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We describe a novel method for continuously transforming two triangulated models of arbitrary topology into each other. Equal global topology for both objects is assumed, extensions for genus changes during metamorphosis are provided. The proposed method addresses the major challenge in 3D metamorphosis, namely specifying the morphing process intuitively, with minimal user interaction and sufficient detail. Corresponding regions and point features are interactively identified. These regions are parametrized automatically and consistently, providing a basis for smooth interpolation. Utilizing suitable 3D interaction techniques a simple and intuitive control over the whole morphing process is offered.

In this paper various algorithms for rendering gaseous phenomena are reviewed. In computer graphics such algorithms are used to model natural scenes containing clouds, fog, flames and so on. On the other hand it has become an important technique in scientific visualization to display three dimensional scalar datasets as cloudy objects. Our emphasis is on this latter subject of so-called {\em direct volume rendering}. All algorithms will be discussed within the framework of linear transport theory. The equation of transfer is derived. This equation is suitable to describe the radiation field in a participating medium where absorption, emission, and scattering of light can occur. Almost all volume rendering algorithms can be shown to solve special cases of the equation of transfer. Related problems like the mapping from data values to model parameters or possible parallelization strategies will be discussed as well.

Line Integral Convolution (LIC) is a powerful technique for generating striking images and animations from vector data. Introduced in 1993, the method has rapidly found many application areas, ranging from computer arts to scientific visualization. Based upon locally filtering an input texture along a curved stream line segment in a vector field, it is able to depict directional information at high spatial resolutions. We present a new method for computing LIC images, which minimizes the total number of stream lines to be computed and thereby reduces computational costs by an order of magnitude compared to the original algorithm. Our methods utilizes fast, error-controlled numerical integrators. Decoupling the characteristic lengths in vector field grid, input texture and output image, it allows to compute filtered images at arbitrary resolution. This feature is of great significance in computer animation as well as in scientific visualization, where it can be used to explore vector data by smoothly enlarging structure of details. We also present methods for improved texture animation, employing constant filter kernels only. To obtain an optimal motion effect, spatial decay of correlation between intensities of distant pixels in the output image has to be controlled. This is achieved by blending different phase shifted box filter animations and by adaptively rescaling the contrast of the output frames.

Segmentation tools in medical imaging are either based on editing geometric curves or on the assignment of region labels to image voxels. While the first approach is well suited to describe smooth contours at subvoxel accuracy, the second approach is conceptually more simple and guarantees a unique classification of image areas. However, contours extracted from labeled images typically exhibit strong staircase artifacts and are not well suited to represent smooth tissue boundaries. In this paper we describe how this drawback can be circumvented by supplementing region labels with additional weights. We integrated our approach into an interactive segmentation system providing a well-defined set of manual and semi-automatic editing tools. All tools update both region labels as well as the corresponding weights simultaneously, thus allowing one to define segmentation results at high resolution. We applied our techniques to generate 3D polygonal models of anatomical structures.

In this thesis we develop new methods for visualizing vector fields which specifically address three design goals: accuracy, performance, and cognition. Our methods will be general-purpose and can be applied to arbitrary vector fields in two- and three-dimensional space. The methodology behind our approach is {\em texture-based visualization}. Texture-based visualization methods imitate techniques known from experimental flow visualization, namely, the observation of randomly dispersed particles or dye injection patterns. Instead of depicting individual lines or symbols, a contiguous high-resolution image or texture is generated. This texture clearly reveals the directional structure of the field. In this way intuitive insight can be obtained and even small details of the field become visible. However, it remains unclear what kind of textures are best suited for our purpose. How do we generate them? Can we apply these methods in three-dimensional space? In particular, we focus on a technique known as {\em line integral convolution} or LIC. This method turns out to be quite versatile and well-suited for visualizing many interesting vector fields. LIC images display the integral curves or field lines of a vector field at high spatial resolution. Although conceptually quite simple, line integral convolution implies a number of interesting mathematical and algorithmic questions.

