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Beamforming performs spatial filtering to preserve the signal from given directions of interest while suppressing interfering signals and noise arriving from other directions.
For example, a microphone array equipped with beamforming algorithm could preserve the sound coming from a target speaker and suppress sounds coming from other speakers.
Beamformer has been widely used in many applications such as radar, sonar, communication, and acoustic systems.
A data-independent beamformer is the beamformer whose coefficients are independent on sensor signals, it normally uses less computation since the coefficients are computed once. Moreover, its coefficients are derived from the well-defined statistical models, then it produces less artifacts. The major drawback of this beamforming class is its limitation to the interference suppression.
On the other hand, an adaptive beamformer is a beamformer whose coefficients depend on or adapt to sensor signals. It is capable of suppressing the interference better than a data-independent beamforming but it suffers from either too much distortion of the signal of interest or less noise reduction when the updating rate of coefficients does not synchronize with the changing rate of the noise model. Besides, it is computationally intensive since the coefficients need to be updated frequently.
In acoustic applications, the bandwidth of signals of interest extends over several octaves, but we always expect that the characteristic of the beamformer is invariant with regard to the bandwidth of interest. This can be achieved by the so-called broadband beamforming.
Since the beam pattern of conventional beamformers depends on the frequency of the signal, it is common to use a dense and uniform array for the broadband beamforming to guarantee some essential performances together, such as frequency-independence, less sensitive to white noise, high directivity factor or high front-to-back ratio. In this dissertation, we mainly focus on the sparse array of which the aim is to use fewer sensors in the array,
while simultaneously assuring several important performances of the beamformer.
In the past few decades, many design methodologies for sparse arrays have been proposed and were applied in a variety of practical applications.
Although good results were presented, there are still some restrictions, such as the number of sensors is large, the designed beam pattern must be fixed, the steering ability is limited and the computational complexity is high.
In this work, two novel approaches for the sparse array design taking a hypothesized uniform array as a basis are proposed, that is, one for data-independent beamformers and the another for adaptive beamformers.
As an underlying component of the proposed methods, the dissertation introduces some new insights into the uniform array with broadband beamforming. In this context, a function formulating the relations between the sensor coefficients and its beam pattern over frequency is proposed. The function mainly contains the coordinate transform and inverse Fourier transform.
Furthermore, from the bijection of the function and broadband beamforming perspective, we propose the lower and upper bounds for the inter-distance of sensors. Within these bounds, the function is a bijective function that can be utilized to design the uniform array with broadband beamforming.
For data-independent beamforming, many studies have focused on optimization procedures to seek the sparse array deployment. This dissertation presents an alternative approach to determine the location of sensors.
Starting with a weight spectrum of a virtual dense and uniform array, some techniques are used, such as analyzing a weight spectrum to determine the critical sensors, applying the clustering technique to group the sensors into different groups and selecting representative sensors for each group.
After the sparse array deployment is specified, the optimization technique is applied to find the beamformer coefficients. The proposed method helps to save the computation time in the design phase and its beamformer performance outperforms other state-of-the-art methods in several aspects such as the higher white noise gain, higher directivity factor or more frequency-independence.
For adaptive beamforming, the dissertation attempts to design a versatile sparse microphone array that can be used for different beam patterns.
Furthermore, we aim to reduce the number of microphones in the sparse array while ensuring that its performance can continue to compete with a highly dense and uniform array in terms of broadband beamforming.
An irregular microphone array in a planar surface with the maximum number of distinct distances between the microphones is proposed.
It is demonstrated that the irregular microphone array is well-suited to sparse recovery algorithms that are used to solve underdetermined systems with subject to sparse solutions. Here, a sparse solution is the sound source's spatial spectrum that need to be reconstructed from microphone signals.
From the reconstructed sound sources, a method for array interpolation is presented to obtain an interpolated dense and uniform microphone array that performs well with broadband beamforming.
In addition, two alternative approaches for generalized sidelobe canceler (GSC) beamformer are proposed. One is the data-independent beamforming variant, the other is the adaptive beamforming variant. The GSC decomposes beamforming into two paths: The upper path is to preserve the desired signal, the lower path is to suppress the desired signal. From a beam pattern viewpoint, we propose an improvement for GSC, that is, instead of using the blocking matrix in the lower path to suppress the desired signal, we design a beamformer that contains the nulls at the look direction and at some other directions. Both approaches are simple beamforming design methods and they can be applied to either sparse array or uniform array.
