The design and understanding of numerical algorithms
plays a key role in building highquality methods
for image processing and computer vision.
This especially holds true for numerical methods
for hyperbolic partial differential equations (PDEs),
as the resolution of discontinuous solution
features easily inspires oscillations in numerical
approximations.
Besides the construction of schemes for specific purposes in
image processing and computer vision, we also perform fundamental
research on numerical methods for hyperbolic PDEs.
Hyperbolic Schemes in Mathematical Morphology
PDEbased mathematical morphology relies on the hyperbolic equations
of dilation/erosion. This holds true for greyscale images
as well as for images composed of matrixvalued data sets (socalled tensor fields).
Click
here
for more information on mathematical morphology.

ShockCapturing Schemes for PDEs in Greyscale Morphology
In [1] we develop a novel flux corrected transport
(FCT) scheme that can accurately cope with the peculiarities of the
morphological processes of dilation and erosion.
Later on, this method has been extended to deal with general convex structuring
elements [2]. In the mentioned works, it is shown
that the method performs better than other PDEbased schemes in the field.
Also, the FCT scheme is competitive to setbased, algebraic solutions.

Schemes for MatrixValued PDEs
For the appropriate numerical treatment of recently modeled
matrixvalued PDEs we devised matrixvalued generalisations of
known realvalued schemes, see especially [3].
Also, the FCTscheme introduced in [1]
for processing grey value images has been successfully extended
to this setting showing
superior resolution of edgelike structures compared with other methods.
We have also shown that our methods can be made locally adaptive
to the data of the evolved image, cf. [4].
Hyperbolic Methods for the Inverse Problem of Shape from Shading
The shape from shading (SFS) task is a classic
inverse problem in computer vision.
It amounts to compute at hand of information
on illumination and reflectance the 3D shape of objects in a single
given image. Modern models for SFS
are described by hyperbolic HamiltonJacobi equations.
For more information on that topic, click
here.

Schemes for Lambertian Shape from Shading
In [5] we introduce an efficient iterative
solver for the stateoftheart Lambertian SFS model, and we
prove conditional stability.
An extensive numerical comparison of recent SFS solvers
including our own approach as well as the optimal control
approaches of Prados et al. and Cristiani et al.
is performed in [6]. The latter work is extended by
[7], where we also analysed for the first time
numerical acceleration techniques like fast sweeping and
cascading multigrid methods for all solvers.

Schemes for NonLambertian Shape from Shading
We extend the Lambertian SFS model of Prados et al. in the
work [8] and propose a numerical solver.
The detailed derivation of the model plus a conditional
stability criterion for an efficient iterative solver
is derived in [9]. In the latter
work, we also perform for the first time in the
computer vision literature a numerical scale
analysis of the underlying PDE that reveals important
information on the relative size of diffusive and specular
reflection terms.

Fast Marching Schemes for NonLambertian Shape from Shading
In [10] we construct a fast marching method
for our new nonLambertian SfS model.
By using this we can solve the SFS task on large images
with computational times of about a minute with usual hardware.
Adaptive Hyperbolic Schemes for Variational Correspondence Problems

A HighResolutiontype Method for Variational Computer Vision Tasks
An Adaptive Derivative Discretisation Scheme
The implementation of variational approaches requires to discretise occurring
derivatives.
Adopting a successful concept from the theory of hyperbolic partial
differential equations, we presented in [13] a sophisticated adaptive derivative discretisation scheme.
It blends central and correctly oriented onesided (upwind) differences based on a smoothness measure.
It can improve the quality of results in the same manner as model refinements.
For more information see our research page on stereo.
Fundamental Research on Hyperbolic Schemes

Implicit Schemes for Hyperbolic PDEs
A novel, mathematically justified theory of implicit monotone methods
is presented in [11].
There it is shown that an Euler implicit time discretisation does not give
automatically unconditional stability in terms of monotonicity
even if the numerical flux function describes a monotone scheme
in the explicit case.

Novel Parallelisation Approach for Fast Marching Methods
In [12] we describe a novel parallelisation
approach that is based on the understanding of the
physical workings of hyperbolic PDEs. It avoids the
traditional domaindecomposition approach and is much easier
to implement than that.

