Matrixvalued data sets (socalled tensor fields) are gaining increasing
importance in digital imaging. This has been triggered by the following
developments:

Diffusion tensor magnetic resonance imaging (DTMRI) by now is an
established tool for medical diagnosis.
DTMRI is a 3D
imaging method that yields a diffusion tensor in each voxel. This
diffusion tensor describes the diffusive behaviour of water molecules
in the tissue. This tensor can be represented by a positive semidefinite 3 x 3
matrix in each voxel.

Tensors have shown their use as a general tool in image analysis,
segmentation and grouping. This also includes widespread applications
of the socalled structure tensor in fields ranging from motion
analysis to texture segmentation.

A number of applications in science and engineering produce tensor
fields: The tensor concept is a common physical description of
anisotropic behaviour, especially in solid mechanics and civil
engineering (e.g. stressstrain relationships, inertia tensors,
diffusion tensors, permittivity tensors).
Diffusion Filters for Tensor Fields
We have proposed various tensorvalued isotropic and anisotropic
diffusion filter concepts:

One class of concepts treats every tensor channel componentwise, in
the spirit of Di Zenzo's approach to multichannel images, here however
all channels share the structural information synthetised in a joint
diffusion tensor (channel coupling). This construction indeed allows
for smoothing preferently along discontinuities of the tensor field,
while smoothing across discontinuities is inhibited. We have shown
that suitable discretisations of these processes preserve the positive
semidefiniteness of the initial tensor fields
[1], [16].

Another class of concepts is based on a novel operatoralgebraic
framework for matrix fields. We have proposed truly matrixvalued
diffusion processes that automatically ensure appropriate channel
interaction, therefore avoiding the need of explicit channel coupling.
In order to solve these novel PDEs we proposed and elaborated
matrixvalued numerical schemes [23],
[24], [31],
[33].
Filtering, Regularisation and Interpolation of Tensor Fields

Novel energy functionals have been presented for the variational
restoration of noisy tensor fields. Their EulerLagrange equations
can be regarded as fully implicit time discretisations of
tensorvalued anisotropic diffusion filters
[1].

Median filters belong to the most popular nonlinear filters for
image denoising. Usual median filters require an ordering of the
input data. This is not possible in the tensorvalued setting.
We have thus expoited another property of the scalarvalued median:
The median minimises the sum of distances to all input data.
This interpretation can be generalised to the matrixvalued case
by considering distances induced by the Frobenius norm. The
resulting median filter is robust under noise and preserves
discontinuities of the tensor field [6].
Extensions and algorithmic aspects are discussed in
[11], [18],
[19].

We have proposed a unified framework for interpolation and
regularisation of scalar and tensorvalued images based on elliptic
PDEs [17]. We showed that a rotationally invariant
interpolation model based on anisotropic diffusion with a diffusion
tensor as in [1], [16]
outperforms interpolants with radial basis functions, it allows
discontinuitypreserving interpolation, and it respects positive
semidefiniteness of the input tensor data.

In [28] we have proposed a discrete variational
approach for the filtering of matrix fields. The model is based on
nonlocal data similarity and smoothness contraints and it can be seen
as the matrixvalued extension of the NDS filter for scalar images
proposed by Mrázek, Weickert and Bruhn in 2006. The
proposed filter contains novel counterparts of Msmoothing and
bilateral filtering for matrix fields. In addition, other existing
methods such as the affineinvariant (Pennec et al. 2006) and the
logEuclidean (Fillard et al. 2007) regularisation of tensor fields
are special cases of the proposed filtering framework.
OperatorAlgebraic Framework for Matrix Fields

For the processing of matrix fields we proposed a general method to
transfer calculus concepts for scalar images to the setting of matrix
fields. The symmetric matrices are regarded as operators generalising
real numbers with a relatively rich underlying algebraic stucture.
This operatoralgebraic framework allows for an efficient transfer of
scalar image processing methodologies such as PDEbased and variational
methods as well as their numerical solution schemes to the
matrixvalued setting [23],
[24], [26].

