EXAMENSARBETEN VID AVDELNINGEN FÖR MATEMATIK, LTH, 2009

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Evaluation and Comparison of Duality-Based A Posteriori Error Estimates (Utvärdering och jämförelse av dualbaserade a posteriori-felestimat)
Student: Aron Wahlberg, F04
Advisor: Achim Schroll
In cooperation with: Anders Logg, Simula Research Laboratory
Date Finished: 2009-08-07
Abstract: Numerical approximations to solutions of differential equations are of great importance in science and engineering. In adaptive finite element methods it is necessary to assess the quality of a numerical approximation. Goal oriented a posteriori error estimates via duality are key tools to automatically control the accuracy of numerical approximations. A few such a posteriori error estimators are developed for the Poisson equation and for Advection-Reaction-Diffusion problems. The error estimators are compared and their behaviour and dependencies are investigated for a few model problems. Specifically, cancellation between elements is of great interest. The studies show that cancellation between elements decreases when the errors are expressed with jump terms over element boundaries.

Human Pose Estimation using Dynamic Programming (Skattning av pose för artikulerade objekt med hjälp av dynamisk programmering)
Student: Linus Färm, F-02
Advisor: Olof Enqvist, Fredrik Kahl
Date Finished: 2009-06-16
Abstract: The goal of this Master's thesis has been to develop a computationally efficient algorithm which is able to find the configuration of articulated objects. The focus of this work has been on recovering the pose of human bodies in three dimensions, but the method proposed is fairly general and can be used for a broad class of deformable objects. The method is based on the pictorial structure representation introduced by Fischler and Elschlager, where an object is represented by a set of parts connected by flexible joints. The parts encode visual properties of the object and the joints encode its deformations. The best match of such a model to an image is found by maximizing a likelihood function that measures both the match of each part and the deformation of their joints. The many degrees of freedom of the objects create very high-dimensional parameter spaces. However, by restricting the model structure, the globally optimal configuration can be found efficiently by the use of dynamic programming and integral images. Results from using the method to find the pose of humans are presented.

Finding Representative Structures in Large Image (Hitta representativa strukturer i stora bildsamlingar)
Student: Oscar Lorentzon, F03 och Nils Lundahl, F04
Advisor: Jan Erik Solem
Date Finished: 2009-06-16
Abstract: The vast amount of images currently available on the internet complicates the task of retrieving relevant results from a text-based search query. In this master's thesis, a way to represent larger image collections by a careful selection of a few representative images is presented. The approach is based on the combination of histograms of SIFT descriptor classes as well as colour content, generating image feature vectors. These were then processed and clustered using principal component analysis and different clustering methods in order to find structures that are especially common. The results are promising, showing that a good selection of representative images can be obtained from a variety of image collections using our methods.

Optimization of Automatic Segmentation Algorithms for Cardiovascular Magnetric Resonance Imaging (Optimering av automatiska segmenteringsalgoritmer för kardiovaskulär MRI)
Student: Peter Holmqvist, E01
Advisor: Fredrik Kahl, Einar Heiberg & Helen Sonesson
In cooperation with: Klinisk Fysiologi, Universitetssjukhuet i Lund
Date Finished: 2009-06-11
Abstract: Magnetic resonance imaging (MRI) is currently the golden standard imaging technique for functional cardiac imaging. MRI provides detailed information about important cardiac parameters of the human heart (such as left ventricular mass, end diastolic volume, end systolic volume and ejection fraction) once the images are segmented into relevant anatomical regions. This segmentation should preferably be performed automatically and hence be independent of the clinician, but still comparable in quality to manual segmentation.

In this thesis such an existing automatic segmentation method (a deformable model) for the hearts left ventricle is improved. This is achieved through optimization of parameters that govern the deformable model, and the implementation of a new algorithm incorporating anatomical /a priori /information about the heart.

Parameter optimization was performed on a large (/n /= 40) training data set of time resolved, three-dimensional MR imagestacks and then evaluated on three different test data set imagestacks, consisting of normals (/n /= 61), patients (/n /= 101) and athletes (/n /= 73). The parameter optimization and new algorithm reduced the mean error in the automatic segmentation compared to manual segmentation by a factor ranging from 2.5 up to 5.5 in the best case, with maintained or improved variability.

MRI can also acquire information about flow in the scanned region. Flow measurements are important since access to both left ventricle segmentation and flow in the connected vessels, enables quantification of valve function and shunt evaluation. Therefore, the optimization method developed was also used to improve vessel segmentation.

Distributed Mobile Computer Vision and Applications on the Android Platform (Distribuerat Mobilt Datorseende och Applikationer på Androidplattformen)
Student: Sebastian Olsson (D-04) och Philip Åkesson (F-05)
Advisor: Carl Olsson
In cooperation with: Epsilon IT
Date Finished: 2009-06-10
Abstract: This thesis describes the theory and implementation of both local and distributed systems for object recognition on the mobile Android platform. It further describes the possibilities and limitations of computer vision applications on modern mobile devices. Depending on the application, some or all of the computations may be outsourced to a server to improve performance.

