Wavelets and Multi-resolution Processing

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Chapter 7

• Wavelets and Multi-resolution Processing

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Background
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Image Pyramids
Total number of elements in a P1 level pyramid
for Pgt0 is
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Example
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Subband Coding

• An image is decomposed into a set of band-limited
  components, called subbands, which can be
  reassembled to reconstruct the original image
  without error.

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Z-Transform

• The Z-transform of sequence x(n) for n0,1,2 is
• Down-sampling by a factor of 2
• Up-sampling by a factor of 2

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Z-Transform (contd)

• If the sequence x(n) is down-sampled and then
  up-sampled to yield x(n), then
• From Figure 7.4(a), we have

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Error-Free Reconstruction

• Matrix expression
• Analysis modulation matrix Hm(z)

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FIR Filters

• For finite impulse response (FIR) filters, the
  determinate of Hm is a pure delay, i.e.,
• Let a2
• Let a-2

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Bi-orthogonality

• Let P(z) be defined as
• Thus,
• Taking inverse z-transform
• Or,

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Bi-orthogonality (Contd)

• It can be shown that
• Or,
• Examples Table 7.1

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Table 7.1
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2-D Case
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Daubechies Orthonormal Filters
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Example 7.2
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The Haar Transform

• Oldest and simplest known orthonormal wavelets.
• THFH where
• F NXN image matrix,
• H NxN transformation matrix.
• Haar basis functions hk(z) are defined over the
  continuous, closed interval 0,1 for k0,1,..N-1
  where N2n.

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Haar Basis Functions
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Example
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Multiresolution Expansions

• Multiresolution analysis (MRA)
• A scaling function is used to create a series of
  approximations of a function or image, each
  differing by a factor of 2.
• Additional functions, called wavelets, are used
  to encode the difference in information between
  adjacent approximations.

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Series Expansions

• A signal f(x) can be expressed as a linear
  combination of expansion functions
• Case 1 orthonormal basis
• Case 2 orthogonal basis
• Case 3 frame

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Scaling Functions

• Consider the set of expansion functions composed
  of integer translations and binary scaling of the
  real, square-integrable function, ,i.e.,
• By choosing j wisely, jj,k(x) can be made to
  span L2(R)

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Haar Scaling Function
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MRA Requirements

• Requirement 1 The scaling function is orthogonal
  to its integer translates.
• Requirement 2The subspaces spanned by the
  scaling function at low scales are nested within
  those spanned at higher resolutions.
• Requirement 3The only function that is common to
  all Vj is f(x)0
• Requirement 4 Any function can be represented
  with arbitrary precision.

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Wavelet Functions
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Wavelet Functions

• A wavelet function, y(x), together with its
  integer translates and binary scalings, spans the
  difference between any two adjacent scaling
  subspace, Vj and Vj1.

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Haar Wavelet Functions
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Wavelet Series Expansion
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Harr Wavelet Series Expansion of yx2
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Discrete Wavelet Transform
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The Continuous Wavelet Transform
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Misc. Topics

• The Fast Wavelet Transform
• Wavelet Transform in Two Dimensions
• Wavelet Packets