M. Breuß, J. Weickert:
A shockcapturing algorithm for the differential equations
of dilation and erosion.
Journal of Mathematical Imaging and Vision, Vol. 25, 187201, (2006).
Revised version of
Technical Report No. 153, Department of Mathematics,
Saarland University, Saarbrücken, Germany, September 2005.

M. Breuß, J. Weickert:
Highly accurate PDEbased morphology for general structuring
elements.
In X.C. Tai et al. (Eds.):
Scale Space and Variational Methods in Computer Vision.
Lecture Notes in Computer Science, Vol. 5567, 758  769,
Springer, Berlin, 2009.

B. Burgeth, M. Breuß, S. Didas, J. Weickert:
PDEbased morphology for matrix fields: Numerical solution schemes.
Tensors in Image Processing and Computer Vision,
Advances in Pattern recognition, pp. 125150.
Springer, London, 2009.
Revised version of
Technical Report No. 220, Department of Mathematics,
Saarland University, Saarbrücken, Germany, September 2008.

L. Pizarro, B. Burgeth, M. Breuß, J. Weickert:
A directional RouyTourin scheme for adaptive matrixvalued morphology.
In M.H.F. Wilkinson and J.B.T.M. Roerdink (Eds.): Proc. Ninth
International Symposium on Mathematical Morphology (ISMM 2009),
Lecture Notes in Computer Science,
vol. 5720, pp. 250 260, Springer, Berlin, 2009.
©
SpringerVerlag Berlin Heidelberg 2009.

O. Vogel, M. Breuß, J. Weickert:
A Direct Numerical Approach to Perspective ShapefromShading
In H. Lensch, B. Rosenhahn, H.P. Seidel, P. Slusallek, J. Weickert (Eds.):
Vision, Modeling, and Visualization 2007.
Saarbrücken, Germany, 91100, November 2007.

M. Breuß, O. Vogel, J. Weickert:
Efficient numerical techniques for perspective shape from
shading.
In A. Handlovicova, P. Frolkovic, K. Mikula, D. Sevcovic (Eds.):
Algoritmy 2009 (Podbanske, Slovakia, March 2009), pp. 1120, Slovak
University of Technology, Bratislava, 2009.

M. Breuß, E. Cristiani, J.D. Durou, M. Falcone, O. Vogel:
Numerical Algorithms for Perspective Shape from Shading.
To appear in Kybernetika (2009).
Revised version of
Technical Report No. 240, Department of Mathematics,
Saarland University, Saarbrücken, Germany, August 2009.

O. Vogel, M. Breuß, J. Weickert:
Perspective shape from shading with nonLambertian reflectance.
In G. Rigoll (Ed.):
Pattern Recognition.
Lecture Notes in Computer Science, Vol. 5096, 517526, Springer, Berlin, 2008.

M. Breuß, O. Vogel, J. Weickert:
Perspective shape from shading for Phongtype nonLambertian surfaces.
Technical Report No. 216, Faculty of Mathematics and Computer Science,
Saarland University, Saarbrücken, Germany, August 2008.

O. Vogel, M. Breuß, T. Leichtweis, J. Weickert:
Fast shape from shading for Phongtype surfaces.
In X.C. Tai et al. (Eds.):
Scale Space and Variational Methods in Computer Vision.
Lecture Notes in Computer Science, Vol. 5567, 733  744,
Springer, Berlin, 2009.
©
SpringerVerlag Berlin Heidelberg 2009.

M. Breuß:
Monotonicity of Implicit Finite Difference Methods for Hyperbolic Conservation Laws.
Technical Report No. 253, Department of Mathematics,
Saarland University, Saarbrücken, Germany, November 2009.

M. Breuß, E. Cristiani, P. Gwosdek, O. Vogel:
A DomainDecompositionFree Parallelisation of the Fast Marching Method.
Technical Report No. 250, Department of Mathematics,
Saarland University, Saarbrücken, Germany, October 2009.

H. Zimmer, M. Breuß, J. Weickert, H.P. Seidel:
Hyperbolic numerics for variational approaches to correspondence
problems.
In X.C. Tai et al. (Eds.): ScaleSpace and Variational Methods in Computer
Vision. Lecture Notes in Computer Science, Vol. 5567, 636647, Springer, Berlin,
2009.
