We consider the problem of enumerating the prime implicants of a given discrete function as a basic task of circuit theory. First, we count PI's for random Boolean functions. Then we use the well known lattice differentiation as a tool for finding implicants. The concept of a peak admits to characterize prime implicants, at least those with no improper domains. The improper case can be reduced to a lower dimensional problem. Since the peak test is local, a parallel algorithm is available. The time and space complexity turns out to be low measured in the input size.

A discrete model of population growth in the spirit of Fibonacci's rabbits, but with arbitrary fixed times for the beginning and the end of fertility and for death is investigated. Working with generating functions for linear recursions we pursue the idea to give asymptotic estimates for the number of existing individuals by means of the powers of one single main root z of the function's denominator. The mathematical problem of mortality is easy to handle. While the outlines of such a paradigm are recognizable in the case of perpetual fertility, there remain open problems with the localization of roots on unexpected "bubbles", if fertility gets lost at a finite time. Therefore, an alternative method of asymptotic approximation via convolution techniques is given.A generalization of this model to realistic situations with age dependent fertility rates is straightforward. Modern computing techniques admit a convenient survey over the existing roots. In competition with continuous models of demography the results seem to clarify the global influence of the demographic data in the so called stable models of demography. This model is basic for prognostics, when - more general - dynamic changes of the demographic parameters occur.

Among the methods proposed for the analysis of functional MR we have previously introduced a model-independent analysis based on the self-organizing map (SOM) neural network technique. The SOM neural network can be trained to identify the temporal patterns in voxel time-series of individual functional MRI (fMRI) experiments. The separated classes consist of activation, deactivation and baseline patterns corresponding to the task-paradigm. While the classification capability of the SOM is not only based on the distinctness of the patterns themselves but also on their frequency of occurrence in the training set, a weighting or selection of voxels of interest should be considered prior to the training of the neural network to improve pattern learning. Weighting of interesting voxels by means of autocorrelation or F-test significance levels has been used successfully, but still a large number of baseline voxels is included in the training. The purpose of this approach is to avoid the inclusion of these voxels by using three different levels of segmentation and mapping from Talairach space: (1) voxel partitions at the lobe level, (2) voxel partitions at the gyrus level and (3) voxel partitions at the cell level (Brodmann areas). The results of the SOM classification based on these mapping levels in comparison to training with all brain voxels are presented in this paper.

Functional magnetic resonance imaging (fMRI) becomes a common method to study task induced brain activation. Using rapid Echo Planar Imaging (EPI) sequences one can obtain a higher MR-Signal under a task condition close by activated areas as a result of susceptibility changes in blood oxygenation (BOLD effect). Beside the commonly used blocked task designs, event-related paradigms gain more importance for activation of higher cognitive functions enabling more sophisticated and complex paradigms. For the analysis of event-related fMRI data one can use statistical tests, in example t-test used by SPM Software. The introduced analysis method based on an artificial neural network algorithm, a self-organizing map (SOM), is capable to distinguish between task related activation, deactivation and baseline patterns from the time series. This is achieved by temporal sorting and projection of all events from one condition into one combined hemodynamic response sampling for each voxel. These responses, having individual patterns can be separated by their pattern features and is done by training of the neural network. After training the SOM consists of a pattern-to-voxel mapping which is superimposed onto either an anatomical or EPI image of the subject for the task evaluation.

Bei der Segmentierung medizinischer Bilder können Algorithmen robuster gestaltet werden, wenn Wissen über die Form dargestellter Strukturen in eine Erkennung einfließt. In unserer Methode wird ein Ballon-Modell hybrid mit einem Point-Distribution-Modell (PDM) verknüpft. Dabei wird in jeder Iteration des Ballon-Modells der zur Kontur ähnlichste zulässige Formprototyp des PDM geschätzt, eine elastische Anbindung führt zu einer Formkraft, die als neuer Anteil der Einflüsse das Ballon-Modell deformiert. Die gegenseitige Annäherung beider Modelle unter gleichzeitigem Einfluß von Bildinformationen führt zu einer robusten Objekterkennung auf artefaktbehaftetem Bildmaterial. Tests auf synthetischen Bildern quantifizieren die Verbesserung einer Segmentierung durch Einsatz von Formwissen. Auf 52 realen Bildern eines sprechenden Mundes konnte die subjektiv bewertete Erkennungsrate von 3,8% auf 80,8% gesteigert werden.

The problem of an instructive and realistic animation and visualization of the shadow- and color-conditions during conjunctions of actively and passively illuminated cosmic objects has found only particularly satisfying solutions so far. As an example, we study a total solar eclipse. There are didactic shortcomings of specialized astronomical software, even though solutions have been given, which are very impressive for experts.Using the possibilities of commercial 3D-animation software, we give an object-oriented partial solution. In order to get correct astronomical representations we model - for different tasks - the object space under cinematic aspects with parameters for spatial and temporal scaling, for illumination and coloring under couplings of varying strength. The adaptation of the parameters to optimal acceptance of the spectator must be done a posteriori.

