Image enhancement using a high sequency ordered hadamard transform filtering (HSHTF)

Comput. & Graphics Vol. 5, pp 2.~-29 © Pergamon Press Ltd.. 1980. Printed in Great Britain

0097.-g493/80t0301-11~23/$~2.00~0

IMAGE ENHANCEMENT USING A HIGH SEQUENCY ORDERED HADAMARD TRANSFORM FILTERING (HSHTF) HASSAN J. EGHBALI Dept. of Computer Science and Engineering, School of Engineering, Shiraz University, Shiraz, Iran

(Received 9 Januao" 1979) Abstract--The enhancement of image where the picture contains additive noise is considered. The procedure of high sequency ordered Hadamard transform filtering HSHTF is utilized to recursively improve the enhancement of the image. In each step the average error between reconstructed and original image is determined. The HSHTF was implemented to a generalized two dimensional Weiner filter to improve the quality of the reconstructed image. Examples are used to illustrate the effectiveness of the procedure,

1. INTRODUCTION

No imaging system will give images of perfect quality. The image quality can be improved by a process known as image enhancement. The problem here is a posteriori, where we have the image whose quality needs to be improved but we do not have any control over how the image was originally produced. There are four kinds of image enhancement techniques(l]: (1) mapping of intensity, (2) eye modeling, (3) sharpening the edges and (4) pseudo color. Mapping of the intensity is a nonlinear operation that takes one gray level into another. The eye modeling is a process that allows the computer to enhance an image for visual consumption by precompensating for the visual system(2]. Edge sharpening is a process that makes it easy for the viewer to distinguish the edges more clearly. Finally the presence of pseudo color will increase the effective viewing dynamic range of the actual gray color form. The spatial frequency mapping is an example of pseudo color enhancement(3]. Many works have been done in the area of enhancement techniques by means of inverse filtering with the assumptions that the system whose output is to be improved is linear and shift invariant. The literature on the area of image enhancement has been reviewed by Huang[4] and Huang, Schreiber aand Tretiak[5]. The restoration of images blurred by motion has been discussed by Slepian[6] and Shack(7]. An optimum linear filter using the least mean square error criterion when noise is presented has been drived by Helstrom[8]. Because of the complexity of the fidelity criterion, it is practically impossible to come up with an optimum restoration for a human observer. MacAdam(9] used a method which allows for the experimenter to search for the optimum restoration by using a digital computer in an iterative manner. Iet f(x, y) represent the original undistorted picture image. The picture f(x, y) is enhanced by performing the two dimensional convolution.

g(x,y)= f:=f: h(x'-x,y'-y)f(x,y)dxdy.

Here h(x', y'), often called the point spread function, is a two dimensional impulse response function with desired enhancement properties and g is the enhanced picture. In the case of a discrete system the picture is a matrix of pixe]s (picture elements) and eqn (1) is approximated by its equivalent discrete form /'4-1 N - I

g(u,v)= ~o J~-o-[(i,j)h(u-i,v-i).

(2)

Andrews(10] and Tretiak and Huang[ll] implemented the method of two dimensional fast fourier transform for the spatial frequency filtering. 2. HADAMARDTRANSFORMO F PICTURE .fix, y)

Let the array f(x, y) represent an original picture over an array of N 2 pixels. Then the two dimensional Hadamard transform F(u, u), of f(x, y) is given by eqn (3)

[F(u, v)] = [H(u, v)][f(x, y)][(H(u, v)]

(3)

where H[u, v] is a Hadamard matrix of order N[12]. The Hadamard matrix is a square array of plus and minus ones whose rows and columns are orthogonal to each other. The number of changes in the sign in each row of the Hadamard matrix is called "sequency". The sequency of the Hadamard matrix is equivalent to a rectangular wave ranging between ± 1. The Hadamard matrix merely performs the decomposition of a function by a set of rectangular waveforms. Figure 1 contains a Hadamard matrix of order N = 23 = 8. There are several equivalent matrices similar to the Hadamard matrix given in Fig. 1 which are of different ordering(13]. The properties of sequency ordered Hadamard transform is used here to enhance the picture f(x, y) with an additive noise intensity n(x, y). Since the Hadamard matrix is a symmetrical matrix, the inverse Hadamard transform can be accomplished by repeated forward transform. A fast algorithm known as fast Hadamard transform (FHT) can be used to calculate the Hadamard transform

(1) 23

24

H. J.

F.~,eAu

Sequencey +

+

+

+

-

+

+

+

+ +

+

+ +

+

+

+

+

-

-

+

+

+

0

+

-

+

-

7

+

+ +

4

3

+

+

+

+

+

-

+

Fig. I. H a d a m a r d

-

1

+

-

+

6

-

+

+

2

+

+

/(x, y) = p(x, y) + ,(x, y).

