Roberto Grosso

The multilevel finite element method for adaptive mesh optimization and visualization of volume data

Multilevel representations and mesh reduction techniques have been used for accelerating the processing and the rendering of large datasets representing scalar- or vector-valued functions defined on complex 2D or 3D meshes. We present a method based on finite element approximations which combines these two approaches in a new and unique way that is conceptually simple and theoretically sound. The main idea is to consider mesh reduction as an approximation problem in appropriate finite element spaces. Starting with a very coarse triangulation of the functional domain, a hierarchy of highly non-uniform tetrahedral (or triangular in 2D) meshes is generated adaptively by local refinement. This process is driven by controlling the local error of the piecewise linear finite element approximation of the function on each mesh element. A reliable and efficient computation of the global approximation error and a multilevel preconditioned conjugate gradient solver are the key components of the implementation. In order to analyze the properties and advantages of the adaptively generated tetrahedral meshes, we implemented two volume visualization algorithms: an iso-surface extractor and a ray-caster. Both algorithms, while conceptually simple, show significant speedups over conventional methods delivering comparable rendering quality from adaptively compressed datasets.

Hierarchical tetrahedral-octahedral subdivision for volume visualization

We present a method for discretizing 3D space in order to make it accessible for handling numerical problems, such as simulation or visualization. Our algorithm generates a hierarchy of 3D meshes. It offers adaptive subdivision, driven by a user-specified local error control. Each 3D mesh consists of tetrahedra and octahedra having minimal numbers of congruence classes. Subdivision is based on a minimal set of rules for regular and irregular refinement. Irregular elements are stored as virtual elements and will be generated only on demand. The hierarchy has a very compact representation. The algorithm generates mesh hierarchies used for efficient, interactive volume visualization algorithms, e.g., isosurface extraction and direct volume rendering. The meshes generated are suited to multilevel, finite element computations as well.

Realtime Isosurface Extraction with Graphics Hardware

In this paper we introduce a method for the display of isosurfaces extracted from unstructured tetrahedral grids. Our algorithm completely runs on the graphics hardware. The tetrahedra are streamed into a vertex program, which extracts the surface for the given isovalue and immediately renders it. The triangles are not stored explicitly but are computed during rendering time, so the user can modify the isovalue with immediate feedback. If the tetrahedra entirely t into video memory, we achieve a throughput of more than nine million tetrahedra per second. Our performance can be further improved by using a hybrid method which pre-selects tetrahedra containing the isovalue. We compare our approach with a pure CPU based implementation which achieves about half the performance of our hardware accelerated method.

Efficient and Reliable Integration Methods for Particle Tracing in Unsteady Flows on Discrete Meshes

In real applications the velocity field of a flow is not available in analytical but in discrete form. One goal of this paper is to analyze particle integration methods for discretized data defined on meshes with regard to numerical efficiency and accuracy. Careful error analysis of the particle tracing process relates the error of velocity interpolation in space and time to the error of the numerical integration. Hence, a fast integration routine which provides accuracy similar to that of interpolation is necessary. This leads to a robust integration routine with adaptive step size control and error monitoring. A second aspect of this work is the treatment of stiff problems. Stiffness occurs m flows with strong shear deformations or vorticity. To detect stiffness in a given flow field, the Jacobian of the velocity field is analyzed. Implicit integration methods are used to handle stiff systems of ordinary differential equations.

Hierarchical meshes for volume data

An algorithm for adaptive refinement of 3D meshes is presented. This algorithm can be applied for the generation of mesh hierarchies used for efficient volume visualization algorithms, e.g. iso-surface extraction or direct volume rendering, as well as for multilevel finite element computations. The aim was to construct an algorithm which generates as few congruence classes as possible. The main idea is to work with consistent partitions of the domain into tetrahedra and octahedra. The refinement consists of regular refinement rules which produce per element type one congruence class. In the case of local mesh refinement, for generating consistent subdivisions a temporary (virtual) closure is done based on suitable irregular refinement rules.

