Recovering the pose of a camera using only visual input has been a well studied problem in computer vision, robotics, and photogrammetry. The underlying assumption has been the existence of corrspondences of image features or the identification of environmental landmarks. Such approaches either in structure from motion or pose estimation or in simultaneous localization and mapping, have been vulnerable to outliers and to the presence of independent motions. We introduce a new framework, where a motion hypothesis can be computed without correspondences using the Radon transform. The Radon transform turns out to become a filter, or more precisely a convolution on groups. The domain of integration is the cross-product of two images and a direct convolution would have a prohibitive computational cost. Using the spherical Fourier transform we have been able to compute the strength of a motion hypothesis in frequency space. Several useful byproducts in image registration result from this framework. The talk will be concluded by applications both in localization and 3d modelling of environments.

In my seminar I shall relate some of the history of the company and summarise the algorithms and software we have developed, in particular our "Gandalf" computer vision library (see gandalf-library.sf.net). There will be extensive demonstrations of mokey and monet. Several outstanding issues relevant to computer vision research will be discussed. If time permits, I will also present a mathematical conundrum in the area of the normalisation of projective quantitities.

We present a method that automatically partitions a single image into non-overlapping regions coherent in texture and colour. An assumption that each textured or coloured region can be represented by a small template, called the seed, is used. Positioning of the seed across the input image gives many possible sub-segmentations of the image having same texture and colour property as the pixels behind the seed. A probability map constructed during the sub-segmentations helps to assign each pixel to just one most probable region and produce the final pyramid representing various detailed segmentations at each level. Each sub-segmentation is obtained as the min-cut/max-flow in the graph built from the image and the seed. One segment may consist of several isolated parts. Compared to other methods our approach does not need a learning process or a priori information about the textures in the image. Performance of the method is evaluated on images from the Berkeley database.

Kernels in the sense of machine learning are two-placed functions whose values can be represented as inner products in some Hilbert space. By this, classification algorithms based on and restricted to linear models can be carried over to non-linear methods by replacing the Gram matrix by a kernel map. Following this strategy in the last decades powerful kernel-based methods like support vector machines, kernel principal component analysis and kernel Fischer discriminant were developped and successfully applied to classification and learning problems in the fields of computer vision, bioinformatics and data mining.

It is interesting that in literature the mechanism of the so-called kernel trick is often motivated heuristically by arguments pointing out that kernels act as a similarity measures without providing any axiomatic framework for this aspect. According to this similarity argument two different data points being similar are mapped close together in the Hilbert space by which the data points are arranged in a more separable manner.

This is the point where fuzzy equivalence relations come in which---based on a relaxed concept for transitivity---provide an axiomatic framework for similarity as a generalization of the classical concept of an equivalence relation. It is worth pointing out that for classical relations the concepts of an equivalence relation and a kernel map are equivalent. Recently it could be demonstrated that all kernels mapping to the unit interval with constant one in its diagonal can be represented as fuzzy equivalence relations in a way that is commonly used in fuzzy systems for representing fuzzy rule bases. At first glance fuzzy systems and kernel-based methods are totally different paradigms. Fuzzy systems are used to model explicit knowledge of features and dependencies, e.g., by means of rule bases. In contrast, kernel-based methods act implicitly in an Hilbert space induced by a feature map for which only the existence has to be guaranteed, that is no explicit knowledge of the feature map is required.

The goal of this contribution is to present recent results and to discuss the synthesis of knowledge-based and kernel-based paradigms.