Segmentation of Stochastic Images using ... - Jacobs University

Chapter 3 SPDEs and Polynomial Chaos Expansions
variables. Fig. 3.1 shows the situation. Instead of the direct computation of the integrals with the
random variable X and the measure Π, we transform the integration into integration over the real
numbers by using the polynomial chaos and the PDF ρ of the underlying random variables.
3.3.1 Wiener Chaos
In his seminal paper [156], Wiener developed the homogeneous (or Wiener) chaos formulated using
Hermite-polynomials in independent Gaussian random variables with zero mean and unit variance.
Let ˜ξ = (ξ 1 ,...) be a vector of independent Gaussian random variables with zero mean, unit variance
and PDFs ρ i , and V n (ξ i1 ,...,ξ in ) be Hermite-polynomials in n random variables. Cameron and
Martin [27] proved that a random variable X with finite second-order moments has the representation
X(ω) = a 0 V 0 +
∞
∑
i 1 =1
a i1 V 1 (ξ i1 (ω)) +
∞ ∞
∑ ∑
i 1 =1 i 2 =1
a i1 i 2
V 2 (ξ i1 (ω),ξ i2 (ω)) + ... . (3.19)
For notational convenience, this expression can be rewritten using multi-index notation
X(ω) = ∑ ∞ α=1 a αΨ α ( ˜ξ (ω)) . (3.20)
The functions V n and Ψ α have a one-to-one correspondence, i.e. every V n appears in the summation
over j, but has a different index. In what follows, we do not denote the dependence of ξ on ω
explicitly to ease notation when no integration over the stochastic space Ω is involved.
The Hermite-polynomials Ψ α form an orthogonal basis of the space L 2 (Ω), i.e.
∫
Ω
Ψ α ( ˜ξ (ω)
)
Ψ β ( ˜ξ (ω)
)dω = 〈Ψ α ,Ψ β 〉 = 〈(Ψ α ) 2 〉δ αβ . (3.21)
For a finite number of basic random variables ξ = (ξ 1 ,...,ξ n ) we simplify (3.21) by using (3.1). The
scalar product 〈 f ,g〉 is
∫
∫
〈Ψ α (ξ ),Ψ β (ξ )〉 = Ψ α (ξ (ω))Ψ β (ξ (ω))dω = Ψ α (x)Ψ β (x)dΠ, (3.22)
Ω
Γ
where Γ = supp(ξ ) ⊂ IR n . It follows from (3.22) that the weighting function w that is needed to get
orthonormal polynomials is
1
w(x) = √ . (3.23)
(2π) n e − 1 2 xT x
This weighting function is the key to understand the good approximation quality of the Hermiteexpansion,
because the weighting function for the Hermite-polynomials is the same as the PDF of
an n-dimensional Gaussian random variable, i.e. w(x) = ∏ i ρ i = ρ(x). Xiu and Karniadakis [160]
investigated this correspondence between the weighting functions for the orthogonal polynomial
basis and the density functions of random variables. Section 3.3.3 summarizes the findings. Thus,
the computation of the scalar product reduces to integration over a subset of IR n . For this, we use a
quadrature rule. Since we are integrating polynomials, the usage of a suitable quadrature rule leads
to exact results up to numerical inaccuracies.
3.3.2 Cameron-Martin Theorem
The Wiener chaos is an abstract representation for random variables, but it is unclear whether it converges
to the desired random variable. The Cameron-Martin theorem [27] fills this gap of knowledge.
We present the theorem in the version proposed in [27], but with the notation used in this thesis.
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