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Description of Complex Objects via the Wavelet Transform

Description of Complex Objects via the Wavelet Transform. Albert Bijaoui Dpt CERGA – UMR 6527 – OCA BP 4229 – 06304 NICE CEDEX 4. Main collaborators Frédéric Rué & Benoît Vandame. A pixel value can be related to different objects. The galaxy L384-350 Superimposed objects

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Description of Complex Objects via the Wavelet Transform

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  1. Description of Complex Objectsvia the Wavelet Transform Albert Bijaoui Dpt CERGA – UMR 6527 – OCA BP 4229 – 06304 NICE CEDEX 4 Main collaborators Frédéric Rué & Benoît Vandame

  2. A pixel value can be related to different objects The galaxy L384-350 Superimposed objects Hierarchical structures

  3. At faint level the isophots are complex, so that it is impossible to define the corresponding fields Detectibility and Structures

  4. The vision depends on the smoothing Smoothing at scale 1 Smoothing at scale 32

  5. The Scale Space • Set of smoothings • Pyramids of Gaussian smoothings • Equivalence of the isotropic diffusion PDE • No linear smoothings • Morphological operators • Anisotropic diffusion equation • Fundamental equation of the image processing • To separate the information between different scales: the differences between smoothings

  6. The wavelet transform • Map a fonction f(x) into a fonction w(a,b) • a : scale - b : position • Four properties: • Linearity : w(a,b)=K(a)<f(x),y((x-b)/a)> • Covariance with translations : f0=f(x-x0) w0(a,b)=w(a,b-x0) • Covariance with dilations : fs=f(sx) ws(a,b)=s-1 w(sa,sb) • Null mean <y(x)>=0

  7. The Flow chart Thresholding Image Wavelet Transform Interscale relation Segmentation Object identification Objects Object Images Image Reconstruction

  8. The à trous algorithm flow-chart

  9. Thresholding and Labelling

  10. Interscale relation and objects An object is a local maximum of the WT

  11. Reconstruction

  12. Example on the IR image of the planetary nebula NGC40 The linear structures are fragmented

  13. Itérations

  14. Cluster of galaxies A521(ROSAT)

  15. MVM on Abel 5121

  16. Structure separation

  17. Conclusion • MVM may be applied to surveys: • Complexity • CPU • Two classes of objects: • Stellar / Quasi stellar objects • Complex structures • MVM is well adapted to describe complex structures • Problem with linear, ringed, wavy structures

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