Mathematical morphology is concerned with the analysis of shapes
of objects in images. This means e.g. that by
morphological filtering it is possible to simplify structures
in an image depending on their scale, or just to keep structures
up to a selected scale, or to enhance flowlike structures.
The field of mathematical morphology arised in the mid of the
sixties, and today it is one of the most successful
areas in image processing.
There exist two basic processes in mathematical morphology:
dilation and erosion. In a dilation process, at each pixel the
given grey value is changed making use of the maximum of grey values
within a certain pixel mask. This means, light structures are
enhanced, and dark structures vanish. In an erosion process this role
of light and dark is switched. All other filters rely on a
suitable combination of dilation/erosion.
On the modelling side, there are two ways
to realise the basic processes dilation/erosion. The classic way is to
define a mask, or a set of masks, over which the maximum/minimum
is computed. In our group we mainly follow an alternative proceeding
using partial differential equations (PDEs). The arising
PDEs are hyperbolic, and they are variations of socalled
Eikonal equations arising e.g. in physics in the field of
geometric optics.
The research efforts in our group proceed in three
main directions.

Establishing a concept of morphology of tensorvalued data,
as arising e.g. in DTMRI (diffusion tensor magnetic resonance imaging).

Construction of highquality numerical schemes for PDEbased
morphology.

Theoretical investigation of the relation between
the scale space concepts of morphological and linear
scale spaces.
Contributions in Morphology for Tensorvalued Data
The processing of matrixvalued data sets (socalled tensor fields)
is becoming an increasingly important task in modern digital imaging
since the tensor concept provides an adequate description of anisotropic
behaviour: Prominent examples are the diffusive behaviour of water molecules
in tissue, visualised by diffusion tensor magnetic resonance imaging (DTMRI),
or the socalled structure tensor used in fields ranging from motion
analysis to texture segmentation. The tensors that arise in these ways have
realvalued entries and are symmetric. For general
information on filtering such data, see also
this webpage.

TensorValued Morphology, AlgebraicGeometric Approaches
Dilation/erosion rely on maximum and minimum operations.
Their generalisation to the tensorvalued setting is not straightforward
due to the lack of suitable ordering relations for symmetric matrices.
In [4] we have proposed such extensions exploiting
analyticalgebraic and geometric properties of symmetric matrices while
preserving positive definiteness of the initial data. Unfortunately the
proposed approaches have their weaknesses when it comes to computational issues.
This led to the development of more appropriate
extensions of dilation and erosion operations for matrixvalued data.

TensorValued Morphology, OrderingBased Approach
In the matrix valued setting there is a partial ordering, the socalled Loewner ordering,
that is suitable for the definition of appropriate maximal and minimal elements of a set of
matrices. The task to determine these extremal elements is already
nontrivial in the case of 2 by 2 matrices [4]
and becomes more intricate
in higher dimensions [5],[6];
it requires tools from convex analysis and computational geometry.
Nevertheless, the dilation and erosion operations inferred from the ordering
approach exhibit the desired rotational invariance,
preservation of positive definiteness, and
continuous dependence on the initial matrix data.
This makes it possible to define morphological
derivatives and shock filters for matrixfields.

The PDEbased Approach
Dilation/erosion can be described by nonlinear hyperbolic first order PDEs as
introduced by R. van den Boomgaard in 1992.
Based on novel mathematical notions for symmetric matrices, in [7]
we were able to find the truly matrixvalued counterpart to these nonlinear
morphological PDEs. For the appropriate numerical treatment of those novel
matrixvalued PDEs we devised matrixvalued generalisations of
known realvalued schemes, see especially [9].
Numerical Contributions

ShockCapturing Schemes for PDEs in Greyscale Morphology
In 1992 R. van den Boomgaard proposed nonlinear
hyperbolic partial differential equations that mimic the dilation and erosion of images
with structuring elements of increasing size. It is quite challenging to
devise numerical schemes that are capable of following a corresponding
evolution of edges adequately, that means, with as little numerical blurring as possible.
In [8] 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 structuring
elements [10]. 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.

Investigating Relations between Fully Discrete and ContinuousScale Morphology
In 2005, Lerallut et al. have proposed a fully discrete means
to incorporate tonal differences into adaptive morphological filters.
In [13] we validate that such morphological amoebas
are a discretisation of socalled selfsnakes, a PDEbased process
somewhat similar to mean curvature motion.

PDEbased Methods for TensorValued Morphology
Relying on the developed PDEs for tensorvalued morphology,
we devised the first highresolutiontype
schemes for morphology of tensor fields [9].
Also, the FCTscheme introduced in [8]
for processing grey value images has been successfully extended
to this setting showing
superior resolution of edgelike structures compared with other methods.