The object recognition methods used are based on local features. These features are extracted and matched against a known set of features in the mobile device or on the server depending on the implementation. In the thesis we describe local features using the popular SIFT and SURF algorithms. The matching is done using both simple exhaustive search and more advanced algorithms such as kd-tree best-bin-first search. To improve the quality of the matches in regards to false positives we have used different RANSAC type iterative methods.

We describe two implementations of applications for single- and multi-object recognition, and a third, heavily optimized, SURF implementation to achieve near real-time tracking on the client.

The implementations are written in the Java language and special considerations have been taken to accommodate this. This choice of platform reflects the general direction of the mobile industry, where an increasing amount of application development is done in high-level languages such as Java.

Finally, we present some possible extensions of our implementations as future work. These extensions specifically take advantage of the hardware and abilities of modern mobile devices, including orientation sensors and cameras.

Trajectory Analysis in Surveillance (Trajektorieanalys inom övervakning)
Student: Erik Lundin (F-03)
Advisor: Karl Åström, Matematikcentrum
In cooperation with:Martin Smedberg, SAAB Microwave Systems
Date Finished: 2009-06-02
Abstract: Surveillance is a steadily increasing phenomenon and the used sensors -- cameras, radars etcetera -- produce more and more data that has to be examined. To sort and value the information, systems are needed that automatically finds interesting behaviors for the operators to have a closer look at. This project aim to examine methods to analyse the traces -- trajectories -- that are created when object move through a surveilled area. Trajectory clustering, anomaly detection and prediction of future behaviour are desirable features of a system to assist an operator. Several methods, based on vector quantisation, hidden Markov modeling and trajectory clustering are studied and two of them are implemented. These implementations aim at test the performance of some interesting methods, and to give hint of what software is capable to do in a surveillance context. The results show that many of the desired features are possible to implement, but more research in the area is needed to take these features closer to a large scale system.

A System for Real Time Gesture Recognition (Ett system för gestigenkänning i realtid)
Student: Daniel Persson, Pi04 och Björn Samvik, Pi04
Advisor: Fredrik Kahl
In cooperation with: TAT
Date Finished:2009-04-27
Abstract: One of the most intuitive and common communication forms for human beings are gestures of different kind. We use them all the time often without even noticing. In this thesis classification of gestures and recognition of hands using a camera are discussed. To find and track the hand the Viola-Jones detector is used. The time series of the trajectories are transformed to angular space which results in scale and translation invariance. These series are then used to classify the gesture to a set of templates using dynamic time warping.

A new method using Viola-Jones detector combined with RAMOSAC which is a feature tracker working with SIFT/SURF features is also tried and evaluated in an attempt to lower the detection error rate and to achieve more robustness to pose variations. The tests show that the system works well but is limited to the lighting and environment for which the algorithms are trained. The performance is real time on a normal PC and has the potential to be optimized to run on a mobile platform.

Automatic Detection and Segmentation in Lymphoma with PET/CT(Automatisk detektion och segmentering av lymfocyter i PET/CT)
Student: Karin Nyström, F03
Advisor: Fredrik Kahl
In cooperation with: Lars Edenbrant, Exini AB
Date Finished: 2009-03-31
Abstract: In this thesis a system for automatic detection and segmentation of malignant lymph nodes in PET/CT is developed and tested. PET/CT is an important tool when determining treatment and the response to treatment for lymphoma patients. The location of the malignant lymph nodes is important, as well as their size and shape. Manual segmentation is time consuming and an automatic method would increase productivity among clinicians.

The PET images are searched for possible malignant areas and these are transfered to the CT image. The areas shape and position in the PET image together with the fact that tumors are even in color in the CT image is the base for the segmentation CT image which is done with graph cuts. Due to computational intensity, the segmentation is done for one transaxial image at a time.

The biggest challenge is that the PET image is blurry and without detail. It is impossible to know, based only on the PET image, which voxels in the CT image are malignant. Only a physician with the knowledge of how a healthy patient looks in CT images can tell what is malignant and not. However, the PET image together with the structural details in the CT image can give a good estimate on how the structure is in detail.

The segmentations of tumors from the system are compared visually to a clinicians detailed statements on five patients. In 40% of the tumors the segmentation was correct, i.e. all malignant tissue is included and no healthy tissue is part of the segmented area.

