AD-AE 569 SOUTHEASTERN MASSACHUSETTS UNIV NORTH DARTMOUTH DEPT

-- ETC F/6 9/2ON A TWO-DIMENSIONAL MAXIMUM ENTROPY SPECTRAL ESTIMATION METHOD--ETC(U)OCT al C H CHEN, G YOUNG NOO14-79-C-0O94

UNCLASSIFIED SMU-EE-TR-81"16

EEEEEEa/i!!!!lBjO

Technical ReportCMU - EE-T- C 1 -116Contract Jiumber Noo014-79-C-0494October 30, 1081

On a Two-DimensionalMaximum ETntropy SectralEstimation Method fcr theTexture-Image Analycic A 7 '1

F 1? "

I?

E DIet L,

byC. H. ChenGia-Kinh Young

Department of Electrical EngineeringSoutheastern M,'assachusett s UniversiyNorth Dartmouth, M assachusetts 02747

, ,- : "Salo; its

distribut~onu Is unlimited.I

*The support of the Statistics and Probability Program ofthe Office of Naval Research on this work is gratefullyacknowledged. The authors also would like to thank -r.N. A. Malik for his kind assistance and Mr. D. Abc,.':jdincfor valuable discussions.

811 12 126

1 rtroeductioin

;lthou it is generally reco'nizeu; that text.re i.7e2

con::ain statistical, spectral nnd structural domain information,

the use of spectral information alone can e cuite efficnve in

the texture-image analysis studies such as texture discrminaticr.

and segmentation. Bajcsy and Liberman [1] expressed the soower

spectrum in polar coordinates, then integrate over r and )k to

obtain the two one-dimensional functions. The location of peak-

in these functions indicates prominant texture coarseness and

directionality. Yleszka et. a!. [2]integrated the power spectrum

within 16 spatial frequency zones which were combinations of fc_-u-

1-octave frequency ranges and four 45 0 orientation sectors. Th, ;

also comn-uted eight "contrast" measures based on the cooccurrence

matrix, and obtained better discrimination than with the power

spectrum measures. Laws [3] computed a number of energy measur:-

by filtering the texture with sets of small linear operators,

then squaring and summing the output of each filter. He report,:

better discrimination with the energy than with the cooccurrenct.

measures.

A fundamental problem with the power spectrum analysis is .!"ic

computational accuracy and computational complexity. For textuv,

study, accurate power spectrum must be computed from the small

image segments. In this case, the two-dimensional Fourier analy;: -

cannot provide sufficient accuracy as the Fourier analysis is mc:'c

accurate with a large number of pixels. The two-dimensional

.................................•-,-.--

-2-

maximum entropy spectral analysis, however, is very suitable for

a small number of pixels. The computatiornal complexity has been

a drawback in using the two-dimensional maximum entropy spectral

estimation procedures. Recently, Lim and MValik [4, 5, 6] have

proposed an efficient iterative algorithm for the two-dimensional

maximum entropy power spectrum estimation suitable for the minic,,,:-

puter implementation. Their method is adapted and generalized f 1

use in our PDB 11/45 minicomputer for the texture-image analysis.

A three-dimensional graphics software is developed for the spec-

tral display at the different viewing positions. Extensive corn-

outer results on the spectral analysis of texture images are alse.

reported.

II. Two-Dimensional Power Spectrum Estimation

To obtain the power spectrum of a two-dimensional signal,

the direct method is to calculate the two-dimensional Fourier

transform of the autocorrelation function, i.e.:

x 1', w2) >-<

The following notations will be used in the report:

x(n 1 , n2): A 2-D random signal ;hose power spectrum we wishto estimate.

Rx (n , n 2): Autocorrelation function of x(n I , n2 )

Rx (n I , n2): An estimate of Rx (n I , n2 )

Px (wit w 2 ) Power spectrum of x(n I , n2 )

; (n1 , n2) Autocorrelation function whose power spectrum asi/2 x (wi t w 2 )

A A set of points (ni , n 2 ) for which R x (n, n2 ) i2known.

F z Discrete time Fourier Transform

F- Inverse discrete time Fourier Transform

From (1), it is important to note that the determination o'

the power spectral density entails complete knowledge of the

generally infinite extent autocorrelation function. For the

finite signal, this method is proved to have ucor' resolution du(

to the truncated and S:3mpled autocorrelation function set.