PDEbased Anisotropic Morphology
Relying on the developed PDEs for greyvalued morphology as well as for
tensorvalued morphology, we developed the concept
of anisotropic morphology for enhancing flowlike, coherent
image structures. Beginning with the investigation in [12],
we refined the modeling as well as the numerics in
subsequent steps [11], [10].
Contributions in Scale Space Theory

Explaining the Logarithmic Relation between Linear and Morphological Systems
In 1994 Dorst/van den Boomgaard and Maragos introduced the
slope transform as the morphological equivalent of the Fourier
transform. As a result an almost logarithmic connection between linear
and morphological systems became apparent.
In [1] we give an explanation for
this relation by revealing that morphology, in essence, is linear
systems theory based on the (max,+)algebra resp. the (min,+)algebra.
The fundamental operations of dilation and erosion turn out to be
convolutions with respect to these algebras.
The socalled Cramer transform as the conjugate of the
logarithmic Laplace transform establishes a homeomorphism between
morphological and Gaussian scale space.
These articles provide a step towards the unification of
linear and morphological scalespace theory.

Families of Morphological Scale Spaces
In [2] morphological dilation and
erosion scale spaces with parabolic structuring functions
are embedded into twoparameter families of scale spaces which
include the Gaussian scale space as a limit case.
The scale space families are obtained by deforming the algebraic
operations underlying the morphological operations.
Aspects of these scale space families such as
continuity, invariance and separability are discussed.
The article enriches the picture of structural analogies
between the classes of scale spaces mentioned above.

B. Burgeth, J. Weickert:
An Explanation for the Logarithmic Connection between
Linear and Morphological Systems.
In L.D. Griffin, M Lillholm (Eds.): Scale Space Methods in Computer Vision.
Lecture Notes in Computer Science, Vol. 2695, Springer, Berlin, 325339, 2003.

M. Welk:
Families of Generalised Morphological Scale Spaces.
In L.D. Griffin, M Lillholm (Eds.): Scale Space Methods in Computer Vision.
Lecture Notes in Computer Science, Vol. 2695, Springer, Berlin, 770784, 2003.

B. Burgeth, M. Welk, C. Feddern, J. Weickert:
Morphological operations on matrixvalued images.
In T. Pajdla, J. Matas (Eds.): Computer Vision  ECCV 2004.
Lecture Notes in Computer Science, Vol. 3024, 155167, Springer, Berlin, 2004.

B. Burgeth, M. Welk, Ch. Feddern, J. Weickert:
Mathematical Morphology on Tensor Data Using the Loewner Ordering.
In J. Weickert, H. Hagen (Eds.):
Visualization and Processing of Tensor Fields.
Mathematics and Visualization, 357367, Springer, Berlin, 2006.
Revised version of
Technical Report No. 160, Department of Mathematics,
Saarland University, Saarbrücken, Germany, December 2005.

B. Burgeth, N. Papenberg, A. Bruhn, M. Welk, C. Feddern, J. Weickert:
Morphology for higherdimensional tensor data via Loewner ordering.
In C. Ronse, L. Najman, E. Decencière (Eds.):
Mathematical Morphology: 40 Years On. Computational Imaging and Vision, Vol. 30,
Springer, Dordrecht, 407–416, 2005.

B. Burgeth, N. Papenberg, A. Bruhn, M. Welk, J. Weickert:
Mathematical Morphology for Tensor Data Induced by the Loewner Ordering in
Higher Dimensions.
Signal Processing, Vol. 87, No. 2, 291308, February 2007.
Revised version of
Technical Report No. 161, Department of Mathematics,
Saarland University, Saarbrücken, Germany, December 2005.

B. Burgeth, A. Bruhn, S. Didas, J. Weickert, M. Welk:
Morphology for Tensor Data: Ordering versus PDEBased Approach.
Image and Vision Computing, Vol. 25, No. 4, 496511, 2007.
Revised version of
Technical Report No. 162, Department of Mathematics, Saarland University,
Saarbrücken, Germany, December 2005.

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.

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.

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,
©
SpringerVerlag Berlin Heidelberg 2009.
 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.

M. Breuß, B. Burgeth, J. Weickert:
Anisotropic continuousscale morphology.
In Proceedings of the 3rd Iberian Conference on Pattern Recognition and
Image Analysis, IbPRIA, June 6–8, 2007, Girona, Spain,
Lecture Notes in Computer Science, Springer, Berlin, 2007.

M. Welk, M. Breuß, O. Vogel:
Morphological amoebas are selfsnakes.
To appear in Journal of Mathematical Imaging and Vision.
Revised version of
Technical Report No. 259, Department of Mathematics,
Saarland University, Saarbrücken, Germany, February 2010.

M. Breuß, J. Weickert:
Highly Accurate Schemes for PDEBased Morphology
with General Convex Structuring Elements.
To appear in International Journal of Computer Vision.
Revised version of
Technical Report No. 236, Department of Mathematics,
Saarland University, Saarbrücken, Germany, May 2009.

B. Burgeth, L. Pizarro, M. Breuß, J. Weickert:
Adaptive continuousscale morphology for matrix fields.
To appear in International Journal of Computer Vision.
Revised version of
Technical Report No. 237, Department of Mathematics,
Saarland University, Saarbrücken, Germany, May 2009.
