Fast reliable interrogation of procedurally defined implicit surfaces using extended revised affine arithmetic.

Techniques based on interval and previous termaffine arithmetic next term and their modifications are shown to provide previous term reliable next term function range evaluation for the purposes of previous termsurface interrogation.next term In this paper we present a technique for the previous termreliable interrogation of implicit surfacesnext term using a modification of previous termaffine arithmeticnext term called previous term revised affine arithmetic.next term We extend the range of functions presented in previous termrevised affine arithmeticnext term by introducing previous termaffinenext term operations for arbitrary functions such as set-theoretic operations with R-functions, blending and conditional operators. The obtained previous termaffinenext term forms of arbitrary functions provide previous termfasternext term and tighter function range evaluation. Several case studies for operations using previous termaffinenext term forms are presented. The proposed techniques for previous termsurface interrogationnext term are tested using ray-previous termsurfacenext term intersection for ray-tracing and spatial cell enumeration for polygonisation. These applications with our extensions provide previous termfast and reliablenext term rendering of a wide range of arbitrary previous termprocedurally defined implicit surfacesnext term (including polynomial previous termsurfaces,next term constructive solids, pseudo-random objects, previous termprocedurally definednext term microstructures, and others). We compare the function range evaluation technique based on previous termextended revised affine arithmeticnext term with other previous termreliablenext term techniques based on interval and previous termaffine arithmeticnext term to show that our technique provides the previous termfastestnext term and tightest function range evaluation for previous termfast and reliable interrogation of procedurally defined implicit surfaces.next term Research Highlights The main contributions of this paper are as follows. ► The widening of the scope of previous termreliablenext term ray-tracing and spatial enumeration algorithms for previous termsurfacesnext term ranging from algebraic previous termsurfaces (definednext term by polynomials) to general previous termimplicit surfaces (definednext term by function evaluation procedures involving both previous termaffinenext term and non-previous termaffinenext term operations based on previous termrevised affine arithmetic)next term. ► The introduction of a technique for representing procedural models using special previous termaffinenext term forms (illustrated by case studies of previous termaffinenext term forms for set-theoretic operations in the form of R-functions, blending operations and conditional operations). ► The detailed derivation of special previous termaffinenext term forms for arbitrary operators

Implicit Blending Revisited

International audienceBlending is both the strength and the weakness of functionally based implicit surfaces (such as F-reps or softobjects). While it gives them the unique ability to smoothly merge into a single, arbitrary shape, it makes implicit modelling hard to control since implicit surfaces blend at a distance, in a way that heavily depends on the slope of the field functions that define them. This paper presents a novel, generic solution to blending of functionally-based implicit surfaces: the insight is that to be intuitive and easy to control, blends should be located where two objects overlap, while enabling other parts of the objects to come as close to each other as desired without being deformed. Our solution relies on automatically defined blending regions around the intersection curves between two objects. Outside of these volumes, a clean union of the objects is computed thanks to a new operator that guarantees the smoothness of the resulting field function; meanwhile, a smooth blend is generated inside the blending regions. Parameters can automatically be tuned in order to prevent small objects from blurring out when blended into larger ones, and to generate a progressive blend when two animated objects come in contact

Multi-scale space-variant FRep cellular structures

Existing mesh and voxel based modeling methods encounter difficulties when dealing with objects containing cellular structures on several scale levels and varying their parameters in space. We describe an alternative approach based on using real functions evaluated procedurally at any given point. This allows for modeling fully parameterized, nested and multi-scale cellular structures with dynamic variations in geometric and cellular properties. The geometry of a base unit cell is defined using Function Representation (FRep) based primitives and operations. The unit cell is then replicated in space using periodic space mappings such as sawtooth and triangle waves. While being replicated, the unit cell can vary its geometry and topology due to the use of dynamic parameterization. We illustrate this approach by several examples of microstructure generation within a given volume or along a given surface. We also outline some methods for direct rendering and fabrication not involving auxiliary mesh and voxel representations

