Implicit Objects for Computer Graphics Jonas Gomes Luiz Velho - - PDF document

Implicit Objects for Computer Graphics

Jonas Gomes Luiz Velho

IMPA

Instituto de Matem´ atica Pura e Aplicada

Motivation

Consolidation of the Area Intense Reseach Activities

– Workshop of Implicit Surfaces

(1995, 1996, 1998)

– SIGGRAPH Courses and Papers – Recent Books

Main Research Area People: jonas, lvelho, lhf, ruben

– Papers – Thesis – Book

References

Introduction to Implicit Surfaces

1997, Morgan-Kaufmann Implicit Objects in Computer Graphics

1992, IMPA / 1998, Springer-Verlag

Outline of the Course

– Basic Concepts (Book)

– Algebraic Varieties and Symbolic Perturbations

(J. Canny)

– PL Approximation and Interval Methods

(L. Velho, L. Figueiredo, J. Gomes)

– Multiresolution Models and Conversion

(L. Velho, J. Gomes)

– Particle Systems: Texturing and Morphing

(Velho, Figueiredo, Gomes, Zonenschein, Wyvil)

– Groups and Papers

Basic Concepts

– Parametric and Implict Forms – Implicit Manifolds

– Properties – Mathematical Characterization – Computational Implementation

– Modification – Combination

Shape Description

g

f f -1 c

where

–

–

* Direct versus Indirect Representation

Examples

2π

θ

f

• ;

• ;
• 2

r p 2 || || = r p

Comparison: Parametric Implicit

– More Popular – Intrinsic Description

– Gaining Acceptance – Extrinsic Description – Unifies Geometric and Volumetric Modeling – Suitable for Constraint Definitions

– Enumeration of Points – Classification of Points

Pros and Cons Parametric Implicit Enumeration +

• Classification
• +

* Complementary Descriptions

Implicit Model

– Space Decomposition – Metrics

– Regularity – Orientability

Intuitive View

• 0.8
• 0.4

• 0.8
• 0.4

• 0.8
• 0.4

volume data function graph

U F

−1

c F (c)

inverse image of

a

c

level curves intersection

Space Decomposition

• f is a Point-Membership Classification Function

• c

+

• f

f

• :

induced by a pseudo-metric

• m-norms

d0 d1 d oo

– Invariance

– Redundancy

• F

– Contours

Regularity

i.e.

• F

–

–

c

* Sards Theorem: Can always satisfy regularity (perturbation)

Orientability

• rF

*

Implicit Representation

– Surfaces and Solids – Canonical Forms

– Mathematical – Computational

Definition

where

• c

f c f < c f=c

– Boundary

– Interior

Canonical Forms

* Important in Applications

Mathematical Characterization

– Normal Curvature at

• v

– Principal Directions (Eigenvectors of

– Principal Curvatures (Eigenvalues of

– Mean Curvature (Trace of

– Gaussian Curvature (Determinant of

Computational Characterization

– Parameters (Local State) – Procedures (System Interface)

– Bounding Box

(domain of

– Range and Value

(

– Point Classification

(implicit function

– Normal

(gradient of

– Curvature

(hessian)

– Object Classes – Instancing

Operations with Implicit Objects

Unary Ops

– Range Operations

– Domain Operations

N-ary Ops

– Blend Operations

– Point-Set Operations

Range Operations

c

• ffset

• c

density

• F

density

Domain Operations

• bs: inverse transformation

– Affine Mappings

– Deformations

• fold, pull

• taper, bend, twist

– Physically Based Deformations

Blend Operations

y = 1 − x

• 1

y = 1/x

• 1

(1− x) + (1− y) = 1

2 2

• P

• F

• 1

Point-Set Operations

– Union

– Intersection

– Difference

– Union

– Intersection

• 1

Computational Methods

– Sampling (root finding) geometry

Intersection with a Line

Dynamic Particles

– Structuring topology

• Domain of

• Solid Region
• Boundary

– PL Approximation – Higher Order Approximations

Modeling

– Object Representation

– Groups

– Auxiliary Structures

– Constructive – Free-Form – Physically Based

– Parametric - Implicit – Exact vs. Approximate

Primitive Implicit Objects

– Plane – Quadrics / Tori – Superquadrics

– Points – Curves – Surfaces

– Scattered Data – Voxel Array

– Hypertextures – Fractals

* Multiresolution

Visualization

– Particles

– Discontinuity Lines – Contour Lines

– Scan Line – Ray Tracing – Polygonal

– Projection – Ray Casting

Texture Mapping

• t

• R

• t

• R

• R

• R

• R

• R

Types of Texture Mapping

– Color – Surface Properties – Displacement – Density

– 2D texture (m

– 3D texture (m

– Parametric

(2D texture)

– Implicit

(3D texture)

Animation

– Particle Systems – Articulated Objects

– Interpolation

• t)F

– Correspondence

Conclusion

Advantages

– Representation of Surfaces & Volumes – Description of Solids, Liquids, Gases – Flexible Formulation for CAD/CAM – Resolution and Levels of Detail – Higher Dimensional Problems

– Computation – Modeling – Visualization – Animation