Segmentation of Three-dimensional Images with Parametric Active Surfaces and Topology Changes

In this paper, we introduce a novel parametric method for segmentation of three-dimensional images. We consider a piecewise constant version of the Mumford-Shah and the Chan-Vese functionals and perform a region-based segmentation of 3D image data. An evolution law is derived from energy minimization problems which push the surfaces to the boundaries of 3D objects in the image. We propose a parametric scheme which describes the evolution of parametric surfaces. An efficient finite element scheme is proposed for a numerical approximation of the evolution equations. Since standard parametric methods cannot handle topology changes automatically, an efficient method is presented to detect, identify and perform changes in the topology of the surfaces. One main focus of this paper are the algorithmic details to handle topology changes like splitting and merging of surfaces and change of the genus of a surface. Different artificial images are studied to demonstrate the ability to detect the different types of topology changes. Finally, the parametric method is applied to segmentation of medical 3D images

Variational methods for solving nonlinear boundary problems of statics of hyper-elastic membranes

A number of important results of studying large deformations of hyper-elastic shells are obtained using discrete methods of mathematical physics. In the present paper, using the variational method for solving nonlinear boundary problems of statics of hyper-elastic membranes under the regular hydrostatic load, we investigate peculiarities of deformation of a circular membrane whose mechanical characteristics are described by the Bidermann-type elastic potential. We develop an algorithm for solving a singular perturbation of nonlinear problem for the case of membrane loaded by heavy liquid. This algorithm enables us to obtain approximate solutions both in the presence of boundary layer and without it. The class of admissible functions, on which the variational method is realized, is chosen with account of the structure of formal asymptotic expansion of solutions of the corresponding linearized equations that have singularities in a small parameter at higher derivatives and in the independent variable. We give examples of calculations that illustrate possibilities of the method suggested for solving the problem under consideration

Procedural function-based modelling of volumetric microstructures

We propose a new approach to modelling heterogeneous objects containing internal volumetric structures with size of details orders of magnitude smaller than the overall size of the object. The proposed function-based procedural representation provides compact, precise, and arbitrarily parameterised models of coherent microstructures, which can undergo blending, deformations, and other geometric operations, and can be directly rendered and fabricated without generating any auxiliary representations (such as polygonal meshes and voxel arrays). In particular, modelling of regular lattices and cellular microstructures as well as irregular porous media is discussed and illustrated. We also present a method to estimate parameters of the given model by fitting it to microstructure data obtained with magnetic resonance imaging and other measurements of natural and artificial objects. Examples of rendering and digital fabrication of microstructure models are presented

Modified Linear Projection for Large Spatial Data Sets

Recent developments in engineering techniques for spatial data collection such as geographic information systems have resulted in an increasing need for methods to analyze large spatial data sets. These sorts of data sets can be found in various fields of the natural and social sciences. However, model fitting and spatial prediction using these large spatial data sets are impractically time-consuming, because of the necessary matrix inversions. Various methods have been developed to deal with this problem, including a reduced rank approach and a sparse matrix approximation. In this paper, we propose a modification to an existing reduced rank approach to capture both the large- and small-scale spatial variations effectively. We have used simulated examples and an empirical data analysis to demonstrate that our proposed approach consistently performs well when compared with other methods. In particular, the performance of our new method does not depend on the dependence properties of the spatial covariance functions.Comment: 29 pages, 5 figures, 4 table

Geometric Properties of Isostables and Basins of Attraction of Monotone Systems

In this paper, we study geometric properties of basins of attraction of monotone systems. Our results are based on a combination of monotone systems theory and spectral operator theory. We exploit the framework of the Koopman operator, which provides a linear infinite-dimensional description of nonlinear dynamical systems and spectral operator-theoretic notions such as eigenvalues and eigenfunctions. The sublevel sets of the dominant eigenfunction form a family of nested forward-invariant sets and the basin of attraction is the largest of these sets. The boundaries of these sets, called isostables, allow studying temporal properties of the system. Our first observation is that the dominant eigenfunction is increasing in every variable in the case of monotone systems. This is a strong geometric property which simplifies the computation of isostables. We also show how variations in basins of attraction can be bounded under parametric uncertainty in the vector field of monotone systems. Finally, we study the properties of the parameter set for which a monotone system is multistable. Our results are illustrated on several systems of two to four dimensions.Comment: 12 pages, to appear in IEEE Transaction on Automatic Contro

Structures of the magnetoionic media around the FR I radio galaxies 3C 31 and Hydra A

We use high-quality VLA images of the Fanaroff & Riley Class I radio galaxy 3C 31 at six frequencies in the range 1365 to 8440MHz to explore the spatial scale and origin of the rotation measure (RM) fluctuations on the line of sight to the radio source. We analyse the distribution of the degree of polarization to show that the large depolarization asymmetry between the North and South sides of the source seen in earlier work largely disappears as the resolution is increased. We show that the depolarization seen at low resolution results primarily from unresolved gradients in a Faraday screen in front of the synchrotron-emitting plasma. We establish that the residual degree of polarization in the short-wavelength limit should follow a Burn law and we fit such a law to our data to estimate the residual depolarization at high resolution. We show that the observed RM variations over selected areas of 3C 31 are consistent with a power spectrum of magnetic fluctuations in front of 3C 31 whose power-law slope changes significantly on the scales sampled by our data. The power spectrum can only have the form expected for Kolmogorov turbulence on scales <5 kpc. On larger scales we find a flatter slope. We also compare the global variations of RM across 3C 31 with the results of three-dimensional simulations of the magnetic-field fluctuations in the surrounding magnetoionic medium. We show that our data are consistent with a field distribution that favours the plane perpendicular to the jet axis - probably because the radio source has evacuated a large cavity in the surrounding medium. We also apply our analysis techniques to the case of Hydra A, where the shape and the size of the cavities produced by the source in the surrounding medium are known from X-ray data. (Abridged)Comment: 33 pages, 25 figures, accepted for publication in MNRA

Viscous regularization and r-adaptive remeshing for finite element analysis of lipid membrane mechanics

As two-dimensional fluid shells, lipid bilayer membranes resist bending and stretching but are unable to sustain shear stresses. This property gives membranes the ability to adopt dramatic shape changes. In this paper, a finite element model is developed to study static equilibrium mechanics of membranes. In particular, a viscous regularization method is proposed to stabilize tangential mesh deformations and improve the convergence rate of nonlinear solvers. The Augmented Lagrangian method is used to enforce global constraints on area and volume during membrane deformations. As a validation of the method, equilibrium shapes for a shape-phase diagram of lipid bilayer vesicle are calculated. These numerical techniques are also shown to be useful for simulations of three-dimensional large-deformation problems: the formation of tethers (long tube-like exetensions); and Ginzburg-Landau phase separation of a two-lipid-component vesicle. To deal with the large mesh distortions of the two-phase model, modification of vicous regularization is explored to achieve r-adaptive mesh optimization