model construction 2001. 2. 14 김 성 남. contents what is modeling? model construction -...

Model ConstructionModel Construction

2001. 2. 14

김 성 남

ContentsContents What is Modeling? Model Construction

- Interactive modeling tools- Scanning tools- Computer vision- Procedural generation • Sweeps • Fractals • Grammars • Particle System • Physically Based Modeling

Summary

ModelingModelingHow do we …

- Represent 3D objects in a computer?- Construct such represents quickly and/or automatically with a computer?- Manipulate 3D objects with a computer?

Model ConstructionModel ConstructionInteractive modeling tools

- CAD programs- Subdivision surface editors

Scanning tools- CAT, MRI, laser, magnetic, robotic arm,…

Computer vision- Stereo, motion, …

Procedural generation- Sweeps, Fractals, Grammars

Interactive Modeling ToolsInteractive Modeling Tools User constructs objects with drawing

programs- Menu commands, direct manipulation, …- CSG, parametric surfaces, quadrics, …

Scanning ToolsScanning ToolsAcquire geometry of objects with

active sensors- CAT / MRI- Laser range scanner- Magnetic sensor- Robotic arm …

Computer VisionComputer VisionInfer 3D geometry from images

- Stereo- Motion- Constraints

Procedural ModelingProcedural ModelingGoal

Describe 3D models algorithmicallyBest for models result from …

- Repeating processes- Self-similar processes- Random processes

Advantages- Automatic generation- Concise representation- Parameterized classes of models

Procedural ModelingProcedural ModelingSweepsFractalsGrammars

SweepSweepUseful for constructing 3D objects that

possess - translational- rotational- other symmetries

P(u) P(u,v)

u

v

v

u

P(u,v)

Generating curve around helico-spiral axis- Model is parameterized Helico-spiral : • z0,λz,r0,Nθ,Δθ

Generating curve : shape, • Nc, λc

definitions Θi+1= Θi + ΔΘ zi+1 = ziλz

SweepSweep

Generate different shells by varying parameters

SweepSweep

GoalFor construction of natural objectswith realistic description- mountain,cloud,tree …

Euclidean equation 보다 빠르고 , 쉽다 .특성

- Infinite detail at every point- Self-similar with infinite resolution

FractalsFractals

Useful for describing natural 3D phenomenon- Terrain- Plants- Clouds- Water- Feathers- Fur …

FractalsFractals

Deterministic self-similar fractals- Parts are scaled copies of original

Statistical self-similar fractals- Parts have same statistical properties as original

FractalsFractals

General procedure- Initiator : start with a shape- Generator : replace subparts with scaled copy of original

Deterministic FractalsDeterministic Fractals

initiator generator

Apply generator repeatedly


nsD = 1s : scaling factorn : number of subparts for subdivision of unit segment.D : fractal similarity dimension


D =ln(1/s)ln n


1DE = 1

1/n s = 1/n, n = 2ns1 = 1

DE = 2 s = 1/n1/2, n = 4ns2 = 1

A

A’ = A/n

DE = 3 s = 1/n1/3, n = 8ns3 = 1

Fractal dimension of an object is alwaysgreater than the Euclidean dimension.


Euclidean dimensionCurve = 1

Fractal dimensionD > 1

D ≈ 1 → smooth curveD = 2 → Peaco curve : 2 dimension2<D<3 → self-intersect

General procedure- Initiator : start with a shape- Generator : replace subparts with a self- similar random pattern

Statistical FractalsStatistical Fractals

Geometric Construction MethodAffine Fractal-Construction MethodRandom Midpoint-Displacement MethodSelf-Squaring Fractal Method

- using to complex space- Mandelbrot set

Self-Inverse Fractal Method


Geometric Construction Method- In general- choose a generator randomly at each step from a set of predefined shapes.- using the twisting,scaling function,…


Affine Fractal-Construction Method- using fractional Brownian motion- random direction and rendom length- Terrains(mountain,valley,ocean,…)- D ≈ 2.15 → realistic mountain


Brownian motion(random walk) in the xy plane

x

y

Random Midpoint-Displacement Method- Fractional Brownian motion → time-consuming


a b

y(a)

y(b)y

xa b

y(a)

y(b)y

xymid

a+b/2

ymid = 1/2 [y(a) + y(b)] + rR : a value from a Gaussian distribution 0 <= r <= |b-a|2H

H = 2 – D, D > 1 is fractal dimension

Self-Squaring Fractal Method- transformation function to point in complex space- z = x + iy, i2 = -1


Self-Inverse Fractal Method- geometric inversion transformation


p

p’

Natural Objects : Terrain,cloud,plant,…


Shape grammars- sets of production rules applied to an initial object to add layers of detail- harmonious with the original shape

L-grammars(graftals)- object : sum of parts- describing plants

GrammarsGrammars

Applying different rules at each step ofthe transformation from initial to final object

Geometric substitution rules to altering

Shape GrammarsShape Grammars

Rule 1 Rule 2 Rule 3

Using for creating plants- object : sum of partsTree(a trunk) → Branches → Leaves

L-Grammars(graftals)L-Grammars(graftals)

Tree = T[T[TL]T[TL][TL]T[T[TL]T[TL][TL][TL]

Display Flude-like property- change over time by flowing,billowing,spattering,expanding,…- clouds,smoke,fire,fireworks,waterfall, water spray,clumps of grass,…

Particle shape can be- spheres,ellipsoids,boxes,…

Particel movement may be controlled byspecified forces such as a gravity field

Particle SystemParticle System

Particle SystemParticle Systemexamples

a clump of grass Waterfall hitting a stone An object disintegratinginto a cloud of particles

Nonrigid object- rope,cloth,rubber ball,…

Interaction of external and internal forcesModeling method

- network of point nodes with flexible connections between the nodes

Spring network

Physically Based ModelingPhysically Based Modeling

Physically Based ModelingPhysically Based Modeling

Examples

SummarySummary Interactive Modeling Tools

- CAD programs- Subdivision surface editors…

Scanning Tools- CAT,MRI,Laser,Magnetic,Robotic arm,…

Computer Vision- Stereo,motion,…

Procedural generation- Sweeps- Fractals- Grammars- Particle Systems- Physically Based Modeling

model construction 2001. 2. 14 김 성 남. contents what is modeling? model construction -...

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