Description of Complex Objects via the Wavelet Transform

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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
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A pixel value can be related to different objects
The galaxy L384-350 Superimposed
objects Hierarchical structures
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Detectibility and Structures
At faint level the isophots are complex, so that
it is impossible to define the corresponding
fields
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The vision depends on the smoothing
Smoothing at scale 1
Smoothing at scale 32
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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

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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)ltf(x),y((x-b)/a)gt
• Covariance with translations f0f(x-x0)
  w0(a,b)w(a,b-x0)
• Covariance with dilations fsf(sx)
  ws(a,b)s-1 w(sa,sb)
• Null mean lty(x)gt0

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The Flow chart
Thresholding
Image
Wavelet Transform
Interscale relation
Segmentation
Object identification
Objects
Object Images
Image Reconstruction
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The à trous algorithm flow-chart
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Thresholding and Labelling
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Interscale relation and objects
An object is a local maximum of the WT
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Reconstruction
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Example on the IR image of the planetary nebula
NGC40
The linear structures are fragmented
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Itérations
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Cluster of galaxies A521(ROSAT)
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MVM on Abel 5121
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Structure separation
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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