Segmentation of Stochastic Images using ... - Jacobs University

Chapter 5 Stochastic Images
❳❳
❳❳ ❳ supp ξ j , j = 1,...n
✘ ✘ ✘ ✘✘ ✘ x i
❳ ❳ ❳ ❳ ❳ supp Pi (x)
Figure 5.1: Sketch of the ingredients of a stochastic image. We discretize the spatial dimensions
using finite elements, but the coefficients of the FE basis functions are random variables.
Every random variable has a support, which spans over the complete image, thus pixels
depend on a random vector.
and an interpolation rule provides the values at positions between neighboring pixels. For fixed α
we call the coefficient f i α a stochastic mode of the pixel i. The set { f i α} i∈I collects the stochastic
modes of all pixels for fixed α. Thus, it is possible to visualize the set as a classical image.
From the polynomial chaos expansion of a stochastic image, we compute stochastic moments of
the image. With the use of the orthogonal set of basis functions we have E(Ψ 1 ) = 1, E(Ψ α ) = 0 for
α > 1 and E(Ψ α Ψ β ) = 0 if α ≠ β. The expected value and the variance of a stochastic pixel are
E( f (x i ,·)) = f i 1 ,
Var( f (x i ,·)) = ∑ N α=2
(
f
i
α
) 2
E
(
(Ψ α ) 2) .
(5.5)
We obtain higher stochastic moments in a similar way. Furthermore, it is possible to visualize the
complete stochastic information of every pixel, e.g. via a visualization of the PDFs of all pixels.
Note that the representation of stochastic images presented here differs from the one discussed by
Preusser et al. [130]. There, a space is used in which every pixel depends on one random variable
only. However, for most of the image acquisition processes and image processing methods the
assumption that the noise is independent for every pixel is not true. To represent these images in the
ansatz space we let every pixel depend on a random vector ξ . Section 5.3 compares both concepts.
The first step, required before the processing of the stochastic images starts, is the identification
of the random variables in the input data. We estimate these random variables from data samples
through the Karhunen-Loève expansion [41]. The Karhunen-Loève expansion, a stochastic version
of the principal component analysis (PCA) [74], determines the eigenvalues and eigenvectors of the
covariance matrix of the data samples and identifies the significant random variables in the data
samples with these eigenvectors and eigenvalues.
5.2 Generation of Stochastic Images from Samples
We obtain the polynomial chaos coefficients of the random variables X ∈ L 2 (Ω) by a maximum
likelihood estimation [41], leading to a representation of X ∈ L 2 (Ω) in the polynomial chaos by
X = ∑ N α=1 a αΨ α (ξ ) . (5.6)
To use the notion of stochastic images developed in the previous sections for image processing, we
need to obtain the coefficients of the representation (5.3) for the image undergoing the analysis. Let
48