Segmentation of Stochastic Images using ... - Jacobs University

Chapter 3 SPDEs and Polynomial Chaos Expansions
Definition The n-th central moment of a random variable X is
M n (X) = E((X − E(X)) n ) . (3.2)
A famous member of this class of moments is the second central moment, the variance:
Var(X) = E ( (X − E(X)) 2) . (3.3)
Later we have to evaluate the relation between random variables. A famous tool for this is the
covariance.
Definition The covariance of two random variables f ,g with finite second order moments is
Cov( f ,g) = E( f g) − E( f )E(g) . (3.4)
In what follows, it will be necessary to have a set of random variables indexed by a spatial position.
This motivates the following definition:
Definition A random field X is a collection of random variables indexed by a spatial position x ∈ IR n :
X = {X x |x ∈ IR n } . (3.5)
Random fields are elements of a tensor product space consisting of functions defined on the Cartesian
product Ω × D. A random field is a function taking two arguments, a random event and a spatial
position. We restrict the investigations to random fields satisfying smoothness assumptions.
Definition Let D ⊂ IR n be the spatial domain of the random field and Ω the sample space. The
tensor space L 2 (Ω) ⊗ H 1 (D) is the space of random fields satisfying u(ω,·) ∈ H 1 (D) almost sure
and u(·,x) ∈ L 2 (Ω), where H 1 (D) is the usual Sobolev space and
{
∫
}
L 2 (Ω) = f : Ω → IR : f (ω) 2 dω < ∞ . (3.6)
Ω
This is a strong limitation which is typically not satisfied for random fields arising in financial problems
[69, 108]. For image processing problems, this space is reasonable, because H 1 -regularity is
typically assumed for classical image processing tasks [17] and L 2 -regularity of the stochastic part
seems reasonable due to the limited energy an image acquisition device detects. Furthermore, the
restriction to random fields with finite variance, i.e. satisfying (3.6), allows a discretization of the
random fields using polynomial chaos expansions. Random fields will play a crucial role in the presentation
of SPDEs, because the locally varying random coefficients of the SPDEs are random fields.
3.2 Stochastic Partial Differential Equations
We introduce SPDEs following [18] and use an elliptic model equation. The deterministic equation is
−∇ · (a∇u) = f
on D
u = g on ∂D ,
(3.7)
where a is a diffusion coefficient, f a source term, g the boundary condition, and D the deterministic
domain this equation holds in. In this equation, we assumed that we perfectly know the diffusion
coefficient a and the right hand side f . In many applications, these quantities are not known exactly,
but a description of the quantities through random fields is possible (see e.g. [8]). Let D be a bounded
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