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.TXT 2 24 4 0 0
Cg a51.625000,51.625000,48
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Frequency Domain 
Steerable Pyramid Filter Design}
.TXT 3 -19 101 0 0
Cg a82.500000,82.500000,31
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Second derivative, k = 3 filter}
.TXT 0 29 5 0 0
Cg a48.625000,48.625000,21
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Ken Castleman 2/17/98}
.TXT 0 30 2 0 0
Cg a18.625000,18.625000,12
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard StPyr_03.MCD}
.EQN 5 -60 6 0 0
{0:N}NAME:16
.EQN 0 10 185 0 0
{0:a}NAME:{0:floor}NAME(({0:N}NAME)/(2))
.EQN 0 19 7 0 0
{0:i}NAME:0;{0:N}NAME
.EQN 0 14 8 0 0
{0:k}NAME:0;{0:N}NAME
.EQN 0 32 96 0 0
{0:j}NAME:\(-1)
.TXT 5 -76 162 0 0
Cg a85.625000,85.625000,67
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Define smooth-edged 
lowpass and highpass functions (raised cosine).}
.EQN 4 50 183 0 0
({0:\q}NAME)[({0:i}NAME,{0:k}NAME):{0:angle}NAME({0:i}NAME+0.001-{0:a}NAME,{0:k}NAME-{0:a}NAME)
.EQN 2 -52 22 0 0
{0:LP}NAME({0:x1}NAME,{0:x2}NAME,{0:x}NAME):{0:if}NAME({0:x}NAME<{0:x1}NAME,1,{1:if}NAME({0:x}NAME>{0:x2}NAME,0,\({0}0.5*(1+{0:cos}NAME({0:\p}NAME*(({0:x}NAME-{0:x1}NAME)/({0:x2}NAME-{0:x1}NAME)))))))
.EQN 2 78 157 0 0
{0:c}NAME:\((8)/(9))
.EQN 1 -26 184 0 0
({0:\r}NAME)[({0:i}NAME,{0:k}NAME):\((({0:i}NAME-{0:a}NAME))^(2)+(({0:k}NAME-{0:a}NAME))^(2))
.EQN 3 -52 28 0 0
{0:HP}NAME({0:x1}NAME,{0:x2}NAME,{0:x}NAME):{0:if}NAME({0:x}NAME<{0:x1}NAME,0,{1:if}NAME({0:x}NAME>{0:x2}NAME,1,\({0}0.5*(1-{0:cos}NAME({0:\p}NAME*(({0:x}NAME-{0:x1}NAME)/({0:x2}NAME-{0:x1}NAME)))))))
.EQN 1 52 111 0 0
{0:sign}NAME({0:x}NAME):{0:if}NAME({0:x}NAME<0,-1,1)
.EQN 0 18 130 0 0
{0:f1}NAME:0*{0:a}NAME
.EQN 0 8 131 0 0
{0:f2}NAME:(5)/(8)*{0:a}NAME
.TXT 5 -77 163 0 0
Cg a86.625000,86.625000,258
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Define the transfer 
functions of the one highpass, two lowpass and three bandpass filters.