This methodology has been succesfully employed to develop
matrixvalued counterparts of nonlinear diffusion processes
[23], [24],
[31], regularisation approaches
[25], [32], inverse
scalespace techniques [33], and PDEbased
mathematical morphology [35],
[38].
Structure Tensors

The structure tensor is a classical tool in image processing
and computer vision (Förstner/Gülch 1987, Rao/Schunck 1990,
Bigün et al. 1991).
It averages orientations by means of Gaussian smoothing of all
tensor channels. This may be regarded as tensorvalued linear
diffusion filtering. In order to allow preservation of
discontinuities, we have replaced this linear diffusion process
by a nonlinear tensorvalued diffusion [1],
[10], creating thus the socalled nonlinear
structure tensor. Applications in the fields of optical flow
estimation [2], texture analysis
[3], and segmentation [4,5],
demonstrate the favourable performance of the resulting nonlinear
structure tensor. A comparison and evaluation of the nonlinear
structure tensor in the context of other adaptive structure tensor
concepts has been presented in [14].

In [1], [7] we have generalised
the concept of image gradient to the tensorvalued setting by
extending Di Zenso's method for vectorvalued data. Each channel
considered as independent scalar image gives rise to a structure
tensor, then these structures tensors are summed up to build a
novel structure tensor for multi channel images that we refer to as
Weickert's structure tensor.

This construction has been refined to a customisable structure
tensor in [21]. There the resulting structure
tensor is a weighted sum of tensors of scalar quantities that are now
not just the channels, but other meaningful scalar quantities derived
from the matrix field. The weights are provided by the user, and
depending on the choice of weights the emerging structure tensor has a
sensitivity for certain features of the matrix field. A special
constellation of the weights turns the customisable structure tensor
into Weickert's structure tensor.

Based on the operatoralgebraic framework we have developed a novel
structure tensor concept for matrix fields, the socalled full
structure tensor [31].
With this tensor we are able to deduce directional information in
matrix fields with the additional possibility to introduce prior
structural knowledge. By a natural reduction operation Weickert's
structure tensor appears as a special case of the full structure
tensor.

It is a known limitation of the traditional structure tensor
(Förstner/Gülch 1987) that it can only represent a single
dominant orientation. We have developed its generalisation
to a higherorder tensor model. The proposed higherorder
structure tensor is able to capture the orientations of more
complex neighborhoods, for example corners, junctions, and multivalued
images. The tensor order allows to specify the maximum complexity the
structure tensor can represent and can be chosen based on the
requirements of a given application [30].
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
[8] 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.

Orderingbased approaches.
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 [15] and becomes more
intricate in higher dimensions [12],
[20]; 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
matrix fields.

PDEbased approaches.
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
[22] 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 [35]. More recently, we have developed
adaptive, anisotropic PDEs for matrixvalued morphology. The
anisotropic evolution is guided by a steering tensor whose
construction relies on the full structure tensor for matrix data. In
order to enable proper directional steering we proposed directional
matrixvalued numerical schemes based on directional finite
differences via interpolation. We demonstrated that these PDEs are
capable, for instance, to fill in missing data and to complete
directional structures [36],
[37], [38].
TensorValued Level Set Methods

Based on Weickert's structure tensor we have proposed tensorvalued
extensions of mean curvature motion, selfsnakes and geodesic active
contours. Our experiments demonstrated that the tensorvalued level
set methods inherit essential properties from their scalarvalued
counterparts and that they are highly robust under noise, both in the
twodimensional [7] and threedimensional setting
[9]. See also the survey [16].

