ASAM Image Processing 20082009

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ASAM - Image Processing2008/2009

• Lecture 6
• The frequency domain

Ioannis Ivrissimtzis 13-Nov-2008
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Overview

• Basis change
• Walsh - Hadamard transform
• Tensor product transforms

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Spatial/Frequency domain

• The analysis and processing of an image can be
  done in different
• domains
• The spatial domain where values correspond to
  pixel intensities.
• The frequency domain.
• Until now we have always worked in the spatial
  domain.

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Basis change

• Example
• Consider two numbers a and b.
• We can represent a and b by themselves, but we
  can also represent
• them by two different numbers, for example,
  (ab)/2 and (a-b)/2.
• Indeed, if we know a and b then we can find
  (ab)/2, (a-b)/2.
• If we know (ab)/2 and (a-b)/2 then we can find
  a and b.

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Basis change

• In matrix language we have

for the first representation and
for the second.
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Basis change

• Terminology

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Basis change

• Why use a more complicated basis?
• In some applications we may have to work with an
  incomplete set of
• coefficients.
• In the previous example, in the first base, the
  first coefficient is a. In
• the second base, the first coefficient is (ab)/2
  which is more
• representative.
• In image transmission we prefer the first 10 of
  the data to give an
• approximation of the whole image, rather than an
  exact description of
• a small part of it.

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Example

• The four matrices below form a basis for the 4x1
  matrices.
• We can write any other 4x1 matrix as a linear
  combination of them, in a
• unique way.

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Example

• Writing a 4x1 matrix in this basis is trivial. We
  have

giving,
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Example

• For example
• We call this base, the natural base.

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Example

• A different basis for the 4x1 matrices
• We can write any other 4x1 matrix as a linear
  combination of the four
• matrices above, in a unique way.

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Basis change

• How do we write a 4x1 matrix in the new basis?

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Basis change

• Let the coefficients be the unknowns
  .

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Example

• We can rewrite the equation as a linear system in
  matrix form.

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Example

• To solve the system we invert the transformation
  matrix.

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Example

• The inverse of this matrix is

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Example
We get,

• This is called the transform of this.

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Example

• The original basis

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Example

• The new basis

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Overview

• Basis change
• Walsh - Hadamard transform
• Tensor product transforms

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The general W-H transform

• The Hadamard matrix Hn of order 2n is defined
  recursively by

and
A sequency ordering of its rows will give the
corresponding Walsh- Hadamard transform.
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Walsh-Hadamard transform

• The Walsh-Hadamard transform of order 8 is given
  by the matrix

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Walsh-Hadamard transform

• The number of sign changes in a row of the matrix
  is called sequency.

0 1 2 3 4 5 6 7
The rows of the Walsh-Hadamard matrix have been
reordered by sequency.
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Walsh-Hadamard transform

• The natural basis for 8x1 matrices

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Walsh-Hadamard transform
The Walsh-Hadamard basis for 8x1 matrices
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Two dimensional W-H transform

• The 2D Walsh-Hadamard transform is the tensor of
  the 1D transform.
• Example Every 4x4 greyscale image can be
  uniquely written in the
• Walsh-Hadamard basis as linear combination of
  these 16 images.

The white squares denote 1s and the black
squares denote -1s.
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Two dimensional W-H transform
How do we compute these sixteen images? Take
the corresponding elements of the 1D basis and
find their tensor product.
(1,-1,1,-1)

• (1,1,-1,-1)

(1,-1,-1,1)

• (1,1,1,1)

• (1,1,1,1)

• (1,1,-1,-1)

(1,-1,-1,1)
(1,-1,1,-1)
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Two dimensional W-H transform
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Overview

• Basis change
• Walsh - Hadamard transform
• Tensor product transforms

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Tensor product transforms

• How can we compute T(F), the W-H transform of a
  2D image F ?
• It may seem that we have to solve a large and
  complicated linear
• system.
• In fact, the 2D W-H transform is computed
  directly by
• where H is the W-H matrix of the 1D transform and
  H' is the transpose
• of H.

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Tensor product transforms

• That is, to find the W-H transform of an image,
  we multiply it with the
• Hadamard matrix from the left and its transpose
  from the right.

The original Image A
The transform of A
The Hadamard matrix H
The transpose of H
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Tensor product transforms

• To see why this happens we first need to
  introduce the notion of
• orthogonality.
• We say that a matrix is orthogonal if its inverse
  is equal to its transpose.

The Hadamard matrices are orthogonal.
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Tensor product transforms

• Consider an image with all its pixels equal to 0,
  except one which has
• value 1. Notice that this is an element of the
  natural basis. We have

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Tensor product transforms

• This is equivalent to the tensor product of the
  two corresponding 1D basis
• images

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Tensor product transforms

• To put it all together, let B be an orthogonal
  matrix corresponding to an
• 1D basis. Then, TB-1 is the transform matrix.
• Let A be a 2D image. We have to show that
  ZTAT' is the transform of
• A corresponding to the tensor product of the 1D
  basis B.
• To see this, let be the (u, v)
  elements of the 2D natural and
• transform basis, respectively.

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Tensor product transforms

• The latter means that the matrix Z gives indeed
  the coefficients for
• writing A in the tensor product basis of B.