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

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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

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Illustration of Decomposition

f = 11+22+33

3 f

3

o

2

1

2

1

Yao Wang, NYU-Poly

EL5123: Fourier Transform

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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

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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

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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

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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

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