Oriented Wavelet

國立交通大學電子工程學系陳奕安

2007.5.9

Outline

BackgroundBackgroundBeyond WaveletBeyond WaveletSimulation ResultSimulation ResultConclusionConclusion

Outline

BackgroundWavelet ReviewWavelet ReviewThe Failure of waveletThe Failure of wavelet

Beyond WaveletBeyond WaveletSimulation ResultSimulation ResultConclusionConclusion

Wavelet Review Signal Decomposition:Signal Decomposition:

Equal temporal and spatial resolutionsEqual temporal and spatial resolutions

““Natural” trade-off of temporal and spatial resolutionsNatural” trade-off of temporal and spatial resolutions

(Wavelet)(Wavelet)

Wavelet Review Wavelet Decomposition:Wavelet Decomposition:

1-D wavelet transform1-D wavelet transform

2-D wavelet transform can be obtained from a2-D wavelet transform can be obtained from a separable extension of 1-D transformseparable extension of 1-D transform

The failure of wavelet

1-D: Wavelets are well adapted to 1-D: Wavelets are well adapted to singularitiessingularities 2-D: 2-D:

Separable wavelets are only well adapted to Separable wavelets are only well adapted to point-sinpoint-singularitygularity

However, in However, in lineline- and - and curve-singularities…curve-singularities…

The inefficiency of wavelet

Wavelet: fails to recognize that Wavelet: fails to recognize that boundary is smooth

New: require challenging New: require challenging non-separable constructions

Outline

BackgroundBackgroundBeyond Wavelet

Curvelet Curvelet Contourlet Contourlet BandeletBandeletOriented WaveleteOriented Wavelete

Simulation ResultSimulation ResultConclusionConclusion

Curvelets Curvelets can be interpreted as a grouping of neCurvelets can be interpreted as a grouping of ne

arby wavelet basis functions into lineararby wavelet basis functions into linear 　　 structurstructures so that they can capture the smooth discontines so that they can capture the smooth discontinuity curve more efficientlyuity curve more efficiently

Curvelets

First, a standard multiscale decomposition is computed, where the low-pass channel is sub-sampled while the high-pass channel is not.

Then, a directional decomposition with a DFB is applied to each high-pass channel.

Contourlet

Contourlet

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

0

1

2

3

Contourlet

0 2

347

01

1

2

34

5

5 6

6

7

ω 2

ω 1

(π ,π )

(-π ,-π )

7

6

5

4

3

2

1

0

Contourlet

0

1

1

2

2

33 44

5

5

6

6

7

7 8

8

16

9

9

10

10

11

11

12

12

13

13

14

1415

15

16

10

0 12

3 4

5

6

7 8

15 161314

11 12

9

Curvelets & Contourlet

Pros:Pros:They do not require a geometric model of the iThey do not require a geometric model of the i

mage. mage.

Cons:Cons:The discrete implementations of curvelet transThe discrete implementations of curvelet trans

forms are currently highly redundant.forms are currently highly redundant.

Bandelets Using separable wavelet basis, if no geometric flowUsing separable wavelet basis, if no geometric flow Using modified orthogonal wavelets in the flow direction, called Using modified orthogonal wavelets in the flow direction, called

bandeletsbandelets Quad-tree segmentationQuad-tree segmentation

Bandelets

BBandeletsandelets use a geometric model to describe th use a geometric model to describe the discontinuities of the image; is theoretically moe discontinuities of the image; is theoretically more efficient than curvelets for compression purpore efficient than curvelets for compression purposes . ses .

They are computationally intensive and have the They are computationally intensive and have the problem of optimization of the bitrate allocation bproblem of optimization of the bitrate allocation between the image geometry description and the etween the image geometry description and the wavelet coefficients. wavelet coefficients.

Oriented Wavelete

Applying the lifting steps of a 1D wavelet Applying the lifting steps of a 1D wavelet transform in the direction of the image contours transform in the direction of the image contours

Oriented Wavelete

Using quincunx multi-resolution sampling, the Using quincunx multi-resolution sampling, the image is filtered along horizontal and vertical or image is filtered along horizontal and vertical or diagonal and anti-diagonal directions. diagonal and anti-diagonal directions.

Oriented Wavelete Horizontal (red) or vertical Horizontal (red) or vertical

(green) filtering directions for the (green) filtering directions for the first decomposition level .first decomposition level .

Diagonal '/' (blue) or anti-diagonal Diagonal '/' (blue) or anti-diagonal '\' (yellow) filtering directions for '\' (yellow) filtering directions for the second decomposition level.the second decomposition level.

Oriented Wavelete Use quad-tree structure to describe the Use quad-tree structure to describe the

geometry of the image leading to an efficient geometry of the image leading to an efficient representation and a simpler rate-distortion representation and a simpler rate-distortion optimization. optimization.

Outline

BackgroundBackgroundBeyond WaveletBeyond WaveletSimulation Result

Image compressionImage compressionDenoising Denoising

ConclusionConclusion

Image compression

Original lena and JPEG Compression (0.25 bpp) Original lena and JPEG Compression (0.25 bpp)

JPEG PSNR 31.8 dB

Image compression

Separable wavelets and oriented wavelets (0.25 bpSeparable wavelets and oriented wavelets (0.25 bpp) p)

Oriented wavelets PSNR 34.3 dB Separable wavelets PSNR 34.3 dB

Denoising

Noisy lena and separable waveletsNoisy lena and separable wavelets

DWT PSNR 29.86 dB Noisy lena PSNR 20.24 dB

Denoising

Noisy lena and separable waveletsNoisy lena and separable wavelets

OWT PSNR 30.41 dB DWT PSNR 29.86 dB

Outline

BackgroundBackgroundBeyond WaveletBeyond WaveletSimulation ResultSimulation ResultConclusion

Conclusion

OWT has similar complexity as the separable OWT has similar complexity as the separable wavelet transform while providing better energy wavelet transform while providing better energy compaction and staying critically sampled.compaction and staying critically sampled.

Filtering along the image contours allows to Filtering along the image contours allows to remove the noise more efficiently than remove the noise more efficiently than anisotropic techniques like the ones based on anisotropic techniques like the ones based on separable wavelets. separable wavelets.

Happy Birthday !!

Outline BackgroundBackground

Wavelet ReviewWavelet Review Failure of waveletFailure of wavelet

Beyond WaveletBeyond Wavelet Curvelet Curvelet Contourlet Contourlet BandeletBandelet Oriented WaveleteOriented Wavelete

Simulation ResultSimulation Result Image compressionImage compression Denoising Denoising

ConclusionConclusion