&& && F( u)! "{ f (x)} = f (x)e # j 2$ u x. f (x)! " #1. F(u,v) = f (x, y) e. f (x, y) = 2D Fourier Transform. Fourier Transform - review.

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1 2D Fourier Transfor 2-D DFT & Properties 2D Fourier Transfor 1 Fourier Transfor - review 1-D: 2-D: F( u)! "{ f (x)} = f (x)e # j 2$ u x % & #% dx f (x)! " #1 { F(u) } = F(u)e j 2$ u x du F(u,v) = f (x, y) e f (x, y) = && && % & #% ( ) dx dy # j 2$ ux+vy j 2$ ux+vy F(u,v)e ( ) du dv 2D Fourier Transfor 2

2 2D FT: Properties Linearity: a f(x,y) + b g(x,y) a F(u,v) + b G(u,v) Convolution: f(x,y) g(x,y) = F(u,v) G(u,v) Multiplication: f(x,y) g(x,y) = F(u,v) G(u,v) Separable functions: Suppose f(x,y) = g(x) h(y), Then F(u,v)=G(u)H(v) Shifting: f(x+x 0, y+y 0 ) exp[2π j (+ x 0 u + y 0 v)] F(u,v) 2D Fourier Transfor 3 Separability of the FT F(u,v) = # # % ( $ ' $ f (x, y)e! j 2" ux dx* e! j 2" vy dy &!# )!# # = $ F(u, y) e! j 2" vy dy!# 2D Fourier Transfor 4

3 Separability (contd.) f(x,y) F(u,y) F(u,v) Fourier Transfor along X. Fourier Transfor along Y. We can ipleent the 2D Fourier transfor as a sequence of 1-D Fourier transfor operations. 2D Fourier Transfor 5 Eigenfunctions of LSI Systes A function f(x,y) is an Eigenfunction of a syste T if T[ f(x,y) ] = α f(x,y) for soe constant (Possibly coplex) α. For LSI systes, coplex exponentials of the for exp{ j2π (ux+vy) }, for any (u,v), are the Eigenfunctions. 2D Fourier Transfor 6

4 Ipulse Response and Eigenfunctions Consider a LSI syste with ipulse response h(x,y). Its output to the coplex exponential is g(x, y) = # $ $ h(x! s, y! t)e j 2" (us+vt ) ds dt!# = $$ h(x, y) e j 2" (ux+vy) e! j 2" (ux +vy ) dx dy j 2" (ux+vy) = H (u,v)e 2D Fourier Transfor 7 2-D FT: Exaple f(x,y) A x X F(u,v) = $ Y # $!# X y f (x, y)e! j 2" (ux+vy) dx dy! j 2" ux! j 2" vy = A$ e dx $ e dy 0 % = AXY sin"ux & ' "ux Y 0 ( ) * % & ' sin"vy "vy ( ) * e! j" (ux +vy ) 2D Fourier Transfor 8

5 Exaple (contd.) 2D Fourier Transfor 9 Exaple2 2D Fourier Transfor 10

6 Discrete Fourier Transfor Consider a sequence {u(n), n=0,1,2,..., N-1}. The DFT of u(n) is N! 1 " v( k) = u( n) W, k = 0, 1,..., N! 1 Where n= 0 W N = e kn N 2#! j N N! 1 1! kn u( n) = " v( k) WN, n = 0, 1,..., N! 1 N k = 0, and the inverse is given by 2D Fourier Transfor 11 2-D DFT Often it is convenient to consider a syetric transfor: v(k) = u(n) = 1 N 1 N N!1 kn " u(n)w N and n=0 N!1 " n=0 v(k)w N! kn In 2-D: consider a NXN iage v(k,l) = 1 N u(,n) = 1 N N!1 N!1 k " u(,n) W l n " N W N, =0 N!1 " k =0 n=0 N!1! k!l n " v(k,l) W N l =0 2D Fourier Transfor 12

