Segmentation of Stochastic Images using ... - Jacobs University

Chapter 2
Image Segmentation and Limitations
In this chapter, we give a short review of the research in mathematical image processing and segmentation
related to the work in this thesis. We focus on PDE based methods for image processing,
because these methods have advantages over other image processing methods:
• They are based on a continuous formulation of images, but the discretization based on finite
differences or finite elements naturally leads to regular grids, characteristic for digital images.
• It is possible to show existence and uniqueness of solutions of PDE based methods using
well-known results from functional analysis.
• Later, we will see that PDE based methods extend naturally to stochastic images, the object
under investigation in this thesis.
The application of PDE models in image processing is a rapidly growing field of research. Many
authors (see [17,130] for an overview) presented methods based on PDEs to solve problems arising in
image processing like denoising, restoration, segmentation, registration, flow extraction, etc. Since
we are interested in segmentation, the presentation focuses on results important for segmentation.
Image segmentation, the separation of an image into object and background, is a repeatedly investigated
problem in image processing. The literature divides the proposed methods into three
categories, based on the user interaction necessary to perform the segmentation:
Automatic segmentation: The user defines segmentation parameters at the beginning only, but
has no possibility to refine the segmentation result.
Semi-automatic segmentation: The user defines initial contours and parameters to optimize the
segmentation result, but again has no chance to refine the result.
Interactive segmentation: The user interactively refines the segmentation result. Thus, this
method computes a segmentation result based on the user input and allows user interaction
afterwards to get new input for the next iteration step.
PDE based image segmentation methods are in all of these segmentation categories. The random
walker segmentation [59] is an interactive segmentation approach, where the user interactively refines
the segmentation result. The level set based segmentation methods [29, 96, 138] are semiautomatic,
because the user has to define an initial contour as the starting point for the algorithm, but
has no chance to influence the segmentation result during the run of the algorithm. The Mumford-
Shah approach [107] is fully automatic. The user defines parameters only, but has no possibility to
define initial contours or to modify the result locally afterwards.
We organized this chapter as follows: First, we present basic definitions needed for the presentation
of PDE based segmentation algorithms. Afterwards, we present five segmentation algorithms:
random walker segmentation, Ambrosio-Tortorelli segmentation, and the level set based segmentation
methods gradient-based segmentation, geodesic active contours and Chan-Vese segmentation.
At the end, we present limitations of classical segmentation algorithms to motivate further investigations
to extend these classical methods and draw conclusions.
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