Shape Analysis and Retrieval

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Shape Analysis and Retrieval

• Spherical Parameterizations

Notes courtesy of Funk et al., SIGGRAPH 2004
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Spherical Parameterizations

• Goal
• Get a one-to-one mapping between points on the
  surface and points on the sphere.

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Spherical Parameterizations

• Start with the EGI

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Spherical Parameterizations

• Limitation
• Different points on the surface can map to the
  same point on the sphere.

Circular Angle
Model Point
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Spherical Parameterizations

• Approach
• Smooth the function, by convolving with a
  Gaussian
• The value of a point is obtained by taking a
  weighted average of the surrounding points.

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Spherical Parameterizations

• Convolving with a Gaussian

Initial Function
Filter
Convolution
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Laplacian Smoothing

• Given a function f define the family of functions
  that you would get by smoothing f by different
  amounts
• Taking derivatives gives

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Laplacian Smoothing

• Taking derivatives gives
• So that smoothing a little amounts to adding the
  second derivative (Laplacian).
• This is important when you want to smooth in a
  space where there is no direct way to convolve.