comlGJity mGsurG (SJT GJistGd or Volmo'orov comlGJity ?GrG comutblG, thGir

vluG s mGsurGs of similrity ?ould bG RuGstionblG. IntuitivGly, thG similrity bGt?GGn im'Gs

doGs not l?ys GRul to thG diffGrGncG in comlGJity. 6his is bGcusG thG contGJt lys n

imortnt rolG GvGn t thG syntctic lGvGl, lthou'h not s much s in thG sGmntic lGvGl.[n obvious ?y of introducin' thG contGJt in thG icturG is to GstimtG thG oint comlGJity of

im'Gs. 6his is still t vGry lo? lGvGl but Gstimtin' thG comlGJity in thG contGJt of othGr

im'G vGrsus thG comlGJity of sin'lG im'G is morG informtivG thn rbitrry comlGJity

vluGs lonG. HGncG ?G rG intGrGstGd in thG distncG tht is dGfinGd s

DSJo,JiT ^ (SJoJiT min( SJoT, (SJiT, S#.#;.#T

ssumin' tht thG oint comlGJity is symmGtric, i.G. (SJ"J#T ^ (SJ#J"T. [lso onG ?nts to

GnsurG tht thG distncG is normli5Gd roritGly.


30/47

[s it ?s notGd bovG thG idGl comlGJity mGsurG doGs not GJist nd Volmo'orov comlGJity

is not comutblG. EnG cn roJimtG thG idGl comlGJity mGsurG in diffGrGnt mnnGrs,

ho?GvGr. -hnnonPs informtion thGory introducGd thG concGt of Gntroy, ?hich is Gsily

GstimtGd from dt. Yntroy cn bG sGGn lso s sttisticl mGsurG of comlGJity. YvGn

thou'h Volmo'orov comlGJity is not comutblG it cn bG roJimtGd usin' comrGssion

bsGd mGthods. (omlGJity cn lso bG GstimtGd from modGl tht roJimtGs thG lo')df of

dt s ?G do in this Gr.

R?l@tiv? ?ntroy @s dist@n"? !?@sr?

&ivGn discrGtG robbility distribution U -hnnonPs Gntroy/SJT is dGfinGd s

HSJT USJT lo' USJT. S#.#;.!T

Yntroy is nturl mGsurG of comlGJity, sincG it GstimtGs thG dG'rGG of uncGrtinty ?ith

rndom vriblGs. IntuitivGly it is Glin'9 6hG morG uncGrtin ?G rG bout n outcomG of n

GvGnt, thG morG comlGJ thG hGnomGnon Sdt, im'G, Gtc.T is.

&ivGn nothGr distribution , thG Vullbc)/GiblGr divGr'GncG is dGfinGd s

!L


31/47

(onditionl Volmo'orov comlGJity VSJ"J#T of strin' J"'ivGn strin' J#is thG lGn'th of shortGst

ro'rm tht roducGs outut J"from inut J#

VSJoJ#T ^ min 9 8SJ#T ^ Jo. S#.#;.$T

Cormli5Gd informtion distncG KLM is bsGd on thG Volmo'orov comlGJity nd is dGfinGd s

N ID ", JiT

mJVSJ"J#T, VSJ#J"T mJVSJ"T, VSJ#T S#.#;.LT

[s Volmo'orov comlGJity is not comutblG, CID nGithGr is comutblG. It cn bG

roJimtGd, ho?GvGr, usin' thG normli5Gd comrGssion distncG SC(DT KLM. C(D

roJimtGs CID by usin' rGl ?orld comrGssor ( nd it is dGfinGd s

ND O JiT

(SJ", J#T min(SJ"T, (SJ#T mJ(SJ"T, (SJ#T S#.#;.>T

6o usG thG C(D for mGsurin' ir)?isG distncGs bGt?GGn im'Gs onG ust comrGssGs im'Gs

sGrtGly nd conctGntGd nd obsGrvGs thG diffGrGncG bGt?GGn thG comrGssion rGsults.

