Segmentation of Stochastic Images using ... - Jacobs University
Segmentation of Stochastic Images using ... - Jacobs University
Segmentation of Stochastic Images using ... - Jacobs University
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Chapter 9<br />
Summary, Discussion, and Conclusion<br />
In this thesis, we presented extensions <strong>of</strong> PDE-based segmentation methods to stochastic images,<br />
i.e. images whose pixels are random variables. The characterization <strong>of</strong> such stochastic images is<br />
based on the recently developed generalized polynomial chaos expansion. With this expansion, we<br />
developed extensions <strong>of</strong> the well-known finite element and finite difference schemes for the discretization<br />
<strong>of</strong> the PDE to the stochastic dimensions, leading to stochastic PDEs. To demonstrate the<br />
power <strong>of</strong> <strong>using</strong> stochastic images, we extended the well-known segmentation methods proposed by<br />
Mumford-Shah and the related approximation by Ambrosio-Tortorelli as well as the random walker<br />
method and three methods based on a level set formulation. The input for the stochastic segmentation<br />
is constructed via computing the leading random variables via a principal component analysis<br />
<strong>of</strong> samples <strong>of</strong> the input scene and a projection on the polynomial chaos basis. Furthermore, we<br />
used the stochastic images and model extensions to perform a sensitivity analysis <strong>of</strong> the methods by<br />
identifying the parameters with random variables.<br />
9.1 Discussion<br />
The work presented in this thesis is a complete framework for the important task <strong>of</strong> error propagation<br />
in mathematical image processing [36, 106]. For every step <strong>of</strong> the mathematical image processing<br />
pipeline (data acquisition, data representation, operator modeling, discretization, solution strategies<br />
and visualization) methods for the solution <strong>of</strong> the particular problems are presented. Besides the<br />
development <strong>of</strong> the framework, theoretical justifications <strong>of</strong> the methods are presented as well. In<br />
particular, these are the extensions <strong>of</strong> the Γ-convergence pro<strong>of</strong> for the stochastic Ambrosio-Tortorelli<br />
model and the pro<strong>of</strong> <strong>of</strong> the existence <strong>of</strong> SPDE solutions used in this thesis.<br />
This thesis applies the error propagation framework to mathematical operators for image segmentation,<br />
but the thesis can also be seen as a case study to demonstrate the applicability <strong>of</strong> the methods<br />
in image processing. Other image processing operators based on a PDE formulation can be extended<br />
by the presented methods easily, because the framework and the implementation <strong>of</strong> all steps around<br />
the operator extension are available. The only step that remains is the stochastic operator extension.<br />
Furthermore, the stochastic parameter study presented in this thesis sensitizes users to be skeptic<br />
about the segmentation results if these are not robust with respect to parameter changes.<br />
9.2 Conclusion<br />
We presented methods for all tasks along the stochastic image processing pipeline, but some <strong>of</strong> the<br />
methods presented in this thesis can be improved to get more stable and more accurate results. For<br />
example, the projection step for the estimation <strong>of</strong> the input distribution (cf. Section 5.2) is based<br />
on a Monte Carlo sampling (it is based on the uncorrelated image samples) and the method has<br />
the poor convergence speed O(1/ √ N) <strong>of</strong> the Monte Carlo method. Stefanou et al. [141] presented<br />
two methods based on an optimization problem. These methods are computationally much more<br />
expensive, but lead to a better convergence speed. Furthermore, the complete stochastic pipeline is<br />
restricted to a few basic random variables, which might be problematic, because image noise has<br />
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