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 Convo lu tion – 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 Ø(x, u). u is typically the freq. index. Yao Wang, NYU-Poly EL5123: Fourier Transform 3 Illustration of Decomposition Φ3 f α3 f = α1Φ1+α2Φ2+α3Φ3 o α2 Φ2 α1 Φ1 Yao Wang, NYU-Poly EL5123: Fourier Transform 4 Decomposition • Ortho-normal basis function 1, u1 u2 (x,u1) *(x,u2 )dx 0, u1 u2 • Forward * Projection of F(u) f (x),(x,u) f (x) (x,u)dx f(x) onto (x,u) • Inverse Representing f(x) as sum of f (x) F(u)(x,u)du (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) j2f0x f (x) e F(u) (u f0 ) 1 f (x) cos(2f x) F(u) (u f ) (u f ) 0 2 0 0 1 f (x) sin(2f x) F(u) (u f ) (u f ) 0 2 j 0 0 1, x x0 sin(2x0u) f (x) F(u) 2x0 sinc(2x0u) 0, otherwise u sin(t) where, sinc(t) t Derive the last transform pair in class Yao Wang, NYU-Poly EL5123: Fourier Transform 7 FT of the Rectangle Function sin(2x u) sin(t) F(u) 0 2x sinc(2x u) where, sinc(t) u 0 0 t f(x) f(x) x =2 x0=1 0 -1 1 x -2 2 x 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 IFT of Ideal Low Pass Signal • What is f(x)? F(u) -u0 u0 u Yao Wang, NYU-Poly EL5123: Fourier Transform 9 Representation of FT • Generally, both f(x) and F(u) are complex • Two representations – Real and Imaginary F(u) R(u) jI(u) – Magnitude and Phase j (u) F(u) A(u)e , where I F(u) I(u) I(u) A(u) R(u)2 I(u)2 , (u) tan 1 R(u) Φ(u) R(u) R • Relationship R(u) A(u)cos(u), I(u) A(u)sin(u) • Power spectrum P(u) A(u)2 F(u) F(u)* F(u) 2 Yao Wang, NYU-Poly EL5123: Fourier Transform 10 What if f(x) is real? • Real world signals f(x) are usually real • F(u) is still complex, but has special properties F * (u) F(u) R(u) R(u), A(u) A(u), P(u) P(u) : even function I(u) I(u),(u) (u) :odd function Yao Wang, NYU-Poly EL5123: Fourier Transform 11 Property of Fourier Transform • Duality f (t) F(u) F(t) f (u) • Linearity Fa1 f1(x) a2 f2 (x) a1F{ f1(x)} a2 F{ f2 (x)} • SliScaling Faf (x) aF{ f (x)} • Translation j2x0u j2u0 x f (x x0 ) F(u)e , f (x)e F(u u0 ) • Convolution f (x) g(x) f (x )g()d f (x) g(x) F(u)G(u) We will review convolution later! Yao Wang, NYU-Poly EL5123: Fourier Transform 12 Two Dimension Fourier Transform • Basis functions (x, y;u,v) e j(2ux2vy) e j2uxe j2vy , u,v ,. • Forward – Transform F(u,v) F{ f (x, y)} f (x, y)e j2 (uxvy)dxdy • Inverse – Transform f (x, y) F 1{F(u,v)} F(u,v)e j2 (uxvy)dudv • PtProperty – All the properties of 1D FT apply to 2D FT Yao Wang, NYU-Poly EL5123: Fourier Transform 13 Example 1 f (x, y) sin 4x cos6y f(x,y) F{sin 4x} sin 4xe j2 (uxvy)dxdy sin4xe j2uxdx e j2vydy sin4xe j2uxdx (v) 1 ( (u 2) (u 2)) (v) 2 j 1 ( (u 2,v) (u 2,v)) 2 j u F(u,v) , x y 0 where (x, y) (x) (y) 0, otherwise v 1 Likewise, F{cos6y} ( (u,v 3) (u,v 3)) 2 Yao Wang, NYU-Poly EL5123: Fourier Transform 14 Example 2 1 f (x, y) sin(2x 3y) e j(2x3y) e j(2x3y) 2 j Fe j(2x3y) e j(2x3y)e j2 (xu yv)dxdy e j2xe j2uxdx e j3ye j2yvdy 3 3 (u 1) (v ) (u 1,v ) [X, Y]=meshgrid(-2:1/16:2,-2:1/16:2); 2 2 f=sin(2*pi*X+3*pi*Y); 3 imagesc(f); colormap(gray) Likewise, Fe j(2x3y) (u 1,v ) Truesize, axis off; 2 Thfherefore, 1 3 3 u Fsin(2x 3y) (u 1,v ) (u 1,v ) F(u,v) 2 j 2 2 v Yao Wang, NYU-Poly EL5123: Fourier Transform 15 Important Transform Pairs 1 sin(2f x 2f y) (u f ,v f ) (u f ,v f ) x y 2 j x y x y 1 cos(2f x 2f y) (u f ,v f ) (u f ,v f ) x y 2 x y