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” In non-technical terms, reasonable function Piece-wise continuous functions

Exceptions Consider g: B → R Ψ 1 : map from medial surface to boundary g. Ψ 1 need not be piecewise continuous but it is still Borel.

Paving W ij i th smooth region of M and j th side

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

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

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

Boundary integrals g a real valued function defined on the boundary For a regional integral, use X r g instead of g.

Integrals over regions radial flow then define

Integrals over regions

Define characteristic function

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

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

Area/Volume of a region Then, area/volume is

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

Expansion of integrals.

Expansion of integrals as moment integrals i th radial moment of g l th weighted integral

Expansions where

Expansions Similarly expand

Skeletal integral expansion Boundary integral Integral over regions

Applications Length Area

Applications Area Volume