Based on the full structure tensor for matrix fields we have proposed
truly matrixvalued PDEs for mean curvature motion and selfsnakes.
Experiments on synthetic and positive semidefinite DTMRI data
illustrated that the matrixvalued methods inherit desirable
properties of their scalarvalued predecessors, e.g. very good
denoising capabilities combined with feature preserving qualities
[23].
Registration of Tensor Fields

A variational framework for the registration of tensorvalued
images has been proposed in [27],
[34]. It is based on an energy functional with
four terms: a data term based on a diffusion tensor constancy
constraint, a compatibility term encoding the physical model linking
domain deformations and tensor reorientation, and smoothness terms for
deformation and tensor orientation. This approach has been specially
designed for the registration of DTMRI data. However, it can be
easily adapted to different tensor deformation models for
registering other types of tensorvalued data.
Organisation of Workshops

1st Workshop on Visualisation and
Image Processing of Tensor Fields,
Dagstuhl, Germany, April 1822, 2004. The workshop led to a book [13] which is the first edited book covering
visualisation and processing of tensor fields.

2nd Workshop on Visualization and Processing of Tensor
Fields.
Dagstuhl, Germany, January 913, 2007. The workshop led to a book [29].

3rd Workshop on New Developments in the Visualization and
Processing of Tensor Fields.
Dagstuhl, Germany, July 1924, 2009.

J. Weickert, T. Brox:
Diffusion and regularization of vector and matrixvalued
images.
In M. Z. Nashed, O. Scherzer (Eds.): Inverse Problems, Image Analysis,
and Medical Imaging. Contemporary Mathematics, Vol. 313, 251268, AMS,
Providence, 2002.

T. Brox, J. Weickert:
Nonlinear matrix diffusion for optic flow estimation.
In L. Van Gool (Ed.): Pattern Recognition.
Lecture Notes in Computer Science, Vol. 2449, Springer, Berlin,
446453, 2002.

M. Rousson, T. Brox, R. Deriche:
Active unsupervised texture segmentation on a diffusion based
feature space.
Technical report no. 4695, Odyssée, INRIA SophiaAntipolis,
France, 2003.
Slightly extended version of the conference paper with the same
title,
Proc. 2003 IEEE Computer Society Conf. on Computer Vision and Pattern
Recognition, Madison, WI, 2003.

T. Brox, M. Rousson, R. Deriche, J. Weickert:
Unsupervised segmentation incorporating colour, texture, and
motion.
In N. Petkov, M. A. Westenberg (Eds.): Computer Analysis of Images
and Patterns. Lecture Notes in Computer Science, Vol. 2756, Springer,
Berlin, 353360, 2003.

T. Brox, M. Rousson, R. Deriche, J. Weickert:
Unsupervised segmentation incorporating colour, texture, and
motion.
Technical report no. 4760, Odyssée, INRIA SophiaAntipolis,
France, 2003.

M. Welk, C. Feddern, B. Burgeth, J. Weickert:
Median filtering of tensorvalued images.
In B. Michaelis, G. Krell (Eds.): Pattern Recognition.
Lecture Notes in Computer Science, Vol. 2781,
Springer, Berlin, 17–24, 2003.

C. Feddern, J. Weickert, B. Burgeth:
Levelset methods for tensorvalued images.
In O. Faugeras, N. Paragios (Eds.):
Proc. Second IEEE Workshop on Variational, Geometric and Level Set
Methods in Computer Vision.
Nice, France, pp. 6572. INRIA, Oct. 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.

C. Feddern, J. Weickert, B. Burgeth, M. Welk:
Curvaturedriven PDE methods for matrixvalued images.
International Journal of Computer Vision, Vol. 69, No. 1, 91103, Aug. 2006.
Revised version of
Technical Report No. 104, Department of Mathematics, Saarland
University, Saarbrücken, Germany, April 2004.

T. Brox, J. Weickert, B. Burgeth, P. Mrázek:
Nonlinear structure tensors.
Image and Vision Computing, Vol. 24, No. 1, 4155, Jan. 2006.
Revised version of
Technical Report No. 113, Department of Mathematics,
Saarland University, Saarbrücken, Germany, 2004.

M. Welk, F. Becker, C. Schnörr, J. Weickert:
Matrixvalued filters as convex programs.
In R. Kimmel, N. Sochen, J. Weickert (Eds.):
ScaleSpace and PDE Methods in Computer Vision.
Lecture Notes in Computer Science, Vol. 3459, Springer, Berlin,
204–216, 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.

