EXAMENSARBETEN VID AVDELNINGEN FÖR MATEMATIK, LTH, 2003

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Non-commutative operator methods for equations of difference and q-difference type
Student: Jakob Willborg, F96
Advisor: Sergei Silvestrov
Date Finished: 2003-12-11
Abstract: Non-commuting operators are of great interest in many different disciplines e.g. applied physics and applied economics besides the pure mathematics. One needs only to point to the fact that some very fundamental theorems in quantum mechanics involve the aspect of non-commuting operators to see the use of this theory. Different generators and defining relations result in different quantum mechanic interpretations. This thesis centers on the application of operator methods for solving differential, difference, q-difference and q,H-difference equations. Methods for solving these equations are developed by considering the commutation relations satisfied by the operators. First some necessary notions are introduced and then a general and effective operator method is presented to solve all of those different types of equations. This operator method is expanded to all such operator equations of order n with constant coefficients. The nature of the operator commutation relations are studied and some results e.g. reordering formulas, covariance commutation relations and more or less explicit formulas for the powers of the operators are presented. Examples are given of solutions to different equations. Both homogenous equations and particular solutions of the non-homogeneous equations are studied. The convergence of the q-deformed difference equation is investigated in some detail with the help of visualizations done numerically with a Matlab program. Finally, D{q,H}-equations are investigated.

On-line handwriting recognition
Student: Jimmy Engström , D98
Advisor: Kalle Åström, Jari Sassi (Ericsson Mobile Platforms)
In cooperation with: Ericsson Mobile Platforms
Date Finished: 2003-12-11
Abstract: During the last 10 years, there has been a few commercial systems that claims to give the user a 'pen and paper feeling'. Most of these systems limited the alphabet to a predefined simpler one that the user had to learn. The reason was mainly that both algorithms and hardware was not developed enough to allow the user to write with the alphabet she uses with pen and paper. During the last years, the market of mobile computers and advanced cellular phones has basically exploded and they are quite common now. When using these small devices, input via keyboard is not usually an option, so good handwriting recognition is increasingly important in this area.
This Masters Thesis focuses on the current research within On-line handwriting recognition (HWR). On line systems are more important in the case of mobile computers, where the system can gain more information than just the image of the text that off-line systems have to use. This makes on-line systems more accurate. The thesis summarizes what methods that are common right now, where the problems are and in what ways it is possible to deal with them. The evaluation was used as a guideline for the implementation of a working prototype of an on-line HWR system based on the Hidden Markov Model.

Modelling Images by a Sum of Parametric Models
Student: Jenny Folkesson, F98
Advisor: Kalle Åström, Matematikcentrum, Gustav Wallmark (Ludesi AB)
In cooperation with: Ludesi AB
Date Finished: 2003-12-12
Abstract: Two dimensional gel electrophoresis is the leading technology for separating different proteins in biological samples. In the analysis of 2D gel electrophoresis images, some of the most vital infor­mation are the locations and the volumes of the protein spots. In this thesis, a method of modelling gel image surfaces that quantifies the proteins and calculates their center coordinates is discussed and implemented. An existing segmentation program is used to mark all the protein spots. Thereafter parametric models are fitted to the spots. It is an iterative scheme, where the largest spot is modelled then subtracted from the image repeatedly until there are no spots remai­ning. Then spots are added back and are refitted along with neighboring spots, meaning every spot will be fitted at least twice. In practice the result is a least squares fit of the parameters of the parametric models to the entire image. The outcome is an artificial image that closely resem­bles the real gel image. The results show that the models can be used to accurately deter­mine the volumes and the coordinates of the proteins. By modelling the image the memory required to store the image is reduced significantly, because there is only need to store a few parameters for every spot instead of every pixel in the image.