Met&od t&at Usin$ I'G @s @n @roBi!@tion for ?ntroy

[ rcticl roJimtion of Gntroy cn bG ttinGd by fiJin' somG modGl ?hich

roJimtGs thG lo')df. QG roosG hGrG to usG this roch, in connGction ?ith thG modGl of

indGGndGnt comonGnt nlysis SI([T, or GRuivlGntly srsG codin'. 6hGsG modGls rG ?idGly

usGd in sttisticl im'G modGllin'. In I([, thG df is roJimtGd s

S#.#;.BT

?hGrG n is thG dimGnsion of thG scG, thG irG linGr fGturGs, collGctGd to'GthGr in thG mtriJ

P. 6hG function & is non)Rudrtic function ?hich mGsurGs thG srsity of thG fGturGs:

tyiclly % < u . ^ u or %


32/47

?hGrG thG GJGcttion is tGn, in rcticG, ovGr thG smlG.

[n intuitivG intGrrGttion of thG Gnsuin' comlGJity mGsurG is lso ossiblG. First, notG tht in

I([, thG vrincG of thG TB is fiJGd to onG. 6hG first tGrm on thG ri'ht)hnd)sidG in S#"T cn thus

bG considGrGd s mGsurG of srsity. In othGr ?ords, it mGsurGs thG non)&ussin sGct of

thG comonGnts, comlGtGly nG'lGctin' thG vrincG)covrincG structurG of thG dt. In fct, this

tGrm is minimi5Gd by srsG comonGnts. Qht is intGrGstin' is tht thG sGcond tGrm doGs

mGsurG thG covrincG structurG. In fct, ?G hvG in I([ thG ?Gll)no?n idGntity

#.#!.##

?hGrG (SBT is thG covrincG mtriJ of thG dt. 6his formul sho?s tht thG sGcond tGrm in

S#.#!.#"T is simlG function of thG dt covrincG mtriJ. In fct, lo' dGt P is mJimum if

thG dt covrincG hs minimum dGtGrminnt. [ minimum dGtGrminnt for covrincG

mtriJ is obtinGd if thG vrincGs rG smll in 'GnGrl, or, ?ht is morG intGrGstin' for our

urosGs, if somG of thG roGctions of thG dt hvG vGry smll vrincGs. -incG in I([, ?G

constrin thG vrincGs of thG comonGnts to bG GRul to onG, only thG lttGr csG is ossiblG.

6hus, our Gntroy mGsurG bGcomGs smll if thG dt is concGntrtGd in subscG of limitGd

dimGnsion.

6hus, this mGsurG of Gntroy ScomlGJityT is smll if thG comonGnts rG vGry srsG, or if thG

dt is concGntrtGd in subscG of limitGd dimGnsion, both of ?hich rG in linG ?ith our

intuition of structurG of multivritG dt.

Qr@"ti"@liti?s %GmGmbGrin' thG idGl comlGJity distncG in YR. # ?G rGsGnt somG rGmrs bout

thG usG of I([ modGl.

[ssumin' tht ?G ?nt to GstimtG thG distncG bGt?GGn t?o im'Gs, ?G GstimtG thG I([ modGl

from both im'Gs sGrtGly nd combinGd.

6hG comlGJity vluG tht ?G 'Gt usin' YR. #" is normli5Gd in similr mnnGr s thG C(D in

YR. >.

In rcticG thG I([ modGl for im'Gs is GstimtGd from dt tht contins lr'G numbGr of

rndomly smlGd im'G tchGs.K;M

/;/; T&e !et&od of "o!%le*ity: -*- !atri"es it& 3inary entriesCHQe apply the model for comple4ity to the simplest e4ample9 the set of all !@! matrices

?ith entries either " or g . (onsider the four distinct matrices that represent all possible matrices

of this type. Qe count arrays that can be transformed into each other by rotations or reflections,

and by chan'in' gs into "s, as similar. 6he four distinct e4amples are labeled S#T throu'h S*T in

Fi'ure #.#*.#.


33/47

Fi$; /;/H;/; For distin"t -


34/47

array S#T S!T S;T

/ife / " # #

(omple4ity ( " B B

Fi$ /;/H;- Vales of L and ' for t&e for -


35/47

6he t?o types of problem reduction must interact in any computation of comple4ity,

since some subblocs may have the full alphabet of symbols ?hile near 'roupin's of cells ?ith a

sin'le symbol mi'ht be present. 6hese subblocs permit a recursive formulation based on the

symbols9 the entire pattern is a ;4; array ?hose elements are !4! subblocs. A measure of the

total pattern comple4ity is based on this subbloc reduction.