7 2D DFT -- PROPERTIES Separability Translation Scaling Periodicity and Conjugate Syetry Rotation convolution 2D Fourier Transfor 13 Separability v(k,l) = 1 k l n " W N " u(,n) W N N = 1 N N!1 =0 N!1 " =0 N!1 n=0 v(,l) W N k For each, v(,l) is the 1-D DFT with frequency values l = 0,1,..., N-1 2D Fourier Transfor 14

8 Separability The DFT of a 2-D array can be obtained by first taking the 1-D DFT of each row (or colun) and then taking the 1-D DFT of each colun (or row). It does not atter if the order of operation is reversed. 2D Fourier Transfor 15 Translation (k '+l n ')! j 2$ N u(! ",n! n ") # v(k,l)e 2D Fourier Transfor 16

9 Displaying the DFT: Scaling v(k,l)= DFT{u(,n)} Display C log[1+lv(k,l)l] Large dynaic range Constant 2D Fourier Transfor 17 In MATLAB f = zeros(30,30); f(5:24,13:17)=1; ishow(f, notruesize ); 2D Fourier Transfor 18

10 In Matlab(2) F =fft2(f); F2 = log(abs(f)); ishow(f2, [-1, 5], notruesize ); colorap(jet); colorbar; 2D Fourier Transfor 19 Displaying the DFT N-1 0 N-1 u Low frequency coponents v 2D Fourier Transfor 20

11 Displaying (again) & Shifting j 2! (k ' +l ' n) u(,n)e N " v(k # k ',l # l ') and u(,n)(#1) +n $ " v k # N 2, l # N % & 2 The origin of the F{u(,n)} can be oved to the center of the array (N X N square) by first ultiplying u(,n) by (-1) +n and then taking the Fourier transfor. Note: Shifting does not affect the agnitude of the Fourier transfor. ' ( ) 2D Fourier Transfor 21 Displaying DFT v(k,l) e -j2π[ k +ln ] / N = v(k,l) A B D C C D B A MATLAB Exaple Low frequency coponents 2D Fourier Transfor 22

12 In Matlab(3): FFTSHIFT F2= fftshift(f); ishow (log(abs(f2),..) 2D Fourier Transfor 23 Another exaple Original iage Its centered DFT agnitude 2D Fourier Transfor 24

13 Periodicity & Conjugate Syetry u(,n) F v(k,l) v(k,l) = v(k+n, l) = v(k, l+n) = v(k+n, l+n) If u(,n) is real, v(k,l) also exhibits conjugate syetry v(k,l) = v* (-k, -l) or v(k,l) = v(-k, -l) 2D Fourier Transfor 25 Rotation (continuous case) If you rotate the iage u(,n) by an angle θ, its F.T also gets rotated by the sae angle. 2D Fourier Transfor 26

14 Rotation 2D Fourier Transfor 27 Average Value u = 1 N!! v(k,l) = 1 N v(0,0) = 1 N n u(,n) = Average!! n!! or u = v(0,0) N n k+l n " j 2# N u(,n)e u(,n) = Nu (Scaled Average) 2D Fourier Transfor 28

15 Convolution (Revisited) Consider 1-D continuous case f (x)! g(x) = # $ "# f (x')g(x " x')dx' Let f (x) % F(u), g(x) % G(u) Then f (x)! g(x) % F(u)G(u) Convolution in Space Multiplication in Frequency 2D Fourier Transfor 29 Discrete Convolution Let us now assue that we discretize f(x) and g(x) into vectors f(n) and g(n) of lengths A and B f(n) g(n) {f(0), f(1),... f(a-1)} {g(0), g(1), g(2),...g(b-1)} (a) DFT and its inverse are periodic functions (b) Convolving two vectors of length A and B gives a vector of diension A+B-1. (Linear convolution) 2D Fourier Transfor 30