Wavelets as an Application of Non-commutative Operator Algebras
Student: Stefan Heimgård, E99
Advisor: Sergei Silvestrov
Date Finished: 2003-12-01
Abstract: In this thesis the theory of wavelets on the integers is developed by considering non-commuting isometric operators on the space of all square summable sequences and the commutation relations satisfied by them. The discrete wavelet transform is defined in terms of those operators, and an effective way to calculate the transform is derived.
Wavelets on the real line are briefly discussed and are connected to discrete wavelets. Motivated by the strong connection between wavelet theory and transfer operators, some different types of transfer operators are examined. The theory is illustrated by a number of concrete examples based on numerical simulations performed in Matlab Wavelet Toolbox. A simple noise reduction and a simple compression algorithm is developed involving, as the main ingredience, the discrete wavelet transform.

Motion Capture for 3D Reconstruction Using Computer Vision
Student: Andreas Alptun,D98 and Daniel Christensen, D98
Advisor: Fredik Kahl, Henrik Stewenius
Date Finished: 2003-10-31
Abstract: 3D Animation is a hot topic in today's film industry. Almost every action related movie uses 3D animated characters in one way or another, not to mention the new game productions. Making these characters move in a smooth and realistic way can be a difficult and time-consuming job. A motion capture system gathers data for a character's movement in a sequence without too much effort and supply data ready to import in a 3D Animation program.
The purpose of this master's thesis is to construct a system like the one proposed above. Low-priced web-cameras will be used for the motion capture. A number of clearly visible markers attached to the human body will be tracked in a three dimensional space. The position of each marker will be calculated and stored for every time frame.
Our solution to this problem functioned well with test sequences but the camera equipment available proved to be insufficient for this task.

The information in a photo taken by a lensless camera
Student: Martin de Maré, F98
Advisor: Kalle Åström, Fredrik Andersson
Date Finished: 2003-10-31
Abstract: The idea behind the work is to replace the lens, in a digital camera, with computer processing. This opens the door to smaller and more affordable cameras as well as offer a way to bypass the limitations of optics. A lensless camera is modeled assuming linear optics and the model is verified with real data. The inverse problem can be divided into the cases of constant depth and varying depth of the scene. The inverse problem of constant depth is one of deconvolution and several algorithms of deconvolution are tested on real and synthetic data. The results are limited primarily by noise and by information loss at the borders. A few approaches to the inverse problem of varying depth are presented as well as suggestions of how to retain the depth of the image including one using the wavelet transform. Some of approaches are tested on real data. The model implies a new way of building cameras, even if the results are in need of improvement.

Three-Dimensional Face Modelling and Analysis
Student: Karl Skoglund, D98
Advisor: Kalle Åström
Date Finished: 2003-10-08
Abstract: Denna rapport beskriver metoder för modellering av mänskliga ansiktens form och struktur i tre dimensioner. Dessa modeller utgör grunden till algoritmer för exempelvis segmentering och igenkänning; viktiga användningsområden inom bildanalys.
Modellerna skapas utifrån en databas med 24 tredimensionella ansikten. Data har insamlats med hjälp av en laserkamera. Innan ansiktena kan användas för modellering och statistisk analys, måste de ges en gemensam representation, en process som kallas registrering. Den metod som beskrivs här använder nio manuellt placerade korresponderande punkter på varje ansikte. Dessa används sedan som stöd för att automatiskt registrera tusentals punkter. Det stora antalet korrespondenser gör att modellerna kan återge ansikten med stor detaljrikedom.
Rapporten beskriver samtliga matematiska metoder, med utförliga presentationer av de mest centrala delarna. Detta följs av en beskrivning av de viktigaste detaljerna ur implementationen. Modellernas möjligheter visas i applikationer såsom ansiktssyntes, automatisk registrering och igenkänning i tvådimensionella bilder.
Resultaten är lovande, men data med högre kvalitet och kvantitet är önskvärt. Ansiktssyntes och automatisk registrering kan utföras tillfredställande, men algoritmen för igenkänning ger sämre resultat. Orsaken till detta är främst en olämplig minimeringsmetod. Avslutningsvis ges ett antal förslag på ut vidgningar och förbättringar av detta arbete.