There are various techniques to estimate the power srectrn

for the one-dimensional signal (7]. One technique that has beer,

recognized as the best due to its high resolution is the M, aximur i

Entropy Method (MEM). The basic idea of this approach is to ex ri' -

polate the autocorrelation function of a random process by maxiiv. :

the entropy H of the corresponding probability density function:1T

H log Px(w) dw (2)w-7

where Px (w) is the power spectrum density. The characteristics

the maximum entropy method are equivalent to the autoregressive

signal modeling H which requires solving a set of linear equa>,i!,,

for the filter coefficients. This can be expressed as:A /

+ (w a e+ _j 2

and the filter coefficients ak are obtained by solving the normzlA A

equations, R (i) - ak R(i-k) (4)1< I

There are different algorithms proposed to find the solutions o"

the normal equations. The most efficient ones are the Levinscn

recursive algorithm and the Burg recursive algorithm [7]. But

for the two-dimensional case, the problem is different since thu.

normal equations become a highly nonlinear problem (4]. For thtu

general form of the two-dimensional case,

)7a. Rx(r-i, s-j) Rx (r, s)

(,)E8 for (r, s) E B

'I- "he set B consists of all points where the filter ,,ask has

non-zero values, and the power spectrum obtained from aij is givenAIIP~ (w1, w2) C

^ L(

n this case, we can see rron, (5) that the size of independent

values of Rx(n I , n,,) requires to solve the above ,ez of equations is1

v, eai:er than the size of the filter mask. For exaJiaile, Fig. 1(a)

,h, the autoregressive filter mask size 3s 3x-, a.,a Fig. 1(c)

'o..vs a larger size of independent values 1.(,, n ) jrequired to

,:-I-vi for a.. in Fig. 1(a) by equation (3>

(a) (1)i i

Fig. 1

CPfarly, the number of correlation points needed is greater than

,h? number of filter coefficients. Since the estimated power

sI", rum given by (6) is completely determined by ths coefficients

1lo1e, it does not posses enough degrees of freedom to solve for

Lte ,-pectrum. Therefore, the normal equations are not linear as

in the one-dimensional case. Many methods have been proposed to

-- . .. , . ..

.,x e'1d the Levinson' s and tne Burg's algorithms in two dimensio:±

iever, those algorithm- are not computationally attractive, an

ni)err is no ruarantee tbat a solution or an approxima~e solution can

.,e nctained. For ins-ca21ce, Burg 131 has proposed an iterative

;oi,.i on which requirer' the inversion of a mattix in each iteration

.. r ~ ne dimension of the matrix is of the order of the number of'

tie ,':iven autocorrelation points. No experimental results using

technique have yet been reported. Wernecke and D'Addario [14]

-v proposed a scheme in which an attempt is made to numerically

Jxi'iize the entropy. The maximization is done by continuously

adju-cing the power spectrum estimate and evaluating the expressicns

i'or ,he entropy and its gradient. The procedure is computationally

snot 7uaranteed to have a solution. Woods

xiresses the Maxim'im Entropy Method as a power series in the

frcqiiency domain and attempts to approximate the f4E PS estimate by1rurvating the power series expansion. Even though such an approach

i as ome computational advantages relative to others, the method is

restricted to ,he class of signals for which the power series expans *)n

it Pnessible.

Based on the reason that the closed form solution of the two-

imc sional ME method is hard to obtain, Lim and Malik developed a

,ri~ iterative algorithm, using adaptive filtering concept. This

i,'irithm can correctly estimate the true spectrum and is computa-

' ;,lly simple due to the utilization of Fast Fourier Transforn

The basic idea of this algorithm is on the no tion that tr.e

vi correlation points in region A is cons;istent an: th corrt_;-

iionditig coefficient sihoula be zero outside region A, and proceed

this iteration repeatedly until the optimal solution is obtained.ATha- is, given Rx(ni, N2 ) for (ni, n 2) E Addetermine Px (W w2)

Asuci Lhat Px (w, VI2 ) has the form

P (w1 w2 )

R xnP, n2) F Px(w, 1)] for (nj, n 2 ) A

A simple flowchart is shown in Fig. 2. We begin with some initial

O~t~mte of (nI , n2), obtain the corresponding correlation function,

corr "u the resulting correlation fanction for (nI , n 2) E A with ti.e

1no-...1- Rx (nI , n 2 ), obtain the corresponding ?\(n I , n2 ) from the*nrrect correlation function, and then replace the resulting X (nl,n )