\par The constants f1 and f2 control the steepness of the cutoffs, and 
c is a required fudge factor.\par The LP(f3,f4) filter is not 
required, but it makes the kernels smaller.}
.EQN 7 -1 133 0 0
({0:L0}NAME)[({0:i}NAME,{0:k}NAME):{0:LP}NAME({0:f2}NAME,{0:a}NAME,({0:\r}NAME)[({0:i}NAME,{0:k}NAME))
.EQN 0 21 144 0 0
({0:H0}NAME)[({0:i}NAME,{0:k}NAME):{0:HP}NAME({0:f2}NAME,{0:a}NAME,({0:\r}NAME)[({0:i}NAME,{0:k}NAME))
.EQN 0 22 132 0 0
({0:L1}NAME)[({0:i}NAME,{0:k}NAME):{0:LP}NAME({0:f1}NAME,({0:a}NAME)/(2),({0:\r}NAME)[({0:i}NAME,{0:k}NAME))
.EQN 0 22 228 0 0
{0:f3}NAME:(6)/(8)*{0:a}NAME
.EQN 0 10 229 0 0
{0:f4}NAME:1.2*{0:a}NAME
.EQN 4 -75 77 0 0
({0:B1}NAME)[({0:i}NAME,{0:k}NAME):{0:c}NAME*{0:LP}NAME({0:f3}NAME,{0:f4}NAME,({0:\r}NAME)[({0:i}NAME,{0:k}NAME))*{0:HP}NAME({0:f1}NAME,({0:a}NAME)/(2),({0:\r}NAME)[({0:i}NAME,{0:k}NAME))*({0:cos}NAME(({0:\q}NAME)[({0:i}NAME,{0:k}NAME)))^(2)*{0:sign}NAME(
{0:cos}NAME(({0:\q}NAME)[({0:i}NAME,{0:k}NAME)))
.EQN 5 0 78 0 0
({0:B2}NAME)[({0:i}NAME,{0:k}NAME):{0:c}NAME*{0:LP}NAME({0:f3}NAME,{0:f4}NAME,({0:\r}NAME)[({0:i}NAME,{0:k}NAME))*{0:HP}NAME({0:f1}NAME,({0:a}NAME)/(2),({0:\r}NAME)[({0:i}NAME,{0:k}NAME))*({0:cos}NAME(({0:\q}NAME)[({0:i}NAME,{0:k}NAME)-({0:\p}NAME)/(3)))^
(2)*{0:sign}NAME({0:cos}NAME(({0:\q}NAME)[({0:i}NAME,{0:k}NAME)-({0:\p}NAME)/(3)))
.EQN 5 0 103 0 0
({0:B3}NAME)[({0:i}NAME,{0:k}NAME):{0:c}NAME*{0:LP}NAME({0:f3}NAME,{0:f4}NAME,({0:\r}NAME)[({0:i}NAME,{0:k}NAME))*{0:HP}NAME({0:f1}NAME,({0:a}NAME)/(2),({0:\r}NAME)[({0:i}NAME,{0:k}NAME))*({0:cos}NAME(({0:\q}NAME)[({0:i}NAME,{0:k}NAME)-(2*{0:\p}NAME)/(3))
)^(2)*{0:sign}NAME({0:cos}NAME(({0:\q}NAME)[({0:i}NAME,{0:k}NAME)-(2*{0:\p}NAME)/(3)))
.TXT 6 0 165 0 0
Cg a87.625000,87.625000,61
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Compute functions to 
show that the constraints are satisfied.}
.EQN 4 0 68 0 0
({0:M1}NAME)[({0:i}NAME,{0:k}NAME):((({0:B1}NAME)[({0:i}NAME,{0:k}NAME)))^(2)+((({0:B2}NAME)[({0:i}NAME,{0:k}NAME)))^(2)+((({0:B3}NAME)[({0:i}NAME,{0:k}NAME)))^(2)
.EQN 0 29 145 0 0
({0:M2}NAME)[({0:i}NAME,{0:k}NAME):((({0:H0}NAME)[({0:i}NAME,{0:k}NAME)))^(2)+((({0:L0}NAME)[({0:i}NAME,{0:k}NAME)))^(2)
.EQN 0 22 147 0 0
({0:M2}NAME)[({0:N}NAME,{0:N}NAME):0
.EQN 0 10 152 0 0
({0:M3}NAME)[({0:i}NAME,{0:k}NAME):({0:M1}NAME)[({0:i}NAME,{0:k}NAME)+((({0:L1}NAME)[({0:i}NAME,{0:k}NAME)))^(2)
.TXT 6 -61 166 0 0
Cg a87.625000,87.625000,61
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Plot L{\fs16\dn 0}(u,v), H{
\fs16\dn 0}(u,v) and the sum of their squared magnitudes}
.EQN 3 0 159 0 0
{0:L0}NAME{1 4 3 215 45 1 21 20 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 6 0 16777215 0 90 3 NO-TITLE}{57}
