1 inzell, germany, september 17-21, 2007 agnieszka lisowska university of silesia institute of...
TRANSCRIPT
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Inzell, Germany, September 17-21, 2007
Agnieszka Lisowska
University of SilesiaInstitute of Informatics
Sosnowiec, [email protected]
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