;,lto a for (n1 , n2 ) 1 A. This completesone iteration and " he correct, a

,, (n1 , n2 ) is a new estimate of 2\ (n, n2

initial estimate of \ (n , n2)1 2Ry(ni , n2 ) =F-IfFf,,,,]]J

y 2 F [(n,correct Ry(nl, n 2 with Rx (n, n2) for (nl, n 2) A

>~(n 1 n 2 ) F=

L .C- N (nI , n 2 ) o for (nI , n2) A

Px(wig w2 ) = F [Ry (nI , n 2 )]

Fig. 2

~owe:,r, to prevent the zero crossing proble when taking -the inverse

of (nl, n)] and F [ Ry(nl, n2 )J due to the correction of

(n and 2 (n n2 ), and also to keep the correlation function

o e positive definite, they modify The procedure by linearly inter-

i~oluting some parameters to prevent divergence of the error and to

n'-la.s e the rate of convergence [4] . T2hs they have added some

rnnstraints to make the practical algorithm as shon.,,i in Fig. 3.

111. Examples

The following are some exampnles of using Lie and ",,alik's

Lo'itom which is implemented in our PDP 11/45 minicomputer. The

in r u signals are Two-dimensional autocorrelation function originated

:r : sinusoids buried in white noise, so that the correlation func-

,ien given has the form of

=(n , n2 ) In, + - a cos(wi r + w 2 n

2 is the white noise power, i2 the number of sinusoids,

S/72 is the power of the ith sinusoid, and wil and wi2 recresent cn

fireqency of the ith sinusoid. Fig. 4 - Fig. 6 are the cases for

L, 1, 3 sinusoids respectively, which correspond to the input data

-n rVL in TableS 1, 2, 3. From these results, we can see this iterativ

procedure can easily predict the true spectrum although the matrix

of mtocorrelation function is not very large.

However, in Lim and Malik's examples and the examples shown, in

1.s. 4-6, the expression used in the autocorrelation-function

.'° )",~lation is Eq. (8) which is an analytical form. This expression

1' t,.-sed on the consideration of continuous and infinite sinusoics.it is values are completely sym:ietric. For the practical two-

....................................c .

-7a -

GIVEN R.(n,.n,).w(nIng), DPT LENGTH *N. k =0.5

0. = 0. P. = 0. C. = 10'

INITIAL ESTIIAATE:R,(n.ng) =R.(0.O).d(n 1.nt)

1___ .6(ni.na)

R'(n,.ns) =IDFTIDA ,.4.) I

YES NO

IYNS CONINUE ITERATIONS

SAND 0,=o WITH LONGER DFT LENGTH

m+ kOP k * R= n,, I

I YES

~NOmlin DF11R (n, n) I

(m,n (DI IN R.(n,.n.) - R(,n) (,

=*(,n, R'(n,.n2 ) + (I- -,.)[R(n,.n,) - N(n,,n.)1 -(,,,.n%)

,\(n.O)IFT - If ____j

No

YES~~~~~ min s j\(,nr.~,n)

(mi DM (,n) + (I M-n)-.FT[, (.,,.nz)(n w(,,,

Fig. 3 A detailed flowchart of the Lim-Malik iterative algorithm for 2-DYE PSE implemiented in the report.UF.EE ASP T~cns- June 1981)

Qat e 1

o i

1 . 1.0 3

A G' a- (v 1/i , ..

13 1 0,-:5,, 3 6.0 1.0 0. C(, .1 10 1

-7 3 6.0 1.0 0.1 ,0.1 2" 10 3 "', 3 (.0 1.0 0.,o.1 110fl 32

?2 3 6.0 1.0 0.2, 0.2 10- 32 21

- __ ____ ____ ____L I

:lona a ;Enal, tis -3- noiogr n u. ,~rml'C

*i- th %%l form ula for the calcula t-*on,- u

oa ie C-4a -l

A~ i: soreu thLat 1h.- 1 L arees ion car. i:he real aoorela-m-o

TO o,."navec us1ecd it to gn eratea simulatec autocorrel!ation

:on o, trle t;.-omn'in l nusoicis, and t -ece_ SejattuOe u

2most te sam e as trima- calculatLed from r --a.(j

_n io Sectoo-n, ,,:will aresent trhe results ofth t'.'-ieriol

* 'xmom ntroy >enco 'ov.'er SCrec-tru:m !storr.aticn oft me ra tr

'has-'ta arel- t aken from the U. S. C.data base. Iesz

'O iar -s Uo-,~ k original textureS are s- "o in F, q

*>< ctu_-u..'il rc2ai, oear but times lar-er in Fig. ma - 1lo 1L!

