CiteULike: Tag implicit-representation

Function representation in geometric modeling: concepts, implementation and applications

A Pasko — 2007-07-11T18:39:01-00:00

The Visual Computer, Vol. 11, No. 8. (12 October 1995), pp. 429-446.

Abstract Concepts of functionally based geometric modeling including sets of objects, operations, and relations are discussed. Transformations of a defining real function are described for set-theoretic operations, blending, offsetting, bijective mapping, projection, cartesian products, and metamorphosis. Inclusion, point membership, and intersection relations are also described. We use a high-level geometric language that can extend the interactive modeling system by input symbolic descriptions of primitives, operations, and predicates. This approach supports combinations of representational styles, including constructive geometry, sweeping, soft objects, voxel-based objects, deformable and other animated objects. Application examples of aesthetic design, collisions simulation, NC machining, range data processing, and 3D texture generation are given.

Compactly supported RBFs in the management of implicit surfaces

Terry Yoo — 2007-07-13T14:05:41-00:00

(2005)

Implicit curves and surfaces in CAGD

CM Hoffmann — 2007-08-01T14:37:55-00:00

Computer Graphics and Applications, IEEE, Vol. 13, No. 1. (1993), pp. 79-88.

The role of implicit curves and surfaces in computer-aided geometric design (CAGD) are described. The ways in which the study of implicit algebraic curves and surfaces draws on algebraic geometry are reviewed. The implicitization of parametric curves and surfaces, parameterization of implicits, and techniques used to circumvent conversions between implicit and parametric representations are discussed

Feature-based volume metamorphosis

Apostolos Lerios — 2006-10-31T17:12:16-00:00

(1995), pp. 449-456.

Using the CW-complex to represent the topological structure of implicit surfaces and solids

John Hart — 2007-07-13T14:04:28-00:00

(2005)

Linear and non-linear geometric object matching with implicit representation

A Leow — 2007-08-07T10:49:27-00:00

Pattern Recognition, 2004. ICPR 2004. Proceedings of the 17th International Conference on, Vol. 3 (2004), pp. 710-713 Vol.3.

This paper deals with the matching of geometric objects including points, curves, surfaces, and subvolumes using implicit object representations in both linear and non-linear settings. This framework can be applied to feature-based non-linear image warping in biomedical imaging with the deformation constrained to be one-to-one, onto, and diffeomorphic. Moreover, a theoretical connection is established between the well known Hausdorff metric and the framework proposed in this paper. A general strategy for matching geometric objects in both 2D and 3D is discussed. The corresponding Euler-Lagrange equations are presented and gradient descent method is employed to solve the time dependent partial differential equations.

Using distance maps for accurate surface representation in sampled volumes

Sarah Gibson — 2005-08-31T10:05:03-00:00

(1998), pp. 23-30.

Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions

Bryan Morse — 2007-07-13T14:10:55-00:00

(2005)

Guaranteeing the topology of an implicit surface polygonization for interactive modeling

Barton Stander — 2007-07-13T14:03:18-00:00

(2005)

Multi-level partition of unity implicits

Yutaka Ohtake — 2007-07-13T18:45:13-00:00

(2005)

A Pseudo-distance for shape priors in level set segmentation

D Cremers — 2006-03-20T10:22:04-00:00

(2003)

We study the question of integrating prior shape knowledge into level set based segmentation methods. In particular, we investigate dissimilarity measures for shapes encoded by the signed distance function.

Fast polygonization of variational implicit surfaces

A Cuno — 2007-07-11T16:59:23-00:00

Computer Graphics and Image Processing, 2004. Proceedings. 17th Brazilian Symposium on (2004), pp. 258-265.

This article presents a simple hierarchical adaptation of the marching cubes algorithm for polygonizing variational implicit surfaces used in modelling and reconstruction applications. The technique relies on placing the normal and boundary constraint points respecting pseudo-Euclidean distance metrics. This procedure makes it possible to quickly prune the space and minimize the number of costly function evaluations and thus converge rapidly to the surface. Timings show that this technique tends to perform faster than Bloomenthal's (1994) continuation polygonizer.

