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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\fonttbl{\f0\fcharset0
\fnil Arial;}}\plain\cf1\fs20 \pard {\fs24\b Frequency Domain 
Steerable Pyramid Filter Design}}
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\fonttbl{\f0\fcharset0
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.TXT 0 31 5 0 0
Cg a48.625000,48.625000,21
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\fonttbl{\f0\fcharset0
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.TXT 0 36 315 0 0
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\fonttbl{\f0\fcharset0
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.EQN 5 -66 6 0 0
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\fonttbl{\f0\fcharset0
\fnil Arial;}}\plain\cf1\fs20 \pard Define lowpass and highpass 
functions (based on the raised cosine) with a smooth transition from 
x1 to x2.}
.EQN 6 0 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)))))))
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{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)))))))
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\fonttbl{\f0\fcharset0
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transfer functions for the highpass, bandpass and two lowpass filters.
\par The constants f1 and f2 control the steepness of the cutoffs; a 
is the folding frequency.\par The LP(f3,f4,f) filter is not required 
for invertibility, and using it makes the bandpass kernel larger.}}
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({0:B1}NAME)[({0:i}NAME,{0:k}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))
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\fonttbl{\f0\fcharset0
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\dn 0}(u,v), H{\fs16\dn 0}(u,v), L{\fs16\dn 1}(u,v), and B{\dn 1}(u,v)}
.EQN 3 0 346 0 0
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3 1 2 0.1
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3 1 2 0.1
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1 5 4 3 0 4 1 0
3 1 2 0.1
.EQN 0 22 349 0 0
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3 1 2 0.1
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\fonttbl{\f0\fcharset0
\fnil Arial;}}\plain\cf1\fs20 \pard Simoncelli's invertibility 
constraint is:}
.EQN 0 29 354 0 0
((({0:H}NAME)[(0)))^(2)+((({0:L}NAME)[(0)))^(2)*(((({0:L}NAME)[(1)))^(2)+((({0:B}NAME)[(1)))^(2))÷1
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\fonttbl{\f0\fcharset0
\fnil Arial;}}\plain\cf1\fs20 \pard Castleman's sufficient conditions are:}
.EQN 0 29 356 0 0
((({0:H}NAME)[(0)))^(2)+((({0:L}NAME)[(0)))^(2)÷1
.EQN 0 15 357 0 0
((({0:L}NAME)[(1)))^(2)+((({0:B}NAME)[(1)))^(2)÷1
.TXT 5 -44 358 0 0
Cg a87.625000,87.625000,64
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\fonttbl{\f0\fcharset0
\fnil Arial;}}\plain\cf1\fs20 \pard Compute functions to determine if 
the constraints are satisfied.}
.EQN 5 0 359 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)
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\fonttbl{\f0\fcharset0
\fnil Arial;}}\plain\cf1\fs20 \pard Plot the constraints.}
.TXT 0 41 377 0 0
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\fonttbl{\f0\fcharset0
\fnil Arial;}}\plain\cf1\fs20 \pard M4 = 1 is Simoncelli's constraint.}
.EQN 3 -41 374 0 0
{0:M2}NAME{1 4 3 215 45 1 21 20 0 1 1 1 4 -1 1 0 1 1 1 4 -1 1 0 1 0 0 2 0 1 6 0 16777215 0 90 2 NO-TITLE}{57}
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3 1 2 0.1
.EQN 0 22 375 0 0
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1 5 4 3 0 4 1 0
3 1 2 0.1
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3 1 2 0.1
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\fonttbl{\f0\fcharset0
\fnil Arial;}}\plain\cf1\fs20 \pard {\fs24\b The Convolution Kernels}}
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\fonttbl{\f0\fcharset0
\fnil Arial;}}\plain\cf1\fs20 \pard Compute the convolution kernels 
using the centered DFT.}
.EQN 1 55 329 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))
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{0:b1}NAME:{0:W}NAME*{0:B1}NAME*{0:W}NAME
.EQN 0 12 310 0 0
{0:l0}NAME:{0:W}NAME*{0:L0}NAME*{0:W}NAME
.EQN 0 13 311 0 0
{0:l1}NAME:{0:W}NAME*{0:L1}NAME*{0:W}NAME
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{0:h0}NAME:{0:W}NAME*{0:H0}NAME*{0:W}NAME
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\fonttbl{\f0\fcharset0
\fnil Arial;}}\plain\cf1\fs20 \pard Plot the bandpass, highpass and 
the two lowpass filter impulse responses.}
.EQN 3 0 227 0 0
{0:h0}NAME{1 4 3 215 45 1 34 29 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}
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1 5 4 3 0 4 1 0
3 1 2 0.1
.EQN 0 35 241 0 0
{0:l0}NAME{1 4 3 215 45 1 34 29 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}
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1 5 4 3 0 4 1 0
3 1 2 0.1
.EQN 33 -35 238 0 0
{0:l1}NAME{1 4 3 215 45 1 34 29 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}
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1 5 4 3 0 4 1 0
3 1 2 0.1
.EQN 0 35 242 0 0
{0:b1}NAME{1 4 3 215 45 1 34 29 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}
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\fonttbl{\f0\fcharset0
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{0:i}NAME${0:k}NAME$({0:l0}NAME)[({0:i}NAME,{0:k}NAME)={0}?_n_u_l_l_
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\fonttbl{\f0\fcharset0
\fnil Arial;}}\plain\cf1\fs20 \pard Round off to four digits and scale 
the values for writing kernel files.}
.EQN 0 65 208 0 0
{0:b}NAME:1
.EQN 6 -66 187 0 0
({0:K1_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)
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({0:K1_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)
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({0:K1_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)
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{0:i}NAME${0:k}NAME$({0:K1_L0}NAME)[({0:i}NAME,{0:k}NAME)={0}?_n_u_l_l_
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{0:i}NAME${0:k}NAME$({0:K1_L1}NAME)[({0:i}NAME,{0:k}NAME)={0}?_n_u_l_l_
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\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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{0:WRITEPRN}NAME({0:K1_L0.arr}NAME):{0:d}NAME
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{0:APPENDPRN}NAME({0:K1_L0.arr}NAME):{0:K1_L0}NAME
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue0;}{\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 4. K. R. Castleman, M.Schulze, and Q. 
Wu, "Simplified Design of Steerable Pyramid Filters," {\i Proc. ISCAS '98,} 
June 3, 1998.\par }