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

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7.2 Background

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FIGURE 7.2• A periodic function may be written as the sum of sines and

cosines of varying amplitudes and frequencies

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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

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7.2 Background If the function is nonperiodic, we can obtain similar

results by letting T → ∞, in which case

Fourier transform pair

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7.3 The One-Dimensional Discrete Fourier Transform

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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

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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

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7.3 The One-Dimensional Discrete Fourier Transform

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• 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

7.3 The One-Dimensional Discrete Fourier Transform

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7.3 The One-Dimensional Discrete Fourier Transform

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7.3 The One-Dimensional Discrete Fourier Transform

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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

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7.4 Properties of the One-Dimensional DFT

e.g.

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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

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• SCALINGwhere 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

7.4 Properties of the One-Dimensional DFT

F

• CONJUGATE SYMMETRY• CONVOLUTION

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7.4 Properties of the One-Dimensional DFT

• THE FAST FOURIER TRANSFORM

2n

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7.5 The Two-Dimensional DFT

• The 2-D Fourier transform rewrites the original matrix in terms of sums of corrugations

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7.5.1 Some Properties of the Two-Dimensional Fourier Transform

• SIMILARITY

• THE DFT AS A SPATIAL FILTER

• SEPARABILITY

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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’)3. Form the element-by-element product of these two

transforms:4. Take the inverse transform of the result:

Put simply, the convolution theorem states or

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7.5.1 Some Properties of the Two-Dimensional Fourier Transform

• THE DC COEFFICIENT

• SHIFTINGDC coefficientDC coefficient

DC coefficientDC coefficient

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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, andfftshift, which shifts a transform

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7.6 Fourier Transforms in MATLAB

e.g.

Note that the DC coefficient is indeed the sum of all the matrix values

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7.6 Fourier Transforms in MATLAB

e.g.

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7.6 Fourier Transforms in MATLAB

e.g.

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7.7 Fourier Transforms of Images

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FIGURE 7.10

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FIGURE 7.11

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FIGURE 7.12

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FIGURE 7.13• EXAMPLE 7.7.2

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FIGURE 7.14• EXAMPLE 7.7.3

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FIGURE 7.15• EXAMPLE 7.7.4

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7.7 Fourier Transforms of Images

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7.8 Filtering in the Frequency Domain• Ideal Filtering

LOW-PASS FILTERING

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FIGURE 7.16

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FIGURE 7.17

D = 15

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7.8 Filtering in the Frequency Domain

>> cfl = cf.*b>> cfli = ifft2(cfl);>> figure, fftshow(cfli, ’abs’)

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FIGURE 7.18D = 5 D = 30

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7.8 Filtering in the Frequency Domain

HIGH-PASS FILTERING

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FIGURE 7.19

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FIGURE 7.20

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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

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FIGURE 7.21

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FIGURE 7.22 & 7.23

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FIGURE 7.24

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FIGURE 7.25

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FIGURE 7.26

>> cfbli = ifft2(cfbl);>> figure, fftshow(cfbli, ’abs’)

>> bl = lbutter(c,15,1);>> cfbl = cf.*bl;>> figure, fftshow(cfbl, ’log’);

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FIGURE 7.27

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7.8.3 Gaussian Filtering

A wider function, with a large standard deviation, will have a low maximum

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FIGURE 7.28

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FIGURE 7.29

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7.9 Homomorphic Filtering

where f(x, y) is intensity, i(x, y) is the illumination and r(x, y) is the reflectance

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7.9 Homomorphic Filtering

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FIGURE 7.32function 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);

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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);

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FIGURE 7.34>>xh=homfilt(x,10,2,0.5,2);

>>imshow(xh/16);

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FIGURE 7.35>> 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);