16 Length of the Convolution A B 2D Fourier Transfor 31 Discrete Convolution: an exaple (Fig 4.36) f() f() h() h() h(-) h(-) D Fourier Transfor 32

17 Discrete conv. (cont.) h(x-) h(x-) 2 x x Range of the DFT=400 2D Fourier Transfor 33 Zero Ibedding In order to obtain a convolution theore for the discrete case, and still be consistent with the periodicity property we need to assue that sequences f( ) and g( ) are periodic with soe period M. Fro (b) it is clear that M> A+B-1 to avoid overlap. Since this period is greater than A or B, the original sequence length ust be increased and this is done by appending zeros at the end. Redefine the extended sequences as # f (n) 0! n! A " 1 f e (n) = $ % 0 A! n! M " 1 # g e (n) = $ g(n) 0! n! B " 1 % 0 B! n! M " 1 2D Fourier Transfor 34

18 f e (n)! c g e (n) = where M "1 # =0 f e ()g e (n " ) c [ ] ( g(n) ) c = g n Modulo M Note: With n expressed as n=n 1 + n 2 N where 0 $ n 1 $ N " 1 n odulo N equals n 1 x od y = x " y % x ( &' y)* x od 0 = x. if y + 0 x % & ( y ) is the integer part of x y 2D Fourier Transfor 35 Theore The DFT of the circular convolution of two sequences of length N is equal to the product of their DFTs. If y(n) = DFT y(n) N!1 " f (n! ) c g(n) then =0 [ ] N = DFT[ f (n)] N DFT[ g(n) ] N A linear convolution of two sequences can be obtained via FFT by ebedding it into a circular convolution. 2D Fourier Transfor 36

19 2-D Convolution These results can be siilarly extended to 2-D signals. Let f(,n) : A x B array g(,n) : C x D array Let M> = A + C -1 N> = B + D -1 For linear convolution using DFT create the extended periodic sequences of period MxN in the 2-D. 2D Fourier Transfor 37 Extended (periodic) Sequences coputing convolution is ore efficient in the frequency doain. # f (, n) 0!! A " 1 % 0! n! B " 1 fe(, n) = $ % 0 A!! M " 1 &% B! n! N " 1 # g(, n) 0!! C " 1 % 0! n! D " 1 ge(, n) = $ % 0 C!! M " 1 &% D! n! N " 1 and the 2- D linear convolution becoes M " 1 ( N " 1 y(, n) = f ( " ', n " n' ) g ( ', n' ) '= 0 ( n'= 0 e c e 2D Fourier Transfor 38

20 Linear Convolution and DFT: Suary y(n) = f(n) * g(n) 1. Let M>= A+B-1 be an integer for which the FFT algorith is available. 2. Define the zero extended sequences f e (n), g e (n). 3. Let F e (k) = DFT { f e (n) } M, G e (k) = DFT { g e (n) } M. Let Y e (k)=f e (k)g e (k) 4. Take the I-DFT of Y e (k) to obtain Y e (n). Then Y (n) = Y e (n) for 0 <= n <= A+B-1 2D Fourier Transfor 39 A note on convolution with iages Note: In any cases involving iages, we deal with square arrays of size N X N. We norally would like to have the resulting convolved output also as an N X N array. 2D Fourier Transfor 40

21 Conv2 (.) in Matlab CONV2 Two diensional convolution. C = CONV2(A, B) perfors the 2-D convolution of atrices A and B. If [a,na] = size(a) and [b,nb] = size(b), then size(c) = [a+b-1,na+nb-1]. C = CONV2(...,'shape') returns a subsection of the 2-D convolution with size specified by 'shape': 'full' - (default) returns the full 2-D convolution, 'sae' - returns the central part of the convolution that is the sae size as A. 'valid' - returns only those parts of the convolution that are coputed without the zero-padded edges, size(c) = [a-b+1,na-nb+1] when size(a) > size(b). 2D Fourier Transfor 41