Segmentation of Stochastic Images using ... - Jacobs University

Chapter 1
Introduction
The development of mathematical methods for image processing became a rapidly growing research
field during the last decades. The fast progress in the speed of widely available computer systems
allowed the numerical implementation of complex models. A specialty is the development of segmentation
algorithms based on partial differential equations (PDEs). The aim of a segmentation
algorithm is the decomposition of an image into the object and the background. Typically, detecting
edges inside an image or meeting a homogenization criterion for the object and the background lead
to a segmentation. Widely used segmentation approaches are the random walker segmentation [59],
the Mumford-Shah segmentation [107] and the related Ambrosio-Tortorelli regularization [14], and
active contour methods based on level set formulations [30,31,82,138]. Besided these segmentation
methods, which will be investigated in this thesis, there are other segmentation methods like region
growing [127], watersheds [136], snakes [76], and graph cuts [25].
Many applications use segmentation methods, e.g. quality control, machine vision, and medical
image processing. For example, the further treatment for cancer patients bases on the segmented
volume of the lesions from images. Fig. 1.1 shows a computed tomography (CT) image of a lung
lesion and the corresponding segmentation mask.
Typically, the segmentation methods act on noisy images (see Figs. 1.1 and 1.2). The image noise
depends on the image acquisition modality (e.g. digital camera, MR, CT, ultrasound), the acquisition
parameters (acquisition time, sound frequency, magnetic field strength), and extrinsic parameters
(illumination, reflection). The acquisition itself is a physical measurement (photon density, time-offlight
of the waves, spin, absorption), and it is good scientific practice to equip this measurement
with information about the measurement error. This last step of quantifying the measurement error
is typically omitted in image processing, leading to a loss of information about the influence of the
input error to the result of the image processing steps. Furthermore, image processing operators,
especially segmentation operators, do not have the ability to propagate this error information to the
result. This is e.g. important in medical application, where physicians decide about the patients’
treatment based on the information extracted from the images.
Figure 1.1: Left: CT image of a lung lesion (the small roundish structure in the middle of the image).
Right: The segmentation mask computed via region growing [127].
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