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Communication & Multimedia C. -H. Hong 112/04/18
Contourlet
Student: Chao-Hsiung HongStudent: Chao-Hsiung HongAdvisor: Prof. Hsueh-Ming HangAdvisor: Prof. Hsueh-Ming Hang
Communication & Multimedia C. -H. Hong 112/04/18
OutlineOutline
Introduction
Curvelet Transform
Contourlet Transform
Simulation Results
Conclusion
Reference
Communication & Multimedia C. -H. Hong 112/04/18
OutlineOutline
IntroductionGoalThe failure of waveletThe inefficiency of wavelet
Curvelet TransformContourlet TransformSimulation ResultsConclusionReference
Communication & Multimedia C. -H. Hong 112/04/18
Goal
Sparse representation for typical image with smooth contoursAction is at the edges!!!
Communication & Multimedia C. -H. Hong 112/04/18
The failure of wavelet
1-D: Wavelets are well adapted to singularities2-D:
Separable wavelets are only well adapted to point-singularityHowever, in line- and curve-singularities…
Communication & Multimedia C. -H. Hong 112/04/18
The inefficiency of wavelet
Wavelet: fails to recognize that boundary is smoothNew: require challenging non-separable constructions
Communication & Multimedia C. -H. Hong 112/04/18
OutlineOutline
IntroductionCurvelet Transform
Key ideaRidgeletDecompositionNon-linear approximationProblem
Contourlet TransformSimulation ResultsConclusionReference
Communication & Multimedia C. -H. Hong 112/04/18
Key IdeaOptimal representation for function in R2 with curved singularitiesAnisotropy scaling relation for curves: width ≈ length2
Communication & Multimedia C. -H. Hong 112/04/18
Ridgelet(1)
Communication & Multimedia C. -H. Hong 112/04/18
Ridgelet(2)
Ridgelet functionsψa, b,θ(x1, x2) = a-1/2ψ((x1cos(θ)+ x2sin(θ) – b)/a)
x1cos(θ)+ x2sin(θ) = constant, oriented at angel θ
Essentially localized in the corona |ω| in [2a, 2a+1] and around the angel θin the frequency domain
Wavelet functionsψa, b,(x) = a-1/2ψ((x – b)/a)
ψa1, b1,a2,b2(x) = ψa1, b1,(x1)ψa2, b2,(x2)
Communication & Multimedia C. -H. Hong 112/04/18
Decomposition
Segments of smooth curves would look straight in smooth windows → can be captured efficiently by a local ridgelet transformWindow’s size and subband frequency are coordinated → width ≈ length2
Communication & Multimedia C. -H. Hong 112/04/18
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 JWavelet: error ≈ O(2-J)Curvelet: error ≈ O(2-2J)
Communication & Multimedia C. -H. Hong 112/04/18
Problem(1)
Translates it into discrete worldBlock-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
Communication & Multimedia C. -H. Hong 112/04/18
Problem(2)
Multiscale and directional decompositionMultiscale decomposition: capture point discontinuitiesDirectional decomposition: link point discontinuities into linear structures
Communication & Multimedia C. -H. Hong 112/04/18
OutlineOutline
IntroductionCurvelet TransformContourlet Transform
Multiscale decompositionDirectional decompositionPyramid Directional Filter BanksBasis Functions
Simulation ResultsConclusionReference
Communication & Multimedia C. -H. Hong 112/04/18
Multiscale Decomposition(1)
Laplacian pyramid (avoid frequency scrambling)
Communication & Multimedia C. -H. Hong 112/04/18
Multiscale Decomposition(2)
Multiscale subspaces generated by the Laplacian pyramid
Communication & Multimedia C. -H. Hong 112/04/18
Directional Decomposition(1)
Directional Filter BankDivision of 2-D spectrum into fine slicesUse quincunx FB’s, modulation, and shearing
Test: zone plate image decomposed by d DFB with 4 levels that leads to 16 subbands
Communication & Multimedia C. -H. Hong 112/04/18
Directional Decomposition(2):Sampling in Multiple Dimensions
Quincunx sampling latticeDownsample by 2Rotate 45 degree
(a) (b) (c)
Communication & Multimedia C. -H. Hong 112/04/18
Directional Decomposition(3):Quincunx Filter Bank
Diamond shape filter, or fan filterThe black region represents ideal frequency supports of the filters
Q: quincunx sampling lattice
Communication & Multimedia C. -H. Hong 112/04/18
Directional Decomposition(4):Directional Filter Bank
At each level QFB’s with fan filters are used
The first two levels of DFB
Communication & Multimedia C. -H. Hong 112/04/18
Directional Decomposition(5):2 Level Directional Filter Bank
H1
H0
F1
F0
H1
H0
H1
H0
F0
F0
F1
F1
d_Q0 Q0
d_Q0 Q0
Q0
Q0
d_Q0 Q1
d_Q0 Q1
d_Q0 Q1
d_Q0 Q1
Q1
Q1
Q1
Q1
Stage 1 Stage 2 Stage 2Analysis Synthesis
Stage 3
Communication & Multimedia C. -H. Hong 112/04/18
Directional Decomposition(8): 3 Level Directional Filter Bank
0 2
347
01
1
2
34
5
5 6
6
7
ω 2
ω 1
(π ,π )
(-π ,-π )
Communication & Multimedia C. -H. Hong 112/04/18
Pyramid Directional Filter Banks
The number of directional frequency partition is decreased from the higher frequency bands to the lower frequency bands
Communication & Multimedia C. -H. Hong 112/04/18
Basis Functions
Communication & Multimedia C. -H. Hong 112/04/18
OutlineOutline
Introduction
Curvelet Transform
Contourlet Transform
Simulation Results
Conclusion
Reference
Communication & Multimedia C. -H. Hong 112/04/18
Simulation Results
Communication & Multimedia C. -H. Hong 112/04/18
OutlineOutline
Introduction
Curvelet Transform
Contourlet Transform
Simulation Results
Conclusion
Reference
Communication & Multimedia C. -H. Hong 112/04/18
Conclusion
Offer sparse representation for piecewise smooth images
Small redundancy
Energy compactness
Communication & Multimedia C. -H. Hong 112/04/18
OutlineOutline
Introduction
Curvelet Transform
Contourlet Transform
Simulation Results
Conclusion
Reference
Communication & Multimedia C. -H. Hong 112/04/18
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
Communication & Multimedia C. -H. Hong 112/04/18
Thank you for your attention!
Any questions?