Reiner Lenz

Reiner Lenz

Many stochastic processes are defined on special geometrical objects like spheres and cones. We describe how tools from harmonic analysis, i.e. Fourier analysis on groups, can be used to investigate probability density functions (pdfs) on groups and homogeneous spaces. We consider the special case of the Lorentz group SU(1,1) and the unit disk with its hyperbolic geometry, but the procedure can be generalized to a much wider class of Lie-groups. We mainly concentrate on the Mehler-Fock transform which is the radial part of the Fourier transform on the disk. Some of the characteristic features of this transform are the relation to group-convolutions, the isometry between signal and transform space, the relation to the Laplace-Beltrami operator and the relation to group representation theory. We will give an overview over these properties and their applications in signal processing. We will illustrate the theory with two examples from low-level vision and color image processing.

Satoshi Oshima, Rica Mochizuki, Reiner Lenz et al.

We use methods from Riemann geometry to investigate transformations between the color spaces of color-normal and color weak observers. The two main applications are the simulation of the perception of a color weak observer for a color normal observer and the compensation of color images in a way that a color weak observer has approximately the same perception as a color normal observer. The metrics in the color spaces of interest are characterized with the help of ellipsoids defined by the just-noticable-differences between color which are measured with the help of color-matching experiments. The constructed mappings are isometries of Riemann spaces that preserve the perceived color-differences for both observers. Among the two approaches to build such an isometry, we introduce normal coordinates in Riemann spaces as a tool to construct a global color-weak compensation map. Compared to previously used methods this method is free from approximation errors due to local linearizations and it avoids the problem of shifting locations of the origin of the local coordinate system. We analyse the variations of the Riemann metrics for different observers obtained from new color matching experiments and describe three variations of the basic method. The performance of the methods is evaluated with the help of semantic differential (SD) tests.

Reiner Lenz

We describe a statistical analysis of the eye tracker measurements in a database with 15 observers viewing 1003 images under free-viewing conditions. In contrast to the common approach of investigating the properties of the fixation points we analyze the properties of the transition phases between fixations. We introduce hyperbolic geometry as a tool to measure the step length between consecutive eye positions. We show that the step lengths, measured in hyperbolic and euclidean geometry, follow a generalized Pareto distribution. The results based on the hyperbolic distance are more robust than those based on euclidean geometry. We show how the structure of the space of generalized Pareto distributions can be used to characterize and identify individual observers.