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
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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
Fourier series
 These are the equations for the Fourier series
expansion of f (x), and they can be expressed in
complex form
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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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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
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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
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 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.
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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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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
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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 TwoDimensional Fourier Transform
• SIMILARITY
• THE DFT AS A SPATIAL FILTER
• SEPARABILITY
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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:
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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
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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
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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 7.18
D=5
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D = 30
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
>> bl = lbutter(c,15,1);
>> cfbl = cf.*bl;
>> figure, fftshow(cfbl, ’log’);
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>> cfbli = ifft2(cfbl);
>> figure, fftshow(cfbli, ’abs’)
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 7.32
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);
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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
>>
>>
>>
>>
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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);