[PDF] 2-D Fourier Transforms Continuous Fourier Transform (FT) 2D





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Lecture 2: 2D Fourier transforms and applications

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2-D Fourier Transforms

Yao Wang

Polytechnic University Brooklyn NY 11201Polytechnic

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)(DTFT) -1D DTFT (review) -2D DTFT

Li C l ti

Li near C onvo l u ti on -1D, Continuous vs. discrete signals (review)- 2D •Filter Design •Computer Implementation

Yao Wang, NYU-PolyEL5123: Fourier Transform2

What is a transform?

•Transforms are decompositionsof a function f(x) into some basis functions

Ø(x, u). u is typically

into some basis functions

Ø(x,

u). u is typically the freq. index.

Yao Wang, NYU-PolyEL5123: Fourier Transform3

Illustration of Decomposition

3 f 3 f = Į 1 1 2 2 3 3 2 o 1 2

Yao Wang, NYU-PolyEL5123: Fourier Transform4

1

Decomposition

•Ortho-normal basis function 1 z f f2121 21
,0, 1 ),(*),(uuuudxuxux •Forward d f f I I P rojection of

Inverse

f d xuxx f uxx f u F P rojection of f(x)onto (x,u)

Inverse

duuxuFxf),()()(

Representing f(x)as sum of

(x,u) for all u, with weight

Yao Wang, NYU-PolyEL5123: Fourier Transform5

F(u)

Fourier Transform

•Basis function 2 f ueux ux j •Forward Transform d f f F F ux j 2

Inverse Transform

d xex f x f F u F ux j 2

Inverse

Transform

f dueuFuFFxf uxj 21

Yao Wang, NYU-PolyEL5123: Fourier Transform6

Important Transform Pairs

f F f uuFxf x f j )()(1)( 2 0 fufuuFxfxf f uu F ex f x f j G S )()(21)()2cos()( 0000 2 0 fufujuFxfxf )()(21)()2sin()( 000 t uxxuuxuFotherwisexxxf S sin( )2sinc(2)2sin()(,0,1)( 0000 t t twhere S sin( )sinc(,

Yao Wang, NYU-PolyEL5123: Fourier Transform7

Derive the last transform pair in class

FT of the Rectangle Function

tttwhereuxxuuxuF )sin()sinc(,)2sinc(2)2sin()( 000 f(x)x 0 =1 f(x)x 0 =2 x1-1 x2-2

Yao Wang, NYU-PolyEL5123: Fourier Transform8

Note first zero occurs at u

0 =1/(2 x 0 )=1/pulse-width, other zeros are multiples of this.

IFT of Ideal Low Pass Signal

•What is f(x)? F(u) uu 0 -u 0

Yao Wang, NYU-PolyEL5123: Fourier Transform9

Representation of FT

•Generally, both f(x) and F(u) are complex •Two representations -Real and Imaginary -Magnitude and Phase j )()()(uj I uRuF )(tan)(,)()()(,)()( 122
u R uIuuIuRuAwhereeu A u F u j I F(u)I I(u) •Relationship u R i A I A R R R(u)

ĭ(u)

•Power spectrum s i n cos uu A u Iquotesdbs_dbs19.pdfusesText_25
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