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In this expository article, we give an introduction into the basics of bootstrap tests in general. We discuss the residual-based and the wild bootstrap for regression models suitable for applications in signal and image analysis. As an illustration of the general idea, we consider a particular test for detecting differences between two noisy signals or images which also works for noise with variable variance. The test statistic is essentially the integrated squared difference between the signals after denoising them by local smoothing. Determining its quantile, which marks the boundary between accepting and rejecting the hypothesis of equal signals, is hardly possible by standard asymptotic methods whereas the bootstrap works well. Applied to the rows and columns of images, the resulting algorithm not only allows for the detection of defects but also for the characterization of their location and shape in surface inspection problems.

This report reviews selected image binarization and segmentation methods that have been proposed and which are suitable for the processing of volume images. The focus is on thresholding, region growing, and shape–based methods. Rather than trying to give a complete overview of the field, we review the original ideas and concepts of selected methods, because we believe this information to be important for judging when and under what circumstances a segmentation algorithm can be expected to work properly.

In this paper we propose a general approach solution method for the single facility ordered median problem in the plane. All types of weights (non-negative, non-positive, and mixed) are considered. The big triangle small triangle approach is used for the solution. Rigorous and heuristic algorithms are proposed and extensively tested on eight different problems with excellent results.

It has been empirically verified that smoother intensity maps can be expected to produce shorter sequences when step-and-shoot collimation is the method of choice. This work studies the length of sequences obtained by the sequencing algorithm by Bortfeld and Boyer using a probabilistic approach. The results of this work build a theoretical foundation for the up to now only empirically validated fact that if smoothness of intensity maps is considered during their calculation, the solutions can be expected to be more easily applied.

This work presents a new framework for Gröbner basis computations with Boolean polynomials. Boolean polynomials can be modeled in a rather simple way, with both coefficients and degree per variable lying in {0, 1}. The ring of Boolean polynomials is, however, not a polynomial ring, but rather the quotient ring of the polynomial ring over the field with two elements modulo the field equations x2 = x for each variable x. Therefore, the usual polynomial data structures seem not to be appropriate for fast Gröbner basis computations. We introduce a specialized data structure for Boolean polynomials based on zero-suppressed binary decision diagrams (ZDDs), which is capable of handling these polynomials more efficiently with respect to memory consumption and also computational speed. Furthermore, we concentrate on high-level algorithmic aspects, taking into account the new data structures as well as structural properties of Boolean polynomials. For example, a new useless-pair criterion for Gröbner basis computations in Boolean rings is introduced. One of the motivations for our work is the growing importance of formal hardware and software verification based on Boolean expressions, which suffer – besides from the complexity of the problems – from the lack of an adequate treatment of arithmetic components. We are convinced that algebraic methods are more suited and we believe that our preliminary implementation shows that Gröbner bases on specific data structures can be capable to handle problems of industrial size.

Background and purpose Inherently, IMRT treatment planning involves compromising between different planning goals. Multi-criteria IMRT planning directly addresses this compromising and thus makes it more systematic. Usually, several plans are computed from which the planner selects the most promising following a certain procedure. Applying Pareto navigation for this selection step simultaneously increases the variety of planning options and eases the identification of the most promising plan. Material and methods Pareto navigation is an interactive multi-criteria optimization method that consists of the two navigation mechanisms “selection” and “restriction”. The former allows the formulation of wishes whereas the latter allows the exclusion of unwanted plans. They are realized as optimization problems on the so-called plan bundle – a set constructed from precomputed plans. They can be approximately reformulated so that their solution time is a small fraction of a second. Thus, the user can be provided with immediate feedback regarding his or her decisions.

An algorithm for automatic parallel generation of three-dimensional unstructured computational meshes based on geometrical domain decomposition is proposed in this paper. Software package build upon proposed algorithm is described. Several practical examples of mesh generation on multiprocessor computational systems are given. It is shown that developed parallel algorithm enables us to reduce mesh generation time significantly (dozens of times). Moreover, it easily produces meshes with number of elements of order 5 · 107, construction of those on a single CPU is problematic. Questions of time consumption, efficiency of computations and quality of generated meshes are also considered.

Calculating effective heat conductivity for a class of industrial problems is discussed. The considered composite materials are glass and metal foams, fibrous materials, and the like, used in isolation or in advanced heat exchangers. These materials are characterized by a very complex internal structure, by low volume fraction of the higher conductive material (glass or metal), and by a large volume fraction of the air. The homogenization theory (when applicable), allows to calculate the effective heat conductivity of composite media by postprocessing the solution of special cell problems for representative elementary volumes (REV). Different formulations of such cell problems are considered and compared here. Furthermore, the size of the REV is studied numerically for some typical materials. Fast algorithms for solving the cell problems for this class of problems, are presented and discussed.

Two-level domain decomposition preconditioner for 3D flows in anisotropic highly heterogeneous porous media is presented. Accurate finite volume discretization based on multipoint flux approximation (MPFA) for 3D pressure equation is employed to account for the jump discontinuities of full permeability tensors. DD/MG type preconditioner for above mentioned problem is developed. Coarse scale operator is obtained from a homogenization type procedure. The influence of the overlapping as well as the influence of the smoother and cell problem formulation is studied. Results from numerical experiments are presented and discussed.

Modeling and formulation of optimization problems in IMRT planning comprises the choice of various values such as function-specific parameters or constraint bounds. These values also affect the characteristics of the optimization problem and thus the form of the resulting optimal plans. This publication utilizes concepts of sensitivity analysis and elasticity in convex optimization to analyze the dependence of optimal plans on the modeling parameters. It also derives general rules of thumb how to choose and modify the parameters in order to obtain the desired IMRT plan. These rules are numerically validated for an exemplary IMRT planning problems.