Computer Graphics:Computer Graphics:ModelingModeling

Franck HétroyFranck Hétroy

Franck.Hetroy@imag.frhttp://evasion.imag.fr/Membres/Franck.Hetroy/

Context We want to represent 3D objects

− Real objects− Virtual/created objects

Several ways for virtual object creation− Interactive by graphists− Automatic from real data

3D scanner, medical angiography, ...− Procedural (on the fly)

Complex scenes, terrain, ... Different uses

− Display, animation, physical simulation, ...

Course overview

Volume-based modeling− Next week

Point-based modeling Surface-based modeling

1. Meshes

2. Parametric surfaces

3. Subdivision surfaces

4. Implicit surfaces

(Tentative) pedagogy:problem-based learning

Objectives:− Learn more things− Remember more things− Know other students

How:− Individual work (papers to read)− Debate to answer a problem

In teams (4 or 5 students)

− Presentation of results

Planning

Today:− Introductive course

Boring => exercises

− Papers to read for next week (individually) Next week:

− I give you a problem− 1 hour to find an answer as a team− Presentation of your solution (10 minutes/team)− Structuring course (1 hour)

Teams (tentative) Team 1:

Reynald Arnerin, Azim Azma, Laurent Belcour, Robert Conceivo Da Silva, Adrien Gomez Espana Pecker

Team 2:Nicolas Esteves, Sylvain Guglielmi, Juan Alberto Lahera

Perez, Mohamed Riadh Trad, Xue Bing

Team 3:Pierre Arnaud, Varun Raj Kompella, Anja Marx, Jorge Pena,

Thibault Serot

Team 4:Antoine Bautin, Adrien Brilhault, Noura Faraj, Stefano

Sclaverano

Today's planning

1.Geometry representations: introduction

2.Point sets

3.Meshes

4.Splines and parametric surfaces

5.Subdivision surfaces

6.Implicit surfaces

Examples

Why not one good representation ?

Multiple applications, different constraints

− Powerful rep. To handle a large class of objects To create complex objects from simple ones

− Intuitive rep. To edit the model To animate some parts of it

− Efficient rep. Memory cost Display/process time cost

Classification: a proposal

Non structured rep.− Point set− Polygon soup

Surface rep.− Mesh− Parametric− Subdivision− Implicit

Volumetric rep.− Voxel line/plane/set− Octree− CSG

Procedural rep.− Fractal− Grammar/L-system− Particle system

Image-based rep.

Today's planning

1.Geometry representations: introduction

2.Point sets

3.Meshes

4.Splines and parametric surfaces

5.Subdivision surfaces

6.Implicit surfaces

Point set representation

Points are samples of the underlying surface 1 point corresponds to 1 surfel (surface

element)− Position− Color− Normal− Radius

Surfel = 2D !

Courtesy M. Zwicker

Surfel

Surfels are designed mostly for rendering

Advantage: no mesh reconstruction necessary

− Time saving No surface connectivity

information

Courtesy M. Zwicker

Point rendering pipeline

Credit: M. Zwicker 2002

Point set

Visibility

Framebuffer

Forwardwarping

Shading Imagereconstruction

Forward warping and shading

Forward warping = perspective projection of each point in the point cloud

Similar to projection of triangle vertices (mesh case)

Shading:− Per point− Conventional models for shading (Phong,

Torrance-Sparrow, reflections, etc.)− Cf. rendering course

Visibility and image reconstruction

Performed simultaneously Discard points that are occluded from the

current viewpoint Reconstruct continuous surfaces from

projected points

Image reconstruction

Goal: avoid holes in the image of the surface Use surfel radius to cover the surface More during the rendering course

Surface approximation

Almost all surface representations are based on points

− Meshes− Parametric rep. (splines)− Implicit rep.

A projection-based surface definition is also possible

− Local polynomial P around each point− Project P(0) onto a local reference plane

Today's planning

1.Geometry representations: introduction

2.Point sets

3.Meshes

4.Splines and parametric surfaces

5.Subdivision surfaces

6.Implicit surfaces

Meshes

Mesh = (V,E,F)− V = set of vertices− E = set of edges− F = connected set of (planar) faces

Not connected = polygon soup Faces can be

− Triangles− Planar quads− Any planar, convex polygon

2-Manifold

Def.: each vertex has a neighborhood on M homeomorphic to a disk

− Continuous bijection, distance does not matter 2-Manifold with boundary: to a [half-]disk 3-Manifold, n-manifold, ... No singularities:

Object topology

Any manifold's topology is defined by a small set of numbers:

− Surface: nb c of connected components + nb g of holes + nb b of boundaries

− Volume: nb of conn. comp. + nb of tunnels + nb of cavities (bubbles) + nb of boundaries

Euler formula for surface meshes:− V-E+F = = 2(c-g)-b

− = Euler characteristic− g = genus

(Easy) Exercise Find the Euler characteristic of the following 6

surfaces:

And for volumes ?

