AO-A094 281 SOUTHEASTERN MASSACHUSETTS UNIV NORTH DARTMOUTH DEPT -- ETC F/6 12/1SOME EXPERIMENTAL RESULTS ON LINEAR ESTIMATION FOR IMAGE ANALYS--ETC(U)JAN 81 C H CHEN R WU, C YEN NOOI-g79-C-O49

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Contract Number N00014-7-C-0494SMU-EE-TR-81-4January 28, 1981

SOME EXPERIMENTAL RESULTS ONLINEAR ESTIMATION FOR IMAGE ANALYSIS*

oc

BYC. H. ChenRong-Hwang WuChihsung Yen

Department of Electrical Engineeringv Southeastern Massachusetts University

North Dartmouth, Massachusetts 02747

II *The support of the Statistics and Probability Program of-- the Office of Naval Research on this work is gratefullyacknowledged.

N8

81 A2 29 0 32

SOME EXPERIMENTAL RESULTS ONLINEAR ESTIMATION FOR IMAGE P:ALYSIS

C. H. ChenRong-Hwang WuChihsung Yen

1. Introduction

In a previous report (ref. 1), a critical comparison is made

on the various statistical image segmentation techniques. Linear

estimation plays an important role in image segmentation which may

be considered as an estimation problem. Included in the linear

estimation methods considered in Ref. 1 are the parameter estimation

for ARMA models, Fisher's linear discriminant, maximum likelihood

and maximum a posteriori estimations for the regions, boundary

estimation via split-and-merge algorithm, and decision-directed

estimation using conditional population-mixture model. In this

report, experimental results are presented for a textured subimage

on the segmentation by maximum likelihood estimation, maximum a

posteriori estimation, and Fisher's linear discrimant. Among the

three methods, the maximum a posteriori estimation performs the

best with 3.96% segmentation error, the Fisher's linear discrimi-

nant is a close second with 4.6% segmentation error and requires

slightly more computation. The performance of the maximum likeli-

hood estimation is much worse even though it requires less computa-

tion than the other two methods.

2. Construction of the Textured Subimage

The construction of the 64x64 textured subimage was kindly

described by Dr. Charles W. Therrien of MIT Lincoln Laboratory who

employed the textured subimage in his study of the linear filtering

-2-

models for image segmentation and classification (Ref. 2). The

desired subimage is constructed from subimages 5 and 9 (counting

from upper left to lower right horizontally) of image B2568-38 of

USC data base by using the following procedure. For each point

(n,m), the following expressions are evaluated.

Bl(n, m) = (n-36) + (m-40)2/50

B2(n, m) = ((n-48)/8)2 + ((m-20)/12)2 - 1

Then if

B .O or B2 -0, take point from subimage 5

if B1>O and B2>O, take point from subimage 9

This results in a new subimage with the texture from subimage 5

above the parabolic boundary and within the elipse and the texture

from subimage 9 everywhere else. The texture from subimage 5 is

denoted as type 1 while the texture from subimage 9 is denoted as

type 2. Thus the segmentation is to classify for each pixel whether

it is from the type 1 or from the type 2 texture. For the results

in Ref. 2, the filters were 4x4 causal (first quadrant) computed by

solving two-dimensional normal equations with an estimated correla-

tion function. Transition probabilities computed in the paper

made use of 5x5 neighborhoods.

By using the lineprinter display with 16 levels, Fig. la, lb

and ic show respectively the original subimages 5, 9 and the cons-

r tructed subimage. Two-level displays of these subimages from the

Tektronix 4010 terminal are shown in Fig. 2. Fig. 3 shows the

ideal segmentation as defined by the parabola and elipse in the

above quations. 0 :,. " r

0 64

0

13Ias

t ito k r "i. vor' '' '"t I-,qr 'r .''i ti'r titnt -- W '-tv~W 'f'~' HH=.~-~ ~ *IFd4' T. 4f-K*WNFiM GIHM: *

H' ----HN~I -Hf HN*- Hr- --- HHC WW- -f: N-IHH* -HMIU N:

4f- *f- HH-HHH --. H; H4WHH~.WN- NHN*eN.