5

matrices or order N :

for spatial frequencies far f r o m the origin, and a relatively low magnitude for frequencies near the origin. Here a high sequency ordered Hadamard transform filtering concept is applied to the problem of image enhancement. Let /(x, y) represent the picture p(x, y) which is corrupted with an additive noise intensity n(x, y). Thus

2j

of a given function or array. The one dimensional Hadamard transform by brute force method requires N 2 operations, where an operation is either an addition or subtraction. The fast Hadamard transform requires N log2 N operations. In 1937, a fast Hadamard transform algorithm was developed by Yates[14]. In 1958, Good[15] described a matrix decomposition technique to perform the Hadamard transform with N log2 N operations. A fast algorithm for Hadamard transform is also described by Shanks[16]. The output coefficients of a typical fast Hadamard transform is called dyadic or paley ordering. To convert from dyadic to sequency ordering, the output coefficients must be decoded by using a gray code to binary decoder. It is possible, by suitable modification, to calculate the sequency..ordered (FHT) without the gray code decoding. One property of a fast matrix factorization is that at the output of the transformation all the coefficients are in bit-reverse order. Therefore it is necessary to reverse the bit of the output coefficients and order them in ascending index order. As Manz[17] suggested, the first step in modification is to take the input data and bit reverse them and arrange them in ascending index order. In the regular (F/IT) it is possible to transform the input data and then bit reverse the transform. 3. ltSHTF A digital computer can be used to perform the two dimensional (FLIT), retaining for outputting only those sequency terms which are to be passed by the filter. The feature that makes this approach feasible is the speed with which the transform can be computed. High spatial frequencies are introduced by the occurance of sharp edges in the original picture. Thus we would expect a low-pass filtering to remove sharp edges and hence produce blurred pictures. A high-pass filter is characterized by a spectrum having a relatively large magnitude

The block diagram of the enhancement operation is shown in Fig. 2. After (FLIT) transformation, the actual scene in the picture is reflected in the value of the weight (spectrum) F~.j. The coefficients F,..j tend to get smaller as the indices i and j increase. To see this, if/(x, y) represents the picture and h,.~(x,y) represents the Hadamard pattern function, then

F,.j = f f /(x, y)h,.j(x, y) dx dy

(5)

and the energy of the object in the picture is given by

E = f f (f(x, y))2 dx dy

(6)

E=F],o+F],z+F~.o+" +F~.j+..

(7)

or

Consequentially F~.j tends to zero as i, ] goes to infinity. Thus the terms with high indices can be ignored. The sequency term F(0, 0) is a measure of the average brightness of a picture. All Hadamard domain samples other than zero sequency samples range betweefi +_N2I/2 where I is the maximum intensity value of the picture. The output coefficients are in matrix form as shown in Fig. 3 for a sequency order N = 8. The (HSHTF) was done recursively in an eight-step algorithm indicated below: 1. Compute the two-dimensional (FLIT) of the recorded image f(x, y). Note that f(x, y) = p(x, y) + n(x, y). 2. Set the coefficients of the highest sequencies as shown in Fig. 3 to zero. 3. Compute the two-dimensional (FHT) of the matrix resulting from Step 2, which is the inverse filter restoration. 4. Calculate the difference and the average error between the original picture p(x, y) and the result of the picture obtained from Step 3.

g (x,y) filter

(,l)

restoration

p(x,y) n(x,y)

Fig. 2. Block diagram of HSHTF.

~ a t o r

Image enhancement using

a high sequency ordered

Hadamard transform filtering

(HSHTF)

25

Sequencies 0

1

2

3

4

5

6

7

0

F0'0 I

F0'I

F0,2

F0,3

F0,4

F0,5

F0,6

F0,7

1

F

1,0

F i,i

F 1,2

F 1,3

F1,4

FI,5

F1,6

FI,7

2

F2,0

F2,1

F2,2

F2,3

F2,4

F2,5

F2,6

F2,7

3

F3,0

F3,1

F3,2

F3,3

F3,4

F3,5

F3,6

F3,7

4

F4,0

F4,1

F4,2

F4,3

F4,4

F4,5

F4,6

F4,7

5

F5,0

FS,I

F5,2

F5,3

F5,4

F5,5

F5,6

F5,7

6

F6,0

F6,1

F6,2

F6,3

F6,4

F6,5

F6,6

F6,7

7

F7,0

F7,1

F7,2

F7,3

F7,4

F7,5

F7,6

F7,7

Fig. 3. Sequency ordered FHT spectrum.