Lastly, a new technique for direction-of-arrival (DOA) estimation based on the annihilating filter is also presented in this dissertation.
It is based on the idea of finite rate of innovation to reconstruct the stream of Diracs, that is, identifying an annihilating filter/locator filter for a few uniform samples and the position of the Diracs are then related to the roots of the filter. Here, an annihilating filter is the filter that suppresses the signal, since its coefficient vector is always orthogonal to every frame of signal.
In the DOA context, we regard an active source as a Dirac associated with the arrival direction, then the directions of active sources can be derived from the roots of the annihilating filter. However,
the DOA obtained by this method is sensitive to noise and the number of DOAs is limited.
To address these issues, the dissertation proposes a robust method to design the annihilating filter and to increase the degree-of-freedom of the measurement system (more active sources can be detected) via observing multiple data frames.
Furthermore, we also analyze the performance of DOA with diffuse noise and propose an extended multiple signal classification algorithm that takes diffuse noise into account. In the simulation,
it shows, that in the case of diffuse noise, only the extended multiple signal classification algorithm can estimate the DOAs properly.
The growing computational power enables the establishment of the Population Balance Equation (PBE)
to model the steady state and dynamic behavior of multiphase flow unit operations. Accordingly, the twophase
flow
behavior inside liquid-liquid extraction equipment is characterized by different factors. These
factors include: interactions among droplets (breakage and coalescence), different time scales due to the
size distribution of the dispersed phase, and micro time scales of the interphase diffusional mass transfer
process. As a result of this, the general PBE has no well known analytical solution and therefore robust
numerical solution methods with low computational cost are highly admired.
In this work, the Sectional Quadrature Method of Moments (SQMOM) (Attarakih, M. M., Drumm, C.,
Bart, H.-J. (2009). Solution of the population balance equation using the Sectional Quadrature Method of
Moments (SQMOM). Chem. Eng. Sci. 64, 742-752) is extended to take into account the continuous flow
systems in spatial domain. In this regard, the SQMOM is extended to solve the spatially distributed
nonhomogeneous bivariate PBE to model the hydrodynamics and physical/reactive mass transfer
behavior of liquid-liquid extraction equipment. Based on the extended SQMOM, two different steady
state and dynamic simulation algorithms for hydrodynamics and mass transfer behavior of liquid-liquid
extraction equipment are developed and efficiently implemented. At the steady state modeling level, a
Spatially-Mixed SQMOM (SM-SQMOM) algorithm is developed and successfully implemented in a onedimensional
physical spatial domain. The integral spatial numerical flux is closed using the mean mass
droplet diameter based on the One Primary and One Secondary Particle Method (OPOSPM which is the
simplest case of the SQMOM). On the other hand the hydrodynamics integral source terms are closed
using the analytical Two-Equal Weight Quadrature (TEqWQ). To avoid the numerical solution of the
droplet rise velocity, an analytical solution based on the algebraic velocity model is derived for the
particular case of unit velocity exponent appearing in the droplet swarm model. In addition to this, the
source term due to mass transport is closed using OPOSPM. The resulting system of ordinary differential
equations with respect to space is solved using the MATLAB adaptive Runge–Kutta method (ODE45). At
the dynamic modeling level, the SQMOM is extended to a one-dimensional physical spatial domain and
resolved using the finite volume method. To close the mathematical model, the required quadrature nodes
and weights are calculated using the analytical solution based on the Two Unequal Weights Quadrature
(TUEWQ) formula. By applying the finite volume method to the spatial domain, a semi-discreet ordinary
differential equation system is obtained and solved. Both steady state and dynamic algorithms are
extensively validated at analytical, numerical, and experimental levels. At the numerical level, the
predictions of both algorithms are validated using the extended fixed pivot technique as implemented in
PPBLab software (Attarakih, M., Alzyod, S., Abu-Khader, M., Bart, H.-J. (2012). PPBLAB: A new
multivariate population balance environment for particulate system modeling and simulation. Procedia
Eng. 42, pp. 144-562). At the experimental validation level, the extended SQMOM is successfully used
to model the steady state hydrodynamics and physical and reactive mass transfer behavior of agitated
liquid-liquid extraction columns under different operating conditions. In this regard, both models are
found efficient and able to follow liquid extraction column behavior during column scale-up, where three
column diameters were investigated (DN32, DN80, and DN150). To shed more light on the local
interactions among the contacted phases, a reduced coupled PBE and CFD framework is used to model
the hydrodynamic behavior of pulsed sieve plate columns. In this regard, OPOSPM is utilized and
implemented in FLUENT 18.2 commercial software as a special case of the SQMOM. The dropletdroplet
interactions
(breakage
and
coalescence)
are
taken
into
account
using
OPOSPM,
while
the
required
information
about
the
velocity
field
and
energy
dissipation
is
calculated
by
the
CFD
model.
