Vladimir V. Savchenko

Function representation in geometric modeling : concepts, implementation and applications

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.

Function Representation of Solids Reconstructed from Scattered Surface Points and Contours

This paper presents a novel approach to the reconstruction of geometric models and surfaces from given sets of points using volume splines. It results in the representation of a solid by the inequality f(x,y,z) ≥ 0. The volume spline is based on use of the Greens function for interpolation of scalar function values of a chosen “carrier” solid. Our algorithm is capable of generating highly concave and branching objects automatically. The particular case where the surface is reconstructed from cross‐sections is discussed too. Potential applications of this algorithm are in tomography, image processing, animation and CAD for bodies with complex surfaces.

Software Tools Using CSRBFs for Processing Scattered Data

Abstract A set of software tools that use compactly supported radial basis functions (CSRBFs) to process scattered data is proposed in this paper. To solve problems concerning the processing of scattered data in such applications as reconstruction of functionally defined geometric objects, surface retouching, and shape modifications, we employ a specially designed C++ software library. Thanks to the efficient octree algorithm used in this study, the resulting matrix is a band-diagonal matrix that permits handling of large data sets in a reasonable time. The method, classes of the software library, time performance of the algorithm, and various examples of the use of the software tools are discussed.

Transformation of functionally defined shapes by extended space mappings

We present a general mathematical framework for transforming functionally defined shapes. The proposed model of extended space mappings considers transformations of a hypersurface in coordinate-function space with its projection onto geometric space. This model covers coordinate space mappings, metamorphosis, and algebraic operations on defining functions, and introduces several new types of transformations, such as function-dependent space mappings and combined mappings. The approach is illustrated by new local deformations created by means of function mappings, feature-based space mapping, offsetting along the normal, thin shell generation, 2D shape blending, and collision-free metamorphosis.

Hybrid system architecture for volume modeling

Abstract A hybrid volume modeling system architecture is proposed. The key architectural decision of the system design is based on the combination of volume representations by voxel data and by real continuous functions (so-called F-rep). We discuss the concept of the volume model, hybrid voxel-function representation, system input in different forms, and possible applications. Several examples illustrate the proposed approach: femur reconstruction and modeling, hybrid metamorphosis, collision detection, and real-time volumetric fly-through.

Real-time 3D Deformations by Means of Compactly Supported Radial Basis Functions

We present an approach to real-time animation of deformable objects. Optimization of algorithms using compactly supported radial basis functions (CSRBF) allows us to generate deformations performed fast enough for such real-time applications as computer games. The algorithm described in detail in this paper uses space mapping technique. Smooth local deformations of animation objects can be defined by only a moderate number of control vectors and locality of deformations can be defined by radius of support. We also present examples of animations and speed benchmarks.

Synthetic carving using implicit surface primitives

Several techniques of computer-aided synthetic carving are presented. We describe both procedural methods for relief carvings and patterned lattices, as well as interactive carving. Different techniques of depth data generation for relief carving are described: polygon-to-function conversion, pattern-dependent interpolation, and ray-casting. All proposed methods are based on using implicit surfaces or, more generally, the function representation of geometric objects.

Reconstructing occlusal surfaces of teeth using a genetic algorithm with simulated annealing type selection

In this paper, we present an application of numerical optimization for surface reconstruction (more precisely: reconstruction of missing parts of a real geometric object represented by volume data) by employing a specially designed genetic algorithm to solve a problem concerning computer-aided design in dentistry. Using a space mapping technique the surface of a given model tooth is fitted by a shape transformation to extrapolate (or reconstruct) the remaining surface of a patients tooth with occurring damage such as a “drill hole.” Thereby, the genetic algorithm minimizes the error of the approximation by optimizing a set of control points that determine the coefficients for spline functions, which in turn define a space transformation. The fitness function to be minimized by the genetic algorithm is the error between the transformed occlusal surface of the model tooth and the remaining occlusal surface of the damaged (drilled) tooth. The algorithm, that is used, is based upon a proposal by Mahfoud and Goldberg. It uses a simulated-annealing type selection scheme, which is applied sequentially (pair-wise, or one-by-one) to the members in the parent generation and their respective offspring generated by mutation-crossover. We outline a proof of convergence for this algorithm. The algorithm is parallel in regard to computing the fitness-values of creatures.

Volume modelling: representations and advanced operations

We present our approach to volume modelling which combines volume representations by voxel data and by continuous real functions. We discuss the main differences between direct volume visualization and modelling with voxel data, questions of conversion between two representations including volume reconstruction from contour data. We illustrate the approach by several advanced operations on a volumetric object: set-theoretic operations, sweeping, hypertexturing, feature-based sculpting, splitting, spatial and temporal transformations.

Surface reconstruction based on compactly supported radial basis functions

In this chapter the use of compactly-supported radial basis functions for surface reconstruction is described. To solve the problem of reconstruction or volume data generation specially designed software is employed. Time performance of the algorithm is investigated. Thanks to the efficient octree algorithm used in this study, the resulting matrix is a band diagonal matrix that reduces computational costs.