J. Weickert, H. Hagen (Eds.):
Visualization and Processing of Tensor Fields.
Springer, Berlin, 2006.

T. Brox, R. van den Boomgaard, F. Lauze, J. van de Weijer, J. Weickert,
P. Mrázek, P. Kornprobst:
Adaptive structure tensors and their applications.
In J. Weickert, H. Hagen (Eds.):
Visualization and Processing of Tensor Fields, 1747, Springer, Berlin, 2006.
Revised version of
Technical Report No. 141, Department of Mathematics,
Saarland University, Saarbrücken, Germany, 2005.

B. Burgeth, M. Welk, C. Feddern, J. Weickert:
Mathematical morphology on tensor data using the Loewner
ordering.
In J. Weickert, H. Hagen (Eds.):
Visualization and Processing of Tensor Fields, 357367, Springer, Berlin, 2006.
Revised version of
Technical Report No. 160, Department of Mathematics,
Saarland University, Saarbrücken, Germany, 2005.

J. Weickert, C. Feddern, M. Welk, B. Burgeth, T. Brox:
PDEs for tensor image processing.
In J. Weickert, H. Hagen (Eds.):
Visualization and Processing of Tensor Fields, 399414, Springer, Berlin, 2006.
Revised version of
Technical Report No. 143, Department of Mathematics,
Saarland University, Saarbrücken, Germany, 2005.

J. Weickert, M. Welk:
Tensor field interpolation with PDEs.
In J. Weickert, H. Hagen (Eds.):
Visualization and Processing of Tensor Fields, 315325, Springer, Berlin, 2006.
Revised version of
Technical Report No. 142, Department of Mathematics,
Saarland University, Saarbrücken, Germany, 2005.

M. Welk, C. Feddern, B. Burgeth, J. Weickert:
Tensor median filtering and Msmoothing.
In J. Weickert, H. Hagen (Eds.):
Visualization and Processing of Tensor Fields, 345356, Springer, Berlin, 2006.

M. Welk, J. Weickert, F. Becker, C. Schnörr, C. Feddern,
B. Burgeth:
Median and related local filters for tensorvalued images.
Signal Processing, Vol. 87, No. 2, 291308, February 2007.
Revised version of
Technical Report No. 135, Department of Mathematics,
Saarland University, Saarbrücken, Germany, April 2005.

B. Burgeth, N. Papenberg, A. Bruhn, M. Welk, J. Weickert:
Mathematical morphology for matrix fields induced by the Loewner ordering in
higher dimensions.
Signal Processing, Vol. 87, No. 2, 277290, February 2007.
Revised version of
Technical Report No. 161, Department of Mathematics,
Saarland University, Saarbrücken, Germany, December 2005.

T. Schultz, B. Burgeth, J. Weickert:
Flexible segmentation and smoothing of DTMRI fields through a customizable
structure tensor.
In G. Bebis, R. Boyle, B. Parvin, D. Koracin, P. Remagnino, A. V. Nefian,
M. Gopi, V. Pascucci, J. Zara, J. Molineros, H. Theisel, T. Malzbender (Eds.):
Advances in Visual Computing. Lecture Notes in Computer Science, Vol. 4291,
Springer, 454464, 2006.
Awarded the ISVC 2006 Best Paper Award.

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.

B. Burgeth, S. Didas, L. Florack, J. Weickert:
A generic approach to the filtering of matrix fields with singular PDEs.
In F. Sgallari, A. Murli, N. Paragios (Eds.),
Scale Space and Variational Methods in Computer Vision.
Lecture Notes in Computer Science, Vol. 4485, 556567,
Springer, Berlin, 2007.

B. Burgeth, S. Didas, L. Florack, J. Weickert:
A generic approach to diffusion filtering of matrixfields.
Computing, Vol. 81, No. 23, 179197, Nov. 2007. Invited Paper.

