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© 2000 Jeff B. Pelz
© 2000 Jeff B. Pelz
© 2000 Jeff B. Pelz
Fourier SynthesisFourier Synthesis
Sinusoidal Functions as “Building Blocks”for Spatial Vision
© 2000 Jeff B. Pelz
““Sinusoidal FunctionSinusoidal Function” ” F(F() = sin() = sin())
sin()
© 2000 Jeff B. Pelz
sin(0) = 0.0sin(0) = 0.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 45 90 135 180 225 270 315 360
© 2000 Jeff B. Pelz
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 45 90 135 180 225 270 315 360
sin(45) = 0.7sin(45) = 0.7
© 2000 Jeff B. Pelz
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 45 90 135 180 225 270 315 360
sin(90) = 1.0sin(90) = 1.0
© 2000 Jeff B. Pelz
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 45 90 135 180 225 270 315 360
sin(135) = 0.7sin(135) = 0.7
© 2000 Jeff B. Pelz
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 45 90 135 180 225 270 315 360
sin(180) = 0.0sin(180) = 0.0
© 2000 Jeff B. Pelz
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 45 90 135 180 225 270 315 360
sin(225) = sin(225) = -- 0.70.7
© 2000 Jeff B. Pelz
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 45 90 135 180 225 270 315 360
sin(270) = sin(270) = -- 1.01.0
© 2000 Jeff B. Pelz
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 45 90 135 180 225 270 315 360
sin(315) = sin(315) = -- 0.70.7
© 2000 Jeff B. Pelz
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
0 45 90 135 180 225 270 315 360
sin(360) = sin(0) = 0.0sin(360) = sin(0) = 0.0
© 2000 Jeff B. Pelz
F(x) = sin(x)F(x) = sin(x)
-1.5
-1
-0.5
0
0.5
1
1.5
sin()
© 2000 Jeff B. Pelz
F(x) = sin(x)F(x) = sin(x)
-1.5
-1
-0.5
0
0.5
1
1.5
© 2000 Jeff B. Pelz
F(x) = sin(x)F(x) = sin(x)
-1.5
-1
-0.5
0
0.5
1
1.5
© 2000 Jeff B. Pelz
F(x) = sin(x)F(x) = sin(x)
-1.5
-1
-0.5
0
0.5
1
1.5
© 2000 Jeff B. Pelz
F(x) = sin(x)F(x) = sin(x)
-1.5
-1
-0.5
0
0.5
1
1.5
-1.5
-1
-0.5
0
0.5
1
1.5
© 2000 Jeff B. Pelz
F(x) = square(x)F(x) = square(x)
© 2000 Jeff B. Pelz
Two-dimensional F(x,y)Two-dimensional F(x,y)
Horizontal position (x)Horizontal position (x)
Ver
tical
pos
ition
(y)
Ver
tical
pos
ition
(y)
© 2000 Jeff B. Pelz
Two-dimensional F(x,y)Two-dimensional F(x,y)
Horizontal position (x)Horizontal position (x)
Ver
tical
pos
ition
(y)
Ver
tical
pos
ition
(y)
© 2000 Jeff B. Pelz
F(x,y) = one of an infinite functions of (x,y)F(x,y) = one of an infinite functions of (x,y)
© 2000 Jeff B. Pelz
One-dimensional F(x) = square(x)One-dimensional F(x) = square(x)
© 2000 Jeff B. Pelz
F(x) = square(x) F(x) = square(x) sin(x) sin(x)
© 2000 Jeff B. Pelz
F(x) = square(x) F(x) = square(x) sin(x) sin(x)
-1.5
-1
-0.5
0
0.5
1
1.5
© 2000 Jeff B. Pelz
-1.5
-1
-0.5
0
0.5
1
1.50
1
3
5
7
9
11
13
15
17
19
21
23
25
sum
F(x) = sin(x)F(x) = sin(x)++11//33 sin(3x) sin(3x)
© 2000 Jeff B. Pelz
-1.5
-1
-0.5
0
0.5
1
1.50
1
3
5
7
9
11
13
15
17
19
21
23
25
sum
F(x) = sin(x)F(x) = sin(x)++11//33 sin(3x) sin(3x) ++11//55 sin(5x) sin(5x)
© 2000 Jeff B. Pelz
-1.5
-1
-0.5
0
0.5
1
1.50
1
3
5
7
9
11
13
15
17
19
21
23
25
sum
F(x) = sin(x)F(x) = sin(x)++11//33 sin(3x) sin(3x) ++11//55 sin(5x) sin(5x) ++11//77 sin(7x) sin(7x)
