BAE 790I BMME 231 Fundamentals of Image Processing Class 14

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BAE 790I / BMME 231Fundamentals of Image
ProcessingClass 14

• Projects
• Unitary Transforms
• Definitions
• Properties
• Cosine transform
• Hadamard transform
• Haar transform
• Karhuen-Loeve transform

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Fourier Basis Imagefrom B(fx,fy) d(fx 11, fy
42)

Real
Imaginary
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Basis Images
An image can be represented as a weighted sum of
basis images.
Dot product or inner product
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Basis Images
In matrix-vector form
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Basis Images
Orthonormality
Completeness
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Basis Images

• Any basis transformation can be written in matrix
  form
• Where a is a vector of basis coefficients
• B is a matrix with basis images lying along its
  rows

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Unitary Transform

• A unitary transform has the property that B is a
  unitary matrix
• In the inverse transform, the conjugate basis
  images lie along columns of the matrix.

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Unitary Fourier Transform

• The Fourier Transform can be a unitary transform

Note that this is not the form we have used in
our programs.
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Fourier transform filtering

• Apply FT
• Filter (D is diagonal)
• Invert FT

The same process can be used with any unitary
transform! It is linear, but not necessarily SI.
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Unitary Transforms

• Some transformations are separable
• May be more efficient transform along one axis,
  then along the other
• Fourier Transform is separable

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Unitary Transforms
Properties

• Energy Conservation

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Unitary Transforms
Properties

• Statistics

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Cosine Transform

• Related to FT, but not same.
• All real B-1 BT
• Separable, fast
• Used in compression, but can filter also.

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Hadamard Transform

• All values 1 or -1
• Very fast, no multiplications
• Not frequency, sequency (number of zero
  crossings)
• Transform is real and symmetric B B-1
• Used in compression and filtering

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Hadamard Basis Images

Sequency 7 x 3
Sequency 31 x 202
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Haar Transform

• Uses base elements of (1 1) and (1 -1) at
  different scales.
• For eight points in 1D, there are eight basis
  functions.
• A DC term
• One at scale 8
• Two at scale 4
• Four at scale 2

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Haar Transform

• In 2D, use separable products of the 1D bases.
• Fast (1, 0, -1 only)
• All real
• Spatial localization Can do spatially-variant
  filtering
• The elementary wavelet

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Haar Basis Images

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Filtering with Haar Basis

Remove all 2x2
Remove all 4x4
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Karhunen-Loeve Transform

• Also Hotelling Transform, Principal Components
• Consider a random process that yields an image n
  and its autocorrelation Rnn
• Find the eigenvalue decomposition of Rnn

Eigenvector k
Matrix of eigenvectors
Diagonal matrix of eigenvalues
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Karhunen-Loeve Transform

• F is unitary.
• The KL transform of n is
• Note

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Karhunen-Loeve Transform

• Important effect on autocorrelation
• The elements of the transformed space are
  uncorrelated!
• The KL transform takes a correlated random
  process and decorrelates it.

Diagonal!
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Karhunen-Loeve Transform

• Alternatively, we can diagonalize the
  autocovariance matrix
• The 1D unitary discrete Fourier transform is the
  KL transform for all periodic random sequences
  (Rnn is circulant).
• Not necessarily true for 2D.