EXAMENSARBETEN VID AVDELNINGEN FÖR MATEMATIK, LTH, 2005

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A New Kind of Two-Parameter Deformation of Heisenberg and Parabose Algebras and Related Deformed Derivative
Student: Fredrik Hagen, F00
Advisor: Sergei Silvestrov
Date finished: 2005-12-14
Abstract: Linear differential and difference type operators solve problems in quantum mechanics.It is a question for research to formulate quantum mechanic models that fit in, i.e. describe physics with these operators. We try to find convenient operators. First we give an introduction to non-commutative algebras, to increase the understanding for different algebras. We concentrate on algebras close to the ones we will be working with later. We propose a new kind of two-parameter (p,q)-deformed Heisenberg and parabose algebras. Corresponding to the two-parameter deformed oscillator, we also introduce a new kind of (p,q)-deformed derivative. We study the structure of Fock-like space of the new(p,q)-deformed oscillators and derive a formal solution for eigenvalue equation of the Hamiltonian. We also consider multiparameter deformation-discretization of derivate and of Heisenberg and parabose algebras, as well as discretizations based on possible nonlinear discretization transformations.

Convexity and Game Theory - Classical Results and Generalizations
Student: Charlotte Soneson, F00
Advisor: Magnus Fontes and Olivier Verdier
Date finished: 2005-12-09
Abstract: Games of different kinds have fascinated humans for thousands of years. Eventually, a mathematical theory for the study of games was developed, mainly by von Neumann. This theory has been developed further and is now widely applicable. It covers the whole range from simple two-person games based on tossing a coin to complex military strategy making. Moreover, convexity plays an important role in the classical theory.This thesis firstly includes an overview of the classical theory. We then present a generalization of the so called Ky Fan inequality, a very useful tool in nonlinear analysis. We also give a generalization of von Neumann's minimax theorem, without convexity assumptions. Finally, we use a notion of geodesic convexity to prove analogies of the geometric version of the Hahn-Banach theorem on the sphere and in the hyperbolic plane.

Connections between Topological Dynamical Systems and Non-Commutative C*-algebras
Student: Johan Öinert, F00
Advisor: Sergei Silvestrov
Date finished: 2005-12-06
Abstract: From a historical perspective mathematics has always been divided into three distinct areas; Algebra, Analysis and Geometry. It turns out that dynamical systems give rise to an interplay between these different areas. For each topological dynamical system we can define the so called transformation group C*-algebra. We have investigated how topological and dynamical properties of a dynamical system are reflected in algebraic properties of its associated transformation group C*-algebra.

Automated detection of malaria in thin blood smears
Student: Gunnar Hansen, F99
Advisor: Niels-Christian Overgaard (Malmö högskola), Per Sennmalm (Cellavision AB)
In cooperation with: Cellavision AB
Date finished: 2005-11-10
Abstract: In this thesis I have deviced a method for automatic detection of malaria in thin blood smears on microscopic slides.The manual detection of malaria is a task both important and time consuming. Even though hundreds of millions are infected by malaria each year, this avenue of research has been virtually unexplored.The approach to malaria detection taken in this thesis is to, through the usage of discriminant analysis, detect malaria through its distinct colour. Morpological reconstruction and active contours are then used to isolate the infected erythrocyte (red blood cell). This approach however, raises questions of the constancy of colour. The imaging systems, as well as the staining procedure used to give the different cytologic objects their colours, are subject to variations in both intensity and hue. The problems caused by these variations as well as certain peculiarities caused by the camera are discussed and dealt with. Malaria infected cells are detected in 135 of 136 cases giving this method a sensitivity of approximately 99%.

Analysis of the Minimal Six-Point Solver for Motion Reconstruction of Calibrated Generalized Cameras
Student: Kévin Falière, Erasmus
Advisor: Kalle Åström, Nicolas Guilbert
Date finished: 2005-07-28
Abstract: This thesis deals with the relative pose problem of calibrated generalized cameras and is primarily intended for autonomous navigation applications.We consider a system with multiple cameras rigidly fixed on a platform with non-overlapping views. This system can be seen as a generalized camera.Then, a random trajectory is carried out for the platform and the purpose is to compute the motion reconstruction. In order to get the good transformation between two different positions of the generalized camera, we use a minimal case solver requiring six corresponding points, i.e. twelve points in the two views. This solver is based on Gröbner bases decomposition methods and is intended to be used as the kernel of a RANSAC scheme.The goal is to investigate under which conditions the solver could potentially be used in practice. Therefore assorted tests are done to delimit the functionality of the system.The theory has been mainly tested on synthetic data. We have notably underlined that because of the noise sensitivity of the solver this algorithm may not perform for practical applications.

