Astrid Franz

Automated model-based vertebra detection, identification, and segmentation in CT images

For many orthopaedic, neurological, and oncological applications, an exact segmentation of the vertebral column including an identification of each vertebra is essential. However, although bony structures show high contrast in CT images, the segmentation and labelling of individual vertebrae is challenging. In this paper, we present a comprehensive solution for automatically detecting, identifying, and segmenting vertebrae in CT images. A framework has been designed that takes an arbitrary CT image, e.g., head-neck, thorax, lumbar, or whole spine, as input and provides a segmentation in form of labelled triangulated vertebra surface models. In order to obtain a robust processing chain, profound prior knowledge is applied through the use of various kinds of models covering shape, gradient, and appearance information. The framework has been tested on 64 CT images even including pathologies. In 56 cases, it was successfully applied resulting in a final mean point-to-surface segmentation error of 1.12+/-1.04mm. One key issue is a reliable identification of vertebrae. For a single vertebra, we achieve an identification success of more than 70%. Increasing the number of available vertebrae leads to an increase in the identification rate reaching 100% if 16 or more vertebrae are shown in the image.

Spine Segmentation Using Articulated Shape Models

Including prior shape in the form of anatomical models is a well-known approach for improving segmentation results in medical images. Currently, most approaches are focused on the modeling and segmentation of individual objects. In case of object constellations, a simultaneous segmentation of the ensemble that uses not only prior knowledge of individual shapes but also additional information about spatial relations between the objects is often beneficial. In this paper, we present a two-scale framework for the modeling and segmentation of the spine as an example for object constellations. The global spine shape is expressed as a consecution of local vertebra coordinate systems while individual vertebrae are modeled as triangulated surface meshes. Adaptation is performed by attracting the model to image features but restricting the attraction to a former learned shape. With the developed approach, we obtained a segmentation accuracy of 1.0 mm in average for ten thoracic CT images improving former results.

Modelling porous structures by repeated Sierpinski carpets

Porous materials such as sedimentary rocks often show a fractal character at certain length scales. Deterministic fractal generators, iterated upto several stages and then repeated periodically, provide a realistic model for such systems. On the fractal, diffusion is anomalous, and obeys the law 〈r2〉∼t2/dw, where 〈r2〉 is the mean square distance covered in time t and dw>2 is the random walk dimension. The question is: How is the macroscopic diffusivity related to the characteristics of the small scale fractal structure, which is hidden in the large-scale homogeneous material? In particular, do structures with same dw necessarily lead to the same diffusion coefficient at the same iteration stage? The present paper tries to shed some light on these questions.

Tsallis and Rényi entropies in fractional diffusion and entropy production

The entropy production rate for fractional diffusion processes using Shannon entropy was calculated previously, which showed an apparently counter intuitive increase with the transition from dissipative diffusion behaviour to reversible wave propagation. Renyi and Tsallis entropies, which have an additional parameter q generalizing the Shannon case (q=1), are shown here to have similar counter intuitive behaviours. However, the issue can be successfully treated in exactly the same manner as with Shannon entropy for q being not too large (i.e., generalizations near the Shannon case), whereas for larger q the Renyi and Tsallis entropies behave in a different way.

The similarity group and anomalous diffusion equations

A number of distinct differential equations, known as generalized diffusion equations, have been proposed to describe the phenomenon of anomalous diffusion on fractal objects. Although all are constructed to correctly reproduce the basic subdiffusive property of this phenomenon, using similarity methods it becomes very clear that this is far from sufficient to confirm their validity. The similarity group that they all have in common is the natural basis for making comparisons between these otherwise different equations, and a practical basis for comparisons between the very different modelling assumptions that their solutions each represent. Similarity induces a natural space in which to compare these solutions both with one another and with data from numerical experiments on fractals. It also reduces the differential equations to (extra-) ordinary ones, which are presented here for the first time. It becomes clear here from this approach that the proposed equations cannot agree even qualitatively with either each other or the data, suggesting that a new approach is needed.

Random walks on finitely ramified Sierpinski carpets

A new algorithm is presented that allows an efficient computer simulation of random walks on finitely ramified Sierpinski carpets. Instead of using a bitmap of the nth iteration of the carpet to determine allowed neighbor sites, neighbourhood relations are stored in small lookup tables and a hierarchical coordinate notation is used to give the random walker position. The resulting algorithm has low memory requirements, shows no surface effects even for extremely long walks and is well suited for modern computer architectures.

Hausdorff dimension estimates for invariant sets with an equivariant tangent bundle splitting

Upper bounds for the Hausdorff dimension of compact and invariant sets of diffeomorphisms are given using a singular value function of the tangent map and the topological entropy under the assumption, that there exists an equivariant splitting of the tangent bundle. This improves previous results for compact uniformly hyperbolic sets of diffeomorphisms satisfying an additional pinching condition. Furthermore, it is shown that the results can be extended to a special class of noninjective maps.

Lung lobe modeling and segmentation with individualized surface meshes

An automated segmentation of lung lobes in thoracic CT images is of interest for various diagnostic purposes like the quantification of emphysema or the localization of tumors within the lung. Although the separating lung fissures are visible in modern multi-slice CT-scanners, their contrast in the CT-image often does not separate the lobes completely. This makes it impossible to build a reliable segmentation algorithm without additional information. Our approach uses general anatomical knowledge represented in a geometrical mesh model to construct a robust lobe segmentation, which even gives reasonable estimates of lobe volumes if fissures are not visible at all. The paper describes the generation of the lung model mesh including lobes by an average volume model, its adaptation to individual patient data using a special fissure feature image, and a performance evaluation over a test data set showing an average segmentation accuracy of 1 to 3 mm.

Clouds, fibres and echoes: a new approach to studying random walks on fractals

Up to now the general approach of constructing evolution differential equations to describe random walks on fractals has not succeeded. Is this because the true probability density function is inherently fractal? When plotted in the appropriate similarity variable, we find a cloud which is not too smooth. Further investigation shows that this cloud has a structure that might be overlooked if one is looking for the usual single-valued probability density function. The cloud is composed of an infinite family of smooth fibres, each of which describes the behaviour of the walk on an infinite echo point class. The fibres are individually smooth and so are naturally amenable to analysis with differential equations.

The pore structure of Sierpinski carpets

In this paper, a new method is developed to investigate the pore structure of finitely and even infinitely ramified Sierpinski carpets. The holes in every iteration stage of the carpet are described by a hole-counting polynomial. This polynomial can be computed iteratively for all carpet stages and contains information about the distribution of holes with different areas and perimeters, from which dimensions governing the scaling of these quantities can be determined. Whereas the hole area is known to be two dimensional, the dimension of the hole perimeter may be related to the random walk dimension.