In
addition
to
this,
the proposed coupled OPOSPM-CFD framework is extended to include the mass transfer. The
proposed framework is numerically tested and the results are compared with the published experimental
data. The required breakage and coalescence parameters to perform the 2D-CFD simulation are estimated
using PPBLab software, where a 1D-CFD simulation using a multi-sectional gird is performed. A very
good agreement is obtained at the experimental and the numerical validation levels.
The present thesis describes the development and the evaluation of a design procedure of inducer with arbitrary meridional and blade shape. This special type of pump impeller, which is usually mounted upstream of a main pump impeller, is employed in many applications demanding the realization of low NPSH values. An inducer basically increases suction performance by producing mostly a small pressure rise while allowing for a greater degree of cavitation, that is the formation of vapor bubbles, at its inlet than a conventional pump impeller. This is achieved by specially designed blade channels promoting the collapse of the produced vapor bubbles.
The main focus of the present thesis is the description of the design method, which enables the generation of the three-dimensional blade geometry. The method is based on a parametric representation of the geometry considering the particular requirements for inducers and the publicly available design practice. Within this approach the sequence of design steps is adapted from the classical design process of mixed flow and radial impellers. As a consequence leading and trailing edge blade angles are determined based on simplifications and certain empirical assumptions for multiple blade sections and are used to design the blade camber curves. Along the camber curves the blade profile is generated following a thickness distribution that has to be prescribed. A special feature of the newly developed method is that arbitrary shaped, asymmetric thickness distributions can be realized.
Due to the detailed description of the design and calculation steps a fully comprehensible procedure is outlined, which covers the development of inducer bladings from an initial set of duty parameters to the final three-dimensional blade geometry.
The components involved in the design procedure are tested by designing two exemplary inducers and they are assessed by comparison with numerical simulations. Functioning of these inducers in the real application is finally demonstrated with water tests.
The main result of this dissertation is a design software for inducers allowing for the design of three-dimensional, asymmetrically profiled bladings. The developed software is free of commercial third-party libraries. As a consequence a program is available that can be modified and extended as desired. As potential future development goals inducers with splitter and tandem blades as well as an integrated design of inducer and impeller are proposed.
Medical cyber-physical systems (MCPS) emerged as an evolution of the relations between connected health systems, healthcare providers, and modern medical devices. Such systems combine independent medical devices at runtime in order to render new patient monitoring/control functionalities, such as physiological closed loops for controlling drug infusion or optimization of alarms. Despite the advances regarding alarm precision, healthcare providers still struggle with alarm flooding caused by the limited risk assessment models. Furthermore, these limitations also impose severe barriers on the adoption of automated supervision through autonomous actions, such as safety interlocks for avoiding overdosage. The literature has focused on the verification of safety parameters to assure the safety of treatment at runtime and thus optimize alarms and automated actions. Such solutions have relied on the definition of actuation ranges based on thresholds for a few monitored parameters. Given the very dynamic nature of the relevant context conditions (e.g., the patient’s condition, treatment details, system configurations, etc.), fixed thresholds are a weak means for assessing the current risk. This thesis presents an approach for enabling dynamic risk assessment for cooperative MCPS based on an adaptive Bayesian Networks (BN) model. The main aim of the approach is to support continuous runtime risk assessment of the current situation based on relevant context and system information. The presented approach comprises (i) a dynamic risk analysis constituent, which corresponds to the elicitation of relevant risk parameters, risk metric building, and risk metric management; and (ii) a runtime risk classification constituent, which aims to analyze the current situation risk, establish risk classes, and identify and deploy mitigation measures. The proposed approach was evaluated and its feasibility proved by means of simulated experiments guided by an international team of medical experts with a focus on the requirements of efficacy, efficiency, and availability of patient treatment.