6he desi'n temperature 6 eRuals the number of different symbols minus one. 6hese

measurements have to be done hierarchically on three different scales. First on the $@$ level,

then on each of four ;@; subblocs, then on each of nine !@! subblocs. It ?ill be useful to

label these subblocs in terms of letters and numbers. /et the inde4 ntae values a, b, c, d,

and mrun from # to B. 6he re'ular ;@; and !@! subdivisions of a $@$ matri4 ?ill be denoted as

follo?s9

a b

c d

# !

*

Fi$; /;/H;; S3divisions of a 0


36/47

?idth. S6he si'nificance of this ?ei'htin' for #fscalin' ?ill be discussed in a separate paperT.

Cote that ?e are not findin' the avera'e over the number of matrices. 6hus, even thou'h there

are four ;@; matrices, their ?idth is #! of the ori'inal $@$ matri4, so ?e divide by ! instead of

*. -imilarly, the ?idth of the !@! matrices is #; of the ori'inal $@$ matri4. Qe ?ill use the

above combination for computin' both 6 and H totals.

Ene could i'nore ?ei'htin' alto'ether, and simply add all contributions from all si5es of

submatrices. Ho?ever, that ?ould se? the numbers so that the smaller elements contribute

much more than the lar'er elements. 3lements of different si5e contribute simultaneously to our

perception of the ?hole, so it is necessary to count them in the proper balance.

6he harmony H is 'enerali5ed from the previous e4ample by includin' measures of

similarity at a distance. Different subblocs may interact ?ith each other. 6his maes it

necessary to count translational symmetry, ?hich did not apply ?hen dealin' ?ith isolated !@!arrays. In addition to the si4 symmetry measures hi, i ^ #,...,$ 'iven in the previous section, ?e

introduce three measures of translational symmetry9

hL^ similarity to another element Syes or no 'ives a # or "T

h>^ relation to another element by a translation, plus a reflection about either the 4)a4is

or the y)a4is Sa 'lide reflectionT.

hB^ relation to another element by a translation, plus a rotation by either gB", )B", or

#>".

Cote that h>and hB?ill sometimes double)count hLin cases of hi'h symmetry. 6hat is

ustified mathematically. 6?o different subblocs may be similar as they are oriented, and also

be similar after a reflection or a rotation. A subbloc may be related by 'lide reflection to another

subbloc, and by 'lide rotation to yet another, ?hich counts as !. SQe do not consider each 'lide

rotation by different multiples of B" separately, because that ?ould lead to more complication

than ?e ?ant in this model. Also, empirical e4periments sho? that 'lide reflections about the

t?o dia'onal a4es do not provide a stron' visual connection, and for that reason they are not

counted hereT.

6he desi'n harmony is defined as the sum of the h i, i ^ #,..., B. 3ach hitaes values " or #,

so H for a 'iven array Sof any si5eT ran'es from " to B. As in the case of 6, these computations

have to be done on three different levels, $@$, ;@;, and !@! S3Ruation S#.#$.*TT, then combined

?ith the appropriate ?ei'hts in 3Ruation S#.#$.


37/47

Si* 0*0 arrays it& for different entries

6he comple4ity model is applied to the si4 $@$ arrays listed in Fi'. #.#$.;. 3ach location

ScellT contains one of four symbols. 6he 'oal is to 'uess the comple4ity accurately after as short

a visual inspection as possible9 the result obtained is a ran)orderin' of the arrays in terms of

decreasin' comple4ity. 6he e4treme scores attainable, ma4imum and minimum, become more

?idely separated as more information is put into the model.

# ! ;

W W W W W W

W W W W W WW W W W W W

W W W W W W

W W W W W W

W W W W W W

W W . . X X

W W . . X X

X X . W W

X X W X W W

W X X

. X XX W W X

X . . X

X X W

X X .

* < $

W . X .

X . W X X

. X . W

. W . W

W X W X X

. W X . W

W W W W W W

W . . . . W

W . X . W

W . X . W

W . . . . W

W W W W W W

W X X W

. .

X W W X

X W W X

. .

