F(u) = f e(t)cos2πutdt + = f e(t)cos2πutdt i f o(t)sin2πutdt = F e(u) + if o(u).

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1 The (Continuous) Fourier Transform Matrix Computations and Virtual Spaces Applications in Signal/Image Processing Niclas Börlin Department of Computing Science, Umeå University, Sweden Given a complex valued function f(t), its Fourier transform is defined as F {f(t)} = F(u) = The inverse Fourier transform is defined as F 1 {F(u)} = f(t) = f(t)e i2πut dt. F(u)e i2πut du. The functions f(t) and F(u) are known as a transform pair. The function f(t) is often described as the signal as a function of time t and F(u) as the spectrum as a function of the frequence u. 1 Transforms of even and odd functions Let f e (t) be an even function and f o (t) be an odd function f e (t) = f e ( t), t, f o (t) = f o ( t), t. Any function may be split into a sum of even and odd functions f(t) = f e (t)+f o (t), f e (t) = 1 2 (f(t)+f( t)), f o(t) = 1 2 (f(t) f( t)). If we use Euler s equation e it = cost + isint, the Fourier transform becomes F(u) = f(t)e i2πut dt = f(t) cos2πutdt i f(t) sin2πutdt. Transforms of even and odd functions The Fourier transform of f(t) = f e (t) + f o (t) becomes F(u) = f e(t)cos2πutdt + f o(t)cos 2πutdt i f e(t)sin2πutdt {z } {z } =0 =0 = f e(t)cos2πutdt i f o(t)sin2πutdt = F e(u) + if o(u). Thus, the Fourier transform of a real, even function is real and even. The Fourier transform of a real, odd function is imaginary and odd. The power spectrum 2 = F e (u) 2 + F o (u) 2 of a real signal is even, since i f o(t)sin2π F( u) 2 = F e( u) 2 + F o( u) 2 = F e(u) 2 + F o(u) 2 = F e(u) 2 + F o(u) 2 = 2. 3

2 The Fourier spectrum of the step function Time and frequency domain symmetry f(x) = { A, x X, 0, x > X. AX = AX sin(πux) πux If we take the (forward) Fourier transform of the frequency function, we get F {F(t)} = F(t)e i2πut dt = [v = u] = = F 1 {F(t)} = f(v) = f( u). F(t)e i2πvt dt f(x) A Thus, for every property of the forward Fourier transform, the same property applies for the inverse Fourier transform, but with a flipped sign. 0 0 X x 0 4/X 3/X 2/X 1/X 0 1/X 2/X 3/X 4/X u 5 The Discrete Fourier Transform (DFT) A discrete sequence is interpreted as a sampling of a continuous signal f(t) = f(t 0 + t t), t = 0,..., N 1. Note that f(t) is commonly used both to describe the continous and the discrete signal, even though t continuous = t 0 + t discrete t. Given the sequence f(t), the Discrete Fourier Transform (DFT) is defined as F(u) = f(t)e i2πut/n. The Inverse Discrete Fourier Transform (IDFT) is defined as f(t) = 1 F(u)e i2πut/n. N The discrete vector [F(0), F(1),..., F(N 1)] T corresponds to a sampling of the continuous spectrum at points F(u u), u = 0,..., N 1, where u = 1 N t. The function F(u) also has a dual meaning with u continuous = u discrete u. Periodicity Observation: The discrete function f(t) is periodic with period N, f(t+n) = F(u)e i2πu(t+n)/n = F(u)e i2πut/n i2πu = corresponding to a period for continuous f(t) of N t. Likewise, the discrete F(u) is periodic with period N, corresponding to a period for the continuous F(u) of N u = N/(N t) = 1/ t. Any analysis of the Fourier spectrum assumes the signal is periodic! Example: A continuous signal f(t) sampled in 0.1s at F s =8kHz has N = 800, t = 1/F s = s. The period of the reconstructed signal is N t = 800 1/8000 = 0.1s. The period of the continuous F(u) is 1/ t = F s! F(u)e i2πut/n 7