This work deals with the simulation of the micro-cutting process of titanium. For this
purpose, a suitable crystal-plastic material model is developed and efficient implemen-
tations are investigated to simulate the micro-cutting process. Several challenges arise
for the material model. On the one hand, the low symmetry hexagonal close-packed
crystal structure of titanium has to be considered. On the other hand, large defor-
mations and strains occur during the machining process. Another important part is
the algorithm for the determination of the active slip systems, which has a significant
influence on the stability of the simulation. In order to obtain a robust implemen-
tation, different aspects, such as the algorithm for the determination of the active
slip systems, the method for mesh separation between chip and workpiece as well as
the hardening process are investigated, and different approaches are compared. The
developed crystal-plastic material model and the selected implementations are first
validated and investigated using illustrative examples. The presented simulations of
the micro-cutting process show the influence of different machining parameters on the
process. Finally, the influence of a real microstructure on the plastic deformation and
the cutting force during the process is shown.
A prime motivation for using XML to directly represent pieces of information is the ability of supporting ad-hoc or 'schema-later' settings. In such scenarios, modeling data under loose data constraints is essential. Of course, the flexibility of XML comes at a price: the absence of a rigid, regular, and homogeneous structure makes many aspects of data management more challenging. Such malleable data formats can also lead to severe information quality problems, because the risk of storing inconsistent and incorrect data is greatly increased. A prominent example of such problems is the appearance of the so-called fuzzy duplicates, i.e., multiple and non-identical representations of a real-world entity. Similarity joins correlating XML document fragments that are similar can be used as core operators to support the identification of fuzzy duplicates. However, similarity assessment is especially difficult on XML datasets because structure, besides textual information, may exhibit variations in document fragments representing the same real-world entity. Moreover, similarity computation is substantially more expensive for tree-structured objects and, thus, is a serious performance concern. This thesis describes the design and implementation of an effective, flexible, and high-performance XML-based similarity join framework. As main contributions, we present novel structure-conscious similarity functions for XML trees - either considering XML structure in isolation or combined with textual information -, mechanisms to support the selection of relevant information from XML trees and organization of this information into a suitable format for similarity calculation, and efficient algorithms for large-scale identification of similar, set-represented objects. Finally, we validate the applicability of our techniques by integrating our framework into a native XML database management system; in this context we address several issues around the integration of similarity operations into traditional database architectures.
This thesis presents a novel, generic framework for information segmentation in document images.
A document image contains different types of information, for instance, text (machine printed/handwritten), graphics, signatures, and stamps.
It is necessary to segment information in documents so that to process such segmented information only when required in automatic document processing workflows.
The main contribution of this thesis is the conceptualization and implementation of an information segmentation framework that is based on part-based features.
The generic nature of the presented framework makes it applicable to a variety of documents (technical drawings, magazines, administrative, scientific, and academic documents) digitized using different methods (scanners, RGB cameras, and hyper-spectral imaging (HSI) devices).
A highlight of the presented framework is that it does not require large training sets, rather a few training samples (for instance, four pages) lead to high performance, i.e., better than previously existing methods.
In addition, the presented framework is simple and can be adapted quickly to new problem domains.
This thesis is divided into three major parts on the basis of document digitization method (scanned, hyper-spectral imaging, and camera captured) used.
In the area of scanned document images, three specific contributions have been realized.
The first of them is in the domain of signature segmentation in administrative documents.
In some workflows, it is very important to check the document authenticity before processing the actual content.
This can be done based on the available seal of authenticity, e.g., signatures.
However, signature verification systems expect pre-segmented signature image, while signatures are usually a part of document.
To use signature verification systems on document images, it is necessary to first segment signatures in documents.
This thesis shows that the presented framework can be used to segment signatures in administrative documents.
The system based on the presented framework is tested on a publicly available dataset where it outperforms the state-of-the-art methods and successfully segmented all signatures, while less than half of the found signatures are false positives.
This shows that it can be applied for practical use.
The second contribution in the area of scanned document images is segmentation of stamps in administrative documents.
A stamp also serves as a seal for documents authenticity.
However, the location of stamp on the document can be more arbitrary than a signature depending on the person sealing the document.
This thesis shows that a system based on our generic framework is able to extract stamps of any arbitrary shape and color.
The evaluation of the presented system on a publicly available dataset shows that it is also able to segment black stamps (that were not addressed in the past) with a recall and precision of 83% and 73%, respectively.