Area Based Stereo Vision
Student: Ulf Holmstedt, F96
Advisor: Anders Heyden
Date Finished: 2003-09-23
Abstract: Stereo vision has been a traditionally interesting and difficult problem in the area of computer vision. In this text, several algorithms for extraction of dense disparity maps are examined. A large issue with these algorithms is the ability to handle real-world piecewise continous surfaces. Using an algorithm which favours continuity while still allowing sharp edges has been developed and employed with good results. Other problems, such as low-texture environments are addressed using multi-resolution schemes but at full resolution, thus keeping detail loss at a minimum. A simple occlusion detection algorithm is run to eliminate obvious bad matches in the final surface.

Planes, Homographies and Augmented Reality
Student: Björn Liljequist, F96
Advisor: Fredrik Kahl, Nicolas Guilbert
Date Finished: 2003-09-19
Abstract: An augmented reality system generates a composite view consisting of a real scene combined with a virtual reality scene generated by the computer. In order to perform augmented reality it is necessary to estimate the camera movements for the studied image sequence. In this thesis a system for estimating camera movements for image sequences where the scene is known to contain significant planar structures is presented. A rather thorough explanation of more or less all the steps involved in the proposed augmented reality system is given as well as an introduction to projective geometry. The system estimates plane induced homographies relating detected corners in pairs of images and then recovers camera movements based on these homographies. Non-linear optimization techniques are used in order to enforce certain constraints on the homographies and to handle the case of focal length varying between images. The major part of the system is fully automatic though initial manual input is required in order to set the scene coordinate system, to identify the planes in the scene and to get an initial estimate of the focal length.
Though not a prioritized subject of this thesis, it is also explained how non-linear optimization techniques can be used to improve results after the entire image sequence has been processed. It is explained how to enforce constraints arising from known angles between planes in the scene and how to limit the camera movement in order to favor smooth solutions. The different features and techniques of the system are evaluated mainly on a qualitative basis and results are generally satisfying though the manual input is found to comprise a significant source of error. The system is able to handle the case of focal length changing during the sequence as well as new planes becoming dominant in the scene.

Very low bit rate wireless video telephony using face coding
Student: Mikael Persson, D98
Advisor: Fredrik Kahl, Henrik Storm (Summus Inc.)
In cooperation with: Summus Inc.
Date Finished: 2003-09-05
Abstract: In this thesis a novel video telephony compression scheme is proposed, implemented and discussed. The scheme generates a talking head sequence from a head and shoulder video telephony sequence. The generated talking head mimics the facial expressions of the individual depicted in the head and shoulder input sequence. The scheme is based on model based coding and more specifically based on an eigenspace approach. The model which is used to represent the objects to be encoded is statistically derived as the principal components of a training sequence depicting the individual performing a wide range of facial expressions. The thesis introduces the concept of eigenfeatures as used in video compression and a method for encoding the facial expressions of the talking head as a number of coefficients defining a linear combination of the eigenfeatures. Using the proposed scheme acceptable video telephony can be achieved at data rates as low as 3-4 kBit/s.

Spherical Harmonics, Precomputed Radiance Transfer and Realtime Radiosity in Computer Games
Student: Manne Öhrström, F97
Advisor: Fredrik Kahl
In cooperation with: Electronic Arts, London
Date Finished: 2003-09-05
Abstract: Soft shadows and interreflections can enhance the visual realism of a scene by giving an accurate perception of the interaction between objects and light. Unfortunately, to incorporate these elements in a computer games rendering pipeline has traditionally been too expensive. In this thesis, we present a lighting implementation that improves the rendering pipeline by including self-shadows and interreflections for dynamic objects. Our approach defines precomputed elements that store the radiance at selected scene locations. These elements are called Lightlets and they store incident radiance in a spherical harmonics frequency representation. Lightlets provide a lighting model compatible with traditional hardware shaders. Lightlets also makes it possible to manually alter the incident radiance in a point, thereby providing a powerful tool for light editing.