5. Set the next highest sequency to zero. 6. Compute the inverse filter restoration. 7. Calculate the error. 8. Compare the average error obtained from Step 7 with the error calculated previously. If this error was greater than the previous one go to Step 5 otherwise restore the picture and stop.

where rs¢ and r# are dyadic crosscorrelation and autocorrelation functions of s and / respectively. The operator ® means the dyadic convolution. In the transform domain

R~ =RIH.

(9)

Thus 4. HSHTF APPLIED TO WEINER FILTI~

A generalized two dimensional Weiner filter is shown in the block diagram of Fig. 4, where fix, y) is a picture produced by a known image and an additive noise intensity n(x, y). H is the dyadic filter in the transform domain. Pichler[18] has shown that the impulse response function of the optimal filter should satisfy the dyadic convolution equation

r,f - rff® h = 0

(8)

Figure 5 shows the block diagram of (HSHTF) as applied to Weiner filter. 5. EXPERIMENTAL RESULTS

Two different images, one artifically created and one taken from the Landsat imagery data have been con-

g

p(x,y) Fig. 4. Generalized Weiner filter.

Sequency 1 >IRestoration~ Conparator~x filteringi

p(x,Y) n(x,y)

H

Fig. 5. Block diagram of HSHTF applied to Weiner filter.

'Y)

H. J. EGHBAL!

26

sidered here to illusterate the result of the (HSHTF) enhancement procedure. Figure 6 shows the gray scale of an artitically created image. A white Gaussian noise with zero mean and three different standard deviations were added to the original image to form the new images which are corrupted with an additive noise. Figure 7 shows the noisy image with noise standard deviation (o9 being 1.0. Figure 8 represents the result of the enhancement operation after 10 iterations. The average error between the original picture and the reconstructed image was 0.0144%. Figures 9 and 10 show the noisy image with o-= 2.0 and the reconstructed image, correspondingly. The calculated average error was 0.0146% after 21 iterations. Figure 11 shows the noisy image with o"= 5.0. The reconstructed image of Fig. tl after enhancement is shown in Fig. 12. The average error after 22 iterations was 0.0178%.

I

-i

qRffgggffgqffg~NBOBffRgffggg~gU~18 8~SROBRMMSfB~gRBMMMMM~MMIMB~M 8RMMMR~MRffXS~g~BNMNMMgM|MMMMM

4

r~ ..

58MMMg~R~f)~

6

8RR~gP~Rgu~ n~Rp~ffP~ggua

7

q to, 11 12 I.~J 1. I-

ggBOBN~RMBBMI~ RBOMBBqgM~IMIB~R

ggggSRRffgq~gg~ ~NSNNg~8~lS~e

~88g ~Bg~ 8Y~ ~RRM~EffRMM~

i~ 29 ?? 22

Rff~KK~Mug~

ffg~Rgggg~

24 ,"5 2~

NfMNM~MMNNIEMi~ RRRN~|MSMIMMM~M ~ffBNMMMMMMI~MMMM~ ggRNMMMMMMMMMM~ ~RggMSffMMMMM~MMM ~MffBN~MMMMMMM~ --gg~gffRffffffffffgRff~g

ffMlOOOM~ffMORON~ff MMgM~Nff~ffB~RRSMg PgEBEOP1L~g~UF~UUglSIIB~SKSOOBBS~ eOSOSO~flOuUFeuu~$O|lOg~OSOOODgg

3O 2I 32

Fig. 8. Enhancedimageof ~g. 7afterl0iteration. g

;~P~3K, mgPl)gF~Nqff~$gffgff$ffffggflNg

L :

2 7

td

l~

~gRg

~

I1 12 t~

1 k

~-

~

~

~

~

1-'.

[

17 I9

~t

"2

24 ??

B88gP~]~

~8~IFIIIlBI~MM "7 2" ?l

Fig.& Anartificiallyc~atedima~.

Fig. 9. Image+noise with ~ = L 0 .

2

t

OgR2N|)~I~vP~eeel~BI~I~RB2e9~

gg~ARARARRo~rt~sRROSDNM£gI~gIIOg ~MIIKBARD~KB~'I'Y~gRNMDOMNM~IIN~g6 OPKAOBRRa~q

4 6 7

12

~g~o ~-

14 1c

~ l ~

17

ggRgl~g~Mg~

MN~ --

l~l - ~

-~gBMMMgHNM~|~IMIM

~gBRRgg~Hlel~e

C 10 I] t? t~ t4 t =.

