Divergence Theorem, flux and applications
Chapter 3, section 4.5, 4.6
Chapter 4, part of section 2.3
Rohit Saboo
Distance Fields
Distance Field (left)
Gradient of distance field (bottom)
Divergence Theorem
Divergence of a vector Flux Standard Divergence Theorem
Flux of a vector field defined on the medial axis
Flux with discontinuities along the medial axis
Average outward flux
Grey : (near) zero flux Black : large negative flux
Modified Divergence Theorem
Γ a region in Ω Γ has regular piecewise smooth
boundaries
Modified Divergence Theorem
Divergence of G
Average outward flux
Medial Volume
Grassfire Flow G
and ~
Average Outward Flux
Average outward fluxZero at non-medial pointsNon-zero at medial points and
computed as shown later.
Modified Divergence Theorem
F is a smooth function, discontinuous at the medial surface.
Define cF : projTM(F) = cF U
Grassfire Flow
F = G = -U projTM(G) = -U
cG = -1
Limiting Flux
Region Γt(x) as t -> 0 x is a point on M
x
Limiting flux
The limiting flux goes to zero everywhere.
Average flux
Limiting value of the average flux
N-dimensional volume : voln (Γt(x))
Invariants at a medial point
is the minimum non-zero values for the different values of U and N at x.
Medial density
Different types of medial points1 dimensional medial axis2 dimensional medial surface
voln-1
Medial density
example
Medial Densities
1/π 1/4
Medial density