HHHH4 H41fH I-C4N'~H-*HH-+N~-NI*~M+HHJ IdE~++E-++ i6418H++flq'-HM ~'F4--+IHNHN-'-4H+HW -+MHWNH-=4~U' H~H 4P*~Hf- -+ A.NHW'*W- H-f,--NN~IHH*-~H-H +H~f~H~--=rHN-W~-H* --- e"~ -=H+NN-~~~--f-.H+NH~~~Y' H HW--N'.HM*'-W*.NH-** *'-N-H:--H-~'-Hf--H--+HHH- :~4 +~~fiJHW~NSN*H~tO: 1H

~ f- H-'.*Hi4~.,--.'~H~~*..f5j+

'-; ~H-H---;H-Hf-HHHEf~.~-H4 '-H-k.NHN- - -f4--f~HH- H=S~I~ H -- ' W~h --'H+ *H+-W I-H 41.1M W.4

t-I -- *. NtH HtH'-'H H 0*H-7:NW -- : IWF-W H+N-*Hi.-+ Hh'* " 4-+-~ NH '- NF f4 - HH+ H *H ;H+M':,k-l 49*. H *HEu

- Hg l -~-- H i -+ I H . -N M H + (0- H - N ' : H - 4" A wU- -* H 4

Subimage 5 _' -~- ~ H--MHWW-Hh6**.=~N~:R~

HM MRH+H'..-NH ,H-HH- - HH~t-.-; HW-fH'W H *+HH:

H HO4:H R ~--H- +-HN-HNH-.MN.fhMH**

* IH-H*HHH -~H--±-HH H'+ +J~. + -H -M *-* 'tS+!FNSM+W~ -ffH -.- ' HN -JHH i +N --- + '.-HWQ-- ,tH

+~.-t>. +v-H-HN+-.*~ 4~'4~~' +NH- H4RA ff+I'J*H~R~u

-,~~~H -NW.I-. HHN+-'- H +.-i~~ +*i NHI1

HNHW-N -~d') N -~--~.-~---*H-- Hf--8-I- +'4-=iH H : H-;=-*:SHNN -

hi-H~'HNNHH+HI-H *M4~ewm ON. NH:H*-*~m*8+M N-+H O wl4-~-IfRH- HHPN+N'.~-~ .N -1 N4:H NN WH4IN-- . +- "+.

r I-HH H HM W)(SIC~--H-;*-H+-d~~.- f N W S+ iH 'H- -HIN. I-+~ --. ~fH4N:'.dH"N.F-WBW*1f-+*+ +N

+ - '"' HN------HP -!-.HfH- HH4M+-.r' N .WI-.N- f*=

-- NM -- H - -

-H$HHff4H-. I--: .6HH1H+:: HH .-.+: -'-=H J-= . **I

+*-HH4kHH*4--- ,4HHH*- - -' H ~ ~ J~H4 *.~~' U!

+F+4HHFRHlh+h4HR' -TH4MHHHO n:,S-7,-Li~'* +-: -WN WM -- M

-HHHHI* *4;-7;- 1tMHH- H *-4'.~ H -

HH H' P; HH4H0'H.'-- -- HHH':- .ifHH+.HIR -4+*4WMR

H H H HH-:-4*H H*- H- VJ..= +~ +.-' - N4Hfl- + ,HIIH*- +.*~* * 4

- I-HRkI-*- p-~ -H*, - -4r -ifHq fft4 N-

Fig. lb HHP

MIRH

-f.fIHH4:-*Hflfl *FR-NWM lf--+

--- Hl~H-- -d 44 H RH-r-- -HHHh4H4HH*+*: -4F *HW4.H~:.,H-4~tHH~- - 4'- - + 4 --- HHFHH4HH +-: - FMH+ *($4fflH4F4MRN4*:=.