2 5 21 21 0 1 1.5 7
1 5 4 3 0 4 1 0
3 1 2 0.1
.EQN 0 22 160 0 0
{0:H0}NAME{1 4 3 215 45 1 21 20 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 6 0 16777215 0 90 3 NO-TITLE}{57}
2 5 21 21 0 1 1.5 7
1 5 4 3 0 4 1 0
3 1 2 0.1
.EQN 0 22 161 0 0
{0:M2}NAME{1 4 3 215 45 1 21 20 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 6 0 16777215 0 90 3 NO-TITLE}{57}
2 5 21 21 0 1 1.5 7
1 5 4 3 0 4 1 0
3 1 2 0.1
.TXT 28 -44 167 0 0
Cg a87.625000,87.625000,78
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Plot the three steerable 
bandpass filters and their sum of squared magnitudes.}
.EQN 3 0 38 0 0
{0:B1}NAME{1 4 3 215 45 1 21 20 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 6 0 16777215 0 90 3 NO-TITLE}{57}
2 5 21 21 0 1 1.5 7
1 5 4 3 0 4 1 0
3 1 2 0.1
.EQN 0 22 39 0 0
{0:B2}NAME{1 4 3 215 45 1 21 20 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 6 0 16777215 0 90 3 NO-TITLE}{57}
2 5 21 21 0 1 1.5 7
1 5 4 3 0 4 1 0
3 1 2 0.1
.EQN 0 22 40 0 0
{0:B3}NAME{1 4 3 215 45 1 21 20 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 6 0 16777215 0 90 3 NO-TITLE}{57}
2 5 21 21 0 1 1.5 7
1 5 4 3 0 4 1 0
3 1 2 0.1
.EQN 0 22 88 0 0
{0:M1}NAME{1 4 3 215 45 1 21 20 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 6 0 16777215 0 90 3 NO-TITLE}{57}
2 5 21 21 0 1 1.5 7
1 5 4 3 0 4 1 0
3 1 2 0.1
.TXT 30 -66 168 0 0
Cg a87.625000,87.625000,90
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Plot L{\fs16\dn 1}(u,v) 
and the sum of its squared magnitude plus that of the three bandpass filters.}
.EQN 2 6 148 0 0
{0:L1}NAME{1 4 3 215 45 1 21 20 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 6 0 16777215 0 90 3 NO-TITLE}{57}
2 5 21 21 0 1 1.5 7
1 5 4 3 0 4 1 0
3 1 2 0.1
.EQN 0 22 149 0 0
{0:M1}NAME{1 4 3 215 45 1 21 20 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 6 0 16777215 0 90 3 NO-TITLE}{57}
2 5 21 21 0 1 1.5 7
1 5 4 3 0 4 1 0
3 1 2 0.1
.EQN 0 22 150 0 0
{0:M3}NAME{1 4 3 215 45 1 21 20 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 6 0 16777215 0 90 3 NO-TITLE}{57}
2 5 21 21 0 1 1.5 7
1 5 4 3 0 4 1 0
3 1 2 0.1
.EQN 27 -48 217 0 0
({0:M4}NAME)[({0:i}NAME,{0:k}NAME):((({0:H0}NAME)[({0:i}NAME,{0:k}NAME)))^(2)+((({0:L0}NAME)[({0:i}NAME,{0:k}NAME)))^(2)*(((({0:L1}NAME)[({0:i}NAME,{0:k}NAME)))^(2)+((({0:B1}NAME)[({0:i}NAME,{0:k}NAME)))^(2)+((({0:B2}NAME)[({0:i}NAME,{0:k}NAME)))^(2)+(((
{0:B3}NAME)[({0:i}NAME,{0:k}NAME)))^(2))
.EQN 0 51 233 0 0
({0:M5}NAME)[({0:i}NAME,{0:k}NAME):(((({0:L1}NAME)[({0:i}NAME,{0:k}NAME)))^(2)+((({0:B1}NAME)[({0:i}NAME,{0:k}NAME)))^(2)+((({0:B2}NAME)[({0:i}NAME,{0:k}NAME)))^(2)+((({0:B3}NAME)[({0:i}NAME,{0:k}NAME)))^(2))