\small Many interesting phenomena in molecular systems like interactions between macro-molecules, protein-substrate docking, or channeling processes in membranes are gouverned to a high degree by classical Coulomb or van-der-Waals forces. The visualization of these force fields is important for verifying numerical simulations. Moreover, by inspecting the forces visually we can gain deeper insight into the molecular processes. Up to now the visualization of vector fields is quite unusual in computational chemistry. In fact many commercial software packages do not support this topic at all. The reason is not that vector fields are considered unimportant, but mainly because of the lack of adequate visualization methods. In this paper we survey a number of methods for vector field visualization, ranging from well-known concepts like arrow or streamline plots to more advanced techniques like line integral convolution, and show how these can be applied to computational chemistry. A combination of the most meaningful methods in an interactive 3D visualization environment can provide a powerful tool box for analysing simulations in molecular dynamics.

Line integral convolution (LIC) has become a well-known and popular method for visualizing vector fields. The method works by convolving a random input texture along the integral curves of the vector field. In order to accelerate image synthesis significantly, an efficient algorithm has been proposed that utilizes pixel coherence in field line direction. This algorithm, called ``fast LIC'', originally was restricted to simple box-type filter kernels. Here we describe a generalization of fast LIC for piecewise polynomial filter kernels. Expanding the filter kernels in terms of truncated power functions allows us to exploit a certain convolution theorem. The convolution integral is expressed as a linear combination of repeated integrals (or repeated sums in the discrete case). Compared to the original algorithm the additional expense for using higher order filter kernels, e.g.\ of B-spline type, is very low. Such filter kernels produce smoother, less noisier results than a box filter. This is evident from visual investigation, as well as from analysis of pixel correlations. Thus, our method represents a useful extension of the fast LIC algorithm for the creation of high-quality LIC images.

We present a new technique for generating surface meshes from a uniform set of discrete samples. Our method extends the well-known marching cubes algorithm used for computing polygonal isosurfaces. While in marching cubes each vertex of a cubic grid cell is binary classified as lying above or below an isosurface, in our approach an arbitrary number of vertex classes can be specified. Consequently the resulting surfaces consist of patches separating volumes of two different classes each. Similar to the marching cubes algorithm all grid cells are traversed and classified according to the number of different vertex classes involved and their arrangement. The solution for each configuration is computed based on a model that assigns probabilities to the vertices and interpolates them. We introduce an automatic method to find a triangulation which approximates the boundary surfaces - implicitly given by our model - in a topological correct way. Look-up tables guarantee a high performance of the algorithm. In medical applications our method can be used to extract surfaces from a 3D segmentation of tomographic images into multiple tissue types. The resulting surfaces are well suited for subsequent volumetric mesh generation, which is needed for simulation as well as visualization tasks. The proposed algorithm provides a robust and unique solution, avoiding ambiguities occuring in other methods. The method is of great significance in modeling and animation too, where it can be used for polygonalization of non-manifold implicit surfaces.

Hyperthermia Treatment Planning in Clinical Cancer Therapy: Modelling, Simulation and Visualization
(1997)

\noindent The speaker and his co-workers in Scientific Computing and Visualization have established a close cooperation with medical doctors at the Rudolf--Virchow--Klinikum of the Humboldt University in Berlin on the topic of regional hyperthermia. In order to permit a patient--specific treatment planning, a special software system ({\sf\small HyperPlan}) has been developed. \noindent A mathematical model of the clinical system ({\it radio frequency applicator with 8 antennas, water bolus, individual patient body}) involves Maxwell's equations in inhomogeneous media and a so--called bio--heat transfer PDE describing the temperature distribution in the human body. The electromagnetic field and the thermal phenomena need to be computed at a speed suitable for the clinical environment. An individual geometric patient model is generated as a quite complicated tetrahedral ``coarse'' grid (several thousands of nodes). Both Maxwell's equations and the bio--heat transfer equation are solved on that 3D--grid by means of {\em adaptive} multilevel finite element methods, which automatically refine the grid where necessary in view of the required accuracy. Finally optimal antenna parameters for the applicator are determined . \noindent All steps of the planning process are supported by powerful visualization methods. Medical images, contours, grids, simulated electromagnetic fields and temperature distributions can be displayed in combination. A number of new algorithms and techniques had to be developed and implemented. Special emphasis has been put on advanced 3D interaction methods and user interface issues.

After a short summary on therapy planning and the underlying technologies we discuss quantitative medicine by giving a short overview on medical image data, summarizing some applications of computer based treatment planning, and outlining requirements on medical planning systems. Then we continue with a description of our medical planning system {\sf HyperPlan}. It supports typical working steps in therapy planning, like data aquisition, segmentation, grid generation, numerical simulation and optimization, accompanying these with powerful visualization and interaction techniques.