# Reiner Lenz

Docent, Senior Lecturer

Personal homepage

Personal homepage

## Contact Information

Mail: reiner.lenz at liu se

Phone: 011 36 32 78

Office: Kopparhammaren G516

## Background

Background I received a Diploma in Mathematics from the Georg-August University in Göttingen, Germany and before moving to Linköping, Sweden I worked in the Image Processing Group at the Institut für Physikalische Elektronik, Stuttgart University. As part of my thesis work I developed some of the first dynamic visualizations of 3D images from CTResources: Shaded human head , Pig head , CLSM (Confocal Laser Scanning Microscopy) , and electron microscopy

Paper PDF.

We also developed a system to present dynamic stereographic 3D views of the 3D image data under interactive control of the user, an image of Olof Palme using the system.

## Research Interests

My main intrest lies in the application of mathematical methods to problems in image processing, pattern recognition and machine learning.#### I) Harmonic Analysis

One common theme is the application of harmonic analysis, ie. the group theoretical generalization of the Fourier transform. This leads to a group theoretical generalization of stationary processes and within this framework one can explain the connection between the symmetry properties of stochastic processes, their principal components and their invariance or co-variance properties (LNCS). A collection of applications of this framework contains the following examples:A) The permutation group In one application the outcome of the stochastic process are RGB vectors. The permutation group S(3) of three elements operates on these vectors by interchanging the color channels. In cases where all permutations appear with equal probabilities one can show that the general theory leads to a description of the color properties in terms of intensity and the two opponent color channels red-green and yellow-blue (VISAPP2010). A more complicated application of the permutation group describes the construction of projection and permutation invariant features in (IJCV98)

B) Dihedral groups Most image sensors arrange their sensor elements either on a square or a hexagonal grid. The symmetry groups of these grids, ie. the transformations that map the grid onto itself are the dihedral groups D(4) and D(6). Filter systems based on the dihedral groups were introduced in (VCIR95)

C) Octahedral groups In 3D the simplest sampling pattern is given by the cubic grid with the octahedral group as symmetry group. Octahedral filter systems are described in (TIP2009)

D) Finite subgroups of the 3D rotation group SO(3) Many image processing and vision problems make essential use of 3D rotations or 3D orientations. Coomputational methods are usually restricted to finite collections and it is therefore interesting to study finite subgroups of the rotation groups. For 2D rotations the results are simple and lead to the discrete Fourier transform (DFT). For 3D rotations it can be shown that there are only finitely many such subgroups. They are described in (ACCV2007) where their application in the processing of reflection properties of materials (BRDFs) are described as illustration.

E) Continuous rotations The group of 2D rotations is the simplest group and leads to the ordinary Fourier transform. Rotations in 3D are more complicated since they do not commute. The Fourier transform in 3D leads to expansions in spherical harmonics. Their usage was introduced in (Pisa85) and (PR1987). A method to recover orientation parameters from filter vectors based on Lie-theory is described in (Josaa1997).

F) Lorentz Groups Positive-valued functions do not span the whole function space. Instead it can be seen that they are all located in a conical region. This motivates the application of Lorentz groups such as SU(1,n) in the study of these functions. One example of such functions that are relevant for image processing are the multispectral distributions of illumination sources or reflectance spectra. Applications of the Lorentz group SU(1,1) to multispectral color distributions are described in (CVPR2007), (JMIV2005) and (AIEP2005)

#### II) Learning and Retrieval

In signal processing filter systems are usually designed. Biological vision systems, on the other hand are the result of evolutionary processes. This leads to the question how technical systems can be constructed by training them on empirical data. Methods based on fourth-order statistics were investigated in (NIPS94) and (Josaa96) where it was shown that filter systems trained on multispectral reflectance spectra from a color atlas have similar properties as the color matching functions used in color science. Harmonic analysis based systems have the property that they provide minimum-mean squared error approximations and that they have natural invariance properties. This makes them ideal candidates for retrieval systems where a large number of images have to be indexed and where retrieval time should be very short.#### III) Non-Euclidean Metrics and Riemann Geometry

The most popular metric by far is the Euclidean metric. For many applications this metric is not flexible enough since the geometrical properties can be different in different parts of the space under consideration. A generalization of Euclidean geometry is the Riemann geometry and the oldest example, used by Riemann himself, is the space of colors. For spaces where the points are given by colors it is well-known that the perceived similarity between different colors is a complicated function of the similarity between the physical stimuli that are the cause of these color perceptions. The situation is even more complicated by the fact that this relation varies between observers. Some of the models describing the properties of color-weak observers are described in (Oshima2009). Other more abstract examples where concepts from Riemann geometry are useful in computer vision and learning are described in (Lenz2010b) and (Lenz2009c). In statistics it is known that the Fisher metric defines a metric in the space of probability distributions. It is known that for many image sources the results of edge-detectors are Weibull distributed and investigations of large image databases show that the parameters of the Weibull distribution can be used for image database retrieval based on the visual apperance of images (Zografos2011)## Current Teaching

HT 2011:TNM089 - Computational Photography

## Publications

**List of all publications**

### Recent Publications

Publications in 2013

Springer New York, page 119-145 - 2013

Computational Color Imaging Workshop (CCIW 2013) - 2013

IEEE Transactions on Image Processing, Volume 22, Number 2, page 621--630 - 2013

Publications in 2012

IEEE Transactions on Visualization and Computer Graphics (TVCG), Volume 18, Number 12, page 2345--2354 - 2012

Publications in 2011

Electronic Letters on Computer Vision and Image Analysis, Volume 10, Number 1, page 24-41 - 2011

Color Research & Application, Volume 36, Number 3, page 210--221 - 2011

Proceedings of the 17th Scandinavian conference on Image analysis (SCIA), page 262--272 - 2011

Proceedings of the 17th Scandinavian conference on Image analysis (SCIA), page 218--227 - 2011

Proceedings of the 17th Scandinavian conference on Image analysis (SCIA), page 579--591 - 2011

Publications in 2010

CGIV 2010: 5th European Conference on Colour in Graphics, Imaging, and Vision, page 353--358 - 2010

Proceedings of the ACM International Conference on Image and Video Retrieval (CIVR) - 2010

CGIV 2010: 5th European Conference on Colour in Graphics, Imaging, and Vision, page 393--398 - 2010

Publications in 2009

Proceedings of the 16th Scandinavian conference on Image analysis (SCIA), page 400--409 - 2009

Scale Space and Variational Methods in Computer Vision, Second International Conference, SSVM 2009, June 1-5, Voss, Norway, page 124--136 - 2009