3D Reconstruction for Traffic Surveillance(3D-rekonstruktion för trafikövervakning)
Student: Hanna Källén, F-04
Advisor: Olof Enqvist och Fredrik Kahl
Date Finished: 2009-03-30
Abstract: In order to get good knowledge of the safety of a traffic intersection, a video camera is placed at the intersection. Video from this camera can then be used to automatically track the vehicles in the intersection. The purpose of this master's thesis is to develop an algorithm for building 3D models of vehicles from such videos. The goal is to use the reconstructions to calculate the distance between road users in the intersection to find out how dangerous the intersection is.
To determine how good a certain reconstruction is, the largest reprojection error is considered, that means that the error is measured in the L1-norm. Between different frames in the video, the vehicles have moved and rotated. Given the rotation, finding the optimal reconstruction is a quasi-convex problem and can be solved efficiently. Since the rotation is not known, a branch and bound algorithm has been implemented. The branch and bound algorithm search through possible rotations and selects the one that gives the best result. To evaluate the algorithm, reconstructions of a bus and two different cars driving through the intersection have been made with good results.

Parameter Identification in a Model of Sedimentation with Compression (Parameteridentifiering i en modell av sedimentering med kompression)
Student: Sebastian Farås, W03
Advisor: Stefan Diehl
Date Finished: 2009-03-06
Abstract: Sedimentation is a commonly adopted method to separate solid particles from a liquid by gravity. Effective operation demands good knowledge of the process dynamics. Here, a published mathematical model describing hindered settling (gravitational settling) and compression is considered. By the conservation law and some constitutive assumptions, the process behaviour is captured by a nonlinear PDE. The main focus is on batch sedimentation, which is performed in a closed vessel and with homogeneous initial concentration. In particular, the inverse problem of identifying model parameters from synthetic and observed data is treated. The investigated method is used on measurements from Ryaverket Wastewater Treatment Plant.

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Monotone Functions and Their Importance for Risk Aversion and Calculation of Risk(Monotona funktioner och deras betydelse för risk aversion och beräkningav risk)
Student: Patrick Remes, D03
Advisor: Sergei Silvestrov
Date Finished: 2009-03-04
Abstract: In this thesis it is shown that monotone functions, as well as matrix monotone functions can be of great value when calculating risk in situations involving choice under uncertainty. Both risk and risk aversion is presented and well explained. Different types of risk aversion are presented.

Connection between risk, risk aversion and monotone functions is made. The Savage axioms, which are "rules" that tell us how we should act in situations under uncertainty, are presented.

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Worst Case Hedges for Derivative Contracts (Värsta falls skattningar för finansiella derivat)
Student: Tor Gillberg, Pi04
Advisor: Magnus Fontes
Date Finished: 2009-02-04
Abstract: There is a wide range of stocks, commodities and banks to invest in on the market. In addition there are numerous bets on future rates and prices. Those bets are called derivatives. A reasonable price of a derivative is the expected profit from the bet. Financial mathematics aim at specifying the prices of derivatives.

Almost all financial models assume the underlying (price or rate) to behave as a certain stochastic process. In this way all future values of the underlying is assigned a probability, and one may calculate the expected profit. This thesis investigates a model assuming nothing of that kind. No assumptions on the underlyings probability distribution are made. Instead one specifies what is not allowed to happen by having bounds on the evolution of the underlying. Such a model was introduced by Paul Wilmott and David Epstein in 1999.

We search for a value spread of a portfolio on one underlying. The portfolio may consist of any simple derivatives as well as the underlying. Assuming the underlying takes the 'worst' possible path allowed by the bounds, one gets a lower value of a contract. This value is called the worst value. Similarly the worst value for a negative share of a contract, a sold contract, is called the best value.

The model is derived in a very general setting so that the underlying may refer to either a stock, rate or commodity price process. The works from Epstein and Wilmott are solely dedicated to interest rate markets, why the result section focuses on a stock market.

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Expected Time to Fixation Due to Mutation, Drift and Selection(Förväntad tid till fixering under mutation, drift och selektion)
Student: Lars Larsson, Pi03
Advisor: Pelle Pettersson, Matematikcentrum
In cooperation with: Torbjörn Säll, Institutionen för cell- och organismbiologi
Date Finished: 2009-01-09
Abstract: The mathematical theory of population genetics was developed during the 1920^th and the 1930^th. Two of the men who invented this theory were Sewall Wright and Sir Ronald A. Fisher. The Wright-Fisher model describes how the frequencies of genes in one generation of a population are derived from the frequencies of genes in the previous generation. The reason for changes in the genome can be mutation, drift and selection. Eventually those changes may result in fixation of a gene meaning that only one type of the gene will remain in the population. The frequencies of the other types have been reduced to zero.

In this thesis we discuss two models based on the Wright-Fisher model. The first model is discrete in time and describes the evolution of genes in the population by Markov chains. The second model is a continuous approximation of the discrete model. The evolution of genes is then described by a diffusion equation.

We use the models to compute the expected time to achieve fixation of a gene. Several fixation times are computed for different values of parameters describing selection, dominance and mutation.

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9 January 2009