3 Interpreting the Fourier spectra load train; dt=1/fs; Y=fft(y); du=1/(n*dt); N=length(y); t=[0:n-1]*dt; u=[0:n-1]*du; plot(t,y); plot(u,abs(y)) f(t) High and low frequencies The periodicity implies that the discrete spectrum satisfies F(u N) = F(u), e.g. for N = 8, the frequencies [0, 1, 2, 3, 4, 5, 6, 7] may also be interpreted as [0, 1, 2, 3, ±4, 3, 2, 1]. w = e i2π/8, w 1 w 2 w 3 w 4 w 5 w 6 w 7 1 w 2 w 4 w 6 w 8 w 10 w 12 w 14 1 w 3 w 6 w 9 w 12 w 15 w 18 w 21 1 w 4 w 8 w 12 w 16 w 20 w 24 w 28 1 w 5 w 10 w 15 w 20 w 25 w 30 w 35 1 w 6 w 12 w 18 w 24 w 30 w 36 w 42 1 w 7 w 14 w 21 w 28 w 35 w 42 w The highest frequency that can be sampled is F s /2. This frequency is called the Nyqvist frequency. 9 Shifting to center u=[0:n-1]*du; plot(u,abs(y)) Unshifted u=([0:n-1]-n/2)*du; plot(u,abs(fftshift(y))) Shifted Run phone demo

(Heisenberg) uncertainty principle Two-dimensional Fourier transform Good localization in time corresponds to poor localization in frequency, and vice versa.

f(t) f(t) Given a complex-valued twodimensional function f(x, y), the two-dimensional continuous Fourier transforms are defined as F {f(x, y)} = F(u, v) = F 1 {F(u, v)} = f(x, y) = f(x, y)e

4 (Heisenberg) uncertainty principle Two-dimensional Fourier transform Good localization in time corresponds to poor localization in frequency, and vice versa. The Gaussian f(t) = e kt2 is an optimal compromize. Discontinuities in one domain causes oscillations in the other. f(t) f(t) Given a complex-valued twodimensional function f(x, y), the two-dimensional continuous Fourier transforms are defined as F {f(x, y)} = F(u, v) = F 1 {F(u, v)} = f(x, y) = f(x, y)e i2π(xu+yv) dxdy, F(u, v)e i2π(xu+yv) dudv. The corresponding discrete transforms are defined as F(u, v) = f(x, y) = 1 NM N 1 x=0 N 1 M 1 y=0 M 1 v=0 f(x, y)e i2π(ux/n+vy/m), F(u, v)e i2π(ux/n+vy/m). 13 Separability The two-dimensional Fourier transform is separable, i.e. F(u, v) = M 1 X x=0 y=0 = x=0 f(x, y)e i2π(ux/n+vy/m) = M 1 X e i2πux/n y=0 f(x, y)e i2πvy/m) = x=0 F(x, v)e i2πux/n. Thus, the two-dimensional Fourier transform may be implemented as a sequence of one-dimensional Fourier transforms in the row and column direction. Real example load mandrill; im=ind2gray(x,map); imshow(im); IM=fft2(im); figure; imshow(rescale(abs(fftshift(im))),256); figure; imshow(rescale(log(1+abs(fftshift(im)))),256); Image F(u, v) log(1 + F(u, v) ) f(x, y) 1D row FT F(x, v) 1D column FT F(u, v) 15

5 Real example 2 load gatlin; X=X(:,80+[1:480]); im=ind2gray(x,map); imshow(im); IM=fft2(im); figure; imshow(rescale(abs(fftshift(im))),256); figure; imshow(rescale(log(1+abs(fftshift(im)))),256); Image F(u, v) log(1 + F(u, v) ) Synthetic example [i,j]=meshgrid(-19:19); im=abs(i)<3 & abs(j)<3; IM=fft2(im); figure; imshow(rescale(abs(fftshift(im))),256); figure; imshow(rescale(log(1+abs(fftshift(im)))),256); Image F(u, v) log(1 + F(u, v) ) 17 Filtering in the frequency domain Calculate Fourier transform F(u, v) = F {f(x, y)} IM=fft2(im); Create filter response H(u, v) H(u,v)=...; Multiply G(u, v) = F(u, v)h(u, v) G=IM.*H; Inverse transform g(x, y) = F 1 {G(u, v)} g=real(ifft2(g)); Ideal lowpass filtering 8 < 1 u 2 + v 2 H 0 H(u, v) = : 0 otherwise Image log(1 + F(u, v) ) H(u, v) log(1 + G(u, v) ) g(x, y) [N,M]=size(IM); uu=[0:floor((m-1)/2),-floor(m/2):-1]/m; vv=[0:floor((n-1)/2),-floor(n/2):-1]/n; [u,v]=meshgrid(uu,vv); 19

h(t) = In the discrete case it is defined as h(t) = 1 N N 1 n=0 f(y)g(t y)dy f(n)g(t n), where f(n) and g(n) must be of the same length and are assumed to be periodic with period N.