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~- HHhH 4H -- *: t =

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+HH *F FIHHft *,-- -*-N Nf4F ffi +H f-'-.- i* :.- I .-+i~--;.HHHH-i-- 44HHH~*4+- :-,~r~4--- ~*H4~0,i4-- *RfiqIi$+: ++

+..~f~L~~H~ I-i -- +-H'. -H~'- HHHI*'-- -±N~- 4~~j 4H =+4IilBI*--

--+ ".;H- +4+ -~. - NS -:

-2c- NHWNNM M'MHMW14-H. H-04 :k ,r4HHNH~-.H H ~ ~ **N. U*~Ja UM*

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*~~,IHHW :HN W 9H1O -d* S.+ =+S *Nf~ N--- N HH-*NHNU--~4H+H4 +H-HN---HM=MW+ H*4NU~ W+A+ 14

~F~I44--N+RH*W+~- -+ ~-~+~-.HNH-**--H M NWH*-**-*e -*H. NUS.

~ ~ ,~ ~ WF.M Sig

Fig. lc HHH~-~4-.H*HNi4HH*HHH"+~H~* *HH-SMConstructed- H~~~'H-ww .W H.--M?-~~H'..*I~AFUWq *+=HRH

Subirnage 0.H-- H+14 N+~ ;+'. :*H.*fNHH**P NNW~A-*.*U

NNHIHFRF*-- -4 W! NNN-04i- h.N4HH*HW-. *-N W*-+HWPIIF.=M +.=-Wff

HHHs-' '- 4+~,---4Hif':HFI*NH*--*-hHfa* :+NW+WHW.*H+: ;NH-* ***

MH~H .- 4* '-~H~*4H~,+ -I-d--.--NW,= * -HNNW. *Fd*HF&fl*H l

H HWWWP&-7

-~+-+- *'-~HNflH ~ -Htj~fl~ :* ~ H +~4NH-OPP+. * +.=FAS* *-*SRF

- ~-H~*-HHM~-- -+[ --+ MR--4~# POPP44~ +HkRf~HWCF* fflWN$iP

+ H~~fIH4HI-'* i~ - -:=HH+ *-HiH-*PN~.ifHH : - 0Vlf +AH4~fl~fi9H + -- 9- 1FiH +wm(4*

-~~ :fNH-4P f-4*=--+-

-HHk.- ---- HM; -- - HHi4.- H,~H~~ HHI~-=-+N

-,HHHhH-fI. -+UHN~.''-FH g1~- - .- W WII-* W.:HHHHJ- HIM=z*4-HN + INS H S

- 2d -

Fig. 2a Subimage 5 (above)in two-level display andhistogram of the subimage(right). There are 16levels in the originalimage.

UFig. 2b Subimage 9 (above)

in two -level display andhistogram of the subimage(right).

rFig. 2c Subimage constructed(two shown above) and thehistogram of the constructedimage (right).

"~ ~ . - - -" - - - -- . ... . i , , ,I ... . .. I I I.. -IlIIII II . . . . .- l

-3-

3. The Maximum Likelihood Estimation

By assuming a first-order autoregressive model, we followed

closely the mathematical development in Ref. 2. The maximum likeli-

hood estimate for the regions is obtained by the minimization of the

espression (10) in Ref. 2. Fig. 4 shows the maximum likelihood (ML)

segmentation obtained by assigning pixels to white and black texture

type 1 and 2 respectively. This result is very close to Fig. 2 (b)

of Ref. 2 which employed a higher order autoregressive model as

described in the previous section. Our first-order model made use

of our earlier work reported in Ref. 3.

4. The Maximum a Posteriori Estimation

The maximum a posteriori estimation which uses the combination

of the Markov chain with a Gaussian autoregressive process is to

minimize the expression (11) in Ref. 2. To begin with, the image

segmentation is considered as an assignment of the pixels to 0 and

1. Such an assignment for a point (n, m) is referred to as its

"::ttae". The state interdependence complicates the computation of

the estimate and an iterative procedure must be employed. The

transition probability that plays a major role in the maximum a

posteriori (MAP) segmentation is counted from the numbers of state

assignment of 0 and 1 for a 5x5 neighborhood. It is noted that

MAP segmentation starts with the ML segmentation result. For

clarity the sequence of lineprinter outputs is shown in Fig. 5.

They include the ML segmentation (Fig. 5a), Iteration 1 (Fig. 5b),

Iteration 4 (Fig. 5c), Iteration 7 (Fig. 5d) and Iteration 11 (Fig. 5e).

