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Zonotopes With Large 2D Cuts
(2009)
We study discrete curvatures computed from nets of curvature lines on a given smooth surface and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.
Functional Magnetic Resonance Imaging inherently involves noisy measurements and a severe multiple test
problem. Smoothing is usually used to reduce the effective number of multiple
comparisons and to locally integrate the signal and hence increase the
signal-to-noise ratio. Here, we provide a new structural adaptive segmentation
algorithm (AS)
that naturally combines the signal detection with noise reduction in one procedure.
Moreover, the new method
is closely related to a recently proposed structural adaptive smoothing
algorithm and preserves shape and spatial extent of activation areas without
blurring the borders.
We present a novel algorithm for automatic parameterization of tube-like surfaces of arbitrary genus such as the surfaces of knots, trees, blood vessels, neurons, or any tubular graph with a globally consistent stripe texture. We use the principal curvature frame field of the underlying tube-like surface to guide the creation of a global, topologically consistent stripe parameterization of the surface. Our algorithm extends the QuadCover algorithm and is based, first, on the use of so-called projective vector fields instead of frame fields, and second, on different types of branch points. That does not only simplify the mathematical theory, but also reduces computation time by the decomposition of the underlying stiffness matrices.
The package fmri is provided for analysis of single run functional
Magnetic Resonance Imaging data. It implements structural adaptive smoothing
methods with signal detection for adaptive noise reduction which avoids blurring
of edges of activation areas. fmri provides fmri analysis from time series
modeling to signal detection and publication-ready images.
In optimal control problems with nonlinear time-dependent 3D PDEs, full 4D discretizations are usually prohibitive due to the storage requirement. For this reason gradient and Newton type methods working on the reduced functional are often employed. The computation of the reduced gradient requires one solve of the state equation forward in time, and one backward solve of the adjoint equation. The state enters into the adjoint equation, again requiring the storage of a full 4D data set. We propose a lossy compression algorithm using an inexact but cheap predictor for the state data, with additional entropy coding of prediction errors. As the data is used inside a discretized, iterative algorithm, lossy compression
maintaining a certain error bound turns out to be sufficient.
This paper presents concepts and implementation of the finite element toolbox Kaskade 7, a flexible C++ code for solving elliptic and parabolic PDE systems. Issues such as problem formulation, assembly and adaptivity are discussed at the example of optimal control problems. Trajectory compression for parabolic optimization problems is considered as a case study.
In high accuracy numerical simulations and optimal control of time-dependent processes, often both many time steps and fine spatial discretizations are needed. Adjoint gradient computation, or post-processing of simulation results, requires the storage of the solution trajectories over the whole time, if necessary together with the adaptively refined spatial grids. In this paper we discuss various techniques to reduce the memory requirements, focusing first on the storage of the solution data, which typically are double precision floating point values. We highlight advantages and disadvantages of the different approaches. Moreover, we present an algorithm for the efficient storage of adaptively refined, hierarchic grids, and the integration with the compressed storage of solution data.
We introduce an algorithm for
diffusion weighted magnetic resonance imaging data enhancement based on structural adaptive smoothing in both space and diffusion direction.
The method, called POAS, does not refer to a specific model for the data, like the diffusion tensor or higher order models.
It works by embedding the measurement space into a space with defined metric and group operations, in this case the Lie group of three-dimensional Euclidean motion SE(3).
Subsequently, pairwise comparisons of the values of the diffusion
weighted signal are used for adaptation.
The position-orientation adaptive smoothing preserves the edges of the observed fine and anisotropic structures.
The POAS-algorithm is designed to reduce noise directly in the diffusion weighted images and consequently also to reduce bias and
variability of quantities derived from the data for specific models.
We evaluate the algorithm on simulated and experimental data and demonstrate that it can be used to reduce the number of applied diffusion gradients and
hence acquisition time while achieving similar quality of data, or to improve the quality of data acquired in a clinically feasible scan time setting.
Structuring of surface meshes is a labor intensive task in reverse engineering. For example in CAD, scanned triangle meshes must be divided into characteristic/uniform patches to enable conversion into high-level spline surfaces. Typical industrial techniques, like rolling ball blends, are very labor intensive.
We provide a novel, robust and quick algorithm for the automatic generation of a patch layout based on a topology consistent feature graph. The graph separates the surface along feature lines into functional and geometric building blocks. Our algorithm then thickens thickens the edges of the feature graph and forms new regions with low varying curvature. Further these new regions - so called fillets and node patches - will have highly smooth boundary curves making it an ideal preprocessor for a subsequent spline fitting algorithm.