Segmentation of Stochastic Images using ... - Jacobs University

Notation
u image function ⊗ tensor product
D image domain ∂D boundary of the domain D
R real numbers S ρ,k,m Kondratiev space
P i finite element hat function H univariate polynomial
I node set of a finite element grid Ψ multivariate polynomial
D c
(
Cantor measure
a
b)
binomial coefficient
BV functions with bounded variation ξ basic random variable
SBV special BV space (D c = 0) δ i j Kronecker delta
GSBV
generalized SBV space
∂ f
∂x
partial derivative
sign sign function ∂ t partial temporal derivative
H d d-dim. Hausdorff measure τ time step size
K
edge set (discontinuities) of an
image
h
spatial grid spacing
φ phase field or level set V finite element space
H 1 (D) Sobolev space H 1 over D S stochastic space, ⊂ L 2 (Ω)
| · | absolute value of real numbers ‖ · ‖ x x-norm
∆ Laplace operator F cumulative distribution function
tanh hyperbolic tangent N normal vector
κ
curvature of level sets or phase
fields
T
tangential vector (of level sets)
∗ convolution operator (·) ′ derivative of univariate function
E expected value W width of the tangential profile of
a phase field
Ω probability (event) space L p Lebesgue spaces
H m
Sobolev spaces
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