communication & multimedia c. -h. hong 2015/6/12 contourlet student: chao-hsiung hong advisor:...

Communication & Multimedia C. -H. Hong 112/04/18

Contourlet

Student: Chao-Hsiung HongStudent: Chao-Hsiung HongAdvisor: Prof. Hsueh-Ming HangAdvisor: Prof. Hsueh-Ming Hang


OutlineOutline

Introduction

Curvelet Transform

Contourlet Transform

Simulation Results

Conclusion

Reference


OutlineOutline

IntroductionGoalThe failure of waveletThe inefficiency of wavelet

Curvelet TransformContourlet TransformSimulation ResultsConclusionReference


Goal

Sparse representation for typical image with smooth contoursAction is at the edges!!!


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…


The inefficiency of wavelet

Wavelet: fails to recognize that boundary is smoothNew: require challenging non-separable constructions


OutlineOutline

IntroductionCurvelet Transform

Key ideaRidgeletDecompositionNon-linear approximationProblem

Contourlet TransformSimulation ResultsConclusionReference


Key IdeaOptimal representation for function in R2 with curved singularitiesAnisotropy scaling relation for curves: width ≈ length2


Ridgelet(1)


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)


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


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)


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


Problem(2)

Multiscale and directional decompositionMultiscale decomposition: capture point discontinuitiesDirectional decomposition: link point discontinuities into linear structures


OutlineOutline

IntroductionCurvelet TransformContourlet Transform

Multiscale decompositionDirectional decompositionPyramid Directional Filter BanksBasis Functions

Simulation ResultsConclusionReference


Multiscale Decomposition(1)

Laplacian pyramid (avoid frequency scrambling)


Multiscale Decomposition(2)

Multiscale subspaces generated by the Laplacian pyramid


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


Directional Decomposition(2):Sampling in Multiple Dimensions

Quincunx sampling latticeDownsample by 2Rotate 45 degree

(a) (b) (c)


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


Directional Decomposition(4):Directional Filter Bank

At each level QFB’s with fan filters are used

The first two levels of DFB


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


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 )


Pyramid Directional Filter Banks

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


Basis Functions


OutlineOutline

Introduction

Curvelet Transform


Simulation Results

Conclusion

Reference


Simulation Results


OutlineOutline

Introduction

Curvelet Transform


Simulation Results

Conclusion

Reference


Conclusion

Offer sparse representation for piecewise smooth images

Small redundancy

Energy compactness


OutlineOutline

Introduction

Curvelet Transform


Simulation Results

Conclusion

Reference


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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communication & multimedia c. -h. hong 2015/6/12 contourlet student: chao-hsiung hong advisor:...

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j slide

ridgelet1 slide

cos x

linear structure slide

failure of wavelet

sin ba x

x ba a1