%Furthermore, to segment colored stamps, this thesis presents a novel feature set which is based on intensity gradient, is able to extract unseen, colored, arbitrary shaped, textual as well as graphical stamps, and outperforms the state-of-the-art methods.
The third contribution in the scanned document images is in the domain of information segmentation in technical drawings (architectural floorplans, maps, circuit diagrams, etc.) containing usually a large amount of graphics and comparatively less textual components. Further, as in technical drawings, text is overlapping with graphics.
Thus, automatic analysis of technical drawings uses text/graphics segmentation as a pre-processing step.
This thesis presents a method based on our generic information segmentation framework that is able to detect the text, which is touching graphical components in architectural floorplans and maps.
Evaluation of the method on a publicly available dataset of architectural floorplans shows that it is able to extract almost all touching text components with precision and recall of 71% and 95%, respectively.
This means that almost all of the touching text components are successfully extracted.
In the area of hyper-spectral document images, two contributions have been realized.
Unlike normal three channels RGB images, hyper-spectral images usually have multiple channels that range from ultraviolet to infrared regions including the visible region.
First, this thesis presents a novel automatic method for signature segmentation from hyper-spectral document images (240 spectral bands between 400 - 900 nm).
The presented method is based on a part-based key point detection technique, which does not use any structural information, but relies only on the spectral response of the document regardless of ink color and intensity.
The presented method is capable of segmenting (overlapping and non-overlapping) signatures from varying backgrounds like, printed text, tables, stamps, logos, etc.
Importantly, the presented method can extract signature pixels and not just the bounding boxes.
This is substantial when signatures are overlapping with text and/or other objects in image. Second, this thesis presents a new dataset comprising of 300 documents scanned using a high-resolution hyper-spectral scanner. Evaluation of the presented signature segmentation method on this hyper-spectral dataset shows that it is able to extract signature pixels with the precision and recall of 100% and 79%, respectively.
Further contributions have been made in the area of camera captured document images. A major problem in the development of Optical Character Recognition (OCR) systems for camera captured document images is the lack of labeled camera captured document images datasets. In the first place, this thesis presents a novel, generic, method for automatic ground truth generation/labeling of document images. The presented method builds large-scale (i.e., millions of images) datasets of labeled camera captured / scanned documents without any human intervention. The method is generic and can be used for automatic ground truth generation of (scanned and/or camera captured) documents in any language, e.g., English, Russian, Arabic, Urdu. The evaluation of the presented method, on two different datasets in English and Russian, shows that 99.98% of the images are correctly labeled in every case.
Another important contribution in the area of camera captured document images is the compilation of a large dataset comprising 1 million word images (10 million character images), captured in a real camera-based acquisition environment, along with the word and character level ground truth. The dataset can be used for training as well as testing of character recognition systems for camera-captured documents. Various benchmark tests are performed to analyze the behavior of different open source OCR systems on camera captured document images. Evaluation results show that the existing OCRs, which already get very high accuracies on scanned documents, fail on camera captured document images.
Using the presented camera-captured dataset, a novel character recognition system is developed which is based on a variant of recurrent neural networks, i.e., Long Short Term Memory (LSTM) that outperforms all of the existing OCR engines on camera captured document images with an accuracy of more than 95%.
Finally, this thesis provides details on various tasks that have been performed in the area closely related to information segmentation. This includes automatic analysis and sketch based retrieval of architectural floor plan images, a novel scheme for online signature verification, and a part-based approach for signature verification. With these contributions, it has been shown that part-based methods can be successfully applied to document image analysis.
For many years real-time task models have focused the timing constraints on execution windows defined by earliest start times and deadlines for feasibility.
However, the utility of some application may vary among scenarios which yield correct behavior, and maximizing this utility improves the resource utilization.
For example, target sensitive applications have a target point where execution results in maximized utility, and an execution window for feasibility.
Execution around this point and within the execution window is allowed, albeit at lower utility.
The intensity of the utility decay accounts for the importance of the application.
Examples of such applications include multimedia and control; multimedia application are very popular nowadays and control applications are present in every automated system.
In this thesis, we present a novel real-time task model which provides for easy abstractions to express the timing constraints of target sensitive RT applications: the gravitational task model.
This model uses a simple gravity pendulum (or bob pendulum) system as a visualization model for trade-offs among target sensitive RT applications.
