Interactive High-Resolution Boundary Surfaces for Deformable Bodies with Changing Topology

An explicit spline representation of smooth free-form surfaces is combined with a hierarchy of meshes to form the basis of an interactive sculpting environment. The environment offers localized hierarchical modeling at different levels of detail, direct surface manipulation, change of connectivity for extrusion and to form holes and bridges, and built-in tangent continuity across the surface where wanted. The free-form surface is represented and can be exported either in NURBS form or as cubic triangular Bezier patches. Key characteristics of the approach are: (1) mesh pieces and surface pieces are related by strictly local averaging rules; (2) refinement rules depend only on direct, coarser-level ancestors and not on adjacent submeshes or patches; (3) submeshes at different levels look alike. The underlying data structure is a single winged-edge structure with additional pointers to support the hierarchy. Multiply refined regions may be directly adjacent to unrefined regions, and mesh fragments at different levels of refinement can be connected. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Curve, surface, solid, and object representations,Splines; I.3.6 [Computer Graphics]: Methodology and Techniques—Graphics data structures and data types 1 Motivation, results and prior work Conventional modeling systems require precise specification of final features, e.g., distance in datum planes, positioning of control vertices with respect to one another, etc. Besides requiring a certain mindset not always compatible with creative, exploratory design, this is modeling at the highest level of refinement: a change in the overall shape of the model entails modifying all detail, however small, associated with the area of change. Not only is this time-consuming, but repeated detail work puts an undue strain on creative, exploratory design. Parameterized modeling, which allows initially precisely instantiated specifications to be varied, has been used in commercial solid modeling systems. A complementary paradigm that supports changes at different levels in the design gonzalez@acm.org yjorg@cise.ufl.edu zSupported by NSF National Young Investigator award 9457806-CCR history is hierarchical refinement. Levels express the designer’s intent by encoding the structure and dependence of geometric features. Several recent modeling environments are based on hierarchical modeling [10, 7, 21, 13]. Localized-hierarchy Surface Splines (LeSS) extend this modeling paradigm by combining localized mesh hierarchies with the ability to change topology by adding a handle or punching a hole. LeSS automatically maintains tangent continuity across the surface where wanted and defines the surface in standard form either as bicubic NURBS or cubic triangular Bezier patches. Non-standard properties of LeSS are: 1. The surface is represented explicitly in piecewise polynomial or spline form, and not as surface patches offset from ancestor surfaces. Finer-level patches replace coarse-level patches. Changes are propagated from coarse to fine inside an underlying mesh hierarchy: fragments (pieces with boundary) of finer meshes are represented as offsets in local coordinate frames of coarser meshes. This has several implications: The evaluation of the spline patches does not require traversal of the hierarchy and is numerically stable. Explicit bounding boxes and convex hulls for the surface are directly available and the moments of the enclosed object can be efficiently computed. The influence of changing coarser levels is transmitted by local reevaluation of the mesh hierarchy. Local topology changes are possible because there is no direct 1-1 correspondence of the coarse to the finer-level surface. 2. LeSS is based on a strictly localized hierarchy: the default position of a finer level mesh fragment before local editing is determined by local averaging rules that depend only on those coarser-level nodes from which the mesh nodes of the fragment are offset and not on adjacent mesh or patch information. This simplifies the data structures since no searches for neighbors at different levels of refinement are necessary. Repeatedly-refined patches can be placed directly next to unrefined patches. (Since the surface representation is explicit, the resolution of ‘crack problems’ where refined and unrefined patches meet can be delegated to the spline display routine.) 3. The representation is homogeneous: control structures everywhere, even across levels, look the same. Together with locality this homogeneity allows connecting arbitrary submeshes. This is useful since not all parts of a complex object must enter a strict dependence relation with all other parts – the designer’s choice of dependence can be only local and later decisions, say to join parts or create a hole (Figure 9), are not prevented by a rigid, global hierarchy.

The problem of calculating the stability of steady state solutions of differential equations is treated. Leading eigenvalues (i.e., having maximal real part) of large matrices that arise from discretization are to be calculated. An efficient multigrid method for solving these problems is presented. The method begins by obtaining an initial approximation for the dominant subspace on a coarse level using a damped Jacobi relaxation. This proceeds until enough accuracy for the dominant subspace has been obtained. The resulting grid functions are then used as an initial approximation for appropriate eigenvalue problems. These problems are solved first on coarse levels, followed by refinement until a desired accuracy for the eigenvalues has been achieved. The method employs local relaxation on all levels together with a global change on the coarsest level only, which is designed to separate the different eigenfunctions as well as to update their corresponding eigenvalues. Coarsening is done using the FAS formulation in a nonstandard way in which the right-hand side of the coarse grid equations involves unknown parameters to be solved for on the coarse grid. This in particular leads to a new multigrid method for calculating the eigenvalues of symmetric problems. Numerical experiments with a model problem that are presented demonstrate the effectiveness of the method proposed. Using an FMG algorithm a solution to the level of discretization errors is obtained in just a few work units (less than 10), where a work unit is the work involved in one Jacobi relaxation on the finest level.

