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Topic 14
THE FREQUENCY DOMAIN
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Basic wave properties
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In a very real sense the A and B coefficients
define or fully describe the square wave
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The set of coefficients for the square wave
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Fourier series in 2 dimensions
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• using the FFT
• The transform works most efficiently when the
  image dimensions are multiples of 2 (e.g. 128,
  256 etc.). It will however work with other
  dimensional ranges.
• The transform generates both an amplitude and
  phase for each frequency component. Both are
  equally important but usually only the amplitude
  component is displayed because the phase is very
  difficult to interpret visually.
• As an alternative to amplitude and phase the
  transform can be expressed as a set of complex
  numbers Z X iY where X is the real part and Y
  the imaginary part. In this case it is usually
  the intensity which displayed.
• The transform is a periodic function of which
  only one cycle is displayed. The transform also
  assumes that the original image is periodic with
  only one cycle visible. Occasionally this can
  cause unwanted artefacts on the processed image.
• Left by itself the FFT will produce a graph in
  which the zero frequency appears in the corners
  and the highest frequency in the centre, MATLAB
  provides a simple function to rectify this.

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Using the Fourier transform in MATLAB
function FFTDemo(SW) To demonstrate the 2D Fast
Fourier Transform SW is the width (pixels) of a
white stripe in a 256 x 256 square A
zeros(256,256) if SW gt 50 SW 50 end LL
128 - round(SW/2) UL 128
round(SW/2) A(1255,LLUL) 1 subplot(1,2,1),
imshow(A) F fft2(A) F1 fftshift(F) F2
abs(F1) subplot(1,2,2), imshow(F2,-1
8,'notruesize') colormap(jet)
colorbar improfile
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The Fourier transform of a thin line
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The Fourier transform of a block
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Designing a low pass filter
The Butterworth filter
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The low pass Butterworth filter in practice
C 50 n 1
C 50 n 5
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function MothDemo(cutoff,order) To demonstrate
the 2D Fast Fourier Transform on a real object A
imread('moth9.gif') B double(A)/255 subplot(
2,2,1), imshow(B) Calculate and display its
Fourier transform F fft2(B) F1
fftshift(F) F2 abs(F1) subplot(2,2,2)
imshow(F2,-1 32)colormap(gray) Get the image
size row col size(B) Now define the
appropriate Butterworh Filter bwlpf
zeros(row,col) centre_row round(row/2) centre_
col round(col/2) for v 1col for u
1row bwlpf(u,v) 1 / (1
(sqrt((u-centre_row)2 (v-centre_col)2)/cutoff)
(2order)) end end bwlpf fftshift(bwlpf) su
bplot(2,2,3) imshow(bwlpf) F F .
bwlpf F_lpf real(ifft2(F)) subplot(2,2,4)
imshow(F_lpf)
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The high pass Butterworth filter
C 50 n 1
C 50 n 5
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