DIP Transform Properties

8/7/2019 DIP Transform Properties

1/11

f I l l " 1111

e de--

" - I

-2

'II

II I

. . . . -

I -

-


2/11

e r _ . _ L I I :each ofh etal

and unita~ DFf of real images exhibit

or O ~ , ~ - Ic-.""""""._ .... A ..,. . - - ' (5 .13)F rom th is , i" can be show n that ,,(k. I ) ' a s only NZindependent teal elements. FD re .. m p e, the' amp.es.b . s ha de d re gio n of .. 'lg . 5 .8 determl'_. the ,co m plete D Ro r un it an l OFT (see probl m 5.10.) .

B asis im :ges. The l lasis images are given by definnoD [see (5 .16) and(5.53~1~

_ ~ .....1i 1 W ' _ , I l . ~ , l: \I=~'I: ~I = = N 'l. .....,..... n " tT:o~dln ion ~Ii cui " con o luti a _ . h T~ fJoremll' me .... of lhe .' o-.dimensional. Clr .ul,~-ronv~lu'ion of to " r:rQS " th e p ~ du ct oj "heir . . .....-,l \vOadlmenslonal CIrcular convolution of to

u((m, . n ) 1 is ,defined a sBE ays him, n) and

Os


3/11

, For

II

lIb".n) 0

M-l ,N-1

'""

rl, A r r - v . h ( m , nl . (b] C reu r. G O nv ol ut 0 0him, nl wltl1 u (m l' n lover N X N reg10n.

,'pre .9 lWo-dimensional clrculu con olution.

."

JI )

ngure 5 .9 s haws themea I g o f c lr cu l f CODVO U 0 t' t e 5 m en anoerec xte ns i6D o f Ii (m.n) is con olved over: an Ii x reg ie 'tb u.(m,n~, 'Th two-d' eoslonaJ. DF1t of h~m - m ' I " - n ')c for fixed m ' , n ' is given by

IJ--

--5) a d u sin g tl ier e w e have used ~S.7.6). aking tJie D -

~receding esult, e . ob ain'DFE{ (m ) =


4/11

I

. aluatin the circLllllir c n oluticn f I i (tt" n) nd ff,(I ll , 1 1)C n be s ' 1 1 ' ith the id f .' 5.9 h (5~82)

P rf e inI--- .

his mean the twodlm sional hnar.onvo tit on 0 (5.1) 2no (/Vl log .) operation. ,lac erreu n or tlo 0' id ~ both id sc (5 77) .~the definition 0.Kro eek r prcdu t. e c bta n

(F )96= 0( )d u ~

.doubly cir ulant and g; is diago I[0] I.. ,~dld": t (m ,n)} ,

Eqn ..(5.83) can be v ,,~ iten as=M

ose lem nts Sire given byI --, here.

(5 .85)Ih t is a .0 bl blo k eireulan a ix i ia 0 l iz .. (. 0 .' .nsiDnalunitary D . From (5.84) and the f .t ransfor . p operry (pag 142.), e concludthat a doub]y block eirculan matrix can be dlagonalized in O(N1 log2 N ) opera-ion, Th eig nvalue of$gi 'en by the o- im e nsion D of h (m " ar tilme as opr t ng on the firs lu f " is b . use the elements oft .

f ir st c ol umn of 9Cace th elemen h (,n, n) m by I xic -r g.di .. lin '- convolution impliesr '11 uon 0 ~r. :pe d ~ d ou b' .I() -. [ o ed I rn Imp 'h t o

,BI ,Qc,k oep,ntz op rh ta n d au I block I 0 .block e ir In' p e . t ,d1me 'ns lo 'na l unital i) ' DFT.

The N x N co ine tran form rnatri ~ J (k )} 1[ , . ' 1 , ~ ,0 caned th '. d iscr ete c asin ,rallSfo,:;" (D' ), is d fin d as1 k=O 7Sn:=;c(k, Ii) _ . '~ ' IN.. .. '2 co m ( 2 1 z i)k. . N 2 - ) (56)'S'l


5/11

The basis vectors of th c 8 x 8 D" are shaw.n in Fig. 5 .1. igu e , .10 show , thecosine iran:: rmoft 1 images, an I i , ' e I own in ' i , g . 5. ,DE,ethat _an - t rans" mc,efficient a - ,' smaUb:t is m s t olf the curgy of th data i pack d in a fe' _ r a ,o s I o m 1 coeffieien IS ~

The 'w o-d im ensional eosine tr.ansform pair is obtained by substi uting,=,. = = , ' , u tS.ll) and ~5.12). e basi "mig of t e 8 x ,o-dlm=nsion I10' in S.2~ l , i l g , j i ' S . _ sh ow s e ,amp :r' . . t o ' f t , : - C o _ . I - _ o e

iges,

150

age : , can l in ', 'The inverseransiO' mati.o n is g iv en by

- I - ~ _ N~1 lk) ( ' k - ) " , , [ T ( l " + . l)k]