Mesh data structures

How to store geometry and connectivity ?− STL-like: store triangles, vertices duplicated

=> no connectivity− Shared vertex data structure (OBJ, OFF file

formats): vertex list, triangles = triples of indices

=> no neighborhood info− Half-edge and variants

=> all is based on oriented edges

Half-edge data structure Three main classes:

− Vertex Coord, [id,] pointer to one

outgoing half-edge

− Half-edge Pointers to the origin vertex, to

the next and to the opposite half-edge, to the incident face

− Face Pointer to one incident half-edge

You can add whatever attributes you want (normal, color, ...)

Example: browsing the 1-ring neighborhood of a vertex

Example: browsing the 1-ring neighborhood of a vertex

Exercise:− Write your own half-edge data

structure: class Vertex class Edge class Face

− Write a procedure browseOneRing(Vertex* v) which returns the 1-ring neighborhood of v as a list.

C++ libraries

CGAL http://www.cgal.org/ Developed by a consortium led by INRIA, lots of stuff Widely used by researchers, tutorials Somehow complicated (genericity)

OpenMesh http://www.openmesh.org/ Developed by Mario Botsch at RWTH Aachen Simpler, clearer Lack of documentation

GTS http://gts.sourceforge.net/ Why not ?

Today's planning

1.Geometry representations: introduction

2.Point sets

3.Meshes

4.Splines and parametric surfaces

5.Subdivision surfaces

6.Implicit surfaces

30

Parametric curves: splines

Motivations : interpolate/approximate points Pk

Easier to give a finite number of “control points”

The curve should be smooth in between

Why not polynomials? Which degree do we need?

u

31

Spline curves

Defined from control point Local control

− Joints between polynomial curve segments

− degree 3, C1 or C2 continuity

Spline curve

Control point

32

Interpolation vs. Approximation

P1P2

P3

P2

P3

P1

33

Splines curves

Mathematical formulation? Curve points = linear

combination of control points

C(u) =∑ Fk(u) Pk Curve’s degree of continuity = degree of

continuity of Fk

Desirable properties for the “influence functions” Fk?

34

Splines curvesSummary of desirable properties

C(t) =∑ Fk(t) Pk,

Interpolation & approximation− Affine invariance ∑ Fk(t) =1− Locality: Fk(t) with compact support− Parametric or geometric continuity

Approximation− Convex hull Fk(t) ≥0 − Variance reduction: no unwanted

oscillation

Tk Tk+1

u

35

Splines curvesMost important models

Interpolation− Hermite curves C1, cannot be local if C2 − Cardinal spline (Catmull Rom)

Approximation− Bézier curves− Uniform, cubic B-spline (unique definition,

subdivision)− Generalization to NURBS

36

Cardinal Spline, with tension=0.5

37

Uniform, cubic B-spline

38

Cubic splines: matrix equation

P1P2

P3 P4 P4

P2

P3

P1

Qi (u) = (u3 u2 u 1) Mspline [Pi-1 Pi Pi+1 Pi+2] t

Cardinal spline B-spline

−−−

−−

=

0020

0101

1452

1331

2

1CatmullM

−−

−−

=

0141

0303

0363

1331

6

1BsplineM

39

Splines surfaces

« Tensor product »: product of spline curves in u and v

Qi,j (u, v) = (u3 u2 u 1) M [Pi,j] Mt (v3 v2 v 1)

− Smooth surface?

− Convert to meshes?

− Locality?

40

Splines surfaces

• Expression with separable influence functions!

Qi,j (u, v) = ∑ Bi(u) Bj(v) Pij

u

v Historic example

Today's planning

1.Geometry representations: introduction

2.Point sets

3.Meshes

4.Splines and parametric surfaces

5.Subdivision surfaces

6.Implicit surfaces

42

Subdivision Curves & Surfaces

Start with a control polygon or mesh progressive refinement rule (similar

to B-spline) Smooth? use variance reduction!

− “corner cutting”

Chaikin

(½, ½)

43

How Chaikin’s algorithm works?

Qi = ¾ Pi + ¼ Pi+1

Ri = ¼ Pi + ¾ Pi+1

Ex: do Chaikin on a square

44

Subdivision Surfaces

Topology defined by the control polygon

Progressive refinement (interpolation or approximation)

Loop

Butterfly Catmull-Clark

= B-spline at regular vertices

45

Example : Butterfly Subdivision Surface

Interpolate Triangular Uniform &

Stationary Vertex insertion

(primal) 8-point

a = ½, b = 1/8 + 2w, c = -1/16 – w

w is a tension parameter

w = –1/16 => surface isn’t smooth

46

Example: Doo-SabinWorks on quadrangles;

Approximates

47

Comparison

Catmull-Clark

(primal)

Doo-Sabin

(dual)

