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Series in Signal and Information Processing, Vol. 32
edited by Hans-Andrea Loeliger






Reto A.  Wildhaber


Localized State Space and Polynomial Filters
with Applications in Electrocardiography


1st Edition 2019. XVIII, 132 pages. € 64,00.
ISBN 978-3-86628-652-8















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 pulse-shaped 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 non-linear, 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 3-D 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; Kostenfunktionen in Form von Polynomen; unüberwachtes Extrahieren von Merkmalen.



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