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+ *($4fflH4F4MRN4*:=.
- ~HHH -iH'- -- - H .;HHH - fW -* -.=+-=~
-H H H -- 4 I- .4 H44.l.- -- +H9TS. - HIB04OPR IF*W=+MH
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+~H H4H 4- + *-- HHHH+- -. H Htfl~fH -+fflfliH+ @Hf*=.
HHHH4~iItH-4-+- -~+NHHHH -+HHW~hHffliHfflHNH+: -0mmm-=++NmN*+
~- HHhH 4H -- *: t =
-,~~~~~~~~~~H~E + i4 H'.'- -!-H -tit4*±++4~~++*F.9ffl*S *Sj-HH---~fHf4HIHHHH*-- ~+---.H H~tH~i .. 3~~N--H U +
+HH *F FIHHft *,-- -*-N Nf4F ffi +H f-'-.- i* :.- I .-+i~--;.HHHH-i-- 44HHH~*4+- :-,~r~4--- ~*H4~0,i4-- *RfiqIi$+: ++
+..~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*
HM----N-HHf. kHMN. Hf.4H4 - NMI- -+ HWHOWflH-WH+O W-=* - A. W*+,#.+jC4 + N -N
*~~,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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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.
S
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