48

49

50

At curve point / regular surface verticesSplines as limit of subdivision

schemes

Loop

Quartic uniform box splines

Catmull-Clark

Cubic uniform B-spline surface

Doo-Sabin

Quadratic uniform B-spline surface

Chaikin

Quadratic uniform B-spline curve

Today's planning

1.Geometry representations: introduction

2.Point sets

3.Meshes

4.Splines and parametric surfaces

5.Subdivision surfaces

6.Implicit surfaces

52

Implicit surfacesDefined by an Implicit Equation

S = { P(x,y,z) / f(x,y,z) = iso }

(f: R3→ R) is the «field function» Surface normal : N = - ∇ f Characterizes a volume! f (x,y,z) > iso

− “in/out test” (used for collisions, ray tracing…)

Smoothness: S and f have same degree of continuity

N

P

F>iso

F<iso

53

History: Solid Geometry

Volumetric primitives S = { P(x,y,z) / f(x,y,z) = iso }

Spheres, ellipsoids

Cylinders, cones

Super-ellipsoids

12

2

2

2

2

2

=++c

z

b

y

a

x

1=++n

n

n

n

n

n

c

z

b

y

a

x

54

Constructive Solid Geometry

Developed for CAGD Solid primitives Boolean operators

− Union (or)− Intersection (and)− Difference (not)

Construction tree

Describes the history of construction in a compact, intuitive way

-

∪

∩

55

Problem: limited shapes

Free form primitives ? S = { P(x,y,z) / f(x,y,z) = iso }

f polynomial (algebric surface), or other smooth function

What should the equation of f be to model this?

How can a user control an implicit shape?

− Intuitive control− Locality− Enable deformations

56

Idea (1982) Blinn Objects “Blobs”

Primitive generated by

points S− f decreasing function of the

distance

Union : f = max (f1,f2)

Intersection : f = min (f1,f2)

Blending : f = f1 + f2

sp

f

Distance

57

Idea (1982) Blinn Objects “Blobs”

Exponential field + Very smooth − No local control − Everything is to be recomputed if a point

moves

Extension to blend primitives of different

sizes

2

)( 2SiPd

i ef−−

=field

Distance to centre

2

2

)(

)(

i

i

SR

SPd

ii ekf−−

=

58

Displaying implicit surfaces?Ray Tracing [Blinn 82]

• Use dichotomy to compute ray/surface intersections

Later extensions• Analytical solutions for intersection• Sphere tracing

– adapt the step size based on Lipschitz constants

Several hours for rendering from a single view-point!

59

Make implicit surfaces local?(1985-1990)

Field function with compact support!

− piece-wise polynomial functions in d(P,Si)2

Metaballs [Nishimura 1985] − if 0<d<1/3 fi = 1- 3 d2

− if 1/3<d<1 fi = 3/2 (1-d2)

Soft Objects [Wyvill et al. 1986]- if 0<d<1 fi = -4/9 d6 + 17/9 d4 – 22/9 d2 +1

field

Distance to centre

60

Improving visualization?(1985-1990)

Marching cubes [Lorensen Cline 87]

– Space grid– Facetize voxels that cross the surface– Mesh can be viewed from different

viewpoints– Extension: file to follow the surface

61

Improving visualization?(1985-1990)

Marching cubes [Bloomenthal 1993-1994]

– Evaluation of implicit surface tilers [– An implicit surface polygonizer (paper +

code in C)

62

Extensions (1990-2000)

Skeleton-based Implicit Surfaces

Idea: Use any primitive Si as a skeleton

S = { P / ∑ fi (P) = iso }

fi decreasing function of d(P,Si)

Intuitive control, deformation, change of topology

fi

Distanceiso

e Point, segments, disc, cylinder

Appendix: internship proposals1. Animated mesh construction

from scans of articulated objects

2. Creation of dynamic hair strands from 2D linesWith Florence Bertails (BiPop team)

3. Classification of non-manifold objects based on topological propertiesWith Jean-Claude Léon

http://evasion.imag.fr/Membres/Franck.Hetroy/Students/internships.html

Proposal 1: animated mesh

Objective: from scans of an object under different poses, create a single animation

Matching between created meshes

Interpolation frames Related MoSIG courses:

Computer Graphics (modeling + animation)

(Computational Geometry) (Computer Vision)

Courtesy L. Belcour, E. Heitz and A. Masson-Sibut

Proposal 2: dynamic hair strands

Objective: from a 2D sketched curve + physical model, create a 3D model of a hair strand

Geometric reconstruction of a 3D curve

Attach physical parameters For realistic animation

Related courses: MoSIG Computer Graphics (animation) (Simple) geometry + mechanics

Proposal 3: non-manifolds

Objective: classify non-manifold surfaces based on topology Maths + algorithms

Applications in CAD, architecture, graphics, ...

Collaboration with Univ. Genova 3 weeks in Italy possible

Related MoSIG course: Computational Geometry

To do for next week

Read the papers Understand the papers Find, read and understand other papers/book

chapters− In case things are not clear

Next week: Problem to solve in teams

− Related to these papers