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Inzell, Germany, September 17-21, 2007

Agnieszka Lisowska

University of SilesiaInstitute of Informatics

Sosnowiec, POLANDalisow@ux2.math.us.edu.pl

Second Order Wedgelets – Efficient Tool in Image

Processing

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Outline

Introduction Geometrical wavelets –

preliminaries Second order wedgelets ... ... and their applications in

Image coding Denoising Edge detection

Summary

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Geometrical wavelets

Wavelets equation (classical wavelets)

1/ 2 1, , 1 2 1 2( , ) ( ( cos sin ))a b x x a a x x b

1/ 2 1, ( ) ( ( ))a b x a a x b

Wavelets equation (geometrical wavelets)

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Beamlet, wedgelet – geometrical wavelets

1, for ( )( , ) , ,

0, for ( )

y b xw x y x y S

y b x

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Wedgelets’ dictionary (Donoho D., 1999)

MW(Si,j) – number of straight wedgelets on Si,j

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Beamlets(Donoho D., Huo X., 2000)

Platelets(Willett R.M., Nowak R.D., 2001)

Modifications of dictionary (1)

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Surflets(Chandrasekaran V. et al., 2004)

Arclets(Führ H. et al., 2005)

Modifications of dictionary (2)

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Conic curves

parabola

ellipse

hyperbola

Second order curves:

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New modification – generalization (2003)

MW(Si,j) – number of straight wedgelets on Si,j

D – the number of bits used to code parameter d

Second Order Wedgelets

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Comparison of different kinds of approximation

a) wavelets b) wedgelets c) second order wed.

Original image and its decompositions:

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Optimal approximation is the solution of the problem:

Optimal image approximation (1)

Solving method:- For every quadtree element the optimal wedgelet

function is found from among the given node

- Using the bottom-up tree pruning algorithm the optimal subtree is found

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Optimal image approximation (2)

Full quadtre

e

Optimal quadtre

e

Bottom-up tree prunning algorithm

Processing of all nodes

Wedgelet ensuring

the smallest

error

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Speed up of computations

1) Find the best wedgelet w1 within the smaller set of beamlets

1)

2) 2) Find the best wedgelet w2 in neighbourhood of w1 (for example +/- 5 pixels)

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Fast computation of second order wedgelet

1) Find the best wedgelet w1

2) Find the best second order wedgelet w2 in neighbourhood of w1 (for example +/- 5 pixels from the wedgelet w1) and changing the value of parameter d

1)

2)

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level 1 level 2 optimal approximation

level 3 level 5 quadtree partition

Optimal image approximation – example (second order wedgelets)

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Image coding

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Image coding with wedgelets

no information – internal node – undecorated node – decorated by straight wedgelet

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Image coding with second order wedgelets

no information – internal node – undecorated node – decorated by straight wedgelet – decorated by curved wedgelet

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Experimental results- coding

Artificial image coding:

Still image coding ->

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Experimaental results

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original image straight wedg. second order wedg.

Experimental results - coding

PSNR: 31.39 dB 31.45 dBNumber of wedg.: 5821 5695

Number of bytes: 14211 14185

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Denoising

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Image denoising

But, in the case of noisy images, instead of F we have Z:

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Experimental results – denoising (1)

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Experimental results – denoising (2)

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Edge detection

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Edge detection - geometry

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Edge detection - multiresolution

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Edge detection – noise resistance

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The adventages of image coding and processing

with the use of second order wedgelets:

Improvement of coding effectiveness (0-25% in the case of artificial images and ~1.44% in the case of still images)

Better denoising effectiveness in comparison to other known methods (up to 0.5dB)

Geometrical multiresolution noise resistant tool in edge detection

Summary

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Main publications

[1] Lisowska A. Effective coding of images with the use of geometrical wavelets, Proceedings of Decision Support Systems Conference , Zakopane, Poland, (2003).

[2] Lisowska A., Extended Wedgelets - Geometrical Wavelets in Efficient Image Coding, Machine Graphics & Vision, Vol. 13, No. 3, pp. 261-274, (2004).

[3] Lisowska A., Bent Beamlets - Efficient Tool in Image Coding, Annales UMCS Informatica AI, Vol. 2, pp. 217-225, (2004).

[4] Lisowska A., Intrinsic Dimensional Selective Operator Based on Geometrical Wavelets, Journal of Applied Computer Science, Vol. 12, No. 2, pp.99-112, (2005).

[5] Lisowska A., Second Order Wedgelets in Image Coding, Proceedings of EUROCON '07 Conference, Warsaw, Poland, (2007).

[6] Lisowska A. Image Denoising with Second Order Wedgelets, Special Issue on "Denoising" of International Journal of Signal and Imaging Systems Engineering, accepted (2007).

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Bibliography

[1] Do M. N., Directional Multiresolution Image Representations, Ph.D. Thesis, Department of Communication Systems, Swiss Federal Institute of Technology Lausanne, November (2001).

[2] Donoho D. L., Wedgelets: Nearly-minimax estimation of edges, Annals of Statistics, Vol. 27, pp. 859–897, (1999).

[3] Donoho D. L., Huo X., Beamlet Pyramids: A New Form of Multiresolution Analysis, Suited for Extracting Lines, Curves and Objects from Very Noisy Image Data, Proceedings of SPIE, Vol. 4119, (2000).

[4] Willet R. M., Nowak R. D., Platelets: A Multiscale Approach for Recovering Edges and Surfaces in Photon Limited Medical Imaging, Technical Report TREE0105, Rice University, (2001).

[5] Zetzsche C., Barth E., Fundamental Limits of Linear Filters in the Visual Processing of Two-Dimensional Signals, Vision Research, Vol. 30, pp. 1111-1117, (1990).

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And finally...

Thank you for your attention

http://www.math.us.edu.pl/al