Application of Fractals in Shape Analysis

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Application of Fractals in Shape Analysis

• Volodymyr Kindratenko
• National Center for Supercomputing Applications

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Topics of Discussion

• What is fractal and what is fractal dimension
• How all this is related to shape analysis
• An example of the application of fractal
  dimension in shape analysis
• Software

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A bit of history

• Monster curves
• Koch snowflake

of sides 3, 3412, 34448, 34i side
length L, L/3, L/9, L/3i perimeter 34i
L/3i 3L(4/3)i
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Other examples

• Cantor bar

• Sierpinski gasket

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What is fractal?

• B. Mandelbrot A rough or fragmented geometric
  shape that can be subdivided in parts, each of
  which is (at least approximately) a reduced-size
  copy of the whole shape (self-similarity).
• Mathematical A set of points whose fractal
  dimension exceeds its topological dimension.

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Topological dimension

• 1D

• 2D

• 3D

SLD Dlog(S)/log(L)
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Hausdorffs definition of Fractal dimension

• spherical ?-dimensional Hausdorff measure

• Hausdorff dimension

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Examples

• Cantor bar
• Fractal dimension ln(2)/ln(3)
• Koch snowflake
• Fractal dimension ln(4)/ln(3)
• Sierpinski gasket
• Fractal dimension ln(3)/ln(2)

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How all this is related to shape analysis

• How long is the coast of Iceland?

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How all this is related to shape analysis

• How long is the coast of Iceland?

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More on Richardson plot
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An example aerosol particles classification

• Spherically shaped fly ash
• Irregularly shaped fly ash
• Irregularly shaped soil dust

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An example aerosol particles classification

• Spherically shaped fly ash

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An example aerosol particles classification

• Irregularly shaped fly ash

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An example aerosol particles classification

• Irregularly shaped soil dust

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Software

• http//www.edv.agrar.tu-muenchen.de/dvs/idolon/ido
  lonhtml/idolon.html
• http//www.trusoft-international.com/benoit.html
• http//algodones.unm.edu/bmilne/frac/clarat/clara
  t.html
• http//www.cee.cornell.edu/mdw/fractech.html
• http//www.ncsa.uiuc.edu/kindr/phd/software.html

ClaraT
Macros for HIN
My software
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Automated Richardson plot analysis
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Automated Richardson plot analysis
Fractal dimension1.2420.0133 Best line is from
0.014 to 0.180