Segmentation of Stochastic Images using ... - Jacobs University

Chapter 5 Stochastic Images
like quality control, the repeated acquisition is possible. However, we cannot apply this technique to
CT data, because the acquisition of CT data uses high-energy radiation [66]. Thus, the acquisition
of multiple samples is unethical for medical applications. Therefore, we present another possibility
for the generation of stochastic images from CT data based on the sinogram, the collection of rays
through the object under different angles and directions [66].
The approach is based on the hypothesis that the sinogram (see Fig. 5.3), the raw data of the
acquisition process (see [21] for details), is free of noise and that the noise and the artifacts in the
final CT images are due to the reconstruction step, which is necessary to transform the sinogram
into the final data set. We use multiple reconstruction techniques and parameter settings to generate
the input samples and use the technique described in the previous section to generate the stochastic
images. The reconstruction techniques range from Fourier based methods to iterative methods with
different settings for the data interpolation and the filter window for the low-pass filtering [154]. For
the computation of the reconstructions, we use CTSim [134], for which source code is available.
Thus, we combine the generation of input samples and the computation of the resulting stochastic
image in one program that runs without user interaction.
Another possibility to generate a stochastic image from the available CT image sample is to use a
noise model.
5.3 Comparison of the Space from [130] and the Space Used in this Thesis
Preusser et al. [130] made a first step for the application of SPDEs in the image processing context.
They proposed to use the space H h,p
still
:= V2 h ⊗ P p ⊂ H 1 (D) ⊗ L 2 (Γ) as ansatz space, where V2
h
is the classical finite element space spanned by multi-linear tent-functions P i and P p the space
spanned by one-dimensional polynomials H 1 ,...,H p . Then, the authors identified a stochastic image
f (x,ξ ) ∈ H h,p
still
with the polynomial chaos approximation:
f (x,ξ ) = ∑ i∈I ∑ p α=1 f i αH α (ξ i )P i (x) . (5.16)
In this representation, every pixel has its own random variable and the pixel is dependent on this
random variable only. Remember, the space used in this thesis uses a limited number of random
variables, but the support of these random variables ranges over the whole image.
An SPDE having stochastic images as input or solution is discretized using the SFEM. The authors
multiplied the equation by a test function of the form H β (ξ i )P i (x) ∈ H 1 (D) ⊗ L 2 (Γ), yielding to a
block system matrix for the unknown polynomial chaos coefficients of the solution.
The ansatz space and the discretization presented by Peusser et al. [130] have drawbacks in comparison
to the space used in this thesis. These drawbacks are listed below:
1. The authors used only test functions of the form H β (ξ i )P i (x), but functions of the form
H β (ξ k )P i (x), k ≠ i, are also elements of the product space H 1 (D) ⊗ L 2 (Γ). This leads to a
much too small system matrix of the SFEM method. Thus, the solution is computed in a
subspace of the tensor product space H 1 (D) ⊗ L 2 (Γ) only.
2. The dependence of pixels on independent random variables allows no propagation of stochastic
information between the pixels. This is a serious problem when dealing with diffusion equations
like in [130], because the diffusion transports stochastic information from a pixel into the
surrounding region. The ansatz space chosen in [130] cannot store this information, because
the neighboring pixels are independent of this specific random variable. Thus, the information
is lost. To be more precise, the solution of the diffusion process of the random variables has
to be projected on the ansatz space and the ansatz space is unable to store this information.
Especially for diffusion equations, stochastic information is lost due to this projection step,
leading to inaccurate results.
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