Valery Adzhiev

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.

Procedural function-based modelling of volumetric microstructures

We propose a new approach to modelling heterogeneous objects containing internal volumetric structures with size of details orders of magnitude smaller than the overall size of the object. The proposed function-based procedural representation provides compact, precise, and arbitrarily parametrized models of coherent microstructures, which can undergo blending, deformations, and other geometric operations, and can be directly rendered and fabricated without generating any auxiliary representations (such as polygonal meshes and voxel arrays). In particular, modelling of regular lattices and cellular microstructures as well as irregular porous media is discussed and illustrated. We also present a method to estimate parameters of the given model by fitting it to microstructure data obtained with magnetic resonance imaging and other measurements of natural and artificial objects. Examples of rendering and digital fabrication of microstructure models are presented.

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.

Cellular-functional modeling of heterogeneous objects

The paper presents an approach to modeling heterogeneous objects as multidimensional point sets with multiple attributes (hypervolumes). A theoretical framework is based on a hybrid model of hypervolumes combining a cellular representation and a constructive representation using real-valued functions. This model allows for independent but unifying representation of geometry and attributes, and makes it possible to represent dimensionally non-homogeneous entities and their cellular decompositions. Hypervolume model components such as objects, operations and relations are introduced and outlined. The frameworks inherent multidimensionality allowing, in particular, to deal naturally with time dependence promises to model complex dynamic objects composed of different materials with constructive building of their geometry and attributes. Attributes given at each point can represent properties of arbitrary nature (material, photometric, physical, statistical, etc.). To demonstrate a particular application of the proposed framework, we present an example of multimaterial modeling - a multilayer geological structure with cavities and wells. Another example illustrating the treatment of attributes other than material distributions is concerned with time-dependent adaptive mesh generation where function representation is used to describe object geometry and density of elements in the cellular model of the mesh. The examples have been implemented by using a specialized modeling language and software tools being developed by the authors.

Hybrid Cellular-functional Modeling of Heterogeneous Objects

An approach to modeling heterogeneous objects as multidimensional point sets with multiple attributes (hypervolumes) is presented. Attributes given at each point represent object properties of arbitrary nature (material, physical, etc.). A proposed theoretical framework is based on a hybrid model of geometry and attributes combining a cellular representation and a functionally based constructive representation of dimensionally non-homogeneous entities. Hypervolume model components such as objects, operations and relations are introduced and outlined. We present examples of modeling a multi-layer geological structure with cavities and wells, time-dependent adaptive mesh generation, and conversion of a 3D implicit complex to the cellular representation.

Function-Based Shape Modeling: Mathematical Framework and Specialized Language

In this survey, we describe the following different aspects of modeling multidimensional point sets (shapes) using real-valued functions of several variables: algebraic system as a formal framework; representation of shapes, operations, and relations using real-valued functions, internal representation of the modeling system; specialized language for function-based modeling, and model extension to point sets with attributes (hypervolumes).

Web-based shape modeling with HyperFun

In this article, we have developed an open system architecture and Web-based shape-modeling applications using HyperFun, a high-level programming language for specifying implicit surfaces and FRep objects. We based our system on empirical modeling principles to allow for implementation of realistic behavior of shape models situated in multiagent environments, providing open-ended exploration and experimentation with shared models. Shape modeling systems based on HyperFun modeling language treat Internet-based shape modeling as a resource and let users collaborate in shape development.

Procedural Function-Based Spatial Microstructures

We propose a new approach to modelling heterogeneous objects containing internal spatial geometric structures with size of details orders of magnitude smaller than the overall size of the object. The proposed function-based procedural representation provides compact, precise, and arbitrarily parametrized models of coherent microstructures, which can undergo blending, deformations, and other geometric operations, and can be directly rendered and fabricated without generating any auxiliary representations (such as polygonal meshes and voxel arrays). In particular, modelling of regular lattices and cellular microstructures as well as irregular porous media is discussed and illustrated. Examples of rendering and digital fabrication of microstructure models are presented.

Surface and Volume Discretization of Functionally Based Heterogeneous Objects

The presented approach to discretization of functionally defined heterogeneous objects is oriented towards applications associated with numerical simulation procedures, for example, finite element analysis (FEA). Such applications impose specific constraints upon the resulting surface and volume meshes in terms of their topology and metric characteristics, exactness of the geometry approximation, and conformity with initial attributes. The function representation of the initial object is converted into the resulting cellular representation described by a simplicial complex. We consider in detail all phases of the discretization algorithm from initial surface polygonization to final tetrahedral mesh generation and its adaptation to special FEA needs. The initial object attributes are used at all steps both for controlling geometry and topology of the resulting object and for calculating new attributes for the resulting cellular representation.

Discretization of functionally based heterogeneous objects

The presented approach to discretization of functionally defined heterogeneous objects is oriented towards applications associated with numerical simulation procedures, for example, finite element analysis (FEA). Such applications impose specific constraints upon the resulting surface and volume meshes in terms of their topology and metric characteristics, exactness of the geometry approximation, and conformity with initial attributes. The function representation of the initial object is converted into the resulting cellular representation described by a simplicial complex. We consider in detail all phases of the discretization algorithm from initial surface polygonization to final tetrahedral mesh generation and its adaptation to special FEA needs. The initial object attributes are used at all steps both for controlling geometry and topology of the resulting object and for calculating new attributes for the resulting cellular representation.