*In a close cooperation, three research groups with quite
different background worked together to develop a new framework
for the detection and accurate quantification of motion, orientation,
and symmetry in images and image sequences. Five internal
meetings and two external workshops were organized. The second
workshop is integrated into the second annual meeting of the
priority program in Teistungen in February 2003 in order to
enhance the exchange of ideas and concepts between the image
processing and time series communities. Within the priority
program, contacts were established or intensified to the groups
of Prof. Maass in Bremen and of Prof. Rumpf in Duisburg. On
the international level, intense contacts with the group of
Prof. Granlund at the University of Link?ping (sabbatical
of Prof. Mester) were very stimulating for this project. Despite
a late start because of difficulties in filling in the position
of the research assistant located in Frankfurt and Heidelberg
and very limited manpower within the project, a number of
new research results could be gained. The most important of
them are summarized in the following. The letters F, L, and
H plus a number stand for the corresponding work packages
in the proposal.*

**Ilumination-invariant motion detection (L1)**
A new approach to the detection of moving objects even under
quickly varying illumination was developed. It is based in
part on our earlier framework for change detection, which
combines a constant false alarm rate (CFAR) test with the
use of a regularizing Markov random field (MRF). The core
of the algorithm is a statistical test for the collinearity
of two vectors observed in noise and has been jointly developed
by ISIP and IAP. The test statistic is derived from a total
least squares concept, and can efficiently be calculated as
the smaller eigenvalue of a symmetric 2 x 2 matrix.

**Orientation estimation (F1, F2, H1)** One
of the essential starting points for this project was the
observation that the estimation of orientation in multidimensional
signals (or estimation of motion direction if one axis represents
time) using differential filter masks has to be formulated
as subspace estimation problem. These are problems where a
rank-deficient matrix is perturbed by an unknown random matrix,
and the null space of the true (but unknown) error-free matrix
has to be estimated. Standard solutions (TLS) exist only for
the case of unstructured i.i.d. noise terms. For the orientation
estimation problem, a general procedure for obtaining the
covariance tensor of the random matrix could be derived. This
allows us now to characterize and compare different filter
sets with one single entity that fully determines the statistical
behavior in the estimation task up to second order. Unfortunately,
the inherent overlap of filter masks when determining structure
tensors from given image data leads to preferred directions
in the estimated direction. For some special cases we could
show that it is possible to optimize the filter mask coefficients
against this directional preferences. This procedure shall
be generalized in the second phase. Furthermore, the conventional
approach of differential tensor-based orientation estimation
has been generalized using higher-order steerable filters,
and the whole class of these approaches could be interpreted
as a Bayesian estimation process that is based on a subdivision
into signal and noise. This provides means for integrating
prior knowledge on motion statistics, as well image and noise
covariance structure, and forms the basis for the investigations
for the 2nd phase. When estimating complex motion scenes,
e.g. with changing illumination or deformable objects, some
independent parameters show errors, others not. Consequently
neither an ordinary (OLS) nor a total least squares (TLS)
approach gives good results. Therefore a combined OLS/TLS
estimation scheme that partitions the error-free and error-laden
parameters was developed. It gave superior results but the
approach is quite complex. It could then be shown that an
approximate TLS scheme in which the errors of the error-free
parameters where set to a low fraction of errors of the other
parameters gives almost as good results. This confirms and
extends work on optimal TLS estimators for structured covariance
matrices performed by the IAP group in Frankfurt.

**General solution for multiple motions (L2) **An
elegant solution to the case of multiple transparent motions
could be found that forms the base for a general theory of
multiple motions. The equation for multiple motions introduced
by Shizawa and Mase can be solved in analogy to the case of
one motion. This solution, however, yields only mixed motion
parameters. The separation of the mixed components, i.e.,
the correct correspondence of components and motion vectors
is found by the observation, that if we interpret the motion
vectors as complex numbers, these numbers behave like the
roots of a complex polynomial. We therefore obtain analytical
solutions for up to four overlaid motions and can use numerical
methods for the case of more than four motions.

**Regularization of motion estimation (H2, L2)**
A new technique for the robust computation of displacement
vector fields in noisy image sequences was introduced. This
orientationenhancing anisotropic diffusion filtering uses
the estimated optical flow field to drive a diffusion process
as a flexible regularization scheme. Doing so, the data is
simultaneously denoised while iteratively refining the estimated
displacement vector field. Moving brightness patterns result
in inclined structures in the spatio-temporal intensity cube.
The basic idea of our method is to smooth the image sequence
by applying a diffusion process whose diffusion tensor allows
anisotropic smoothing by acting mainly along the direction
of these structures, thus enhancing the signal. One of the
major problems in anisotropic diffusion application is to
find an appropriate stopping criterion. For optical flow the
reliability of the estimate can be determined by a simple
normalized confidence measure. It further turned out that
isotropic nonlinear diffusion and anisotropic diffusion correspond
to isotropic and directional statistical models, respectively.
Thus the diffusion formulation determines what image statistics
to compute while the Bayesian formulation provides optimal
parameters for the diffusion model. Exploiting this relationship
results in a fully automatic algorithm in which all parameters
are learned from training data. The resulting anisotropic
diffusion algorithm has many of the benefits of Bayesian approaches
along with a well-behaved numerical discretization. By replacing
the optical flow equation by the brightness constraint of
Shizawa and Mase and using the mixed motion parameters, the
standard regularization approach proposed by Horn and Schunck
with a global smoothness could be extended to multiple motions.
This leads to a Euler-Lagrange system of differential equation
that is linear in the mixed motion parameters (that would
be nonlinear in the motion parameters themselves).

** Test sequences (H4)** In the beginning,
no test sequences suited for complex motion analysis were
available at all. Therefore a first set of real-world test
image sequence with ground truth for transparent motion were
taken using optical translation tables for precise motion
control for use by all research groups of the LOCOMOTOR project.
Because of the severe limitation manpower in this project,
this work is only in its beginning and will be continued in
the second phase.