The HybridTree: Mixing skeletal implicit surfaces, triangle meshes, and point sets in a free-form modeling system

Remi Allegre — 2006-05-10T10:10:17-00:00

Graphical Models, Vol. 68, No. 1. (January 2006), pp. 42-64.

In this paper, we present a hybrid modeling framework for creating complex 3D objects incrementally. Our system relies on an extended CSG tree that assembles skeletal implicit primitives, triangle meshes and point set models in a coherent fashion: we call this structure the HybridTree. Editing operations are performed by exploiting the complementary abilities of implicit and polygonal mesh surface representations in a complete transparent way for the user. Implicit surfaces are powerful for combining shapes with Boolean and blending operations, while triangle meshes are well-suited for local deformations such as FFD and fast visualization. Our system can handle point sampled geometry through a mesh surface reconstruction algorithm. The HybridTree may be evaluated through four kinds of queries, depending on the implicit or explicit formulation is required: field function and gradient at a given point in space, point membership classification, and polygonization. Every kind of query is achieved automatically in a specific and optimized fashion for every node of the HybridTree.

Some notes on radial basis functions and thin plate splines

John Hart — 2007-07-13T14:01:55-00:00

(2005)

Modelling with implicit surfaces that interpolate

Greg Turk — 2007-07-13T14:09:48-00:00

(2005)

An extended level set method for shape and topology optimization

SY Wang — 2008-05-01T18:44:26-00:00

Journal of Computational Physics, Vol. 221, No. 1. (20 January 2007), pp. 395-421.

In this paper, the conventional level set methods are extended as an effective approach for shape and topology optimization by the introduction of the radial basis functions (RBFs). The RBF multiquadric splines are used to construct the implicit level set function with a high level of accuracy and smoothness and to discretize the original initial value problem into an interpolation problem. The motion of the dynamic interfaces is thus governed by a system of coupled ordinary differential equations (ODEs) and a relatively smooth evolution can be maintained without reinitialization. A practical implementation of this method is further developed for solving a class of energy-based optimization problems, in which approximate solution to the original Hamilton-Jacobi equation may be justified and nucleation of new holes inside the material domain is allowed for. Furthermore, the severe constraints on the temporal and spatial discretizations can be relaxed, leading to a rapid convergence to the final design insensitive to initial guesses. The normal velocities are chosen to perform steepest gradient-based optimization by using shape sensitivity analysis and a bi-sectioning algorithm. A physically meaningful and efficient extension velocity method is also presented. The proposed method is implemented in the framework of minimum compliance design and its efficiency over the existing methods is highlighted. Numerical examples show its accuracy, convergence speed and insensitivity to initial designs in shape and topology optimization of two-dimensional (2D) problems that have been extensively investigated in the literature.

Fast Polygonization of Variational Implicit Surfaces

Alvaro Claudio — 2005-08-19T08:59:43-00:00

This article presents a simple hierarchical adaptation of the Marching Cubes algorithm for polygonizing variational implicit surfaces used in modelling and reconstruction applications. The technique relies on placing the normal and boundary constraint points respecting a pseudo-Euclidean distance metrics. This procedure makes it possible to quickly prune the space and minimize the number of costly function evaluations and thus converge rapidly to the surface. Timings show that this...

Shape transformation using variational implicit functions

Greg Turk — 2007-07-13T14:08:40-00:00

(2005)

Level Set Methods and Their Applications in Image Science

Richard Tsai — 2007-08-14T22:03:45-00:00

this article, we discuss the question "What Level Set Methods can do for image science". We examine the scope of these techniques in image science, in particular in image segmentation, and introduce some relevant level set techniques that are potentially useful for this class of applications. We will show that image science demands multi-disciplinary knowledge and flexible but still robust methods. That is why the Level Set Method has become a thriving technique in this field

Creating surfaces from scattered data using radial basis functions

R Schaback — 2007-08-30T14:40:05-00:00

(1995)

. This paper gives an introduction to certain techniques for the construction of geometric objects from scattered data. Special emphasis is put on interpolation methods using compactly supported radial basis functions. x1. Introduction We assume a sample of multivariate scattered data to be given as a set X = fx 1 ; : : : ; xN g of N pairwise distinct points x 1 ; : : : ; xN in IR d , called centers, together with N points y 1 ; : : : ; yN in IR D . An interpolating curve, surface, or...