O. Christiansen, J. Lie, B. Burgeth:
A dual operator algebraic method for regularization of matrix
valued images.
Accepted for publication in Inverse Problems and Imaging, 2007.

B. Burgeth:
Partial Differential Equations for Scale Space Analysis and
Matrix Field Processing.
Habilitation Thesis. Faculty of Mathematics and Computer Science,
August 2008. Available upon request.

S. Barbieri, M. Welk, J. Weickert:
Variational registration of tensorvalued images.
Proc. CVPR Workshop »Tensors in Image Processing and Computer
Vision«, Anchorage, Alaska, USA, 23 June 2008, pages 16.

L. Pizarro, B. Burgeth, S. Didas, J. Weickert:
A generic neighbourhood filtering framework for matrix fields.
In D. Forsyth, P. Torr, A. Zisserman (Eds.):
Computer Vision – ECCV 2008. Lecture Notes in Computer Science,
Vol. 5304, 521532. Springer, Berlin, 2008.

D. H. Laidlaw, J. Weickert (Eds.):
Visualization and Processing of Tensor Fields: Advances and Perspectives
Springer, Berlin, 2009.

T. Schultz, J. Weickert, H.P. Seidel:
A higherorder structure tensor.
In D. H. Laidlaw, J. Weickert (Eds.):
Visualization and Processing of Tensor Fields: Advances and Perspectives.
Springer, Berlin, 263279, 2009.
Revised version of
Research Report MPII20074005, MaxPlanckInstitut für
Informatik, Saarbrücken, Germany, July 2007.

B. Burgeth, S. Didas, J. Weickert:
A general structure tensor concept and coherenceenhancing diffusion
filtering for matrix fields.
In D. H. Laidlaw, J. Weickert (Eds.):
Visualization and Processing of Tensor Fields: Advances and Perspectives.
Springer, Berlin, 305323, 2009.
Revised version of
Technical Report No. 197, Department of Mathematics,
Saarland University, Saarbrücken, Germany, July 2007.

S. Setzer, G. Steidl, B. Popilka, B. Burgeth:
Variational methods for denoising matrix fields.
In D. H. Laidlaw, J. Weickert (Eds.):
Visualization and Processing of Tensor Fields: Advances and Perspectives.
Springer, Berlin, 341360, 2009.

J. Lie, B. Burgeth, O. Christiansen:
An operator algebraic inverse scale space method for
symmetric matrix valued images.
In D. H. Laidlaw, J. Weickert (Eds.):
Visualization and Processing of Tensor Fields: Advances and Perspectives.
Springer, Berlin, 361376, 2009.

S. Barbieri, M. Welk, J. Weickert:
A variational approach to the registration of tensorvalued images.
In S. AjaFernandez, R. de LuisGarcia, D. Tao, X. Li (Eds.):
Tensors in Image Processing and Computer Vision, pages 5977,
Springer, London, 2009.
Revised version of
Technical Report No. 221, Department of Mathematics,
Saarland University, Saarbrücken, Germany, September 2008.

B. Burgeth, M. Breuß, S. Didas, J. Weickert:
PDEbased morphology for matrix fields: Numerical solution schemes.
In S. AjaFernandez, R. de LuisGarcia, D. Tao, X. Li (Eds.):
Tensors in Image Processing and Computer Vision, pages 125150,
Springer, London, 2009.
Revised version of
Technical Report No. 220, Department of Mathematics,
Saarland University, Saarbrücken, Germany, September 2008.

B. Burgeth, M. Breuß, L. Pizarro, J. Weickert:
PDEdriven adaptive morphology for matrix fields.
In X.C. Tai, K. Mørken, M. Lysaker, K.A. Lie (Eds.):
Scale Space and Variational Methods in Computer Vision.
Lecture Notes in Computer Science, Vol. 5567, 247258, Springer, Berlin, 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.):
Proceedings of Ninth International Symposium on Mathematical Morphology,
Lecture Notes in Computer Science, Vol. 5720, 250260, Springer, Berlin, 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.