We consider jobs as objects in a pendulum system, and the target points as the central point.
Then, the equilibrium state of the physical problem is equivalent to the best compromise among jobs with conflicting targets.
Analogies with well-known systems are helpful to fill in the gap between application requirements and theoretical abstractions used in task models.
For instance, the so-called nature algorithms use key elements of physical processes to form the basis of an optimization algorithm.
Examples include the knapsack problem, traveling salesman problem, ant colony optimization, and simulated annealing.
We also present a few scheduling algorithms designed for the gravitational task model which fulfill the requirements for on-line adaptivity.
The scheduling of target sensitive RT applications must account for timing constraints, and the trade-off among tasks with conflicting targets.
Our proposed scheduling algorithms use the equilibrium state concept to order the execution sequence of jobs, and compute the deviation of jobs from their target points for increased system utility.
The execution sequence of jobs in the schedule has a significant impact on the equilibrium of jobs, and dominates the complexity of the problem --- the optimum solution is NP-hard.
We show the efficacy of our approach through simulations results and 3 target sensitive RT applications enhanced with the gravitational task model.
This thesis is concerned with the modeling of the solid-solid phase transformation, such as the martensitic transformation. The allotropes austenite and martensite are important for industry applications. As a result of its ductility, austenite is desired in the bulk, as opposed to martensite, which desired in the near surface region. The phase field method is used to model the phase transformation by minimizing the free energy. It consists of a mechanical part, due to elastic strain and a chemical part, due to the martensitic transformation. The latter is temperature dependent. Therefore, a temperature dependent separation potential is presented here. To accommodate multiple orientation variants, a multivariant phase field model is employed. Using the Khachaturyan approach, the effective material parameters can be used to describe a constitutive model. This however, renders the nodal residual vector and elemental tangent matrix directly dependent on the phase, making a generalization complicated. An easier approach is the use of the Voigt/Taylor homogenization, in which the energy and their derivatives are interpolated creating an interface for material law of the individual phases.
Numerical Godeaux surfaces are minimal surfaces of general type with the smallest possible numerical invariants. It is known that the torsion group of a numerical Godeaux surface is cyclic of order \(m\leq 5\). A full classification has been given for the cases \(m=3,4,5\) by the work of Reid and Miyaoka. In each case, the corresponding moduli space is 8-dimensional and irreducible.
There exist explicit examples of numerical Godeaux surfaces for the orders \(m=1,2\), but a complete classification for these surfaces is still missing.
In this thesis we present a construction method for numerical Godeaux surfaces which is based on homological algebra and computer algebra and which arises from an experimental approach by Schreyer. The main idea is to consider the canonical ring \(R(X)\) of a numerical Godeaux surface \(X\) as a module over some graded polynomial ring \(S\). The ring \(S\) is chosen so that \(R(X)\) is finitely generated as an \(S\)-module and a Gorenstein \(S\)-algebra of codimension 3. We prove that the canonical ring of any numerical Godeaux surface, considered as an \(S\)-module, admits a minimal free resolution whose middle map is alternating. Moreover, we show that a partial converse of this statement is true under some additional conditions.
Afterwards we use these results to construct (canonical rings of) numerical Godeaux surfaces. Hereby, we restrict our study to surfaces whose bicanonical system has no fixed component but 4 distinct base points, in the following referred to as marked numerical Godeaux surfaces.
The particular interest of this thesis lies on marked numerical Godeaux surfaces whose torsion group is trivial. For these surfaces we study the fibration of genus 4 over \(\mathbb{P}^1\) induced by the bicanonical system. Catanese and Pignatelli showed that the general fibre is non-hyperelliptic and that the number \(\tilde{h}\) of hyperelliptic fibres is bounded by 3. The two explicit constructions of numerical Godeaux surfaces with a trivial torsion group due to Barlow and Craighero-Gattazzo, respectively, satisfy \(\tilde{h} = 2\).
With the method from this thesis, we construct an 8-dimensional family of numerical Godeaux surfaces with a trivial torsion group and whose general element satisfy \(\tilde{h}=0\).
Furthermore, we establish a criterion for the existence of hyperelliptic fibres in terms of a minimal free resolution of \(R(X)\). Using this criterion, we verify experimentally the
existence of a numerical Godeaux surface with \(\tilde{h}=1\).