Homogenisation of reaction-diffusion equations in porous media
Student: Anders Hansson, F98
Advisor: Magnus Fontes
In cooperation with: Comsol AB, Stockholm
Date Finished: 2003-08-25
Abstract: A reaction-diffusion equation arising from the analysis of porous electrodes of electrochemical cells is considered. The mathematical challenges are that the current-overvoltage dependence is exponential (Butler-Volmer kinetics), and that the solution domain is perforated. This latter fact poses a serious impediment to numerical treatment in so far as the finite-element method would require a very large number of elements to yield accurate solutions in representative cases. Using a homogenisation pro-cess based on asymptotic expansions, we derive a homogeneous model, which describes the large-scale behaviour of the electrode and is accessible with today's computing resources. The thesis also includes existence and uniqueness theorems for the reaction-diffusion problem before homogenisation.

Mass Flow Calculation and Characterization of Turbulent Gas Emissions Imaged by Gas Correlation Spectroscopy
Student: Björn Ardö, F98
Advisor: Kalle Åström, Hans Edner (Atomfysik), Tomas Christiansson (GasOptics AB)
Date Finished: 2003-09-02
Abstract: The aim of this thesis is to calculate the total mass flux of gas leaks filmed by a gas correlation camera. The gas used was ethene released from a hose, resulting in a turbulent jet. Methods to find the source of the leak in the gas image and to characterize the main shape of it have been developed and tested with good results. One method using block matching in time with a Markov field was evaluated with promising, but not fully satisfying, results. A method that tries to generalize the existing method used in lidar measurements was developed and tested. This method has not yet proven to be usable.

Automatic object reconstruction from turntable sequences.
Student: Rikard Stymne, F94 and Magnus Mathiasson, F97
Advisor: Magnus Oskarsson, Kalle Åström
Date Finished: 2003-06-12
Abstract: In this thesis an automatic system for 3D reconstruction of real objects is presented. The objects are placed on a turntable and filmed while rotating. Possible applications include CAD and production of animations and computer games. The only input to the system is the image sequence (in jpeg format) from the camera and the output is a 3D model (in VRML format). No camera calibration, no user interaction and no special equipment is needed. Features on the object are tracked. The camera parameters are then determined by using a method based on fitting conics to the paths of the features. All such conics lie on planes perpendicular to the rotation axis and the geometry of this single axis motion can be recovered unambiguously given at least three conics. The complete 3D structure may be determined to within a two parameter family and a solution to escape this ambiguity is proposed. When the geometry of the arrangement is completely known, the construction of the 3D model can begin. First a volumetric model (a representation of what parts of space are inside the object and what are not) is generated by an algorithm using the silhouettes of the object. This model is transformed to a surface model by an algorithm generating a triangle mesh covering it. The last step is to apply texture to the model. This is done by taking small segments of the images in the sequence and placing them on the surface. A number of image sequences were produced to test the system. The result shows that the quality of the resulting models is promising but many requirements must be met for the system to work. For example the lighting must be good and the object can not be too smooth and uniform in colour.
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Detection of gesture paths using stereoscopic vision
Student: Ludvig Linge, D98, and Karl-Anders Johansson, E98
Advisor: Henrik Stewenius, Fredrik Kahl
Date Finished: 2003-06-12
Abstract: This thesis presents a method, as well as a prototype system, for detecting and identifying hand gestures from a sequence of discrete motions in 3D using ordinary web cameras for stereoscopic vision. The system does not require the user to wear any supporting equipment such as specialized gloves, nor does it require any expensive hardware. The problem is broken down into gesture detec-tion, triangulation, stereo correlation, object tracking, image segmentation and image correction.
Example applications are developed to evaluate the performance of the method and the prototype system. It is concluded that the example applications and the proposed system perform satisfactory in a controlled indoor environment.