.EQN 5 -51 219 0 0
({0:M4}NAME)[({0:N}NAME,{0:N}NAME):0
.EQN 0 10 220 0 0
({0:M5}NAME)[({0:N}NAME,{0:N}NAME):0
.TXT 0 13 221 0 0
Cg a63.625000,63.625000,34
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard M4 = 1 is Simoncelli's 
constraint.}
.EQN 3 -23 222 0 0
{0:M4}NAME{1 4 3 215 45 1 21 20 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 6 0 16777215 0 90 3 NO-TITLE}{57}
2 5 21 21 0 1 1.5 7
1 5 4 3 0 4 1 0
3 1 2 0.1
.EQN 0 23 223 0 0
{0:M5}NAME{1 4 3 215 45 1 21 20 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 6 0 16777215 0 90 3 NO-TITLE}{57}
2 5 21 21 0 1 1.5 7
1 5 4 3 0 4 1 0
3 1 2 0.1
.TXT 26 -24 158 0 0
Cg a85.500000,85.500000,55
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Compute the convolution 
kernels using the centered DFT.}
.EQN 5 24 93 0 0
({0:W}NAME)[({0:i}NAME,{0:k}NAME):(1)/({0:N}NAME+1)*{0:exp}NAME(-{0:j}NAME*2*{0:\p}NAME*({0:i}NAME-{0:a}NAME)*({0:k}NAME-{0:a}NAME)/({0:N}NAME+1))
.EQN 6 -25 135 0 0
{0:b1}NAME:{0:W}NAME*({0:j}NAME*{0:B1}NAME)*{0:W}NAME
.EQN 0 16 136 0 0
{0:b2}NAME:{0:W}NAME*({0:j}NAME*{0:B2}NAME)*{0:W}NAME
.EQN 0 17 137 0 0
{0:b3}NAME:{0:W}NAME*({0:j}NAME*{0:B3}NAME)*{0:W}NAME
.EQN 0 16 138 0 0
{0:h0}NAME:{0:W}NAME*{0:H0}NAME*{0:W}NAME
.EQN 0 14 139 0 0
{0:l0}NAME:{0:W}NAME*{0:L0}NAME*{0:W}NAME
.EQN 0 13 153 0 0
{0:l1}NAME:{0:W}NAME*{0:L1}NAME*{0:W}NAME
.TXT 4 -76 170 0 0
Cg a87.625000,87.625000,49
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Plot the three bandpass 
filter impulse responses.}
.EQN 2 7 95 0 0
{0:b1}NAME{1 4 3 215 45 1 20 18 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 6 0 16777215 0 90 3 NO-TITLE}{57}
2 5 21 21 0 1 1.5 7
1 5 4 3 0 4 1 0
3 1 2 0.1
.EQN 0 21 140 0 0
{0:b2}NAME{1 4 3 215 45 1 20 18 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 6 0 16777215 0 90 3 NO-TITLE}{57}
2 5 21 21 0 1 1.5 7
1 5 4 3 0 4 1 0
3 1 2 0.1
.EQN 0 21 141 0 0
{0:b3}NAME{1 4 3 215 45 1 20 18 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 6 0 16777215 0 90 3 NO-TITLE}{57}
2 5 21 21 0 1 1.5 7
1 5 4 3 0 4 1 0
3 1 2 0.1
.TXT 24 -48 171 0 0
Cg a86.625000,86.625000,63
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Plot the highpass and 
the two lowpass filter impulse responses.}
.EQN 2 6 142 0 0
{0:h0}NAME{1 4 0 215 45 1 20 18 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 6 0 16777215 0 90 3 NO-TITLE}{57}
.EQN 0 21 154 0 0
{0:l0}NAME{1 4 0 215 45 1 20 18 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 6 0 16777215 0 90 3 NO-TITLE}{57}
.EQN 0 21 155 0 0
{0:l1}NAME{1 4 0 215 45 1 20 18 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 0 1 1 1 0 -1 1 6 0 16777215 0 90 3 NO-TITLE}{57}