6 Gaussian lowpass filtering Gaussian highpass filtering H(u,v) = e k (u 2 +v 2 ) Image log(1 + F(u, v) ) H(u, v) log(1 + G(u, v) ) g(x, y) H(u,v) = 1 e k (u 2 +v 2 ) Image log(1 + F(u, v) ) H(u, v) log(1 + G(u, v) ) g(x, y) 21 Convolution The convolution of two complex-valued functions f(t), g(t), denoted f g is defined as in the continuous case. h(t) = In the discrete case it is defined as h(t) = 1 N N 1 n=0 f(y)g(t y)dy f(n)g(t n), where f(n) and g(n) must be of the same length and are assumed to be periodic with period N. Convolution Example Convolve the signal f(n) = [0,0,1, 1,4,4,0,0], n = 0,..., 7 with g(n) = [1,0, 1], n = 0, 1,2. Convolution sum for t = 0 is h(0) = 1/8(f(0)g(0) + f(1)g( 1) f(7)g( 7)). Flip g(n) about n = 0: g(n) = [1,0,0,0, 0,0, 1,0]. Convolve: 8h(0) = P f(n) g(0 n)= = 0 8h(1) = P f(n) g(1 n)= = 0 8h(2) = P f(n) g(2 n)= = 1 8h(3) = P f(n) g(3 n)= = 1 8h(4) = P f(n) g(4 n)= = 3 8h(5) = P f(n) g(5 n)= = 3 8h(6) = P f(n) g(6 n)= = 4 8h(7) = P f(n) g(7 n)= = 4 In reality, the padded zeros are of course not multiplied. 23

Convolution theorem The convolution theorem states that convolution in one domain corresponds to a multiplication in the other, i.e. Verification: f(t) g(t) F(u)G(u), f(t)g(t) F(u) G(u).

7 Convolution Example The convolution of f(n) = [0, 0, 1, 1, 4, 4, 0, 0] and g(n) = [1, 0, 1] is thus h(n) = [0, 0, 1, 1, 3, 3, 4, 4]/8. Convolution with the filter kernel g(n) finds the positive edges shifted one step to the right, i.e. the center of g(n). Convolution theorem The convolution theorem states that convolution in one domain corresponds to a multiplication in the other, i.e. Verification: f(t) g(t) F(u)G(u), f(t)g(t) F(u) G(u). F(u) = F {f(n)} = [10, i, 3 3i, i, 0, i, 3 + 3i, i] G(u) = F {g(n)} = [0, 1 + 1i, 2, 1 1i, 0, 1 + 1i, 2, 1 1i] H(u) = F(u)G(u) = [0, i, 6 6i, i, 0, i, 6 + 6i, i] h(n) = F 1 {H(u)} = [0, 0, 1, 1, 3, 3, 4, 4] 25 Edge detection example load gatlin; X=X(:,80+[1:480]); im=ind2gray(x,map); imshow(im); hx=[-1,-1,-1;0,0,0;1,1,1]; Ex=conv2(im,hx); hy=hx ; Ey=conv2(im,hy); figure; imshow(rescale(-abs(ex))); figure; imshow(rescale(-abs(ey))); figure; imshow(rescale(-sqrt(ex.ˆ2+ey.ˆ2))); Image I (I hx) (I hy) I hy 2 + I hx 2 Polynom multiplication The Fourier transform may be used to multiply polynomials! (x 2 1)(x 1) = x 3 x 2 x + 1. Convolution of a = [0, 0, 1, 0, 1] and b = [0, 0, 0, 1, 1] is [0, 1, 1, 1, 1]. 27

G52IVG, School of Computer Science, University of Nottingham

G52IVG, School of Computer Science, University of Nottingham Image Transforms Fourier Transform Basic idea 1 Image Transforms Fourier transform theory Let f(x) be a continuous function of a real variable x. The Fourier transform of f(x) is F ( u) f ( x)exp[ j2πux]