The Fourier transform

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• Lecture 13
• The Fourier transform
• Image compression using (DCT) discrete
• cosine transform
• Texture analysis

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One-dimensional, continuous Fourier
transformation I

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One-dimensional, continuous Fourier
transformation II

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One-dimensional discrete Fourier transform

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Ambiguity of periodic sampling

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Two-dimensional discrete Fourier transform I

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Two-dimensional discrete Fourier transform II

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Fast Fourier transform (FFT) Computational
advantages are obtained if the the side length M
and N are integer powers of 2 The algorithm base
on Fast Fourier transform has a complexity
of (NlogN)(MlogM) rather than (MN)2

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Visualization of the Fourier transform In the
following slides the capitial I stands for an
image, F stands for the Fourier transform of,
IF stands for the inverse Fourier transform of,
and stands for the modulus of the interior

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Visualisation of the 2 D Fourier transform
I F(I) I
F(I)


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Manipulation in the frequency domain
Blurring Edge detected
I1 I2 IF(F(I1) I2)

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Visualisation of template matching using cross
correlation
I1 I2 IF(F(I1)F(I2)

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Image restoration Let an image I0 be blurred in
such a way that the blurred image I0 is a
convolution of I0 and a mask Imask Sometimes the
blurred image and the mask is given, but the
original image is not. Using Fourier transform
one has F(I0) F(I0)F(Imask). Thus I0
IFF(I0)/F(Imask) However F(Imask) may have
amplitudes close to zero, so noise waves with
directions and wave lengths of such amplitudes
will be amplified. So the technique has severe
limitations.

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• Applications of Fourier transforms in image
  processing and analysis
• Image restoration
• Template matching based on cross
• correlation
• Convolution where the mask to be
• convolved with the image is large
• Texture analysis

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• Image compression
• Loss-free compression
• Two so called entropy coding methods
• Hufman coding
• Arithmetic coding
• Runlength coding, predictor base coding, etc.
• Example Give each word in a language a numerical
  
• code, the most frequently used words short codes.
  Then
• a text can be compressed to about half the size
  compared
• to the representation with 7-8 bits per
  character.
• Lossy (cheat-the-eye) compression
• 8x8 discrete cosine transform (old JPEG )
• Wavelet transform (new JPEG)
• Extension to motion pictures MPEG
• ) Joint Photographic Expert Group

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The forward discrete cosine transform
The inverse discrete cosine transform
In JPEG N8
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Texture analysis
Can we classify regions wringles (forhead and
cheeks), hair, dress, uniform background?

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Statistical approach to texture
analysis Co-occurrence matrix Matrix
M0..cmax-10..cmax-1 of floating point
elements Graytones c(x,y) 0 .. cmax-1 Input
is the considered N N subimage and a
displacement vector D connecting a pixel pair
under consideration Pseudo code Set all elements
of M to zero Repeat for all pixel positions P1
P2 P1D Increment matrix element
Mc(P1) c(P2) by (1/N2) Usually cmax is
between 4 and 16, and N 32-64. Each
displacement vector D defines a co-occurrence
matrix. Usually D is (1,1), (1,0) or analogous
vectors


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• Numerical texture descriptors
• M is normalized, that
• Si Sj Mij 1
• 1) Maximum probability max(Mij)
• 2) Element difference moment of order k
• Si Sj (i-j)k Mij
• 3) Inverse element difference moment of order k
• Si Sj Mij/(i-j)k
• 4) Uniformity Si Sj Mij2
• 5) Entropy -Si Sj Mij log(Mij)

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Spectral approach to texture analysis Let C(kx,
ky) be the Fourier transform of c(x,y) using a N
x N subimage (N is a power of 2 so that FFT can
be used) kx and ky is in the range -1/(2N). .
1/(2N) Let A(ik) be average amplitudes of C
along rings of radius k sqrt(kx2ky2) (ik)Dk
and width Dk, where ik is an integer. Radial
moments of order q mq S ik A(ik)ikq are good
descriptors for texture analysis. C(0,0) is the
average graytone, so mq/C(0,0) is invariant to
scaling of the greytone.