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S

Series in Signal and Information Processing, Vol. 32
edited by HansAndrea Loeliger
Reto A. Wildhaber
Localized State Space and Polynomial Filters
with Applications in Electrocardiography
1^{st }Edition 2019. XVIII, 132 pages. € 64,00.
ISBN 9783866286528
Abstract
Signals observed in
biological systems originate most often from multiple, superimposed sources:
some of those sources are producing continuously wandering baseline signals and
are only vaguely known. By contrast, other signal sources are pulseshaped and well
characterized, for example, signals from neuronal or muscular cells, observed
in electrocardiogram (Ecg),
electromyogram (Emg), and electroneurogram
(Eng) signals, and the like. The challenges addressed
in this thesis are to robustly identify and detect events in such biological
signals, to separate sources from others, and to extract features. As
biological signals take many shapes, the methods must be chosen carefully.
In this work, we contribute
two robust signal processing methods, whereby both methods are based on
localized, linear or nonlinear, models: the first method uses localized
autonomous linear state space models to identify and to detect events with
certain characteristics; the second method applies localized cost functions of
general polynomial forms to solve complex optimization problems. In this
context, localization considers the signals only within a weighted window of
finite or infinite length. Both methods, linear state space models and
polynomial cost functions, are modest in their computational complexity and,
hence, suitable for practical applications in wearable and, in particular, implantable
medical devices.
This thesis is driven by a
project in cardiology which conducts research in esophageal
electrocardiography. As part of this project, we develop novel esophageal
catheters with 3 dimensional electrode arrangements to localize cardiac events
in 3D space. The project has the overall goal to offer a superior, minimally
invasive device for improved arrhythmia diagnostics.
In the first part of this
thesis, we reconstruct the electrical field of the heart, as observed in the
esophagus. Therefore, we apply our methods on measurements from an esophageal
catheter solely. After depicting the results graphically as a 2 dimensional
map, denoted as esophageal
isopotential map, we briefly discuss the medical
implications of the obtained new modality. Furthermore, we derive a method to
estimate the cardiac depolarization sequence on the heart surface (epicardium)
of the left atrium. Solving such problems is also known as the inverse problem
of electrocardiography and leads to the technique of electrocardiographic
imaging (Ecgi). The methods used to solve the
problems in these first parts are derived in the following, second part of this
thesis.
In the second part, we
derive the methods required to solve the problems introduced in the first part.
We introduce autonomous linear state space models (Alssms)
and supplement them with local windows, which are generated by their own Alssms. We also combine multiple such Alssms,
each localized by its own window, to generate more versatile models. We
likewise superimpose multiple models of different time scales to discriminate
signals of different temporal spread, and we join multiple models with adjacent
windows to detect onsets of, or transitions between, multiple, alternating
signal sources. Such joined models are either applied at particular time
indices of interest, or repetitively over a whole signal leading to a sliding
window filter. Such filters do not necessarily have scalar outputs, but rather
provide signals of feature vectors. Applying appropriate transformations to
such feature vectors simplifies unsupervised feature detection. Despite all
these modifications and extensions on Alssms, we
strictly preserve the inherent recursive computation rules of the involved
linear state space models, and, thus, their efficient computations.
Further in the second part,
we introduce cost functions of general polynomial forms and apply them to solve
optimization problems. We expand their fields of application by introducing
localization. To apply our method on a particular problem, we first project a
given signal to a localized feature space, i.e., we locally approximate the signals
by a given class of functions. Then, any further processing is executed in this
lowerdimensional feature space. To efficiently handle
problems of increased complexity, we also provide a new calculus. This calculus
simplifies the manipulation of cost terms of polynomial forms.
In the third and last part,
we apply our methods and provide solutions to the problems introduced. Finally,
we conclude with a list of additional practical examples that we have already
published, and which successfully apply our methods.
Keywords: Linear state space models; event detection and
estimation; polynomial cost functions; unsupervised feature extraction.
Stichworte: Lineare Zustandsraummodelle; Ereignisdetektion und Ereignisschätzung;
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