Fast and Adaptive Finite Element Approach for Modeling Brain Shift

Objective: In this paper we introduce a finite element-based strategy for simulation of brain deformation occurring during neurosurgery. The phenomenon, known as brain shift, causes a decrease in the accuracy of neuronavigation systems that rely on preoperatively acquired data. This can be compensated for with a computational model of the brain deformation process. By applying model calculations to preoperative images, an update within the operating room can be performed. Methods: One of the crucial concerns in the context of developing a physical-based model is the choice of governing equations describing the physics of the phenomenon. In this work, deformation of brain tissue is expressed in terms of a 3D consolidation model for a linearly elastic and porous fluid. The next crucial issue is ensuring stable calculations within the chosen model. For this purpose, we developed a special technique for generating the underlying geometry for the simulation. With this technique an unstructured grid consisting of regular tetrahedra is created, whereupon time-dependent finite element simulation is performed in an adaptive manner. Results: We applied our algorithm to preoperative MR scans and investigated the value of the method. Due to the adaptivity of the method, only 5-10% of the computing time was needed as compared to traditional finite element approaches based on a uniformly subdivided grid. The results of the experiments were compared to the corresponding intraoperative MR scans. A close match between the computed deformation of the brain and the displacement resulting from the intraoperative data was observed. Conclusion: A model-based approach for the simulation of brain shift is presented. In this computational model the brain tissue is described as an elastic and porous material using Biot consolidation theory. Validating experiments conducted with MR data provided promising results.

Progressive Iso-Surface Extraction from Hierarchical 3D Meshes

A multiresolution data decomposition offers a fundamental framework supporting compression, progressive transmission, and level‐of‐detail (LOD) control for large two or three dimensional data sets discretized on complex meshes. In this paper we extend a previously presented algorithm for 3D mesh reduction for volume data based on multilevel finite element approximations in two ways. First, we present efficient data structures which allow to incrementally construct approximations of the volume data at lower or higher resolutions at interactive rates. An abstract description of the mesh hierarchy in terms of a coarse base mesh and a set of integer records offers a high compression potential which is essential for an efficient storage and a progressive network transmission. Based on this mesh hierarchy we then develop a new progressive iso‐surface extraction algorithm. For a given iso‐value, the corresponding iso‐surface can be computed at different levels of resolution. Changing to a higher or coarser resolution will update the surface only in those regions where the volume data is being refined or coarsened. Our approach allows to interactively visualize very large scalar fields like medical data sets, whereas the conventional algorithms would have required at least an order of magnitude more resources.

Estimating Mechanical Brain Tissue Properties with Simulation and Registration

In this work a new method for the determination of the mechanical properties of brain tissue is introduced. Young’s modulus E and Poisson’s ratio ν are iteratively estimated based on a finite element model for brain shift and on the information contained in pre- and intraoperative MR data after registration. In each iteration, a 3D dataset is generated according to the displacement vector field resulting from a numerical simulation of the intraoperative brain deformation. This reconstruction is parametrized by elastic moduli of tissue. They are automatically varied in order to achieve the best correspondence between the grey value distribution in the reconstructed image and the intensity entropy in the MR image of the brain undergoing deformation. This work contributes to the difficult problem of defining correct mechanical parameters to perform reliable model calculations of brain deformation. Proper boundary conditions that are crucial in this context are also addressed.

Multiresolution and hierarchical methods for the visualization of volume data

Abstract As three-dimensional data sets resulting from simulations or measurements become available at ever growing sizes the need for visualization tools which allow the inspection and the analysis of these data sets at interactive rates is increasing. One way to deal with the complexity is the compression of the data in such a way that the number of cells which have to be processed by the visualization mapping is reduced. Since this compression will be lossy, it is up to the user to choose between quality or speed. The decision will usually be made interactively requiring fast access to a complete hierarchy of representations of the data set at various levels of resolution. Two different approaches and visualization algorithms based upon them are presented in this paper: wavelet analysis deriving a hierarchy of coarser representations from the original data set and multilevel finite elements generating successively refined tetrahedral grids from an initially coarse triangulation.

Chaos and Graphics: Optimal rotation alignment of 3D objects using a GPU-based similarity function

In this paper, we address the challenging task of finding the best alignment between two 3D objects by solving a global optimization problem in the space of rotations SO(3). The objective function to be optimized is a newly developed rotation-variant similarity measure, which is obtained directly from the objects geometry and is entirely implemented on the GPU. By exploiting the modern GPUs parallel architecture, we can process considerably greater amounts of data than a CPU implementation can do in the same amount of time. This allows us to create a similarity measure which combines speed and accuracy. The actual problem of rotation alignment is then solved by finding the global maximum of this similarity function in the space of rotations. A special rotation representation allows for an efficient local optimization on the manifold SO(3). Furthermore, unwanted local maxima can be avoided by a heuristic global optimization procedure which exploits rotational symmetry. Due to this common sense heuristics, the global search can be gradually reduced to a lower-dimensional problem up to a 1D line search to handle objects with high rotational symmetry. We show that our method is superior to existing normalization techniques such as PCA and provides a high degree of precision despite remarkably short runtimes.