x y 2D rectangular function 2D sinc function Yao Wang, NYU-Poly EL5123: Fourier Transform 16 Properties of 2D FT (1) • Linearity Fa1 f1(x, y) a2 f2 (x, y) a1F{ f1(x, y)} a2 F{ f2 (x, y)} • Translation j2 (x0u y0v) f (x x0 , y y0 ) F(u,v)e , j2 (u0 xv0y) f (x, y)e F(u u0 ,v v0 ) • Conjugation f * (x, y) F * (u,v) Yao Wang, NYU-Poly EL5123: Fourier Transform 17 Properties of 2D FT (2) • Symmetry f (x, y) is real F(u,v) F(u,v) • Convolution – Definition of convolution f (x, y) g(x, y) f (x , y )g(, )dd – Convolution theory f (x, y) g(x, y) F(u,v)G(u,v) We will describe 2D convolution later! Yao Wang, NYU-Poly EL5123: Fourier Transform 18 Separability of 2D FT and Separable Signal • Separability of 2D FT F2{ f (x, y)} Fy{Fx{ f (x, y)}} Fx{Fy{ f (x, y)}} – where Fx, Fy are 1D FT along x and y. – one can do 1DFT for each row of original image, then 1D FT along each column of resulting image • Separable Signal – f(x,y) = fx(x)fy(y) – F(()u,v) = Fx(()u)Fy(()v), • where Fx(u) = Fx{fx(x)}, Fy(u) = Fy{fy(y)} – For separable signal, one can simply compute two 1D transforms! Yao Wang, NYU-Poly EL5123: Fourier Transform 19 Example 1 f (x, y) sin(3x)cos(5y) 1 f (x) sin(3x) F (u) (u 3/ 2) (u 3/ 2) x x 2 j 1 f (y) cos(5y) F (v) (v 5/ 2) (v 5/ 2) y y 2 F(u,v) Fx(u)Fy(v) 1 3 5 3 5 3 5 3 5 (u ,v ) (u ,v ) (u ,v ) (u ,v ) 4 j 2 2 2 2 2 2 2 2 u v Yao Wang, NYU-Poly EL5123: Fourier Transform 20 Example 2 1, | x | x0 ,| y | y0 f (x, y) 0, otherwise F(u,v) 4x0 y0 sinc(2x0u)sinc(2y0v) x 2 u x0 = 2 y0 = 1 v -1 1 y -2 w/ logrithmic mapping Yao Wang, NYU-Poly EL5123: Fourier Transform 21 Rotation • Let x r cos, y r sin, u cos, v sin. • 2D FT in polar coordinate (r, θ) and (ρ, Ø) 2 F(,) f (r, )e j2 (r cos cos r sin sin )rdrd 00 f (r, )e j2r cos( )rdrd • Property f (r, 0 ) F(, 0 ) Yao Wang, NYU-Poly EL5123: Fourier Transform 22 Example of Rotation Yao Wang, NYU-Poly EL5123: Fourier Transform 23 Fourier Transform For Discrete Time Sequence (DTFT) • One Dimensional DTFT – f(n) is a 1D discrete time sequence – Forward Transform F(u) i s peri o dic in u, w ith per io d j2un F(u) f (n)e of 1 n – ITfInverse Transform 1/ 2 f (n) F(u)e j2undu 1/ 2 Yao Wang, NYU-Poly EL5123: Fourier Transform 24 Properties unique for DTFT • Periodicity – F(u) = F(u+1) – The FT of a discrete time sequence is only considered for u є (-½ , ½)andu=½), and u = +½isthe½ is the highest discrete frequency • Symmetry for real sequences f (n) f * (n) F(u) F * (u) F(u) F(u) F(u) is symmetric Yao Wang, NYU-Poly EL5123: Fourier Transform 25 Example 1, n 0,1,..., N 1; f (n) 0, others N 1 1 e j2uN sin 2u(N / 2) F(u) e j2nu e j (N 1)u j2u sin 2u(1/ 2) n0 1 e f(n) 0 N-1 n N=10 There are N/2 zeros in (0, ½], 1/N apart Yao Wang, NYU-Poly EL5123: Fourier Transform 26 Two Dimensional DTFT • Let f(m,n) represent a 2D sequence • Forward Transform F(u,v) f (m,n)e j2 (munv) m n • Inverse Transform 1/ 2 1/ 2 f (m,n) F(u,v)e j2 (munv)dudv 1/ 2 1/ 2 • Properties – Periodicity, Shifting and Modulation, Energy Conservation Yao Wang, NYU-Poly EL5123: Fourier Transform 27 Periodicity F(u,v) f (m,n)e j2 (munv) mn • F(u,v) is periodic in u, v with period 1, i.e., for all integers k, l: – F(u+k, v+l) = F(u, v) • To see this consider e j2 (m(uk )n(vl)) e j2 (munv)e j2 (mk nl) e j2 (munv) In MATLAB, frequency scaling is such that 1 represents maximum freq u,v=1/2.