View Control using Camera Motion Estimation
Student: Tobias Antonsson, E00 and Per Holmström, E00
Advisor: Rikard Berthilsson, Håkan Ardö
Date finished: 2005-06-17
Abstract: In this master thesis we propose methods for estimating camera movements in real time and on highly constrained platforms. These put sever restrictions on the computational complexity of the methods.Three different image motion estimation methods are studied. These are the Block matching algorithm, a differential technique for calculating optical flow, and the KLT-Tracker algorithm. Block matching and the KLT-Tracker are two methods for tracking features from frame to frame in image sequences. Optical flow is used to estimate the complete motion field of moving scenes. The motion estimation algorithms have successfully been implemented and evaluated on PC working on real image streams.Many of today's new handheld devices, such as mobile phones and PDA's are equipped with an integrated camera. By applying advanced vision methods on the images captured by the camera a whole field of new applications opens up. This is exploited, where we in this thesis aim to be able improve the user interface of these devices by estimating the three-dimensional movement of the device.As a consequence of this the block matching method was implemented on a camera equipped PDA running Windows CE operating system. Furthermore, a series of plausible applications based on the knowledge of the movement of the device are discussed and implemented with promising results.

Curve representation of fingerprints
Student: Erik Wittrup, D99
Advisor: Kalle Åström, Björn Nordin (Precise Biometrics)
In cooperation with: Precise Biometrics
Date finished: 2005-06-03
Abstract: Ever since the 19th century, fingerprints have been used for recognition of people. The usage has continuously grown and today, automatic fingerprint recognition is a very large business.There exists a variety of different ways to store and match fingerprints. The method most widely used is the one that principally uses the minutia points as characteristics. However, ISO has recently come with another proposal of how to store fingerprints. This proposal suggests that the fingerprint will be stored by replacing the ridges by one pixel wide curves, each approximated with a number of polygon elements of fix size.The purpose of this thesis is to investigate how well this ISO-method works to match and store fingerprints.The results show that the performance is not as good as a real world system would require. On the other hand one sees that the investigated matching method works very well by itself, but the supporting systems, as image restoring and alignment, fails. These have to be considerably improved. This implies a very promising future for the ISO-method. Probably, with a better image restoring method and a more stable aligning method, the ISO-method can get a very high performance. However, there is still much to do.

Obstacle Detection Using Robotic Vision
Student: Bengt Petersson, E97
Advisor: Anders Heyden
Date finished: 2005-05-11
Abstract: An attempt to test an algorithm for robot navigation with only one camera. To analyze an algorithm and see if it can be used as a detection algorithm for obstacle detection for a robot equipped only with one camera.

Segmentation of the Brain in SPECT Images Using Active Shape Models
Pupular
Student: Johan Baldetorp, E00, and Simon Ristner, E00
Advisor: Kalle Åström, Anders Ericson, Johan Karlsson
Date finished: 2005-04-29
Abstract: This thesis presents a method for automatically finding the surface of the brain in a 3DSPECT image. The method consists of two major parts, the building of a model from a training set and the segmentation of a 3D image using the model. The training set contained 16 brain images. From these images 16 point clouds were manually created. Then a surface, represented by atriangulation of landmarks, was fitted to each point cloud. A correspondence between the landmarks of the different surfaces was obtained using ICP. The training set was then aligned using Procrustes Analysis and a statistical shape model was built using PCA. That model was used to automatically segment the brain in new SPECT images using the so called Active Shape algorithm. This segmentation method proved to be successful on this kind of images and the results are displayed in the end of the thesis along with an example of how this can be used in hospitals.

Automatic Image Rectification for Stereoscopic Display on Mobile Telephones
Student: Daniel Ragnarsson, F98
Advisor: Kalle Åström, Rikard Berthilson (Cognimatics), Johan Gulliksson (Sony Ericsson)
In cooperation with: Cognimatics, Sony Ericsson
Date finished: 2005-05-04
Abstract: Recent research on displays for mobile phones has produced so called stereoscopic displays that are capable of showing a separate image for each eye. This makes it possible to display three dimensional images on mobile phones. This thesis investigates the possibility of producing such stereoscopic images automatically from two images taken with the same phone. Given the internal camera parameters, it is possible to do a Euclidean reconstruction of a scene using only images as input and allowing general camera motion. A system was developed in Matlab that performs automatic rectification of stereo images in a calibrated environment. The rectified images are suitable for stereoscopic display. The system includes feature extraction and matching, robust epipolar geometry estimation and construction and application of rectifying homographies. The thesis describes the system, in theory and in practice and gives examples of its performance. Dependencies on the input images are discussed, along with plausible constraints on the acquisition of images. The target platform is a mobile telephone, which puts strict requirements on efficiency and memory consumption. For this reason, vital parts of the system were also developed in ANSI C for testing in mobile telephone environments.The system was tested on numerous image pairs, with promising results, demonstrating that it is indeed possible to make stereoscopic images automatically from two images. However, the current method for construction of rectifying homographies was found not to be sufficiently robust to general camera motion. Several ways to improve the system are given as suggestions for future work.