The algebraic multigrid (AMG) method is an efficient solver for linear systems arising in finite element analyses. The AMG method is applicable at a matrix level, different from the geometric multigrid solvers. This paper proposes a combination of the parallel processing technique and the AMG method as a fast solver for electromagnetic field analyses. While the AMG method consists of a setup phase and a solution phase, parallel processing of the former phase is difficult. We present the use of long-range interpolation instead of the conventional direct interpolation for improvement of the parallel efficiency of the AMG setup phase. A magnetostatic analysis and an eddy-current analysis show the solver performance. The numerical results show that parallelized AMG is a fast solver and has sufficient scalability, as compared with the conventional solver.

A method is presented to include irregular domain boundaries in a geometric multigrid solver. Dirichlet boundary conditions can be imposed on an irregular boundary defined by a level set function. Our implementation employs quadtree/octree grids with adaptive refinement, a cell-centered discretization and pointwise smoothing. Boundary locations are determined at a subgrid resolution by performing line searches. For grid blocks near the interface, custom operator stencils are stored that take the interface into account. For grid block away from boundaries, a standard second-order accurate discretization is used. The convergence properties, robustness and computational cost of the method are illustrated with several test cases. New version program summaryProgram Title: AfivoCPC Library link to program files:https://doi.org/10.17632/5y43rjdmxd.2Developer's repository link:https://github.com/MD-CWI/afivoLicensing provisions: GPLv3Programming language: FortranJournal reference of previous version: Comput. Phys. Commun. 233 (2018) 156–166. https://doi.org/10.1016/j.cpc.2018.06.018Does the new version supersede the previous version?: Yes.Reasons for the new version: Add support for internal boundaries in the geometric multigrid solver.Summary of revisions: The geometric multigrid solver was generalized in several ways: a coarse grid solver from the Hypre library is used, operator stencils are now stored per grid block, and methods for including boundaries via a level set function were added.Nature of problem: The goal is to solve Poisson's equation in the presence of irregular boundaries that are not aligned with the computational grid. It is assumed these irregular boundaries are defined by a level set function, and that a Dirichlet type boundary condition is applied. The main applications are 2D and 3D simulations with octree-based adaptive mesh refinement, in which the mesh frequently changes but the irregular boundaries do not.Solution method: A geometric multigrid method compatible with octree grids is developed, using a cell-centered discretization and point-wise smoothing. Near irregular boundaries, custom operator stencils are stored. Line searches are performed to locate interfaces with sub-grid resolution. To increase the methods robustness, this line search is modified on coarse grids if boundaries are otherwise not resolved. The multigrid solver uses OpenMP parallelization.

An aggregation/disaggregation iterative algorithm for computing stationary probability vectors of stochastic matrices is analysed. Two convergence results are presented. First, it is shown that fast, global convergence can be achieved provided that a sufficiently high number of relaxations is performed on the fine level. Second, local convergence is shown to take place with just one relaxation performed on the fine level. The convergence proofs are general and require no assumptions on the magnitude of off-diagonal elements (blocks). Furthermore, a relationship between the errors on the fine and on the coarse level is described. To illustrate the theory, the results of some numerical experiments are presented. © 1998 John Wiley & Sons, Ltd.

A novel method combing feature constraint with multilevel strategy to improve simultaneously the registration accuracy and speed is proposed for non-parametric image registrations. To images between which the local difference is large, integrating feature constraint constructed with local structure information of images into objective function of image registration improves the registration accuracy. When applying feature constraint under multilevel strategy, parameter searching is prevented from entrapped into local extremum by using the optimization result on coarser levels as the starting points on finer levels; meanwhile traditional optimization methods without demanding intelligent optimization algorithms which consume more time can find the accurate registration parameter on finer levels, so registration speed is improved. Experimental results indicate that this method can finish fast and accurate registration for images between which there exists large local difference.

Multiresolution data representations provide an indispensable tool for the compression, progressive transmission, and visualization of scientific data. Wavelet transforms based on B-spline scaling functions are frequently used to obtain continuous surface- and volume representations at multiple levels of resolution. Starting with a fine-resolution data set, wavelet decomposition provides a sequence of coarser approximations based on B-spline scaling functions and a set of wavelet coefficients containing the geometric differences with respect to the finer levels. The inverse transform can be used to reconstruct the finer levels of resolution from these wavelet coefficients within linear computation time. Biorthogonal wavelet transforms facilitate local computation of decomposition and reconstruction. We survey the most relevant results regarding wavelets and present different biorthogonal wavelet constructions based on highly efficient and simple-to-use lifting operations. Our approaches are suitable for lossy and lossless compression of large-scale data sets.