© 2000 Jeff B. Pelz
-1.5
-1
-0.5
0
0.5
1
1.50
1
3
5
7
9
11
13
15
17
19
21
23
25
sum
F(x) = sin(x)F(x) = sin(x)++11//33 sin(3x) sin(3x) ++11//55 sin(5x) sin(5x) ++11//77 sin(7x) sin(7x)
++11//99 sin(9x) sin(9x)
© 2000 Jeff B. Pelz
-1.5
-1
-0.5
0
0.5
1
1.50
1
3
5
7
9
11
13
15
17
19
21
23
25
sum
F(x) = sin(x)F(x) = sin(x)++11//33 sin(3x) sin(3x) ++11//55 sin(5x) sin(5x) ++11//77 sin(7x) sin(7x)
++11//99 sin(9x) sin(9x) ++11//1111 sin(11x) sin(11x)
© 2000 Jeff B. Pelz
-1.5
-1
-0.5
0
0.5
1
1.50
1
3
5
7
9
11
13
15
17
19
21
23
25
sum
F(x) = sin(x)F(x) = sin(x)+ + 11//33 sin(3x) sin(3x) + + 11//55 sin(5x) sin(5x) + + 11//77 sin(7x) sin(7x)
+ + 11//99 sin(9x) sin(9x) + + 11//1111 sin(11x) sin(11x) + + 11//1313 sin(13x) sin(13x)
© 2000 Jeff B. Pelz
-1.5
-1
-0.5
0
0.5
1
1.50
1
3
5
7
9
11
13
15
17
19
21
23
25
sum
F(x) = sin(x)F(x) = sin(x)++11//33 sin(3x) sin(3x) ++11//55 sin(5x) sin(5x) ++11//77 sin(7x) sin(7x)
+ + 11//99 sin(9x) sin(9x) ++ …... …... ++11//2525 sin(25x) sin(25x)
© 2000 Jeff B. Pelz
-1.5
-1
-0.5
0
0.5
1
1.50
1
3
5
7
9
11
13
15
17
19
21
23
25
sum
F(x) = F(x) = 11sin(sin(11x)x)++11//33 sin( sin(33x) x) ++11//55 sin( sin(55x) x) ++11//77 sin( sin(77x)x)
+ + 11//99 sin( sin(99x) x) ++ …... …... ++11//2525 sin( sin(2525x) x) [freq components][freq components]
© 2000 Jeff B. Pelz
-1.5
-1
-0.5
0
0.5
1
1.50
1
3
5
7
9
11
13
15
17
19
21
23
25
sum
FF(() = () = (1 ,1 , 11))+(+(11//3 3 ,3,3) ) +(+(11//5 5 ,, 55) ) + (+ (11//7 7 ,, 77) + () + (11//99 ,, 99) ) ++ …... …...
© 2000 Jeff B. Pelz
FF(() = ) = 1, 0, 1, 0, 11//33, 0, , 0, 11//5 5 , 0, , 0, 11//7 7 , 0, , 0, 11//99, …..., …...
-1.5
-1
-0.5
0
0.5
1
1.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1 3 5 7 9 11 13 15 17 19 21 23 25
frequency
pow
er
“Spatial domain”
“Frequency domain”
© 2000 Jeff B. Pelz
sin(x)
-1.5
-1
-0.5
0
0.5
1
1.5
0 1 2 3 4 5 6 7
-20
0
20
40
60
80
100
120
140
1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115 121 127
Fourier PairsFourier Pairs
© 2000 Jeff B. Pelz
1/5*sin(9x)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 1 2 3 4 5 6 7
-5
0
5
10
15
20
25
30
1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115 121 127
Fourier PairsFourier Pairs
© 2000 Jeff B. Pelz
sin(x) +1/5*sin(9x)
-1.5
-1
-0.5
0
0.5
1
1.5
0 1 2 3 4 5 6 7
-20
0
20
40
60
80
100
120
140
1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115 121 127
Fourier PairsFourier Pairs
© 2000 Jeff B. Pelz
Fourier PairsFourier Pairsarbitrary
-4
-2
0
2
4
6
0 1 2 3 4 5 6 7
0
50
100
150
200
250
1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115 121 127
© 2000 Jeff B. Pelz
Fourier PairsFourier Pairs
© 2000 Jeff B. Pelz
Fourier PairsFourier Pairs
© 2000 Jeff B. Pelz
Fourier PairsFourier Pairs
© 2000 Jeff B. Pelz
Fourier PairsFourier Pairs
© 2000 Jeff B. Pelz
Fourier PairsFourier Pairs
© 2000 Jeff B. Pelz
Low-pass FilteringLow-pass Filtering
© 2000 Jeff B. Pelz
High-pass FilteringHigh-pass Filtering
© 2000 Jeff B. Pelz
Frequency ChannelsFrequency Channels
© 2000 Jeff B. Pelz
© 2000 Jeff B. Pelz
2D Gabor Functions2D Gabor Functions
© 2000 Jeff B. Pelz