Quantification of Myocardial Deformation Using Phase Contrast Magnetic Resonance Imaging
Student: Erik Bergvall, F99
Advisor: Fredrik Andersson, Gunnar Sparr, Peter Cain (Klinisk fysiologi), Håkan Arheden (Klinisk fysiologi)
In cooperation with: Klinisk fysiologi, Lund University
Date Finished: 2003-06-10
Abstract: In this thesis a method for estimating two-dimensional myocardial kinematics is investigated. This method, named Fourier tracking, uses data from phase contrast magnetic resonance imaging providing velocity vector fields. Position calculation by integrating the velocity field is a difficult task due to the presence of noise and the fact that spatio-temporal resolution is limited. As the heart cycle is periodic, Fourier techniques are used for analysis. Integration of velocity data is made in the frequency domain by an iterative scheme. The method has been implemented and compared to a common differential equation solver with almost identical result. A key difference is the execution time of the methods where Fourier tracking is significantly faster. A preliminary validation versus data acquired by tagged magnetic resonance imaging has been made showing good results. The method extends naturally to three-dimensional motion analysis and the rapid development of magnetic resonance imaging technology will further increase the potential for clinical use.

High Resolution Methods for Laser Bathymetry
Student: Claes Johannesson, F98
Advisor: Kalle Åström, Fredrik Andersson,, Andreas Axelsson (AHAB)
In cooperation with: AHAB
Date Finished: 2003-05-15
Abstract: I detta examensarbete har en ny metod för djupbestämning utifrån vågformer från laserbatymetrisystem utvecklats och testats.
De vågformspulser som erhålles från bottenreflektioner kan lämpligen modelleras med viktade histogram över djupfördelningen kring mätområdet. Utifrån strukturen hos dessa histogram har ett linjärt ekvationssystem härletts för rekonstruktion av bottennivån. Tidigare metoder har använt ett distinkt värde hos varje mätning för bestämning av ett djup, medan ovanstående metodik istället dels kopplar samman de olika mätningarna och dels utnyttjar mer än ett värde i varje mätning. Förhoppningsvis kan detta användas för robustare rekonstruktion samt förhöjd upplösning.
Modelleringar har gjorts främst på tvådimensionella bottnar men även, i mindre omfattning, på tredimensionella. Resultaten är lovande och metoden verkar både robust och ge hög upplösning. Det återstår dock vidare utveckling och utvärdering innan en definitiv slutsats om metodens praktiska användbarhet kan dras.

Facedetection and determination of orientation of digital images.
Student: Josefin Ripa, E97
Advisor: Kalle Åström, Olof Barr, Anders Eriksson
Date Finished: 2003-04-14
Abstract: Digitalkameran har idag ersatt den vanliga kameran i många fall. Det finns därför anledning att se över vilka metoder som kan vara av intresse i hanteringen av databaser med digitala bilder. I det här examensarbetet studeras två olika uppgifter: Automatisk rättvändning av bilder och ansiktsdetektion. Båda dessa problem går under konceptet klassifikation. I arbetet har stödvektormaskiner (SVM) och förstärkningsalgoritmer (boosting) använts för att konstruera klassificerare. Flera sådana klassificerare används i tillsammans i en kaskad för att detektera ansikten. Algoritmen studerar delbilder av olika storlek och position. Varje sådan delbild klassificeras sedan som ansikte eller icke-ansikte. Algoritmerna har utvärderats på några exempelsamlingar där ansiktenas antal och position varit kända. Utvärderingen har gett lovande resultat.

Automatic Positioning of Landmarks for Shape Analysis
Student: Johan Karlsson, D98
Advisor: Anders Ericsson, Kalle Åström
Date Finished: 2003-02-14
Abstract: Shape segmentation and shape analysis is an interesting and promising research area with applications in various fields, for example medical image analysis. The usual way of building a model of the shape of some set of objects is to start by placing a number of corresponding landmarks on the shapes. Then the shapes can be analyzed based on the modes of variation of these landmarks. Recently different methods have emerged for placing such landmarks automatically and in some sense optimally. This masters thesis concerns the implementation of algorithms flr automatic landmark placement in 2D and 3D in Matlab. Correspondences between landmarks is achieved by optimizing the parameterizations of the shapes (and thereby the positions of the landmarks) and evaluating the resulting models by looking at the variances of the modes of variation. The algorithms are tested on real data and it is shown that the resulting models are better than models built from equally spaced landmarks and in some aspects better than models built by hand.

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