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Chapter 7: The Fourier Transform 7.1 Introduction • The Fourier transform allows us to perform tasks that would be impossible to perform any other way • It is more efficient to use the Fourier transform than a spatial filter for a large filter • The Fourier transform also allows us to isolate and process particular image frequencies 1 7.2 Background 2 FIGURE 7.2 • A periodic function may be written as the sum of sines and cosines of varying amplitudes and frequencies 3 7.2 Background Fourier series These are the equations for the Fourier series expansion of f (x), and they can be expressed in complex form 4 7.2 Background If the function is nonperiodic, we can obtain similar results by letting T → ∞, in which case Fourier transform pair 5 7.3 The One-Dimensional Discrete Fourier Transform 6 7.3 The One-Dimensional Discrete Fourier Transform • Definition of the One-Dimensional DFT This definition can be expressed as a matrix multiplication where F is an N × N matrix defined by 7 7.3 The One-Dimensional Discrete Fourier Transform Given N, we shall define e.g. suppose f = [1, 2, 3, 4] so that N = 4. Then 8 7.3 The One-Dimensional Discrete Fourier Transform 9 7.3 The One-Dimensional Discrete Fourier Transform • THE INVERSE DFT If you compare Equation (7.3) with Equation 7.2 you will see that there are really only two differences: 1. There is no scaling factor 1/N 2. The sign inside the exponential function has been changed to positive. 3. The index of the sum is u, instead of x 10 7.3 The One-Dimensional Discrete Fourier Transform 11 7.3 The One-Dimensional Discrete Fourier Transform 12 7.4 Properties of the One-Dimensional DFT • LINEARITY This is a direct consequence of the definition of the DFT as a matrix product Suppose f and g are two vectors of equal length, and p and q are scalars, with h = pf + qg If F, G, and H are the DFT’s of f, g, and h, respectively, we have • SHIFTING 13 Suppose we multiply each element xn of a vector x by (−1)n. In other words, we change the sign of every second element Let the resulting vector be denoted x’. The DFT X’ of x’ is equal to the DFT X of x with the swapping of the left and right halves 7.4 Properties of the One-Dimensional DFT e.g. 14 7.4 Properties of the One-Dimensional DFT Notice that the first four elements of X are the last four elements of X1 and vice versa 15 7.4 Properties of the One-Dimensional DFT • SCALING F where k is a scalar and F= f If you make the function wider in the x-direction, it's spectrum will become smaller in the x-direction, and vice versa Amplitude will also be changed • CONJUGATE SYMMETRY • CONVOLUTION 16 7.4 Properties of the One-Dimensional DFT • THE FAST FOURIER TRANSFORM 2n 17 7.5 The Two-Dimensional DFT • The 2-D Fourier transform rewrites the original matrix in terms of sums of corrugations 18 7.5.1 Some Properties of the TwoDimensional Fourier Transform • SIMILARITY • THE DFT AS A SPATIAL FILTER • SEPARABILITY 19 7.5.1 Some Properties of the TwoDimensional Fourier Transform • LINEARITY • THE CONVOLUTION THEOREM Suppose we wish to convolve an image M with a spatial filter S 1. Pad S with zeroes so that it is the same size as M; denote this padded result by S’ 2. Form the DFTs of both M and S’ to obtain (M)and (S’) 3. Form the element-by-element product of these two transforms: 4. Take the inverse transform of the result: 20 Put simply, the convolution theorem states or 7.5.1 Some Properties of the TwoDimensional Fourier Transform • THE DC COEFFICIENT • SHIFTING DC coefficient DC coefficient 21 7.5.1 Some Properties of the TwoDimensional Fourier Transform • CONJUGATE SYMMETRY • DISPLAYING YRANSFORMS fft, which takes the DFT of a vector, ifft, which takes the inverse DFT of a vector, fft2, which takes the DFT of a matrix, ifft2, which takes the inverse DFT of a matrix, and fftshift, which shifts a transform 22 7.6 Fourier Transforms in MATLAB e.g. Note that the DC coefficient is indeed the sum of all the matrix values 23 7.6 Fourier Transforms in MATLAB e.g. 24 7.6 Fourier Transforms in MATLAB e.g. 25 7.7 Fourier Transforms of Images 26 FIGURE 7.10 27 FIGURE 7.11 28 FIGURE 7.12 29 FIGURE 7.13 • EXAMPLE 7.7.2 30 FIGURE 7.14 • EXAMPLE 7.7.3 31 FIGURE 7.15 • EXAMPLE 7.7.4 32 7.7 Fourier Transforms of Images 33 7.8 Filtering in the Frequency Domain • Ideal Filtering LOW-PASS FILTERING 34 FIGURE 7.16 35 FIGURE 7.17 D = 15 36 7.8 Filtering in the Frequency Domain >> cfl = cf.*b >> cfli = ifft2(cfl); >> figure, fftshow(cfli, ’abs’) 37 FIGURE 7.18 D=5 38 D = 30 7.8 Filtering in the Frequency Domain HIGH-PASS FILTERING 39 FIGURE 7.19 40 FIGURE 7.20 41 7.8.2 Butterworth Filtering Ideal filtering simply cuts off the Fourier transform at some distance from the center It has the disadvantage of introducing unwanted artifacts (ringing) into the result One way of avoiding these artifacts is to use as a filter matrix, a circle with a cutoff that is less sharp 42 FIGURE 7.21 43 FIGURE 7.22 & 7.23 44 FIGURE 7.24 45 FIGURE 7.25 46 FIGURE 7.26 >> bl = lbutter(c,15,1); >> cfbl = cf.*bl; >> figure, fftshow(cfbl, ’log’); 47 >> cfbli = ifft2(cfbl); >> figure, fftshow(cfbli, ’abs’) FIGURE 7.27 48 7.8.3 Gaussian Filtering A wider function, with a large standard deviation, will have a low maximum 49 FIGURE 7.28 50 FIGURE 7.29 51 7.9 Homomorphic Filtering where f(x, y) is intensity, i(x, y) is the illumination and r(x, y) is the reflectance 52 7.9 Homomorphic Filtering 53 FIGURE 7.32 function res=homfilt(im,cutoff,order,lowgain,highgain) % HOMFILT(IMAGE,FILTER) applies homomorphic filtering % to the image IMAGE % with the given parameters u=im2uint8(im/256); u(find(u==0))=1; l=log(double(u)); ft=fftshift(fft2(l)); f=hb_butter(im,cutoff,order,lowgain,highgain); b=f.*ft; ib=abs(ifft2(b)); res=exp(ib); 54 FIGURE 7.33 >>i=imread(‘newborn.tif’); >>r=[1:256]’*ones(1,256); >>x=double(i).*(0.5+0.4*sin((r-32)/16)); >>imshow(i);figure;imshow(x/256); 55 FIGURE 7.34 >>xh=homfilt(x,10,2,0.5,2); >>imshow(xh/16); 56 FIGURE 7.35 >> >> >> >> 57 a=imread('arch.tif'); figure;imshow(a); a1=a(:,:,1); figure;imshow(a1); >> a2=double(a1); >> ah=homfilt(a2,128,2,0.5,2); >> figure;imshow(ah/14);