Skeletal Integrals

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Skeletal Integrals

• Chapter 3, section 4.1 4.4
• Rohit Saboo

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Skeletal Integral

• A skeletal structure (M,U)
• A multi-valued function h M ? R
• We consider integral of h over M

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Skeletal Integrals

• h belongs to a class of Borel measurable
  functions
• In non-technical terms, reasonable function
• Piece-wise continuous functions

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Exceptions

• Consider g B ? R
• ?1 map from medial surface to boundary
• g . ?1 need not be piecewise continuous
• but it is still Borel.

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Paving

• Wij ith smooth region of M and jth side

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Definitions

• is a skeletal integral
• Medial when (M,U) satisfied the partial Blum
  conditions
• Integrable when finite
• Xr characteristic function of region R
• Integral on a region given by

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Medial measure

• To correct for non-orthogonality of spokes
• E.g. small for branches due to surface bumps.

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Conversion to medial integrals

• Boundary integrals
• Volume integrals
• Applications in measuring volume and surface area.

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Boundary integrals

• g a real valued function defined on the boundary
• For a regional integral, use Xr g instead of g.

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Integrals over regions

• radial flow
• then
• define

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Integrals over regions
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Integrals over regions

• Define characteristic function

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Sample application

• Length/surface area of boundary parts
• g 1
• So is also 1
• Using

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Area/Volume of a region

• Again, define g as 1, and
• For n 2
• For n 3

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Area/Volume of a region

• Then, area/volume is

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Gauss Bonnet formula

• Lets us know how accurate our discrete
  approximations are
• Euler characteristic
• 2 2g, g is the genus ( of holes)

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Expansion of integrals.
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Expansion of integrals as moment integrals

• ith radial moment of g
• lth weighted integral

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Expansions

• where

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Expansions

• Similarly expand

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Skeletal integral expansion

• Boundary integral
• Integral over regions

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Applications

• Length
• Area

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Applications

• Area
• Volume