Abstracts of Ph.D. theses in mathematics

summary:Leischner, Pavel: Spatial imagination development of the secondary school pupils. Mašíček, Libor: Diagnostics and sensitivity of robust models. Duintjer Tebbens, Erik Jurjen: Modern methods for solving linear problems. Matonoha, Ctirad: Numerical realization of trust region methods. Duda, Jakub: Delta convexity, metric projection and negligible sets. Smrčka, Michael: Choquet's theory in function spaces. Hanika, Jiří: Search problems and bounded arithmetic. Pawlas, Zbyněk: Asymptotics in stochastic geometry. Bodlák, Karel: Methods of stereology and spatial statistics in applications. Čapek, Václav: M-smoothers Zvára, Petr: Prediction in non-linear autoregressive processes. Blanda, Jiří: Pricing of life insurance products Finfrle, Pavel: Model for calculation of liability value arising from life insurance. Finěk Václav: Orthonormal wavelets and their applications. Stanovský David : Left distributive left quasigroups. Koblížková, Michaela: Polyhedra and secondary school mathematics. Krýsl, Svatopluk: Invariant differential operators for projective contact geometries. Šmíd, Dalibor: Properties of invariant differential operators. Šmíd, Martin: Notes on approximation of stochastic programming problems. Komárková, Lenka: Change point problem for censored data. Kechlibar, Marian: Commutative algebra and cryptography

Stereology as the 3D tool to quantitate lung architecture

Stereology is the method of choice for the quantitative assessment of biological objects in microscopy. It takes into account the fact that, in traditional microscopy such as conventional light and transmission electron microscopy, although one has to rely on measurements on nearly two-dimensional sections from fixed and embedded tissue samples, the quantitative data obtained by these measurements should characterize the real three-dimensional properties of the biological objects and not just their "flatland" appearance on the sections. Thus, three-dimensionality is a built-in property of stereological sampling and measurement tools. Stereology is, therefore, perfectly suited to be combined with 3D imaging techniques which cover a wide range of complementary sample sizes and resolutions, e.g. micro-computed tomography, confocal microscopy and volume electron microscopy. Here, we review those stereological principles that are of particular relevance for 3D imaging and provide an overview of applications of 3D imaging-based stereology to the lung in health and disease. The symbiosis of stereology and 3D imaging thus provides the unique opportunity for unbiased and comprehensive quantitative characterization of the three-dimensional architecture of the lung from macro to nano scale

Abstracts of theses in mathematics

summary:Krejčíř, Pavel: The theory and applications of spatial statistics and stochastic geometry. Vítek, Tomáš: Detection of changes in econometric models. Hliněný, Petr: Contact representations of graphs. Kliková, Alice: Finite volume -- finite element solution of compressible flow. Hrach, Karel: Bayesian analysis of models with non-negative residuals. Svatoš, Jan: M-estimators in the linear model for nonregular densities. Ševčík, Petr: Extremal martingale measures in finance. Hlávka, Zdeněk: Robust sequential estimation. Holub, Štěpán: Equations in free monoids. Klaschka Jan: Mathematical methods of state change assessment in medical research. Unzeitigová, Vladimíra: Mathematical models of health insurance for commercial insurance companies -- embedded value of accident insurance. Friesl, Michal: Bayesian estimation in exponent competing risks and related models with applications to insurance. Fiala, Jiří: Locally injective homomorphisms. Kaplický, Petr: Qualitative properties of solutions of systems of mechanics. Ghoneim, Sobha: Selfdistributive rings and near-rings

Computation of Lacunarity from Covariance of Spatial Binary Maps

We consider a spatial binary coverage map (binary pixel image) which might represent the spatial pattern of the presence and absence of vegetation in a landscape. ‘Lacunarity’ is a generic term for the nature of gaps in the pattern: a popular choice of summary statistic is the ‘gliding-box lacunarity’ (GBL) curve. GBL is potentially useful for quantifying changes in vegetation patterns, but its application is hampered by a lack of interpretability and practical difficulties with missing data. In this paper we find a mathematical relationship between GBL and spatial covariance. This leads to new estimators of GBL that tolerate irregular spatial domains and missing data, thus overcoming major weaknesses of the traditional estimator. The relationship gives an explicit formula for GBL of models with known spatial covariance and enables us to predict the effect of changes in the pattern on GBL. Using variance reduction methods for spatial data, we obtain statistically efficient estimators of GBL. The techniques are demonstrated on simulated binary coverage maps and remotely sensed maps of local-scale disturbance and meso-scale fragmentation in Australian forests. Results show in some cases a fourfold reduction in mean integrated squared error and a twentyfold reduction in sensitivity to missing data. Supplementary materials accompanying the paper appear online and include a software implementation in the R language

On random tomography with unobservable projection angles

We formulate and investigate a statistical inverse problem of a random tomographic nature, where a probability density function on $\mathbb{R}^3$ is to be recovered from observation of finitely many of its two-dimensional projections in random and unobservable directions. Such a problem is distinct from the classic problem of tomography where both the projections and the unit vectors normal to the projection plane are observable. The problem arises in single particle electron microscopy, a powerful method that biophysicists employ to learn the structure of biological macromolecules. Strictly speaking, the problem is unidentifiable and an appropriate reformulation is suggested hinging on ideas from Kendall's theory of shape. Within this setup, we demonstrate that a consistent solution to the problem may be derived, without attempting to estimate the unknown angles, if the density is assumed to admit a mixture representation.Comment: Published in at http://dx.doi.org/10.1214/08-AOS673 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Flaw reconstruction in NDE using a limited number of x-ray radiographic projections

One of the major problems in nondestructive evaluation (NDE) is the evaluation of flaw sizes and locations in a limited inspectability environment. In NDE x-ray radiography, this frequently occurs when the geometry of the part under test does not allow x-ray penetration in certain directions. Other times, the inspection setup in the field does not allow for inspection at all angles around the object. This dissertation presents a model based reconstruction technique which requires a small number of x-ray projections from one side of the object under test. The estimation and reconstruction of model parameters rather than the flaw distribution itself requires much less information, thereby reducing the number of required projections. Crack-like flaws are modeled as piecewise linear curves (connected points) and are reconstructed stereographically from at least two projections by matching corresponding endpoints of the linear segments. Volumetric flaws are modeled as ellipsoids and elliptical slices through ellipsoids. The elliptical principal axes lengths, orientation angles and locations are estimated by fitting a forward model to the projection data. The fitting procedure is highly nonlinear and requires stereographic projections to obtain initial estimates of the model parameters. The methods are tested both on simulated and experimental data. Comparisons are made with models from the field of stereology. Finally, analysis of reconstruction errors is presented for both models