Segmentation of Stochastic Images using ... - Jacobs University

Chapter 7 Stochastic Level Sets
where t is the time, ω a stochastic event, and y(t,ω) the path of a particle on the interface. Using the
chain rule, we get the stochastic version of the advection equation
φ t (t,x,ω) + v(t,x,ω) · ∇φ(t,x,ω) = 0 , (7.2)
where v = ∂y(t,ω)
∂t
is the speed of the level set propagation. The speed decomposes in a component in
the normal direction N and in the tangential directions T of the interface:
where v N and v T are
v(t,x,ω) = v N (t,x,ω) + v T (t,x,ω) , (7.3)
v N (t,x,ω) = (v(t,x,ω) · N(t,x,ω))N(t,x,ω) resp. v T (t,x,ω) = v(t,x,ω) − v N (t,x,ω) . (7.4)
Note that the decomposition is dependent on the stochastic event ω, because for every realization
ω ∈ Ω of the level set φ we get a different normal N(t,x,ω) and a different decomposition of the
stochastic quantity v(t,x,ω). Substituting (7.3) and (7.4) into (7.2) and using the relations
v T (t,x,ω) · ∇φ(t,x,ω) = 0 and v N (t,x,ω) · ∇φ(t,x,ω) = v n (t,x,ω)|∇φ(t,x,ω)| , (7.5)
where v n is the speed in the normal direction, yields the stochastic extension of the level set equation:
φ t (t,x,ω) + v n (t,x,ω)|∇φ(t,x,ω)| = 0 . (7.6)
As already mentioned, the discretization of this deterministic equation uses methods for hyperbolic
conservation laws, e.g. upwinding schemes. To the best of the authors knowledge, there is no accurate
and fast upwinding scheme for SPDEs available. To avoid the use of a numerical upwinding
scheme for hyperbolic SPDEs, we modify the stochastic level set equation in the spirit of Sun and
Beckermann [143]. We start with a decomposition of the speed v n into a component independent and
a component dependent on the interface curvature κ:
v n (t,x,ω) = a(t,x,ω) − b(x,t,ω)κ(t,x,ω) . (7.7)
The curvature κ is expressed using the level set φ, this is a standard approach for deterministic level
sets [138], and rewritten using the quotient rule:
( )
∇φ(t,x,ω)
κ(t,x,ω) = ∇ · N(t,x,ω) = ∇ ·
|∇φ(t,x,ω)|
(
) (7.8)
1
(∇φ(t,x,ω)) · ∇(|∇φ(t,x,ω)|)
=
∆φ(t,x,ω) − .
|∇φ(t,x,ω)|
|∇φ(t,x,ω)|
The previous modeling is valid for sufficiently smooth level set functions. If we prescribe a special
behavior of the level set in the normal direction of the level set, quantities like the gradient or the
curvature can be computed easily. For the special choice of the level set function
( ) n(t,x,ω)
φ(t,x,ω) = −tanh √ , (7.9)
2W
where n is the distance to the interface in the normal direction and W ∈ IR an additional parameter
controlling the width of the tangential profile, we get for the norm of the gradient:
|∇φ(t,x,ω)| = − ∂φ(t,x,ω)
∂n
This is because the derivative of the hyperbolic tangent is
= 1 − φ(t,x,ω)2 √
2W
. (7.10)
(tanhx) ′ = 1 − tanh 2 x . (7.11)
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