itus corres pond in- est-im.ated snectrum shown in Fi. b -Fi.IC

o' c - Fig-. 16c and Fi. d -i.16d.

Fig. 6b shows tnat the p -iture has one main frequency componen:

v -ar (0.01, 0.22), one smrall do compjonent and one frequency a:

<,0.5) and other very sm.-all ripples. B3ecause of -the three-

!n11iusional display, .Fig. -1o and Fig-. Cd cannot accurately indicate

rh (0.01, 0.22) point, b ut. w-e can see the entire power spectrum

rivindis .riobution in Figr. 0Cc and Fi Ig. td. Fig-. 9b shows th.,at

-tt' iain frequency comooncrt is approxim,:ately equal To (0.02,0.126)

-todI ther smnall ripples. Comparing Figs. b and 9l, we can sea clearly

ldifference in pow,.,tr smectrum for different textures.

Fig. 10-I shows_ that the textureu contains a main f'requency

'-:,'nent around (G.C, 0.0) andl oth-er small amulit udo components.

1nas a rl~f c C&:uuco i. n a c

an uc c ca., zt-- ,- -,v

~e dc I : I:o cc 0 m li- 1d and -I.

are (.~z .I Uc ccc,- rr- C

-)n,:h, cc an cc e r 1e ci o,, e1, d~ ?-

enomnorc . ac.. ---a. c- -,-, o C- 2

I c, nree~e o arly 1 eric c Co-c . cc

I i zc. 14 - I e -o - L o~n b e -L 'r&daESa Eca c -c, ana i

-;.c cate ---l e --E5a! -,o'icr '

I~-cussion

or a real tvio-di.mersional siLccai, this aicoritha can accur2aIely

-1-t the main f--equency corp.-onentswitin a moder-,a-.e auccorrefunction (A-i) matr-x and short DFI- lengt. owve, orcc

f requency term,,s with smnaller amplitude, it mu.,st takIe a larzer

!;jaturix and lonp-er DFT length to discriminate, since the larger

A'> 'atrix will add more irnfcrmatior, about the si~rial. But this

<ilcake Loo muon comoutational time. Als-o the spec-tral -ea'-s th:-at

..~ cosecanot_ ~'eovedI' tis al;or-thm w,,ith- the m.Oderate

ACF matrix and DII- lengtuh. 10 understand and -to ilove'' ec

ral resolution caicability of the algorith m, %vc will1 an~alyze the

Krspectrum. h~e are now continuing the research by generat-ing a n

I'2-i~> soffiI al L ,. I I,'

.c~~~e- AR:o~ Lcu LyJa r L:~ ~j IIi

o~ rn =

:cor u c x e aCIC -- Ci eI L:, r

i''~ onal -' 'L I-\eIC.:.e C . li Y>c

CuCCI~- ;. C......iorn :or u i fe rL ~ * x

n:' :e s!-,ll im a e s e c nciSeQCCLL.

r> exiure discr- r%-ia:r a eI. e C:2CC :I.

u;.ency o-aa Cc. . > 1 C:.IICa : 2~~c:CJa~ul

'IiL al&aori1hn is rfrcsentlV -he eC~ CC aailat, e fuC 4l.CC c >

.1 -menta-uion. -lh C Luoe ~ n sz1cctral r, 3oluCIIc. %w li

h.2~Iis algori-lhm even more 1powerful for t1-he -tcxtuure-a -e

Bajcsy anid L.Lieb-er!nan, -fxtor c a d In as aC"Cbi(,muuter Grann'-ics anq 7aowessig vol ,t',1>.