Boosting Auto-Focus Object Selection in Mobile Phone Digital Cameras
Student: Anders Ericson, F99, and Mats Pehrson, F99
Advisor: Kalle Åström, Rikard Berthilson (Cognimatics), Johan Gulliksson (Sony Ericsson)
In cooperation with: Cognimatics, Sony Ericsson
Date finished: 2005-04-08
Abstract: In digital cameras today the algorithm for selecting a focus point is far from optimal. In many cameras the centre point in the image is selected by default, or there may be a grid of focus areas. Ideally the algorithm that selects the focus area would be much more flexible, it should be able to detect for example faces, people and other objects that are the focus of attention in an image. In this thesis we have developed a detector capable of detecting interesting objects in an image. A database of interesting areas were collected by ourselves and volunteers. These areas were then used to train a classifier using the well established boosting algorithm AdaBoost. To improve the detection speed, thedetector uses a cascaded structure which focuses computation on promising regions of an image. The results are reasonably good but our evaluation shows that it is not good enough to be confused with a human.

Residual Stress Determination by Automatic Analysis of Interferometric Images
Student: Frida Boström, D00
Advisor: Kalle Åström, Christer Persson (Materialteknik)
In cooperation with: Materialteknik
Date finished: 2006-02-07
Abstract: One semi-destructive method for determination of a residual stress field in a material is performed by relaxing the material, by introducing a blind drill hole of a non-significant size, and by measuring the displacements that the relaxation cause to the material surface. By superposing sets of two holographic images of the material surface, from before and after relaxation, and capturing the resulting interferometric intensity images on a CCD, the surface displacement field is recorded in the form of intensity maps corresponding to phase difference data. A phase differences map can be extracted from a number of interferometric images, through an arcus tangent function. The function wraps the phase data to the interval (-?, ?], which calls for unwrapping in order to remove the produced 2? phase discontinuities, and find the true phase differences. In this work two algorithms for path following, two-dimensional phase unwrapping of noisy phase data are implemented and evaluated. One of the methods handles phase discontinuities by placing branch cuts that prohibits the unwrapping path to cross the discontinuities. The other one is based on a phase local derivative variance quality map, which guides the unwrapping in the direction of the best quality data. The results are varying, depending on the image set used, but generally the latter algorithm produce better results. From the unwrapped, true phase difference map, the out-of-plane displacements map can be derived if the direction of the sensitivity vector of the optical acquisitioning system is known. The relationship between out-of-plane displacements and the stress field that has been relaxed can be described by finite element modeling.Modern computer technology has enabled the processing of full-field displacement data, which provides the possibility for robust stress determination from noisy data. In this work a method for least-squares estimation of a stress component in a uniaxial stress field, from an out-of-plane displacement map and a finite element model of the stress state, is presented. The method is based on multi-dimensional Newton-Raphson fixed-point iteration, and use adaptive-size steps to reach a minimum. The finite element model depends on information of size and placement of the drill hole, and the fixed point iteration requires a good quality initial estimate of those parameters. A method for automatic drill-hole localization and size estimation will therefore also be presented. The method is based on enhancing the drill hole features, while suppressing influences from noise. This is made by filtering the interferometric images with a large negated Gaussian kernel and by identifying a local maximum corresponding to the drill-hole center. The least-squares method does not produce perfect results. Though, this is mainly because of erroneous out-of-plane displacement maps, coming from that the phase shift in-between the interferometric images are non-correct and from that the intensity vector of the acquisitioning system is not known, but estimated. Though, based on visual impressions of the results, the author of this work is convinced that with good displacement maps, the performance of the lest-squares algorithm will be drastically improved.

