This article addresses the model-based reconstruction and prediction of distributed phenomena
characterized by partial differential equations, which are monitored by sensor networks. The novelty of the
proposed reconstruction method is the systematic approach and the integrated treatment of uncertainties,
which occur in the physical model and arise naturally from noisy measurements. By this means it is possible
not only to reconstruct the entire phenomenon, even at non-measurement points, but also to reconstruct the
complete density function of the state characterizing the distributed phenomenon. In the first step, the
partial differential equation, i.e., distributed-parameter system, is spatially and temporally decomposed
leading to a finite-dimensional state space form. In the next step, the state of the resulting lumped-parameter
system, which provides an approximation of the solution of the underlying partial differential equation,
is dynamically estimated under consideration of uncertainties. By using the estimation results, several
additional tasks can be achieved by the sensor network, e.g. optimal sensor placement, optimal scheduling,
model improvement, and system identification. The performance of the proposed model-based reconstruction
method is demonstrated by means of simulations.

In order to assist surgeons during minimally invasive interventions
on the beating heart, it would be helpful to develop a robotic surgery
system, which synchronizes the instruments with the heart surface,
so that their positions do not change relative to the point of interest
(POI). The synchronization of the robotic manipulators requires an
estimation of the heart surface motion. In this paper, a modelbased
motion estimation of the heart surface is presented. The motion of
a partition of the heart surface is modelled by means of a thin or
thick vibrating membrane in order to represent the epicardial surface
or the connected epicard and myocard. The membrane motion is described
by means of a system of coupled linear partial differential equations
(PDEs), whose 3D-input function is assumed to be known. After spatial
discretization of the PDE solution space by the Finite Spectral Element
Method, a bank of lumped systems is obtained. A Kalman filter is
used to estimate the state of the lumped systems by incorporating
noisy measurements of the heart surface. Measurements can be the
position or velocity of sonomicrometry-based sensors or of certain
landmarks, which are tracked by optical sensors. With the model-based
estimation it is possible to reconstruct the entire partition of
the heart surface even at non-measurement points and thus at each
POI.

In order to assist surgeons during surgery on moving organs, e.g.
minimally invasive beating heart bypass surgery, a master-slave system
which synchronizes surgical instruments with the organ\'s motion is
desired. This synchronization requires reliable estimation of the
organ\'s motion. In this paper, we present a new approach to motion
estimation based on a state motion model for a partition of the heart\'s
surface. Its motion behavior is described by a partial differential
equation whose input function is assumed to be periodic. An estimator
is used on one hand to predict future model states based on reconstruction
of the input function and on the other hand to incorporate noisy
spatially discrete measurements in order to improve state estimation.
The model-based motion estimation is evaluated using a simple heart
simulator. Measurements are obtained by reconstructing 3D position
of markers on a pulsating membrane by means of a stereo camera system.

This paper addresses a systematic method for the reconstruction and
the prediction of a distributed phenomenon characterized by partial
differential equations, which is monitored by a sensor network. In
the first step, the infinite-dimensional partial differential equation,
i.e. distributed-parameter system, is spatially and temporally decomposed
leading to a finite-dimensional state space form. In the next step,
the state of the resulting lumped-parameter system, which provides
an approximation of the solution of the underlying partial differential
equations, is dynamically estimated under consideration of uncertainties
both occurring in the system and arising from noisy measurements.
By using the estimation results, several additional tasks can be
achieved by the sensor network, e.g. optimal sensor placement, optimal
scheduling, and model improvement. The performance of the proposed
model-based reconstruction method is demonstrated by means of simulations.

Offene oder minimal invasive Operationen am schlagenden Herzen erfordern
von dem Chirurgen eine hohe Konzentrationsfähigkeit über
einen längeren Zeitraum. Daher ist es für den Chirurgen sehr
hilfreich durch ein robotergestütztes Chirurgiesystem unterstützt
zu werden, das die Instrumente im Interventionsareal mit der Herzbewegung
synchronisiert. Um eine Bewegungskompensation durchzuführen,
muss ein Mechanismus gefunden werden, der aufgrund einer Prädiktion
der Herzbewegung die Instrumente nachführt. Für die Prädiktion
der Herzbewegung wird in diesem Artikel ein Verfahren zum Entwurf
eines stochastischen 3D-Bewegungsmodells für die Herzoberfläche
gezeigt. Ein Schätzer nimmt dieses Modell als Grundlage und verwendet
die verrauschten Sensormessungen von Landmarken der Herzoberfläche
um die Herzoberflächenbewegung zu prädizieren.

A primary challenge for the reconstruction of continuous-time, continuous-amplitude
distributed parameter systems is the inclusion of recent discrete-time,
discreteamplitude, spatially discrete measurements. Hence, a systematic
method for data processing is required that also handles incomplete
and noisy data, e.g. data from a sensor network. This article presents
two approaches to the reconstruction of distributed parameter systems
that can be described by linear partial differential equations (PDEs)
and involve one or several discrete measurement points. In both approaches,
the linear PDE is first converted into a bank of linear lumped systems
by means of modal analysis. In addition, a measurement equation relating
state and (sensor) data is derived. In the second step, a Kalman
filter (KF) is used to dynamically estimate the state of the lumped
systems, which provides an approximation of the solution of the underlying
PDE. The first approach uses Fourier Analysis. The second approach
uses Fourier Analysis and the collocation method. The approaches
are both demonstrated for a simple linear inhomogeneous PDE, the
one-dimensional heat equation.