At Iteration 11, the sm.11 false region in the upper left corner

starts to disappear. It is noted that such false region was not

- *

Fig. 3 Ideal segmentation

Fig. 4 ML segmentation

~ Fig. 6a MAP segmentation with 14 iterations

~ Fig. 6b Smoothing of Fig. 6a

Note: Learning samples for ML and MAP segmentations arebased on 480 pixels selected from each type oftexture. The following are parameters computedfor each texture. Type 1 has mean of 108 andType 2 has mean of 81.

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-4 -

removed in Ref. 2 even after 16 iterations. At Iteration 14, the

Tektronix display is shown in Fig. 6a which is then smoothed by

using a procedure due to Sklansky with the result shown in Fig. 6b.

The percentage segmentation error of Fig. 6b is computed as 3.96%.

It is important to note that for all linear estimation methods

considered in this report, the supervised learning samples are

selected from regions of type 1 and type 2.

5. Fisher's Linear Discriminant

Our initial effort of using Fisher's linear discriminant

follows the procedure employed in Ref. 4. The features are the

gray levels of the pixels. Fig. 7a is a tabulation of the scattered

matrices for Type 1(target) and Type 2 (background) textures. The

two scatter matrices are nearly proportional by a factor of 3. The

Fisher's linear discriminant simply cannot classify the two types

of textures. A new feature selection procedure is used that extracts

three features x(i), i=1,2,3 for a 3x3 neighborhood

B C

D E F

G H I

by computing

X(1) = E

x(2) = (A+B+C+D+E+F+G+H+I)/9

x(3) = (B+D+F+H) - 4E

The learning samples are taken from a 24x28 area with each type of

texture region. With each 3x3 neighborhood represented by the

feature vector (x(1), x(2), x(3)), the two scatter matrices as

tabulated in Fig. 7b are quite different. The Fisher's linear

discriminant can then successfully segment the subimage as shown in

-A-

u t. A ' '' 1''! -'1 4' flF

. P ., ii.....I . . . ~ . *~*~

~7. *.y *'i.

* j...~ ~,-

'C I-.

ii

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

1.1? *':i1

*1 *~.

Fig. 7b

1 .1 -I. '~

I -.: -

-'7'..

K fI'.- .

-. 11 -I *N

~1 ---------------------------------- :-j-

-F

' -r- I X. * I -;

.1 ,.

- ka -

- *3

C-- -. aflJ -

- 4b -

Fig. 8a Segmentation by Fisher'slinear discriminant

Fig. Tb Sobel gradient operationon Fig. ba. (thresholdis ).162)

Fig. 8c Robor t. ui'idijnt operationon Fic. ta. (thresholdis i)

'4]h.4

-5-

Fig. Sa. The small false region, however, cannot be removed. The

results of Sobel and Robert gradient operations on Fig. 8a are

shown in Fig. 8b and Fig. 8c respectively. Due to the false

region, the percentage segmentation error of Fig. 6a is 4.64%. The

Fisher's linear discriminant requires slightly more computation time

because of the scatter matrix and feature projection computations.

As a concluding remark, it is interesting to note that while

both the maximum a posteriori estimation and the Fisher's linear

discriminant are feasible linear estimation methods for image analy-

sis, they employ quite different procedures for utilizing the statis-

tical cw,:, x, al information.

Acknowledgment We thank Dr. Therrien for helpful communications.

The results reported in Sections 3 and 4 are due to R. H. Wu and

the results reported in Section 5 are due to C. Yen.

References

1. C. H. Chen, "A comparative evaluation of statistical imagesegmentation techniques," Technical Report SMU-EE-TR-81-3,January 19bl.

2. C. W. Therrien, "Linear filtering models for texture classifica-tion and segmentation," 'roc. of the 5th International Conferenceon Pattern Recognition, pp. 1132-1135, Dec. 1980.

3. C. H. Chen and J. Chen, "A comparison of image enhancementtechniques," Technical Report SM1-EE-TR-8, Dec. 1979. Alsoappeared in the Iroc. of 1980 IEEE International Conference onCybernetics and Society.

4. C. H. Chen and C. Yen, "Statistical image segmentation usingFisher's linear discriminant," submitted for publication, 1980.

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