,Jeszka, D. Lyeir and A. 'oc.oc," ~:.uuoi .Y a,xture fosuostr -ze'ra-iri c~a-Sj- &t, ~z~r on

.'.- stems,>.an anc. Cybernet'-os;, vol. T---, r Z-2z5 17.

aws, "I1ex-lrc enecDy in ,.(s,;Kvaiia an A.emiannual !c'c,-.i i' -,r-," TUniv e rS ' f. Cau 'e4 o(a

CT7pI Re-port pp.n1jl 17

S. LT and N. A. 'alit , "A ntr.e- -,i Gr n a aixi murr en-.roL~ ,Dv,,r eca1est '--aliar,

Li:.% and ". A. l 0-.tsc-' c-ineratv t2, 1, toe ai~m~ c 2. S

a-r 'th ,rorccedln-gs o: 7Tn e 2n d 1n tern a:c nc y~su: c nauter Aid ed 'Seisn Araly sis and Si s crim na-Lona., AuL. 191

Li a*,,d N. A. :aiik, "Some -craoer-,es a: woc-':-'o-!lximumen-zrory row~er spectrum estimnatlon, Fr o- AlP >rkshr-

-i --octal -s-imaion, Aug. 1931l, vol. 1

GChi-lder-s, -">,oder. Suectral A'nalysis", Ne,'YokI

.akhoal, 'Li*near rred ictiori: a tutorial reviev;," -Proc. zz1~. 63, c.561-zf, 1>f75.

1. Cr-r-enheim and P.-. caer,"Digital Sig-nal Processing,>glewood Cli'+-, N. ., -rnce al,1975.

.:.A. Cadzow and K. OFgino, "'lao) dimensional spectral es-Lrmatton,"Trans-. Accust. , Speech, Signal Processing, ASSP-29,(3),

396-0-1, June 11

1t. .V.."oads, "T-vo-dtmernstonal ",arkov spectral estimation," IEE'ans. Inform. 'Theory, vol. IT-22, PP. 552-559, Sept. 1976.

I .' V.~iocstno,*. . ±iesand A. L. Vickers,, "Texture atscrt-.r;,.attox. based upon: an assumed stochastic texture modiel," IEans. on Patternr.AnalySis and 7>achine Intelligernce, vol. PAM".-3,a.5, *557-53l, Sc.ld

3. .P. BurE, unpub'L!-lish.ed notes, 1974.

id .j. 'vornecke and L. R. D'Addario, "I,,aximu.m- entropy imag,,ericonstru..ction," IEEE T~rans. Comput. , vol. C-2b, pp. 351-364,Ajnil 1977.

inalysis," Asiron. Astrophysics, vol. cL4., u.~ 19?7.

10 .L. ,.arzetta, "Iodmninllinear T~reu~l*.--or: a';:--occrrela-; ion arrays, mini,:mum phase icreuiction error itrao':oefficient- array," hilL ran.L. Acoust. ,Se, J- -roce-sanr,,,ScSP-2",, 'op. ?25-3-, Dec. 1sQ

71 ic* AN

A.

AA

xidi,

Sau cdccj'n-- _ tthpcrsc

P,

Fiq Itc Irc-i~~in~ iv'IA. Ky , c c XtlL' ~~ \ ~ '1( *~~ U Lcn~: at u Ad& ~

Pro.,',~0.) uj-i

5(a)'U.cntc~x map of >

7, 4A

S, W7j-

1. ZIP,

5() 'I -, si n l d s)

Tl,7e(t._ -Qtat (Q 0

~~ ~' ~ ~ of viwut (.50.)

ZowxPao :

, IM 4 -

Yv '" y

A prVI

A's IUL

1t.6(c) or-cctcsc~c1 c:S 'a

9~~7.~~j4_it.~ ~ VP at(G. 5, 0.3

-L6

Fig. 7 The original test texture data(taken from USC data base ). Each dataformat is 64x64. The right one insecond row is reconstructed from the leftand ceniter pictures of the second row.

* * a

j~~ ~ ~ ~ L7VV --Li Lm

rict. 8 (a) The original test data. Fig. 8 (b) The contour map %,.ith .I d13=3 w thmain frequency around (0.01, 0.22).

'iv I

Fig. 8 (C) Thrxee-dimensional display at Fig. 8(d) Three-dimensional display a!-(0.0,0.0) point of view. (0.5,0.5) point of view.

- 18 -

Y-ig. 9(a) The original test data. Fig. 9(b) The contour map with AdB=3 withmain frequency around (0.02, 0.126).