Modern techniques for implicit modeling

James O'Brien — 2007-07-13T14:07:46-00:00

(2005)

Global and local optimization using radial basis function response surface models

Dale Mcdonald — 2008-06-06T11:42:43-00:00

Applied Mathematical Modelling, Vol. 31, No. 10. (October 2007), pp. 2095-2110.

The focus of this paper is the optimization of complex multi-parameter systems. We consider systems in which the objective function is not known explicitly, and can only be evaluated through computationally intensive numerical simulation or through costly physical experiments. The objective function may also contain many local extrema which may be of interest. Given objective function values at a scattered set of parameter values, we develop a response surface model that can dramatically reduce the required computation time for parameter optimization runs. The response surface model is developed using radial basis functions, producing a model whose objective function values match those of the original system at all sampled data points. Interpolation to any other point is easily accomplished and generates a model which represents the system over the entire parameter space. This paper presents the details of the use of radial basis functions to transform scattered data points, obtained from a complex continuum mechanics simulation of explosive materials, into a response surface model of a function over the given parameter space. Response surface methodology and radial basis functions are discussed in general and are applied to a global optimization problem for an explosive oil well penetrator.

Two-dimensional Potential Fields for Advanced Implicit Modeling Operators

L Barthe — 2007-07-24T11:50:29-00:00

pp. 23-33.

Current methods for building models using implicit volume techniques present problems defining accurate and controllable blend shapes between implicit primitives. We present new methods to extend the freedom and controllability of implicit volume modeling. The main idea is to use a free-form curve to define the profile of the blend region between implicit primitives. The use of a free-form implicit curve, controlled point-by-point in the Euclidean user space, allows us to group boolean composition operators with sharp transitions or smooth free-form transitions in a single modeling metaphor. This idea is generalized for the creation, sculpting and manipulation of volume objects, while providing the user with simplicity, controllability and freedom in implicit modeling. ACM CSS: I.3.5 Computational Gemoetry and Object Modeling—Curve, surface, solid, and object representations

Introduction to Implicit Surfaces, First Edition (The Morgan Kaufmann Series in Computer Graphics)

Jules Bloomenthal — 2006-05-01T11:36:59-00:00

(01 August 1997)

Implicit surfaces offer special effects animators, graphic designers, CAD engineers, graphics students, and hobbyists a new range of capabilities for the modeling of complex geometric objects. In contrast to traditional parametric surfaces, implicit surfaces can easily describe smooth, intricate, and articulatable shapes. These powerful yet easily understood surfaces are finding use in a growing number of graphics applications. This comprehensive introduction develops the fundamental concepts and techniques of implicit surface modeling, rendering, and animating in terms accessible to anyone with a basic background in computer graphics. + provides a thorough overview of implicit surfaces with a focus on their applications in graphics + explains the best methods for designing, representing, and visualizing implicit surfaces + surveys the latest research With contributions from seven graphics authorities, this innovative guide establishes implicit surfaces as a powerful and practical tool for animation and rendering.

Hierarchical 3D surface reconstruction based on radial basis functions

P Dalmasso — 2006-08-17T09:24:32-00:00

3D Data Processing, Visualization and Transmission, 2004. 3DPVT 2004. Proceedings. 2nd International Symposium on (2004), pp. 574-579.

Volumetric methods based on implicit surfaces are commonly used in surface reconstruction from uniformly distributed sparse 3D data. The case of nonuniform distributed data has recently deserved more attention, because it occurs frequently in practice. This work describes a volumetric approach to surface reconstruction from nonuniform data which is suitable for the reconstruction of surfaces from images, in particular from multiple views. Differently from volumetric methods which use both 3D surface points and surface normals, the approach does not use the surface normals because they are often unreliable when estimated from image data. The method is based on a hierarchical partitioning of the volume data set. The working volume is split and classified at different scales of spatial resolution into surface, internal and external voxels and this hierarchy is described by an octree structure in a multiscale framework. The octree structure is used to build a multiresolution description of the surface by means of compact support radial basis functions (RBF). A hierarchy of surface approximations at different levels of details is built by representing the voxels at the same octree level as RBF of similar spatial support. At each scale, information related to the reconstruction error drives the reconstruction process at the following finer scale. Preliminary results on synthetic data and future perspectives are presented.