~ l ~ l - -

zz~

17 t~ I£ ?q II~{~ffR~ 22 23

2~

2~

gMlR~ffOffff~ffR~NvffffKff~?-

-~n~qoRe~eeeu~n ~fffffgff~fflqgglll~9

88~n~kPF ~kkP~fF~T

R~ff~EgR~ffqRMHSff~

2~ ~8FF~8£a~w~$~88~800888~888§

?!

flggffqffSffffBgffffgggUffggg~|~gS~gqHSM 22

Fig. 7. Aaificiallycreatedimage+noise with e = l . 0 .

Fig. 10. EnhancedimageofFig. 9after20iteration.

Image enhancement using a high sequency ordered Hadamard transform filtering (HSHTF) BRO~Bw g 8 - ~ 8 ~

euOgOg~tg~wRl~-llS

4

lie

~wug~WIl~811gele-e

6

-q80u

g

10

- l f f - B N f f ~ 8~ Off~lgff -~18

12 1~ 1~"

I-R -g ~81 W EB-

17

~P~Ke~IF{B

I

In

ff~lffff;D

g~gBe~g8

g~u~-

~

8

W

-ll~teff~l~ ~ -lellll

24 2~ 2~

l~l ~I~B~ ~I~W~B~P ~IB£ 8l

2°

-~gegPIIIF~

--

-ASMI$

{8~-

~N~N38~ I

u

!~-

-3II1-1IOe

~ ~ 8-~ ~ B

9 ---I-~$~-

21 22

gDg-il-i

8 ~eOWggBg-O e l g~

~lg -

--| gW~e -B N

8

N Og~-8

~egqg

~ffleffuffff-eN~BN~ ~w~-E~--~8

l

23

-

l

a

--E

B~B

£llllgl

I 2

|geHggElgggg

gglEffffff~N~g~

c

glg~lg~lllll

giJig;ReNffgffffffq~

gEg~gK~ge~pywk~ffff~glggff~ffp~N~

ggJgffffS~ffff~SRS~

fTlfuuWUgggBB~DD

l~ 14

WW~W wuSu

.... ....

1~

wWs~

....

21

22 23

'l:t,laMWuugogB~a

~www .... ~WUIIOIFEP~ w~u~B~PF~F~ WWWH~I~IFIBF ~KwWIIlg~FF~ ;~eOIIPIF~B~ F@~P~ISFFFF ~F~IIPIE~t

Illl liB] oOIe 11111

~lll~vuwIg~lN~gg 0811WWNWEIIO~NIg IIeIW~WWII|INNg8 81llNWUWllIO~Neg eNs~lmllff~l~BllI ~eOOIlff~ff~IOIe

ffeRffllJlff~eNIIIe

2r.

~F~eFmnF@~

26,

8BOIIIII~£~F

28 29 30 21

OIEIIIIIf~ .... 8~Be~effeff3N~N3 EBP~gEI~NWW~SffffffffffffuWWWWWWW ~RP~I3I~wWWWwWWN~BR~ffffSwWWWWWWw EPB~RIkwwWWwwwe8883ffffgwuWwuUuu R~p~3~|wWww~www~ff~ffff~ff~uuwwu~

7 (; I0 !I !2 12

15 lu 17 lL 1 ~. 2C 21 22

~

~

1

~

--

~

-

8~BPflt'~ffffBffeffNN8

al ~e

-F-

-

-

-

I

~

~

~ q ~

~W~T

[~--

~ -~-~-

~-~lll|~

~ 8 ~ ; P ~ --,~lBg~B3 ~ - ~ P E ~

Fig. 15 EnhancedimageofFigll4after3iteration.

m

~. . . . . . . .

S~@~%gqSP~

-Wt-~-~-IIW~

?2

1

~[~.-U~SI|~N~--

3g@ffOgOl@N)$IOII ....

Fig. 12. EnhancedimageofFig. llafler22iteration. ,"

~v--~B~B

~ll--~?~&~I~I~W~

--~-

22

~--8~

-':~-

~ggg|~gggggl.

IP I0 20

~

Fig. 14. L~ndsalim~erydala+noise with G=I.O.

egBB~[g~eqq~.~ORggggREff~ffgg888

wk~U~Me~ wuwwwwu~?

