Review Questions

1
Review Questions

• Jyun-Ming Chen
• Spring 2001

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Wavelet Transform

• What is wavelet?
• How is wavelet transform different from Fourier
  transform?

• Wavelets are building blocks that can quickly
  decorrelate data. Wim Sweldens
• Wavelets are optimal bases for compressing,
  estimating, and recovering functions - David
  Donoho
• Both try to represent a function in other basis
  (transform into other domain) and hope this
  transformation can reveal some insights. Yet,
  unlike Fourier transform, wavelet can choose many
  different basis.

3
On Details of Wavelet Transform

• Describe the concepts of filter banks
• Analysis
• Synthesis
• MRA (multi-resolution analysis)

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Formal Definition of an MRA

• An MRA consists of the nested linear vector space
  such that
• There exists a function f(t) (called scaling
  function) such that is
  a basis for V0
• If and
  vice versa
• Remarks
• Does not require the set of f(t) and its integer
  translates to be orthogonal (in general)
• No mention of wavelet

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Details (cont)

• The roles of scaling functions and wavelets
• Basis functions in V and W
• Refinement (two-scale) relations

• Graphing by cascading
• Computing wavelet coefficients (orthogonal)

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Important Properties of Fourier Transform

• Linearity
• Time shifting
• Time scaling

• Parsevals theorem

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On Wavelet Coefficients
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Orthogonal MRA
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Biorthogonal MRA
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Semiorthogonal MRA

• Common property
• Differences
• if orthogonal scaling functions (and wavelets)
  of the same level are orthogonal to each other
• If semiorthogonal, wavelets of different levels
  are orthogonal (from nested space)

Dual and primal are the same
11
Comparison of Different Types of Wavelet
Transforms

• Whats the advantage of orthgonality?
• Why choose to design biorthogonal wavelets
  (instead of orthogonal wavelets)?

12
On Lifting

• What kind of wavelet transform does lifting
  produce?
• What are the advantages of lifting?

• In-place computation
• Easy inversion
• Extensible to 2nd generation wavelets
• More efficient computations

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Details of Lifting

• Types of predictors
• Interpolating
• Average-interpolating
• B-spline
• Types of Update
• Number of vanishing moments of the wavelets

• Characteristics of the transform
• MRA order
• Dual MRA order
• Polynomial reproducibility and vanishing moments
• Cascading algorithms
• Lifting theory
• why it ensures biorthogonality
• Exact reconstruction guaranteed

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Applications of Wavelet Transform

• Denoising
• Compression
• Progressive Transmission
• Geometric simplification
• MR Editing

• Feature recognition
• Graphics related

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On Variations of Wavelet Transform

• What is continuous wavelet transform?
• What is fast wavelet transform?
• What is wavelet packet?

• What types of information does each one reveal?

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• Derivatives of phi, psi?