-.1 '}: - ~ '''\;' , ' 1 1 . 1 ' . cos ~~2-~, 0 ' " , ' ~ "-1 (S S9 1 ' . '

I. I', e cosi 'e trasform is rea, and orthogon Ihat i~: ; > -1= Cf"

SeI15 8


6/11

1 ( 0 , ) CD.,s~n'eIransfo'rm eHBmp~es,ofmonocnrOml l lm-ages;

. . T he cosin e tran sto rm ISa fa st (Ian form ... he 0 ine tram form a f a ector af Nilem ents ca _ .e cal _J ~ _ d in O l l( _ I O g 2 ) op atiens 18 In .. -poim IT:[19]. To ..ho thl . de ie a ne e uene u b .re rdermg t .0 andodd elements , e l f II(II) a s

u~.) = = f'( 2 1 1 )I i ( . - n - ) - I 1 ( 2 +)

Nlow, \ 1 & 1 split the uimmation t erm in (5~87) into even and odd terms and use(5 "9 1 ) 1 to obtain , ' 1 2 + ] .

v(k) ~ .. I t ) ! u (2 ) c o s [ - ' _ ~ ' ~ ~. . . . 1 0

t5 .9 t)

Ch : _ng ing the inde . 0 su mation m t h oIcolmbinin,g t rrns. . obtain

~v(k~=aCk) 2 : u(n) eo f1 l'(4 n+ l)k]

n::O t 2N I

C ' 1 2 ) - 1

+ E u(n~ O- 1-I

~mag __Tran


7/11

" , , - 1 1 ) ~ - j n' ] =:N t , {u (n )J ]

which pr v th _ p ... .- ['1t, (_.' I -) '~ Ii ted s it- ,U I " : , ' or I,n',,

~ ' ( 2 n )1 ; : U (2 n) ~ Re ..'. rll( I e ) V k ) e J 1 T k l 2 N J , f ' 2, .k /k= O

The odd data points ar ~obtam. d by noting that

Therefore, if we c,alculate the . -point in erse FFr of rhequenceo.(k)v(k) .'.p(j'R 'k/2 ), we can ,al. obtain th e in, e_ ' _ _ in 0 , ' Ig )o era 'on' . Dire, t.al .o rithm tha not r UI- -. a , in te rm ed ia te 1. . . . . t ha cemnlex ari thm tie: I " v id . , , ar " 1 . pos Ibl-, [1']. 1 1 11 - ' . om put .t ienal complei it y of. th .. direct as w II as the - based methods is about thesame,

'. 1lli. cosi ietr nsfo m l as excelh rr nerat com ae io n 0- hih , J t..data , . h is is due to th . fe llowm g p 0zertiec . 'T he l 'ia < 's vectors of th e cosine transform (th ,t is , rows of') a. he eigen-,ecton 0 ' t h ~ 1 sym _ etric t rr id !, , " a n - I( ma"ri:,.: - : ' f i n ' "1 a .......(1

(5~9.5)..1,,..' -,

-a o1"",

- - - - -- - - -\ .


8/11

110.. .

ft,gurf! s~3 Examples Q~ Hadamard Ira n fo rm s,un ges 0 tb .ad' . arc t anform ,8 e ShO'D n igs. 5. ,ndS,.2 . Eampls oft o -d ll1 1e nio na l Hd am a rd i ranstonns ot im s .... are shown i Fig t: '

2. The - I adamard n ansform I a fast transform Th o ne- dim en sio nal tran s-orma io n ,o 'S. lU)iI::an be im pl nen tec i O( og addi In a d IubtractionsSince the Hadam ard trans orm contain.s onl , alue no mu lt ip bc a-t ioncre re q ired in the tr. nsfann calculation. ,oeD '.I, the number 0add'ions or . traction re ired can b _educe om ...1o about ....1 0 1 : 2 This is due otbe act tha an b ."'. i.' 0- a p o duct of:n ..parse matric ..tb,at IS"

c _ ) H- dame O J Irn_ form af monochram

H=

1 10 01 t ~ I ~II! '" '

1_ - V '2 -----1 --1 00 1 0 1

'"I III 0 0

_ T - -1. . . . . . - (5. 1)

(5 .115

01 0 0.. 1

f ! ! ! . . . . . . .. ~'1 1


9/11

Since ~ co , ins ani w o nonzerc term s t th e t, :: H~ - fiB ~. ~H, n - le '82 '

1 1 ' 1 '7 #"4. (5.U1)

c an b e a cc om p lis he d I. operating n tim es on u, Due to . ' ! . s rueonly _ additions or subtraction -_are required each tim e -- operate on av e cto r, g iv in g a to ta l of Nn ~ /\, ~Dg2N additions OIJi subtractions.3 . The natural order o f the Hadam ard transform coefficient turns out to b

equal to th e bit re' ersed gray code epresentation of its qu ncy z ] thsequency s has the bina epres n arion b; b 11 - ] b, and ithe corre p ond . .ing g ray code is g il g'l I I : I , th en th e b it- re verse d representation glg 2 . . . 8~gives th e natura- order. Table 5.1 show s: the conversion of sequeney s ton atu ra l o rd er h, an . ice v ersa , fo r - = 8i ne ..rat,

g,=b.~)b~ I I t-, , ,r 1. g - b P i:1-h = gn l - .t I

andg.= h,.-k ib ; ; ; ; : , g ffib _n -1. . . f - ,. . .