17

-

II

EIgRffKOE~I|WW~BeEIOBff~P~eN8

G lO

1~

~--

l~ueq ffg~w-~BBW 3 e l B l ~ g ~ f f ff~g 88 Y~8©lffoeg~ OeN3B

-

Fig II I m a g e + n o k e w i t h ~ = i O 2

27

....

~9~-

W~ W ~ I ~ - ~

~ m - ~

Fig. 13. Landsat imagery data.

Fig)6Landsatimage+noise with ~ = 2 . 0

-

28

H.J. EGHBALI

--:~4 r, 6 7

g 10 11 12 13 l~ l (, 17 lg 20 21 2Z ?3

--~--

-{P~EE-

L

MB~l~'lqPBBE~=w~ME~M~ME|MffMMBSBM

..... { I m l I i i t + + + +

?

T~eNPEre

,11 C++MStIIMMN+

- CE~O m-Elmlmm

--

WllMlilC.Miiffi-~ ~|~I~|8NNIM I m -IM~ISIM~M~

~' i~

zz~v--~r--

....

:+t:Z,~t~Ii~ ~ 8 ~8q8

W{--I-t~leMlOigO~I~ul~fflOIIiiMN$

lltI~ItEt£111gWltl~lwK~It~llI!

i~,

MK|uI~wF~EtMISIMMM=-IM8£ £M++#IIIIMMMII{{

I~ I~

IKMI£1

~-

~M~ffM-ffSMg M~EIONMMN

-Ill,MiNI

IIIIMMSM~I

.~M~KIE{F-~iiMI~Iill

$ ~MMSMM

i~ 2~ 21

-~-~ - -¢~

-~Itl ~I~

I~IO$SNN

£E ++~+m

16

u:E:

i~ 19

-~tM ~ l ~ I { - { {

.: M - g ~ M B P I : B E -

~--

~l

~

:

78 79

31

{

iIiM

MS|M~INI~

E~ ~

~

--t . . . . ~ ! -

-I-N

-~--

mt-m

{1*1~!m$$

~

t

2~

EPtMMMSIPS--

3O 3t 32

P~P~MMPMMM~INMMIMMM~NMMMMM~M~ ~MW~MP~MM~MMIM~NN~N~NKMMMMMM~

N

M-

+~+m

-

pp~M~Suk~p~KSMSK~K~K~N~M~

-

[~

PPM~Pa~

~8~SSK~MKD89q8

14

~q

l ~ 1c

~PMBMM~MMI}% ~M~MMMM~

--~B~MIMM~MMMEEE~ ~M~MMMM~MMM~

,

~Y~MM~SZBt~

~BBYEEB~BBBBE

9;

Y~MMSMMP~M~

--M~MM~MMM~MB~88

=N-g=N-mI-~ w-ff:l~

It

~I

MDmMNN

MMMM8

MMBIIMMM

£11~ltiU

BB~

MCMN~IINN 8 8M~MNM

N(l'III-

IBIMI -+-+M

~ £t

~INffNffffN~NMSMMM~ ~MNNffffNff~NNMNNMMM ~M~ff~ffMNNggffN~NM T?MMMMII~M~ONMMNMM T~M~NNff~NMMN~NqMN~

Fig. 20. Enhanced image of Fig. 9 using generalized Weiner filter.

Mg-~l~8

t~m-B+F-F*W~mBm+lI~-

:~PSNIMMMSI~g~8 ~=MMMM~MIgONMMMM~

g~ff~MMMM~M-~gBMS~g-PPPBMM~MMM--

.~ MB l l ~ f f - | - ~ l 8+M EM-,IMMffMII +M i + +~+I~+M lfl-8~@]

....

PPE~P~-P~P~B~PM~E-§g~M~PM~NT~ ggKM~M~M~P~f

tMIMMMMNM

Fig. 17. EnhancedimageofFig. 16after~iteration.

9 It

~ u

23

12

77 2P 2g 30 31

2,,

MM~--~--

wSIF{E~II~e~I|WMM--eMM~I~INMM

2q

13 14

~tEE~BP~B~UBH~Bg --flMEE~8~MMqS~M

88 IM

iMt-UMI£1m

Fig. 18. Landsatimage+noise with~=5.0. Fig. 21. Enhance image of Fig. 8 using HSHTF applied to Weiner filter.

lO

*l~!{~m-lg~fE~Mt-

17

++Wtt+t+to+++lSMtm--

20

I~wmr-r

"= ~6 27

-

28 2~

E~E~¢tMwtw

ff-

-ItIMitll

::EW+¢llll II~:llll

I'lI--IfI¢-IlSill

.-" - I -I{: - -~'-

(.-

([

-{

-ift{{~-~t-£1tlll --Wi-~{llgIIlll :-£w: "~, - l l l l l l l l -tt~I..-.lW-II

Fig. 19. Enhanced image of Fig. 18 after 10 iteration.

