2-D Fourier Transforms
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
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