Commuting Differential Operators and Commuting Elements in Non-Commutative Algebras
Student: Christian Svensson, F99
Advisor: Sergei Silvestrov
Date finished: 2005-02-04
Abstract: An algebra is, very informally, a vector space in which you can multiply vectors in a way that "makes sense". The theory of commuting elements in non-commutative algebras has many important applications, for example in the operator algebras that are studied in quantum mechanics. Besides, the sets of objects considered possess rich structures in themselves, relevant in many different branches of mathematics that they are therefore linking together. For example, commutation of two elements is an algebraic relation. However, in the algebra of ordinary differential operators, commutation of two elements can be described completely in terms of satisfaction of a certain number of differential equations involving the coefficient functions of the operators considered. These equations give again interesting algebraic properties of the coefficient functions of commuting operators. This is a simple example of the beautiful interplay between algebra and analysis that appears when studying the topic of this thesis. One of our main objectives is to find necessary and sufficient conditions for commutation of elements of different degrees of generality, in different algebras. We start off by considering the algebra of ordinary differential operators and find a number of such conditions there. Also, in 1922 it was proved that for any pair of commuting such operators, you can build up a matrix using a certain fixed algorithm, that has the property that its determinant is, possibly after division by some function, a nonzero complex polynomial of two commutative variables that actually annihilates these operators. The proof uses the theory of ordinary differential equations, so here we search a purely algebraic argument for this. Quite recently, a conjecture was made that this matrix construction for two commuting ordinary differential operators, is actually sound construction in algebras of operators more general than the algebra of usual differential operators. We prove this conjecture by algebraic means in some special cases. Finally, we also consider commutation conditions in another algebra, indicating that it can actually be described in terms of the theory of dynamical systems.

Symmetric Spaces
Student: Peter Holmelin, F99
Advisor: Sigmundur Gudmundsson
Date finished: 2005-01-31
Abstract: In this text we study the differential geometry of symmetric spaces. We describe how a symmetric space (M,g) can be seen as a homogeneous space G/K, the quotient of its isometry group G and a isotropy group K at a point. We study the one-to-one correspondence between symmetric spaces and symmetric pairs. Furthermore we investigate the expressions for curvature on a symmetric space. Finally we describe the notion of dual symmetric spaces. To illustrate how well symmetric spaces lend themselves to explicit calculations we calculate the curvature of the real Grassmann manifold and find their dual space.
On the q-Deformed Gelfand-Dickey Hierarchy, the q-DeformedVolterra Equation and Their Generalizations
Student: Roland Mezöfi, E00
Advisor: Sergei Silvestrov
Date finished: 2005-01-28
Abstract: The theory of solitons, i.e. waves travelling infinite distances in shallow water without changing its shape, started as the Korteweg-deVries equation was discovered. In this thesis we will shortly introduce the theory developed by Israil Gelfand and Leonid Dickey. Starting with the KdV equation, methods of finding exact solutions of a whole hierarchy of nonlinear differential equations have been developed. Lately some generalizations have been made by studying q-deformed versions of those equations, based on discretizations using q-difference operators. We will look more closely into these matters. Finally, we will extend the q-deformed Volterra equation into its sigma-twisted version.

Pan Tilt Zoom Surveillance
Student: Adam Eliasson, D02 and Henrik Svensson, D00
Advisor: Magnus Oskarsson, Daniel Elvin (Axis)
In cooperation with: Axis Communications AB
Date finished: 2005-01-24
Abstract: The object of this project is to combine the large field of view of a wide angle lens camera with the high detail of a Pan Tilt Zoom (PTZ) camera, forming a surveillance system able to span a big area with high detail. The goal of this thesis is to make a correct mapping from the static camera's coordinate system to the PTZ camera's. The two cameras are unable to move in relation to one another apart from the PTZ camera's intended movement.The static camera is to search for interesting objects in its field of view through a simple image differentiation of two consecutive frames. When such an object is found the camera is to compute where in space the PTZ should look, recompute those angular values to pan tilt values that the PTZ camera understands and further more tell it to move there.The PTZ camera will locally try to follow objects recognized by the static camera and after receiving new pan and tilt values decide if they are of such an interest that they are to be followed.Different mapping techniques i.e. how the static camera computes where the PTZ should look, are to be developed and experimented on. Evaluation of the mappings' complexity and how well they behave when executed on the cameras are also to be investigated in the objective to see if an easy to handle/install commercial product can be produced.This thesis project was performed at Axis Communications AB.

Design and Implementation of a Region of Interest Analysis System for Eye-tracking Studies
Student: Lisa Mossfeldt, F98 och Jonas Tillander, Linköpings Tekniska Högskola
Advisor: Gunnar Sparr, Carl-Magnus Fahlcrantz (STFI-Packforsk AB)
In cooperation with: STFI-Packforsk AB , Stockholm
Date finished: 2005-01-20
Abstract: Eye-tracking may be a valuable tool when investigating how people assess print quality in images. Eye-tracking data is generally analysed by calculation of various statistics linked to specific regions of the observed image, so called Region of Interests (ROI). This thesis presents a complete system for ROI analysis. It enables the user to draw and edit ROIs based on closed polygons within images and calculates statistics of eye-tracking data in these areas. To make the analysis process more effi-cient, algorithms for automatic definition of ROIs have been developed and evaluated. The first model is based on edge detection in grey scale images. The second model operates by segmenta-tion of the principal component hue, an alternative way of representing colours where shifts in lightness are suppressed. The most promising results are given by the second model but further improvements are required when the image consists of objects with large local colour variance.

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