Space-Time Transfinite Interpolation of Volumetric Material Properties

The paper presents a novel technique based on extension of a general mathematical method of transfinite interpolation to solve an actual problem in the context of a heterogeneous volume modelling area. It deals with time-dependent changes to the volumetric material properties (material density, colour and others) as a transformation of the volumetric material distributions in space-time accompanying geometric shape transformations such as metamorphosis. The main idea is to represent the geometry of both objects by scalar fields with distance properties, to establish in a higher-dimensional space a time gap during which the geometric transformation takes place, and to use these scalar fields to apply the new space-time transfinite interpolation to volumetric material attributes within this time gap. The proposed solution is analytical in its nature, does not require heavy numerical computations and can be used in real-time applications. Applications of this technique also include texturing and displacement mapping of time-variant surfaces, and parametric design of volumetric microstructures

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 parameterised 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

Repairing triangle meshes built from scanned point cloud

The Reverse Engineering process consists of a succession of operations that aim at creating a digital representation of a physical model. The reconstructed geometric model is often a triangle mesh built from a point cloud acquired with a scanner. Depending on both the object complexity and the scanning process, some areas of the object outer surface may never be accessible, thus inducing some deficiencies in the point cloud and, as a consequence, some holes in the resulting mesh. This is simply not acceptable in an integrated design process where the geometric models are often shared between the various applications (e.g. design, simulation, manufacturing). In this paper, we propose a complete toolbox to fill in these undesirable holes. The hole contour is first cleaned to remove badly-shaped triangles that are due to the scanner noise. A topological grid is then inserted and deformed to satisfy blending conditions with the surrounding mesh. In our approach, the shape of the inserted mesh results from the minimization of a quadratic function based on a linear mechanical model that is used to approximate the curvature variation between the inner and surrounding meshes. Additional geometric constraints can also be specified to further shape the inserted mesh. The proposed approach is illustrated with some examples coming from our prototype software

Accurate geometry modeling of vasculatures using implicit fitting with 2D radial basis functions

Accurate vascular geometry modeling is an essential task in computer assisted vascular surgery and therapy. This paper presents a vessel cross-section based implicit vascular modeling technique, which represents a vascular surface as a set of locally fitted implicit surfaces. In the proposed method, a cross-section based technique is employed to extract from each cross-section of the vascular surface a set of points, which are then fitted with an implicit curve represented as 2D radial basis functions. All these implicitly represented cross-section curves are then being considered as 3D cylindrical objects and combined together using a certain partial shape-preserving spline to build a complete vessel branch; different vessel branches are then blended using a extended smooth maximum function to construct the complete vascular tree. Experimental results show that the proposed method can correctly represent the morphology and topology of vascular structures with high level of smoothness. Both qualitative comparison with other methods and quantitative validations to the proposed method have been performed to verify the accuracy and smoothness of the generated vascular geometric models

N-ary implicit blends with topology control

International audienceConstructive implicit surfaces are attractive for modeling and animation because they seamlessly handle shapes with complex and dynamic topology. However, the way they merge shapes is difficult to control. This paper introduces a solution: an improved blend operator that provides control over how topology changes are handled. It is based on a correction applied to the standard blending operator: the sum. Building on summation preserves the n-ary nature of the blend, providing the simplicity of arbitrary (e.g. flat) construction trees and segmentation invariance. The correction is based on projection to a reference case in the variation-space defined by the field and the norm of its gradient. It provides a single parameter, allowing for tuning behavior to achieve effects ranging from avoiding topological combination, through merging only during overlap, to merging at a distance. Dynamic adjustment of the parameter allows for context-dependent effects. Applications range from skeleton-based modeling, where shapes keep the topology of their skeleton, to objects that change topology during animation, with controllable merging. We illustrate the latter with Manga-style hair, where merging depends on the angle between hair wisps