2-D Fourier Transforms - New York University
2-D Fourier Transforms
Yao Wang Polytechnic University, Brooklyn, NY 11201 With contribution from Zhu Liu, Onur Guleryuz, and Gonzalez/Woods, Digital Image Processing, 2ed
Lecture Outline
? Continuous Fourier Transform (FT)
? 1D FT (review) ? 2D FT
? Fourier Transform for Discrete Time Sequence (DTFT)
? 1D DTFT (review) ? 2D DTFT
? Linear Convolution
? 1D, Continuous vs. discrete signals (review) ? 2D
? Filter Design ? Computer Implementation
Yao Wang, NYU-Poly
EL5123: Fourier Transform
2
What is a transform?
? Transforms are decompositions of a function f(x) into some basis functions ?(x, u). u is typically the freq. index.
Yao Wang, NYU-Poly
EL5123: Fourier Transform
3
Illustration of Decomposition
f = 11+22+33
3 f
3
o
2
1
2
1
Yao Wang, NYU-Poly
EL5123: Fourier Transform
4
Decomposition
? Ortho-normal basis function
(
x,
u1 )
*
(x,
u2
)dx
01,,
u1 u2 u1 u2
? Forward
F (u) f (x),(x,u)
f (x)*(x,u)dx
Projection of f(x) onto
(x,u)
? Inverse
f (x) F (u)(x,u)du
Representing f(x) as sum of
(x,u) for all u, with weight
F(u)
Yao Wang, NYU-Poly
EL5123: Fourier Transform
5
Fourier Transform
? Basis function
(x,u) e j2ux , u ,.
? Forward Transform
F (u) F{ f (x)} f (x)e j2uxdx
? Inverse Transform
f (x) F 1{F (u)} F (u)e j2uxdu
Yao Wang, NYU-Poly
EL5123: Fourier Transform
6
Important Transform Pairs
f (x) 1 F (u) (u)
f (x) e j2f0x F (u) (u f0 )
f (x) cos(2f0x)
F (u)
1 (u
2
f0 ) (u
f0)
f (x) sin(2f0x)
F (u)
1 2j
(u
f0)
(u
f0)
f
(
x)
1,
x x0
0, otherwise
F
(u)
sin(2x0u) u
2x0
sinc(2x0u)
where, sinc(t) sin(t) t
Derive the last transform pair in class
Yao Wang, NYU-Poly
EL5123: Fourier Transform
7
FT of the Rectangle Function
F (u)
sin(2x0u) u
2x0
sinc(2x0u)
f(x)
x0=1
where,
sinc(t)
sin(t) t
f(x)
x0=2
-1
1
x
-2
2x
Note first zero occurs at u0=1/(2 x0)=1/pulse-width, other zeros are multiples of this.
Yao Wang, NYU-Poly
EL5123: Fourier Transform
8
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