W X X W

Fi$re /;/H;H; Si* 0


38/47

6he computations for 6 and H are strai'htfor?ard, and all details are 'iven in the

Appendi4. 8nlie the simple !@! case treated earlier, this is not an e4haustive classification of

all possible $@$ arrays ?ith four different entries. Qe ust pic a sample of arrays to sho? ho?

the method ?ors in practice. 6he matrices chosen have very different internal structure that

illustrates various possibilities.

Nefore readin' further, ?e su''est that the reader study the above matrices and ran)

order them in terms of decreasin' ( and /. %emember that ( measures the intensity of desi'n

complications. In art, ( measures the level of visual e4citement, ?hich often arises from chaotic

aspects of a desi'n. 6he life / measures the de'ree of or'ani5ed comple4ity in a desi'n: the

visual interest comes from the de'ree to ?hich elements interact coherently. 6he name life is

chosen because continuin' to increase / mathematically brin's one closer to the structure of

livin' or'anisms. 6his comparison is useful for the test that ?e propose. 6he reader can decide?hich of the above si4 arrays most resembles somethin' that could be or'anic, then ran)order

them in / based on this impression.

6his e4ample reRuires the follo?in' amended definition of ( and /, instead of 3Ruation

S#.#$.#T9

/ 6 H , ( ^ 6 S .< ##.L #$.; #

Harmony H


39/47

6hese measures are in accord ?ith our perception of pattern comple4ity. ost readers

?ill a'ree that these ranin's correspond to ?hat they have already concluded from direct

observation. 6he si4 e4amples are decreasin' in comple4ity in the same order as our feelin's. Qe

demonstrate here a stron' correlation bet?een the subconscious process of perception and a

simple Ruantitative model. Eur model can be refined by incorporatin' more and more input, but

even at this sta'e, it is remarably accurate in predictin' our emotional response to a desi'n.

3ven thou'h the desi'n temperature of array S$T is hi'h, the number of internal

symmetries or'ani5es the comple4ity so that ( is lo?ered and / is raised. (ontrast this to the

hi'h)6, lo?)H array S*T ) it has very little internal or'ani5ation, ?hich raises ( and lo?ers /.

Array S$T sho?s ho? ( essentially differs from 6. (ould one not sip the additional

complications of measurin' symmetries in this model and simply compute 6 as the comple4ity

of a desi'nZ 6he ans?er is no, because the ranin' in decreasin' 6 is *, $, ;,


40/47

/;/H; T&e analysis of e*istin$ !et&ods of t&e i!a$e "o!%le*ity deter!ination

Met&od Des"ri%tion Advanta$es Disadvanta$es

/; Met&od 3ased on

Entro%y

6his method 'ive us an 'lobl l'orithm

tht cn outut sin'lG vluG for Gch

filtGrGd im'Gs, ?hilG rGsGrvin' its clss of

comlGJity.

6his roch lGds to thG usG of stndrd

distncG functions, ?hich rGsGct thG usul

roGrtiGs of idGntity, symmGtry nd

trin'ulr inGRulity, u'mGntGd by Gntroic

functions.

-; G?@r@n"? of

$r@y l?v?l

6hG GrncG of 'ry lGvGl cn rGflGct thG

'ry lGvGl.

6hG morG informtion in thG im'G, thG

morG comlGJ in thG im'G ?ill be ?ill be

beneficial on the ima'e comple4ity

determination.

6hG informtion Gntroy is usGd to

dGscribG thG informtion continGd

in 'ry lGvGl only.

; G?@r@n"? of

?d$?s

6hG Runtity nd thG comlGJity of thG

tr'Gt cn bG GJrGssGd ?ith Gd'Gs.

Qith this method ?G cn tG dvnt'G of

Gd'G Gntroy to chrctGri5G thG GrncG

of thG tr'Gt. 6hG GrncG of thG tr'Gt

cn rGflGct thG im'GPs comlGJity.

H; R@ndo!n?ss ofi!@$? t?Btr?s

For GJmlG, somG im'Gs hvG thG smGinformtion Gntroy nd Gd'G Gntroy but

thGy hvG diffGrGnt im'G comlGJity.

Informtion Gntroy nd Gd'G Gntroy cnPtGJrGss thG scG distribution of 'ry lGvGl.

Fot this method ?G should hvGsomG othGr chrctGristic to

GJrGss thG scG distribution of

'ry lGvGl.