The Landsat imagery data shown in Fig. 13 has been considered to further illustrate the effect of the (HSHTF) enhancement. Figure 14 represents the Landsat image with mean zero and standard deviation of 1.0. Figure 15 represents the enhanced image of Fig. 14 after 3 iterations. The average error at this point was 0.0122%. Figure 16 shows the noisy image with o" = 2.0. After 7 iterations, the enhanced image is shown in Fig. 17 with the average error of 0.0176%. Figure 18 indicates the Landsat image plus noise with o" = 5.0. After l0 iterations the enhanced image is shown in Fig. 19. The calculated error for this enhanced image is 0.029%. The generalized Weiner filtering was implemented to the picture of Fig. 7. The result is shown in Fig. 20. When the processes shown in the block diagram of Fig. 5 were also implemented to the picture of Fig. 9, the result was an enhanced picture which was improved over using

Image enhancement using a high sequency ordered Hadamard transform filtering (HSHTF) the generalized Weiner filter alone as shown in Fig. 21. Note that this process takes 18 iterations. 6. CONCLUSION The sequency ordered properties of the Hadamard transform was used to develop an implementable technique for the image enhancement. A high sequency Hadamard transform filter was used to recursively enhance a noisy image. Its performance was tested through application to a number of examples. The (HSHTF) was implemented to the Weiner filter. The result was an improved enhanced picture. It is noteworthy to mention that the process of (HSHTF) is relatively fast. This is because of the fast method used to calculate the Hadamard transform.

REFERENCES

1. H. C. Andrews. Digital image restoration: A survey-IEEE computer society, 36--45 (May 1974). 2. T. G. Stockham, Jr.. Image processing in the context of visual model. Proc. IEEE 60(7), 828-842 (July 1972). 3. H. C. Andrews, A. G. Tescher and R. R. Kruger, Image processing by digital computer. IEEE spectrum 9('7), 20-32 (July 1972~. 4. T. S. Huang. Image enhancement: A review, opt-Electron.. Vol. 1, pp. 49-59 (1%9~.

29

5r T. S" Huang, W. F, Schreiber and O. J. Tretiak, Image processing. Proc. IEEE 59, 1586--1609(Nov. 1971). 6. D. Slepian. Restoration of photographs blurred by image motion. Bell system Tech. J. 46, 2353-2362 (Dec. 1%7). 7. R V. Shack, The influence of image motion on the photographic image. Proc. NASAIEQC seminar, sp-193, p.q. NASA publishers. Cambridge, Mass. (Dec. 1%8). 8. C W. Helstrom. Image evaluation by the method of least squares. J, Opt. Soc. Am. 57, 297-303 (Mar. 1%71. 9. D P, MacAdam. Digital image restoration by constrained deconvolution. 3. Opt. Soc. Am, 60, 1617-1627 (Dec. 1970). 10. H. C. Andrews, Computer Techniques in Picture Processing. Academic Press. New York (1970), I I. O. J, Tretiak and T. S. Huang, Notes of Image Processing Workshop, Inst. Technology, Cambridge, Mass. (June 1970). 12. W. K. Pratt, J. Kane and H. C. Andrews, HadamardTransform Image Coding. Proc. IEEE, 57(1), 58--68 (Jan, 1968). 13. N. Ahmed and K. R. Kao, Orthogonal Transforms for Digital Signal Processing. Springer-Verlag, New York I1975). 14. F Yates, The design and analysis of factorial experiments. Harpened, Imperial Bureau of Soil Analysis ('1937). 15, 1. J, Good, The interaction algorithm and practical Fourier analysis. J. Roy Statist. Soc., Series B, Vol. 20, pp. 361-372 (1958~. 16. J. L. Shanks, Computation of Fast Walsh Fourier Transform. IEEE Trans. on Comp. Short Notes (May 1%9). 17. J. W. Manz, A sequency-ordered fast Walsh transform. IEEE Trans. on Audio and Elect., Vol. Au20, No. 3 (Aug. 1972~. 18. F. Pichler, Walsh Fourier synthesis of optimal filters. Archly Der Electrischen, Band 29, Ubentreguny (1970).