Contourlet

1
Contourlet

• Student Chao-Hsiung HongAdvisor Prof.
  Hsueh-Ming Hang

2
Outline

• Introduction
• Curvelet Transform
• Contourlet Transform
• Simulation Results
• Conclusion
• Reference

3
Outline

• Introduction
• Goal
• The failure of wavelet
• The inefficiency of wavelet
• Curvelet Transform
• Contourlet Transform
• Simulation Results
• Conclusion
• Reference

4
Goal

• Sparse representation for typical image with
  smooth contours
• Action is at the edges!!!

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The failure of wavelet

• 1-D Wavelets are well adapted to singularities
• 2-D
• Separable wavelets are only well adapted to
  point-singularity
• However, in line- and curve-singularities

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The inefficiency of wavelet

• Wavelet fails to recognize that boundary is
  smooth
• New require challenging non-separable
  constructions

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Outline

• Introduction
• Curvelet Transform
• Key idea
• Ridgelet
• Decomposition
• Non-linear approximation
• Problem
• Contourlet Transform
• Simulation Results
• Conclusion
• Reference

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Key Idea

• Optimal representation for function in R2 with
  curved singularities
• Anisotropy scaling relation for curves width
  length2

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Ridgelet(1)
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Ridgelet(2)

• Ridgelet functions
• ?a, b,?(x1, x2) a-1/2?((x1cos(?) x2sin(?)
  b)/a)
• x1cos(?) x2sin(?) constant, oriented at angel
  ?
• Essentially localized in the corona ? in 2a,
  2a1 and around the angel ?in the frequency
  domain
• Wavelet functions
• ?a, b,(x) a-1/2?((x b)/a)
• ?a1, b1,a2,b2(x) ?a1, b1,(x1)?a2, b2,(x2)

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Decomposition

• Segments of smooth curves would look straight in
  smooth windows ? can be captured efficiently by a
  local ridgelet transform
• Windows size and subband frequency are
  coordinated ?
• width length2

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Non-Linear Approximation

• Along a smooth boundary, at the scale 2-j
• Wavelet coefficient number O(2j)
• Curvelet coefficient number O(2j/2)
• Keep nonzero coefficient up to level J
• Wavelet error O(2-J)
• Curvelet error O(2-2J)

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Problem(1)

• Translates it into discrete world
• Block-based transform have blocking effects and
  overlapping windows to increase redundancy
• Polar coordinate
• Group the nearby coefficients since their
  locations are locally correlated due to the
  smoothness of the discontinuity curve
• Gather the nearby basis functions at the same
  scale into linear structure

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Problem(2)

• Multiscale and directional decomposition
• Multiscale decomposition capture point
  discontinuities
• Directional decomposition link point
  discontinuities into linear structures

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Outline

• Introduction
• Curvelet Transform
• Contourlet Transform
• Multiscale decomposition
• Directional decomposition
• Pyramid Directional Filter Banks
• Basis Functions
• Simulation Results
• Conclusion
• Reference

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Multiscale Decomposition(1)

• Laplacian pyramid (avoid frequency scrambling)

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Multiscale Decomposition(2)

• Multiscale subspaces generated by the Laplacian
  pyramid

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Directional Decomposition(1)

• Directional Filter Bank
• Division of 2-D spectrum into fine slices
• Use quincunx FBs, modulation, and shearing
• Test zone plate image decomposed by d DFB with 4
  levels that leads to 16 subbands

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Directional Decomposition(2)Sampling in
Multiple Dimensions

• Quincunx sampling lattice
• Downsample by 2
• Rotate 45 degree

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Directional Decomposition(3)Quincunx Filter Bank

• Diamond shape filter, or fan filter
• The black region represents ideal frequency
  supports of the filters
• Q quincunx sampling lattice

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Directional Decomposition(4)Directional Filter
Bank

• At each level QFBs with fan filters are used
• The first two levels of DFB

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Directional Decomposition(5)2 Level Directional
Filter Bank
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Directional Decomposition(8) 3 Level
Directional Filter Bank
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Pyramid Directional Filter Banks

• The number of directional frequency partition is
  decreased from the higher frequency bands to the
  lower frequency bands

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Basis Functions
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Outline

• Introduction
• Curvelet Transform
• Contourlet Transform
• Simulation Results
• Conclusion
• Reference

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Simulation Results
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Outline

• Introduction
• Curvelet Transform
• Contourlet Transform
• Simulation Results
• Conclusion
• Reference

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Conclusion

• Offer sparse representation for piecewise smooth
  images
• Small redundancy
• Energy compactness

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Outline

• Introduction
• Curvelet Transform
• Contourlet Transform
• Simulation Results
• Conclusion
• Reference

31
Reference

• M. N. Do and Martin Vetterli, The Finite
  Ridgelet Transform for Image Representation,
  IEEE Transactions on Image Processing, vol. 12,
  no. 1, Jan. 2003.
• M. N. Do, Directional Multiresolution Image
  Representations, Ph.D. Thesis, Department of
  Communication Systems, Swiss Federal Institute of
  Technology Lausanne, November 2001

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Thank you for your attention!

• Any questions?