J; -*- !atri"esQe apply the model for comple4ity to the 6hese calculations 'ive us an important 6his method is used more on the


41/47

it& 3inary

entries

simplest e4ample9 the set of all !@!

matrices ?ith entries either " or g . (onsider

the four distinct matrices that represent all

possible matrices of this type.

Ruantity in visuali5ation9 it measures the

difference bet?een or'ani5ed and

disor'ani5ed comple4ity.

temperature estimation measures

the de'ree of internal contrast: the

density of differentiations: the

smallness of subdivisions.

0; Met&od t&at

Usin$ I'G @s @n

@roBi!@tion

for ?ntroy

[ rcticl roJimtion of Gntroy cn bG

ttinGd by fiJin' somG modGl ?hich

roJimtGs thG lo')df.

Qe can usG this roch, in connGction

?ith thG modGl of indGGndGnt comonGnt

nlysis SI([T, or GRuivlGntly srsG

codin'.

6hGsG modGls rG usGd onli in

sttisticl im'G modGllin'.


42/47

; Met&od t&at

Usin$ &ierar"&y

to $enerali5e to

&i$&er

di!ensions

6his method usin' $4$ arrays. Ene basic

component of problem reduction is

'eometric9 in addition to the entire field of

;$ cells ?e consider smaller arrays ?hich

are similar to the ori'inal array.

6his model introduced a useful

distinction for discussin'

comple4ity in theoretical terms.

; 0*0 arrays it&

for different

entries

6he 'oal is to 'uess the comple4ity

accurately after as short a visual inspection

as possible9 the result obtained is a ran)

orderin' of the arrays in terms of decreasin'

comple4ity. 6he e4treme scores attainable,

ma4imum and minimum, become more

?idely separated as more information is put

into the model.

$@$ case sho?ed the considerable po?er of

the model. Qe are in fact measurin' the

or'ani5ational entropy Sde'ree of disorderT,

?hich is the ne'ative of the de'ree of

connections established via visual

symmetries.

6his method is also used more on

the temperature estimation

measures the de'ree of internal

contrast: the density of

differentiations: the smallness of

subdivisions.

K; I!a$e%artitionin$

In this method, ?e present a 'reedy

al'orithm ?hich partitions an ima'e in

Ruasi)homo'eneous re'ions. 6he optimal

partitionin' al'orithm is C+)complete. 6o

do this partition, a natural approach could

consider the above channel as the startin'

point for the ima'e partitionin', desi'nin' a

pi4el clusterin' al'orithm ?hich minimi5es

the loss of I.

6his process can be interpreted in the fol)

lo?in' ?ay9 the choice of the partition

?hich ma4imi5es the I increases the

chances of 'uessin' the intensity of a pi4el

chosen randomly from the no?led'e of the

re'ion it pertains to.

6he al'orithm of this method not

'enerates a partitionin' tree for a

'iven probability of error3e.


43/47

/;/J; 'on"lsion6he many advanta'es of our approach include that9 it sho?s sensitivity to habitat features

at the community level: it is ine4pensive, simple and accessible to all: it allo?s for the

monitorin' of forests at multiple scene scales in both space and time: it can provide additional

information of ecolo'ical relevance to sensor net?ors: it can be added to the ba' of samplin'

devices of most field protocols: and the use of structural comple4ity as an 3E is practically and

theoretically attractive. 6he list of disadvanta'es includes that9 it is a methodolo'ical approach in

its infancy that ?ill reRuire confirmation from other systems: photo'raphic settin's ?ill have to

be fully standardi5ed and their calibration addressed Se.'., ima'e resolution versus e4tentT: and

I& estimates ?ill need to be correlated to other measures of plant architecturesuch as canopy

closure, canopy cover and vertical structureto further interpret themechanisms beyond our

present definition of structuralcomple4ity in an ima'e. onitorin' forest dynamics at a hi'h

resolution in spaceand time offers the possibility of discernin' the ecolo'icalsi'nature of these

systems. -i'nature variations could provideinformation on the inte'rity and stability of

ecolo'icalprocesses, both 'lobally and locally. 6he detection of localdisturbances assessed by a

chan'e in structural comple4itycould help alert ecolo'ists and 'uide their actions to sites?here

the inte'rity is threatened. Ny revisitin' the same sites?ee after ?ee one Ruicly reali5es ho?

dynamic anecosystem may be. +hototropism, floodin' events, sprin'and fall phenolo'y, 'ro?th

and senescence, flo?erin' time,'ra5in' and disease perturbations, fallin' trees, and 'apdynamicsare some of the many processes that structure theforest habitat on a relatively short temporal

?indo?. A holisticapproach capable of inte'ratin' these processes in time and space ?ould

certainly benefit scientists and decision maers.


