Resumé
ResuméThe topic of Ole Fogh Olsen's thesis is segmentation of grey-scale images. A segment consists of similar neighboring image points. This apparently simple concept is difficult to pin down precisely, even if it is often obvious for a human in a particular application. Interpretation of images by segmentation is often used in practice. Examples are segmentation of satellite images into forest, cultivated areas, lakes, etc and segmentation of medical images from scanning devices which supplies 3-dimensional data. In this case the image is 3-dimensional, but the segmentation concept still holds.
The problem is thus how to program a computer to perform a reasonable segmentation in absence of a precise definition. For this purpose the image is represented by an ensemble of images. Each image is the original image convolved by a Gaussian. This simulates the measurement which would have been made by a sensor with the gaussian as point spread function. The variances of the Gaussians are chosen within a certain range. The advantage of this representation is that details in the coarse images have been suppressed (at the expense of localization and distortion of the remaining features). Thus segments can be identified at coarse scale and tracked by succesively lowering the scale to be located precisely at the finest scale.
The problem of segmentation is addressed within the framework of linear Gaussian scale space. The image at a given scale is partitioned according to a dissimilarity measure. Each minimum in the measure corresponds to exactly one segment. The segment is defined as the catchment basin belonging to the minimum. The gradient squared is used as the measure of dissimilarity.
An essential part of the scale-space paradigm is to study the family of images indexed by scale as a family and not as a distinct set of images. The structure of the segments across scale are analysed exploiting the established duality between the minima in the dissimilarity measure and the segments. Catastrophe theory is used to analyse the structure of the minima across scale. The generic catastrophe events for the minima of the gradient squared are proved to be the fold and the cusp catastrophes.
The generic events of annihilation, creation, split and merge for the minima are used to analyse possible linking schemes for the segments. The implemented linking scheme produces a hierarchical structure. Two segments are merged when the saddle connecting the two corresponding minima is annihilated. A segment is divided when a border is created within the segment. The creation of a border corresponds to the creation of a saddle connecting the resulting two minima.
The duality between minima and catchment basins is exploited to make a robust implementation of the linking scheme. The actual detection and tracking of segments is region based. A semi-automatic segmentation tool based on this uncommitted segmentation scheme has been implemented. Experimental results obtained using the segmentation tools on artificial, camera and medical data are presented. The experimental data are two and three dimensional.
The singularity analysis and the implementation is performed for two and three dimensional images. The theory can be generalised to higher dimensions.
This thesis also includes an introduction to scale-space theory, differential invariants, differential description of local forms for two and three dimensional functions, single scale and multi-scale segmentation methods, Morse theory and Catastrophe theory.
The thesis treats a subject, image segmentation, which is of great importance. In applications ad hoc methods are often used. Development of a method for segmentation using catastrophe theory is technically difficult, but the resulting method is easy to apply. After completion of the thesis the method has been succesfully used in medical applications. There exists another system which is regularly used by Utrecht University Hospital for segmentation of medical images. Ole's theoretical results simplifies the necessary algorithms significantly, and are amenable to generalization to other similarity measures.
The thesis is well written and can be used as an introduction to the application of catastrophe theory to image processing.
Vejleder: Peter Johansen