In the last few years more and more University Hospitals as well as private hospitals changed to digital information systems for patient record, diagnostic files and digital images. Not only that patient management becomes easier, it is also very remarkable how clinical research can profit from Picture Archiving and Communication Systems (PACS) and diagnostic databases, especially from image databases. Since images are available on the finger tip, difficulties arise when image data needs to be processed, e.g. segmented, classified or co-registered, which usually demands a lot computational power. Today's clinical environment does support PACS very well, but real image processing is still under-developed. The purpose of this paper is to introduce a parallel cluster of standard distributed systems and its software components and how such a system can be integrated into a hospital environment. To demonstrate the cluster technique we present our clinical experience with the crucial but cost-intensive motion correction of clinical routine and research functional MRI (fMRI) data, as it is processed in our Lab on a daily basis.

Die funktionelle Magnetresonanztomographie (fMRT) des Gehirns ermöglicht die Lokalisation funktioneller Abläufe im Gehirn. Erhöther Blutfluß als Folge von erhöhter neuronaler Aktivität in einem aktivierten Hirnareal ist mit Hilfe sehr schneller MR-Bildgebung meßbar. Durch geeignete Aktivierungsparadigmen über einen Zeitraum ist es möglich, aktivierte von nicht aktivierten Regionen zu trennen. Zur Trennung werden die gemessenen Zeitreihen (MR-Signal in der Zeit) für jedes Voxel auf dessen Korrespondenz mit dem Aktivierungsparadigma überprüft. Neben den bisher verwendeten statistischen Verfahren ist es auch möglich eine Separierung der Zeitreihen mittels einer selbstorganisierenden Merkmalskarte (SOM, Self- Organizing Map) zu erreichen, ohne dabei ein Modell der hämodynamischen Antwort des Gehirns zu verwenden. Die nur sehr geringe Anzahl an aktivierten Voxel setzt eine Vorauswahl geeigneter Kandidaten für das Training des SOM Neuronalen Netzwerkes voraus. Die hier beschriebene Methode wählt die geeignetsten Lernkandidaten anhand des Vorwissens über die Periodizität des Aufgabenparadigmas unter Verwendung des Fourierspektrums bzw. des Frequenzperiodogramms jeder Zeitreihe aus, wodurch eine erhebliche Verbesserung der Klassifikationsleistung der SOM erreicht wird.

Functional magnet resonance imaging (fMRI) has become a standard non invasive brain imaging technique delivering high spatial resolution. Brain activation is determined by magnetic susceptibility of the blood oxygen level (BOLD effect) during an activation task, e.g. motor, auditory and visual tasks. Usually box-car paradigms have 2-4 rest/activation epochs with at least an overall of 50 volumes per scan in the time domain. Statistical test based analysis methods need a large amount of repetitively acquired brain volumes to gain statistical power, like Student's t-test. The introduced technique based on an self-organizing neural network (SOM) makes use of the intrinsic features of the condition change between rest and activation epoch and demonstrated to differentiate between the conditions with less time points having only one rest and one activation epoch. The method reduces scan and analysis time and the probability of possible motion artifacts from the relaxation of the patients head. Functional magnet resonance imaging (fMRI) of patients for pre-surgical evaluation and volunteers were acquired with motor (hand clenching and finger tapping), sensory lice application), auditory (phonological and semantic word recognition task) and visual paradigms (mental rotation). For imaging we used different BOLD contrast sensitive Gradient Echo Planar Imaging (GE-EPI) single-shot pulse sequences (TR 2000 and 4000, 64x64 and 128x128, 15-40 slices) on a Philips Gyroscan NT 1.5 Tesla MR imager. All paradigms were RARARA (R = rest, A = activation) with an epoch width of 11 time points each. We used the self-organizing neural network implementation described by T. Kohonen with a 4x2 2D neuron map. The presented time course vectors were clustered by similar features in the 2D neuron map. Three neural networks were trained and used for labeling with the time course vectors of one, two and all three on/off epochs. The results were also compared by using an Kolmogorov-Smirnov statistical test of all 66 time points. To remove non-periodical time courses from training an auto-correlation function and bandwidth limiting Fourier filtering in combination with Gauss temporal smoothing was used. None of the trained maps, with one, two and three epochs, were significantly different which indicates that the feature space of only one on/off epoch is sufficient to differentiate between the rest and task condition. We found, that without pre-processing of the data no meaningful results can be achieved, because of the huge amount of the non-activated and background voxels represents the majority of the features and is therefore learned by the SOM. Thus it is crucial to remove unnecessary capacity load of the neural network by selection of the training input, using auto-correlation function and/or Fourier spectrum analysis. However by reducing the time points to one rest and one activation epoch either strong auto-correlation or an precise periodical frequency is vanishing. Self-organizing maps can be used to separate rest and activation epochs of with only a 1/3 of the usually acquired time points. Because of the nature of the SOM technique, the pattern or feature separation, only the presence of a state change between the conditions is necessary for differentiation.Also the variance of the individual hemodynamic response function (HRF) and the variance of the spatial different regional cerebral blood flow (rCBF) is learned from the subject and not compared with a fixed model done by statistical evaluation. We found that reducing the information to only a few time points around the BOLD effect was not successful due to delays of rCBF and the insufficient extension of the BOLD feature in the time space. Especially for patient routine observation and pre-surgical planing a reduced scan time is of interest.