44/47

2i3lio$rafie:

/; Hon'hai u, -tefan Qinler, P IA&3 (E+/3@I6 ACD -+A6IA/ICFE%A6IECPP S 3(3 Department, 8niversity of Illinois at 8rbana)(hampai'n, 8-A

Advanced Di'ital -ciences (enter SAD-(T, 8niversity of Illinois at 8rbana)(hampai'n,

-in'apore !"##T.http9vinta'e.?inlerbros.net+ublicationsRome4!"#;si.pdf

-; +erreira Da -ilva, incent (ourboulay, +ascal 3straillier PIA&3 (E+/3@I63A-8%3 NA-3D EC I-8A/ A663C6IECPPSatthieu. I333 International

(onference on Ima'e +rocessin' ) I(I+, -ep !"##, Nru4elles, Nel'ium. to be published,

!"##T.

https9hal.archives)ouvertes.frhal)""$#LL!


45/47

; (hihman . , Nondaro ., Danilova . , &olu5ina A. , -helepin . P(omple4ity ofima'es9 e4perimental and computational estimates comparedS+avlov Institute of

+hysiolo'y, %ussian Academy of -ciences !"#!T.

http9???.ncbi.nlm.nih.'ovpubmed!;"B**

K; ario I'nacio (hacn ur'ua, Alma Delia (orral -en5 and %afael -andoval%odr'ue5 (hihuahua PA Fu55y Approach on Ima'e (omple4ity easurePPSInstitute of

6echnolo'y, D-+ ision /aboratory !""BT.http9???.scielo.or'.m4pdfcysv#"n;v#"n;a$.pdf

/.; eaceslav /. +eru PDetermination of the ima'e comple4ity feature patternreco'nitionPPS!"";T.

http9???.math.mdnrofdo?nloads.phpZfile^filescsmv##)n;v##)n;)Spp!;#)!*LT.pdf

//; %oRue arin, 3va Enaindia, Alberto Nu'arin P(urrent 6opics in Artificial Intelli'encePPS##th (onference of the -panish Association for Artificial Intelli'ence !""


46/47

/H; 8niversitatea PNabes)NolyaiPP, Faculutatea si Informatica P+relucrarea -emanlelor IIPPhttp9???.cs.ubbclu.rojperNC+rel_Im'!"II.pdf

/J; Ioan Ispas P odelare 7i modele matematice =n recunoa7terea obiectelor 7i clasificareaautomat0 a ima'inilorPP S(atedra de atematic0)Informatic0, 8niversitatea +etru aior,

6xr'u ure7T.

http 9 ??? .upm .ro facultati _ departamente stiinte _ litere conferinte situl _ inte'rare _ europ

eana /ucrari Ispas .pdf

/0; (onstantin 3%6AC P +%3/8(%A%3A -I ACA/IqA IA&ICI/E% P!""#.http9ima'.pub.rorocursuriarchivecarte_pai.pdf

/; V z{|z}zz~( {zz https 9 ru .?iipedia .or' ?ii D "B A D " N3 D " NN D " N( D " N3 D " N ; D " N3 D #>" D " N3

D " N ! D #># D " NA D " N " D #> F _ D #># D " NN D " N3

D " N $ D " ND D " N3 D #># D #>! D #> (

/; 2.[. , .. 2G}GG~ P zz{z}G JG {zz|GJ zG |{z}z |G P.http 9 i5vestia .asu .ru !"#;#)# info ) comp 6heCe?sEfA-8 )!"#;)#)#) info ) comp )"B.pdf

/K; . . {z~, . [. z P zGG {zz zG z|z~|GJ ~ PS z{G| GG z| |. [. [.

G~ U[, z~, Uz z{ ~ Gz{{G} #


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--; \]^_` ?_ bZ[ z { ~Gz~z}z z~ z|{z}z {zzG ~G{J zGz~ z| {G zGPP

https 9 habrahabr .ru post !!BL

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