Image Based Information Processing

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Image Based Information Processing

• Lecturer Steve Maybank
• School of Computer Science and Information
  Systems
• sjmaybank_at_dcs.bbk.ac.uk
• Spring 2009
• Week 4The Fourier Transform and the Discrete
  Cosine Transform

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History

• Jean Baptiste Joseph Fourier, born 1768 in
  Auxerre, France.
• Discovers that any continuous function can be
  written as a sum of sines and cosines.
• Publishes memoir (1807) and book La Théorie
  Analytique de la Chaleur (1822).
• First applications to find solutions to the heat
  equation which describes heat flow in a solid.
• Mathematicians take 100 years to adjust to this
  discovery.

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Sines and Cosines
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Decomposition of a Function
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Discrete Cosine Transform

• Any discrete 1-dimensional image with n pixels
  can be written exactly as the sum of n cosines.
  Eg for x0, x1, x 2, there exist numbers a0, a1
  , a2 such that

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Matrix for the Discrete Cosine Transform

• The DCT coefficients a of a 1-dimensional image x
  with n pixels are defined by aMx where M is an
  nxn matrix.
• The entries of M and M-1 are cosines.
• The inverse transform is xM-1a, thus x is a
  linear combination of cosines.

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Properties of the DCT

• Does not require complex numbers, because M
  contains real numbers only.
• Linear if aMx and bMy then
• r as bM(r xs y)
• Preserves the length of vectors,
• aMxx
• Easily invertible, because M-1 MT.

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DCT for 2-Dimensional Images

• If I is a 2-dimensional nxn image then the DCT of
  I is the nxn matrix A defined by
• AMIMT
• Note that IMTAM

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Why Use the DCT?

• If an image contains uniform regions then only a
  few DCT coefficients are large.
• Most of the image information is recorded by a
  small number of coefficients

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Extreme Example
(and similarly for any nxn matrix with all
entries equal to 1)
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Example 1 Horizontal Edge
Absolute values of DCT coefficients
8x8 image
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Example 2 Diagonal Edge
Absolute values of DCT coefficients
8x8 image
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Example 3 Random Image
Absolute values of DCT coefficients
8x8 image
There is no advantage in finding the DCT
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First Few Components of a DCT Basis
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JPEG Compression

• The image is divided into 8x8 blocks and the DCT
  is applied to each block.
• The smaller DCT coefficients are set to zero.
• The more coefficients that are set to zero, the
  larger the compression.

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Fourier Transform

• The image is expressed as a sum of sines and
  cosines.
• The Fourier coefficients, F, of a 2-dimensional
  image I are given by
• FUIUT
• The matrix U contains complex values.
• The length is preserved, FI.

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FT and Filtering

• Low pass filtering reduce the high frequency
  Fourier coefficients in F.
• Effect smoothes sharp edges, reduces high
  frequency noise.
• High pass filtering reduce the low frequency
  coefficients in F.
• Effect emphases edges, increases contrast.

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Images
Magnitudes of the Fourier coefficients. High
frequencies at The centre.
Original image (Microsoft)
Grey level image
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Low Pass Filter Example
There are 240,000 Fourier coefficients of which
228,000 are set to zero. Note the blurring.
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High Pass Filter Example
Twenty Fourier coefficients out of 240,000 (!!)
are set to zero. Regions where the grey level
gradient is high are emphasised.
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The FT and Linear Filtering

• The FT can be used to implement linear filtering
  efficiently (convolution theorem).

FT
Iimage Mmask
rij(I), rij(M)
FT-1
Products rij(I)rij(M)
Linearly filtered image