We propose two improvements to the Reitzinger and Schoberl algebraic multigrid (AMG) method for solving the eddy current approximations to Maxwell's equations. The main focus in the Reitzinger/Schoberl method is to maintain null space properties of the weak $\nabla \times \nabla \times$ operator on coarse grids. While these null space properties are critical, they are not enough to guarantee h-independent convergence of the overall multigrid method. We illustrate how the Reitzinger/Schoberl AMG method loses h-independence due to the somewhat limited approximation property of the grid transfer operators. We present two improvements to these operators that not only maintain the important null space properties on coarse grids but also yield significantly improved multigrid convergence rates. The first improvement is based on smoothing the Reitzinger/Schoberl grid transfer operators. The second improvement is obtained by using higher order nodal interpolation to derive the corresponding AMG interpolation operators. While not completely h-independent, the resulting AMG/CG method demonstrates improved convergence behavior while maintaining low operator complexity.

Scalable implementation of the parallel multigrid method on massively parallel computers

Algebraic multigrid (AMG) is an $\mathcal{O}(n)$ solution process for many large sparse linear systems. A hierarchy of progressively coarser grids is constructed that utilize complementary relaxation and interpolation operators. High-energy error is reduced by relaxation, while low-energy error is mapped to coarse-grids and reduced there. However, large parallel communication costs often limit parallel scalability. As the multigrid hierarchy is formed, each coarse matrix is formed through a triple matrix product. The resulting coarse-grids often have significantly more nonzeros per row than the original fine-grid operator, thereby generating high parallel communication costs on coarse-levels. In this paper, we introduce a method that systematically removes entries in coarse-grid matrices after the hierarchy is formed, leading to an improved communication costs. We sparsify by removing weakly connected or unimportant entries in the matrix, leading to improved solve time. The main trade-off is that if the heuristic identifying unimportant entries is used too aggressively, then AMG convergence can suffer. To counteract this, the original hierarchy is retained, allowing entries to be reintroduced into the solver hierarchy if convergence is too slow. This enables a balance between communication cost and convergence, as necessary. In this paper we present new algorithms for reducing communication and present a number of computational experiments in support.

A biologically inspired model of emotion eliciting from visual stimuli

We propose a coarse-to-fine approach for localizing eye and lip corners, which is accurate and robust, and can be executed in real-time. Given an image, we first detect the face and the rough initial positions of eyes and lip in the coarsest level. In the middle level, SUSAN corner detection is applied to obtain candidate corners, from which we select a best one as the facial feature corner. Finally, in the finest level, the corner is further refined by Normalized Cross-Correlation (NCC) template matching. We have tested our algorithm on the database containing over 5000 faces with various poses, expressions, occlusions and light conditions. The experimental results show that our method can successfully localize the facial feature corners with more than 20FPS, which is very appealing.

We present an approach for analyzing the visual aesthetic property of a handwritten document page which matches with human perception. We formulate the problem at two independent levels: (i) coarse level which deals with the overall layout, space usages between lines, words and margins, and (ii) fine level, which analyses the construction of each word and deals with the aesthetic properties of writing styles. We present our observations on multiple local and global features which can extract the aesthetic cues present in the handwritten documents.

Nighttime vehicle detection is part of an intelligent transportation system for road safety, driving navigation, and surveillance at night. However, previous nighttime vehicle detection methods only deal with a single class or two classes of vehicles. This paper presents an effective system based on cascade feature selection and a coarse-to-fine mechanism for detecting preceding multiclass vehicles at night. First, we train a coarse-level classifier using contrast features. Second, we combine a cascade selection framework with feature mutual correlation, similarity and regions’ overlap to select a set of image regions, where we can extract effective features. Three features, including local binary pattern, histogram of oriented gradients, and four direction features, are extracted from the selected regions for training a fine-level multiclass vehicle classifier. During the detection stage, we utilize a coarse-to-fine mechanism. In the coarse level, a multiscale sliding window is classified by the coarse-level classifier to find regions of interest (ROIs) that are likely to be vehicles. In the fine level, these ROIs are identified by the trained multiclass vehicle classifier. Evaluations on a Hong Kong nighttime multiclass vehicle dataset show that our proposed system successfully detects a car, taxi, bus, and minibus within nighttime images under different scenes. Quantitatively, our proposed system obtains 95.48% detection rate at 0.055 false positives per image, outperforming some state-of-the-art detection approaches including two current nighttime vehicle detection methods, and two deep learning-based methods.

A fast and efficient hybrid block-matching motion estimation algorithm is proposed in this paper. The fact that motion characteristics are relatively stable in a GOP (group of pictures) is utilized to reduce the computation burden without quality degradation. For P-frames, a multi-resolution block-matching algorithm is used. At the coarsest level, edge information is used to reduce searching positions before an exhaustive search. At the finer levels, only the several candidates chosen at the upper level are verified. For B-frames, MAE (mean absolute error) is used not only for the matching criteria, but also for classifying the macroblocks with the motion information extracted from P-frames. Concurrently, the search window size is adjusted using the contribution of different motion range to the improvement of the picture quality with motion compensation. When the decoded I-frame and P-frames are used as the reference, experimental results show that reconstructed digital frames have even better quality than the full search algorithm in terms of PSNR, under MPEG-1 coding environment.