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## Chapter 7: The Fourier Transform 7.1 Introduction

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**Chapter 7:The Fourier Transform7.1 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**FIGURE 7.2**• A periodic function may be written as the sum of sines and cosines of varying amplitudes and frequencies**7.2 Background**• These are the equations for the Fourier series expansion of f (x), and they can be expressed in complex form • Fourier series**7.2 Background**• If the function is nonperiodic, we can obtain similar results by letting T → ∞, in which case • Fourier transform pair**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**7.3 The One-Dimensional Discrete Fourier Transform**• Given N, we shall define • e.g. suppose f = [1, 2, 3, 4] so that N = 4. Then**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: • There is no scaling factor 1/N • The sign inside the exponential function has been changed to positive. • The index of the sum is u, instead of x**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 • 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**Notice that the first four elements of X are the last four elements of X1 and vice versa**7.4 Properties of the One-Dimensional DFT**• SCALING • 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 F • CONJUGATE SYMMETRY • CONVOLUTION**7.4 Properties of the One-Dimensional DFT**• THE FAST FOURIER TRANSFORM 2n**7.5 The Two-Dimensional DFT**• The 2-D Fourier transform rewrites the original matrix in terms of sums of corrugations**7.5.1 Some Properties of the Two-Dimensional Fourier**Transform • SIMILARITY • THE DFT AS A SPATIAL FILTER • SEPARABILITY**7.5.1 Some Properties of the Two-Dimensional Fourier**Transform • LINEARITY • THE CONVOLUTION THEOREM • Suppose we wish to convolve an image M with a spatial filter S • Pad S with zeroes so that it is the same size as M; denote this padded result by S’ • Form the DFTs of both M and S’ to obtain (M)and (S’) • Form the element-by-element product of these two transforms: • Take the inverse transform of the result: • Put simply, the convolution theorem states • or**7.5.1 Some Properties of the Two-Dimensional Fourier**Transform • THE DC COEFFICIENT • SHIFTING DC coefficient DC coefficient**7.5.1 Some Properties of the Two-Dimensional 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**7.6 Fourier Transforms in MATLAB**e.g. Note that the DC coefficient is indeed the sum of all the matrix values**FIGURE 7.13**• EXAMPLE 7.7.2**FIGURE 7.14**• EXAMPLE 7.7.3**FIGURE 7.15**• EXAMPLE 7.7.4**7.8 Filtering in the Frequency Domain**• Ideal Filtering • LOW-PASS FILTERING**FIGURE 7.17**D = 15**>> cfl = cf.*b**>> cfli = ifft2(cfl); >> figure, fftshow(cfli, ’abs’) 7.8 Filtering in the Frequency Domain**FIGURE 7.18**D = 5 D = 30**7.8 Filtering in the Frequency Domain**• HIGH-PASS FILTERING**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**>> bl = lbutter(c,15,1);**>> cfbl = cf.*bl; >> figure, fftshow(cfbl, ’log’); >> cfbli = ifft2(cfbl); >> figure, fftshow(cfbli, ’abs’) FIGURE 7.26**7.8.3 Gaussian Filtering**A wider function, with a large standard deviation, will have a low maximum