Fig. 9 (c) Three-dimensional display at Fig. 9(d) Threc-dimensional display at(0.0,0.0) point of view. (0.5,0.5) point of view. 1I

-19

j 'u'VLP

,,rA-* ~*J *

"E' IE aIJrsrPj-

I *j DeDIFFPW;

.7F5

Fici. 10(c) Three-dimansional display at Fig. 10(d) Thrc-diensional display aL(0-0,0.0) point of view. (0. , 0. 5) point of vienw.

r 16_-5

riq.11() Th orginl tet dta.Fig. 11(b) The contour map with _IdB=1.0l~iq 11a) Te oiginl tst dta.with main frequencies around (0.0,0.0)

and (0.06,0.125).

Fig. 11 (c) Three-dimensional display at Fig. 11(Cd) Three-dimensional display at(0.0,0.0) point of view. (0.5,0.5) point of view.

-, ME1 -S OTUPMPU

:,o4 IFEIC

F f*12(a) The original test data. Fig. 12(b) The contour mao with .AdB=2 .with main freauency around (0.03,0.07).

Fiq. 12((e) Three-dimensional display at Fig. 12(d) Three-dimecnsionlal display at

(0.0,0.0) point of view. (0.5,0.5) point of view.

'-22

1. N

. S I

IT N. A .

Fi. 13(a) Thetedimdsioal ispa astute Fig. 13(CD) The-dle oournrk1 isplay at

(0.0,0.0) a ain jFig. (2a).5,0.5)aa frequncy arun (oiew..

IT

-23-

-.- 750. ......

......................... ... . .....

" 2L.4

1u. 5(a) The originial test data. ric-. 15(b) The contour map) with _'B=-3.with main frequcncy around (0.065,0.20).

---------------------------------.

Fig. 15(c) Three-dimensional display at Fig. 15(d) Three-dimensional display at(0.0,0.0) point of view. (0.5,0.5) point of view.

a d a , .0

wit mai frqunc arud0315

r,.-DK P.. PON OF VIW (0-0,0-0

nij.a. 16c The diesoa dipa at ig 1(d Theedienionl isla a

(0.0,0.0 a pon of view (0505 pon ofviw

Pic. 17(bI) hc-hnso1UiQ:(0.0,0.0) point of view%,.

-:-.*~~ Fig. 17(,~) Throo-diiansiona1 isia':ct(0.5,0.5) IxTh-t of vicv.

UnflaSjss fi ed* EW2RT OCCU'. ' v : ' > ;" - :""::. :''<-. ,i" ..f)

ON A TWO-DIMENSIONAL MAXIMUV. ENTROPY !'crrnica1 mortSPECTRAL ESTIMATION METHOD FOR THE TEXTURE-IMAGE ANALYSIS -

C. H. Chen NO0014-79-C-04-Gia-Kinh Young

G GAN, 7 l- : At : ',V Z ~ kz 4 *0 A

Electrical Engineering DepartmentSoutheastern Massachusetts University NR 042-422North Dartmouth, Mass. 02747 [

-'C - C F F ' Z~ N ~% V A 5 0 - . .

Statistics and Probability Program October 30, 1981Office of Naval Research, Code 436Arlington, Virginia 22217

Unclassified

n-. , - 7 o' V

DSJ.. ':' .N 5 '. "r: ;.Peor

'S~~ ~~ '_'Il-< r r- j

* ' _ -'-TY N A RY %Cni~ . n .,vr ad : .],r o tr~ rckr, r

Texture images, 2-D maximum entropy spectral analysis, spectral Ifeatures, texture discrimination, minicomputer implementation,fast Fourier transform, autocorrelation function matrix, powers ectrum, three-dimensional graphical display.

A , Or ,rf .1.. H! ....... r- .c-rsf 1 b.-y cr,

The two-dimensional maximum entropy spectral estimation methodproposed by Lim and Malik is studied and modified for spectralanalysis of texture images. The computational efficiency ofthe algorithm makes it feasible for minicomputer implementation,to extract effective spectral domain texture features fortexture discrimination. Extensive computer results arepresented.

1 A7. 7 'oN 2* ' ., S ,sc :-T Unclassified

-. ' -.--

C L S : r A '" : N C " G ' * . , Xf* ont e r . ,

IU