.TXT 27 -48 186 0 0
Cg a86.625000,86.625000,71
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Round off to four digits 
and scale the values for writing kernel files.}
.EQN 0 65 187 0 0
{0:b}NAME:1
.EQN 7 -66 188 0 0
({0:K3_L0}NAME)[({0:i}NAME,{0:k}NAME):({0:floor}NAME(10000*{0:Re}NAME(({0:l0}NAME)[({0:i}NAME,{0:k}NAME))+0.5))/({0:b}NAME)
.EQN 0 29 189 0 0
({0:K3_L1}NAME)[({0:i}NAME,{0:k}NAME):({0:floor}NAME(10000*{0:Re}NAME(({0:l1}NAME)[({0:i}NAME,{0:k}NAME))+0.5))/({0:b}NAME)
.EQN 0 29 190 0 0
({0:K3_H0}NAME)[({0:i}NAME,{0:k}NAME):({0:floor}NAME(10000*{0:Re}NAME(({0:h0}NAME)[({0:i}NAME,{0:k}NAME))+0.5))/({0:b}NAME)
.EQN 6 -58 191 0 0
({0:K3_B1}NAME)[({0:i}NAME,{0:k}NAME):({0:floor}NAME(10000*{0:Re}NAME(({0:b1}NAME)[({0:i}NAME,{0:k}NAME))+0.5))/({0:b}NAME)
.EQN 0 29 192 0 0
({0:K3_B2}NAME)[({0:i}NAME,{0:k}NAME):({0:floor}NAME(10000*{0:Re}NAME(({0:b2}NAME)[({0:i}NAME,{0:k}NAME))+0.5))/({0:b}NAME)
.EQN 0 29 210 0 0
({0:K3_B3}NAME)[({0:i}NAME,{0:k}NAME):({0:floor}NAME(10000*{0:Re}NAME(({0:b3}NAME)[({0:i}NAME,{0:k}NAME))+0.5))/({0:b}NAME)
.EQN 6 -58 193 0 0
{0:i}NAME${0:k}NAME$({0:K3_L0}NAME)[({0:i}NAME,{0:k}NAME)={0}?_n_u_l_l_
.EQN 0 19 194 0 0
{0:i}NAME${0:k}NAME$({0:K3_L1}NAME)[({0:i}NAME,{0:k}NAME)={0}?_n_u_l_l_
.EQN 0 21 195 0 0
{0:i}NAME${0:k}NAME$({0:K3_H0}NAME)[({0:i}NAME,{0:k}NAME)={0}?_n_u_l_l_
.EQN 1 24 212 0 0
{0:d}NAME:({2,2}ö{0:a}NAMEö{0:N}NAME+1ö{0:a}NAMEö{0:N}NAME+1)
.EQN 6 -64 196 0 0
{0:i}NAME${0:k}NAME$({0:K3_B1}NAME)[({0:i}NAME,{0:k}NAME)={0}?_n_u_l_l_
.EQN 0 19 197 0 0
{0:i}NAME${0:k}NAME$({0:K3_B2}NAME)[({0:i}NAME,{0:k}NAME)={0}?_n_u_l_l_
.EQN 0 21 211 0 0
{0:i}NAME${0:k}NAME$({0:K3_B3}NAME)[({0:i}NAME,{0:k}NAME)={0}?_n_u_l_l_
.EQN 0 24 213 0 0
{0:d}NAME={0}?_n_u_l_l_
.TXT 7 -64 198 0 0
Cg a85.625000,85.625000,103
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard Write kernel files for 
input to the WiT 2-D convolution operator (Use b = 10,000 for unscaled 
kernels).}
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue128;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard 1.  E. P. Simoncelli, 
and W. T. Freeman, "The Steerable Pyramid: A Flexible Architecture for 
Multi-Scale Derivative Computation," {\i Proc. ICIP-95}: 444-447, 
1995.\par 2. P. J. Burt, and E. H. Adelson, "The Laplacian Pyramid as 
a Compact Image Code," \par {\i IEEE Trans}. {\b C-31}:532-540, 1983.
\par 3. E. P. Simoncelli, W. T. Freeman, E. H. Adelson, and D. J. 
Heeger, "Shiftable Multiscale Transforms," \par {\i I}{\i EEE Trans}. {
\b IT-38}(2):587-607, 1992.\par }
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