A shape design system using volumetric implicit PDEs

Haixia Du — 2007-07-13T14:13:38-00:00

(2005)

Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations

Stanley Osher — 2005-08-07T19:40:32-00:00

Journal of Computational Physics, Vol. 79 (1988), pp. 12-49.

We devise new numerical algorithms, called PSC algorithms, for following fronts propagating with curvature-dependent speed. The speed may be an arbitrary function of curvature, and the front can also be passively advected by an underlying flow. These algorithms approximate the equations of motion, which resemble Hamilton-Jacobi equations with parabolic right-hand-sides, by using tech-niques from the hyperbolic conservation laws. Non-oscillatory schemes of various orders of accu-racy are used to ...

3D scattered data interpolation and approximation with multilevel compactly supported RBFs

Yutaka Ohtake — 2006-05-02T02:41:09-00:00

Graph. Models, Vol. 67, No. 3. (May 2005), pp. 150-165.

Implicit modeling with PDE-based techniques

Haixia Du — 2007-07-13T14:12:17-00:00

(2005)

Step Function Representation of Solid Models and Application to Mesh Free Engineering Analysis

Ashok Kumar — 2007-07-10T15:02:13-00:00

Journal of Mechanical Design, Vol. 128, No. 1. (2006), pp. 46-56.

Numerical methods for solving boundary value problems that do not require generation of mesh to approximate the analysis domain have been referred to as mesh-free methods. While many of these are "mesh less" methods that do not have connectivity between nodes, a subset of these methods uses a structured mesh or grid for the analysis that does not conform to the geometry of the domain of analysis. Instead the geometry is represented using implicit equations. In this paper we present a method for constructing step functions of solids whose boundaries are represented using implicit equations. Step functions can be used to compute volume integrals over the solid that are needed for mesh free analysis. The step function of the solid has a unit value within the solid and zero outside. A level set of this step function can then be defined as the boundary of the solid. Boolean operators are defined in this paper that enable step functions of half-spaces and primitives to be combined to construct a single step function for more complex solids. Application of step functions to analysis using nonconforming mesh is illustrated.

A New Robust And Fast Implicit Polynomial Fitting Technique

Cem Ünsalan — 2008-03-23T00:36:48-00:00

In this paper Fourier series expansion of polar representation of an object contour is used to obtain robust and fast implicit polynomial fittings for star shaped objects in 2D and 3D. The new fitting algorithm is compared with existing implicit polynomial fitting algorithms with respect fitting performance and timing requirements.

Sub-Voxel Topology Control for Level-Set Surfaces

S Bischoff — 2005-09-26T19:16:32-00:00

Computer Graphics Forum, Vol. 22, No. 3. (September 2003), pp. 273-280.

Active contour models are an efficient, accurate, and robust tool for the segmentation of 2D and 3D image data.In particular, geometric deformable models (GDM) that represent an active contour as the level set of an implicitfunction have proven to be very effective. GDMs, however, do not provide any topology control, i.e. contours maymerge or split arbitrarily and hence change the genus of the reconstructed surface. This behavior is inadequate insettings like the segmentation of organic tissue or other objects whose genus is known beforehand. In this paperwe describe a novel method to overcome this limitation while still preserving the favorable properties of the GDMsetup. We achieve this by adding (sparse) topological information to the volume representation at locations whereit is necessary to locally resolve topological ambiguities. Since the sparse topology information is attached to theedges of the voxel grid, we can reconstruct the interfaces where the deformable surface touches itself at sub-voxelaccuracy. We also demonstrate the efficiency and robustness of our method.

A computing method for incompressible flows bounded by moving walls

James Viecelli — 2008-04-30T17:37:12-00:00

Journal of Computational Physics, Vol. 8, No. 1. (August 1971), pp. 119-143.

A modified type of Marker and Cell computing method is presented for solving problems in incompressible hydrodynamics. The method is applicable to time dependent problems in two spatial dimensions or three spatial dimensions with axial symmetry. Details are presented for calculation of arbitrarily shaped curved wall boundaries and flexible moving wall boundaries. Example problems with moving walls and free surfaces are given.