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Skeletal Integrals

Skeletal Integrals. Chapter 3, section 4.1 – 4.4 Rohit Saboo. Skeletal Integral. A skeletal structure (M,U) A multi-valued function h: M → R We consider integral of h over M. Skeletal Integrals. h belongs to a class of “Borel measurable functions”

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Skeletal Integrals

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  1. Skeletal Integrals Chapter 3, section 4.1 – 4.4 Rohit Saboo

  2. Skeletal Integral • A skeletal structure (M,U) • A multi-valued function h: M → R • We consider integral of h over M

  3. Skeletal Integrals • h belongs to a class of “Borel measurable functions” • In non-technical terms, reasonable function • Piece-wise continuous functions

  4. Exceptions • Consider g: B → R • Ψ1 : map from medial surface to boundary • g . Ψ1 need not be piecewise continuous • but it is still Borel.

  5. Paving • Wij ith smooth region of M and jth side

  6. 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

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

  8. Conversion to medial integrals • Boundary integrals • Volume integrals • Applications in measuring volume and surface area.

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

  10. Integrals over regions • radial flow • then • define

  11. Integrals over regions

  12. Integrals over regions • Define characteristic function

  13. Sample application • Length/surface area of boundary parts • g = 1 • So is also 1 • Using

  14. Area/Volume of a region • Again, define g as 1, and • For n = 2: • For n = 3:

  15. Area/Volume of a region • Then, area/volume is

  16. Gauss Bonnet formula • Let’s us know how accurate our discrete approximations are • Euler characteristic • = 2 – 2g, g is the genus (# of holes)

  17. Expansion of integrals.

  18. Expansion of integrals as moment integrals • ith radial moment of g • lth weighted integral

  19. Expansions where

  20. Expansions • Similarly expand

  21. Skeletal integral expansion • Boundary integral • Integral over regions

  22. Applications • Length • Area

  23. Applications • Area • Volume

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