Lecture 2: 2D Fourier transforms and applications
Fourier transforms and spatial frequencies in 2D. • Definition and meaning 1D Fourier Transform. Reminder transform pair - definition. Example.
2D Fourier Transform
2D Fourier Transforms. – Generalities and intuition. – Examples. – A bit of theory. • Discrete Fourier Transform (DFT). • Discrete Cosine Transform (DCT)
2-D Fourier Transforms
Continuous Fourier Transform (FT) 2D FT. • Fourier Transform for Discrete Time Sequence ... Transforms are decompositions of a function f(x).
Fourier transform in 1D and in 2D
Fourier tx in 1D computational complexity
2D Fourier Transforms
The Nyquist theorem says that the original signal should lie in an ?? dimensional space before you down-sample. Otherwise information is corrupted (i.e. sig-.
Affine transformations and 2D Fourier transforms
18 feb 2020 age and frequency domains of a 2D Fourier transform. ... TABLE I. Examples of corresponding affine-transformation matrices arranged ...
Two-Dimensional Fourier Transform Theorems
Recall: the general equation of a curve in a plane is c(x y)=0. 5. Separability of 2D Delta Function. Proof: ? 1.
2D and 3D Fourier transforms
4 mar 2020 The Fourier transform of a 2D delta function is a constant. (4) and the product of two rect functions (which defines a square region in the ...
The 2D Fourier Transform The analysis and synthesis formulas for
continuous Fourier transform are as follows: • Analysis Separability of 2D Fourier Transform. The 2D analysis formula can be written as a.
2D Discrete Fourier Transform (DFT)
2D DFT. • 2D DCT. • Properties. • Other formulations. • Examples Fourier transform of a 2D set of samples forming a bidimensional sequence.
2-D Fourier Transforms
Yao Wang
Polytechnic University Brooklyn NY 11201PolytechnicUniversity
Brooklyn
NY 11201With 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 DTFTLi C l ti
Li near C onvo l u ti on -1D, Continuous vs. discrete signals (review)- 2D •Filter Design •Computer ImplementationYao Wang, NYU-PolyEL5123: Fourier Transform2
What is a transform?
•Transforms are decompositionsof a function f(x) into some basis functionsØ(x, u). u is typically
into some basis functionsØ(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 2Yao Wang, NYU-PolyEL5123: Fourier Transform4
1Decomposition
•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 weightYao 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 2Inverse Transform
d xex f x f F u F ux j 2Inverse
Transform
f dueuFuFFxf uxj 21Yao 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 x2-2Yao 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 0Yao 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)(,)()()(,)()( 122u 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)
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