(5-118)

f .119)6 , . = = C n

give t~e forward and reve'r:se conversion formulas for (he sequ - ncy and naturalordenng.4 . ~e H adam ard transfo rn ras good to /er - good energ com paction fo rhighly correlated image. Lei {u(n), 0 s 11! - I} be a s ta ti on ar y random

TABLE 5.1 Na ural' Ordering ~ersus Sequ neV Ol 'der i1ng of Hadama dTrans fo rm , Cae Jelents l O l r N = B(iray code of. o Sorrev r. e binaryepre ntation

" 8281 - hihth]Seque cy biryrepr senlatl'on

bJ lh 6.~ l a tu ' al orderh

Ar present onII I h 1 1 . Sequeneysu1234567

000O O J010011001 1 0 1110Ill]

0001000101 1 00 0 1101O l l111

00 01110111000 0 11 1 1 00101 - , I

n7J41Ii

158

_ ._ . - --. --~=--- - -

-


10/11

aee.

r

.. Ii. . .

I '", .

sequence w i . t . n autocorrelation l' en), 0 sn s N - I. T he '. " .~ f h- "._pected. e n e r ~ y :8cked in .th e first N t 1 J sequency ordered H a ~ : : : : d ~ a~s~o:'coefficients IS given b [23] _

(NI2I- I)

v'kD I,1:=0

1 J J A .' [HRH~]Q .:i= = _ . , ' . - ~ _ k, k ,1(5.120)

~S.121)where fJt are the fir.st Nf.1}sequency ordered elements D". N ote that the V,are sim pl~ the mean square v alues o f ' the transform coe ffic ie nts [HuI i ! ~ Thes i,g nific an 'c e o ~ 'th is result is ' th,at C . C~12 / ) depends on th e first 2 ! auto-correlations onl.~.. Fo~ j = = I"~the fractional energ,~ pa .cKed in l ' l l , e first Nl2s.eque,ncy. ordered coefficients will be (1 + rfl)/lr(O)t2 and d e p e n d, o n l~ u .p o nthe one-step correlation p ~ 1 \ ' (: I ) / r ( O ) . . 'fbus for p= 0.9S, 97.5% of the totale 'Defgy ' is conce,n' lra ted in hair ,of the transform coefficients. T :he result of(5.120) is useful in ca'iculati'ngthe e'nerg~ compaction effi~ienc' of the'Hadamard t'f8Dsfo~m.

ExampleS.]Conside~fthe covariance m atri ., R of (2 68) fo n N = 4~Using th e definition of Hz we'obtain

Sequencyo3

o. - ~D di 1 [8 RH-~ Jj. = I. lagona . - ' 2 , " , 2J - 4 " 1

2

5. . THE H~A TRANSFORMTh. - . ' . ' . . . 'ontinuous inter:val ~ , e [0 , 1], and forte Haan func tions h 1 c ( t x ) are define.d,on a C . be t~nuev decomposed asI t = = 0 . a,1 J, _ 1, . here N = 1 : l 1 n ' rphe I ,oteger Itcan e un ,. ' / I , ' ., ,n .. L IW, '. _ . ~ , . . .5 . 2 2 )k=W+q-l .


11/11

F_gtlfl 5 . , 1 , 4 , Haar transform of th e 2 5 6 1 X 2 5 6image S . h D in Fig. 5 . , 1 6 . .

I. 1 '1 - , e. la ' r transform is real and orthogo _,I. ThereforeHt=H- , -

5 '0'1 Y H , - " S " A , I N T TR'A"I - '5 '0 - ' R - -11 . III . - " ~ IL-H II: 111""_ ' - -- -, . .

~.' r-5 . 5 S la It rans 'o rm, D f. th ' 15 6)( . 1 6Un , shown 10 'Ig ..5.6a.

101 1 1 0 II 0 I I 0b I I -fA b Ia n n I~ I ' If -'.,n II _- ----'r--

- ~ - r ~ m - 2r- 0 ~ 1'.(1)-2I I I,~~--:-J----l-O-=1;:-1"---o I I l . f II I I....... n I ~ I b n a " II 'n Wg___ ,,"---, I ,--,-~---o II : 112 ) :I I 0 ! -It~5~126)

2. The -'aar trans nrm is a er.y fast transterm .. 1 , - " 0 an -, xl ctor It can beimplc'mented i I Q I ( . operatic ..

,II The 'basis vectors , of the Haar matFi . are s equeney o rde red .4 . The a at transform has poor cner.gy compaction for im ages.

TheN x Sla t t rans. foml matnce ' are defined b y, . be recursion

1III '" ,c;,v'2

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DIP Transform Properties