CST 494/598 Intro to Applied Computer Graphics Anshuman Razdan [email protected].

CST 494/598

Intro to Applied Computer Graphics

Anshuman [email protected]

A Razdan 3

Problem Statement

The physics of how light interacts with the environment is very complicated we need (lots of) simplification

Ingredients:models to describe light sources

models to describe material properties(surface properties)

rendering algorithms

A Razdan 4

Overview

Important Rendering MethodsHardware-based rendering: based on rasterization

Software-based rasterization

Ray tracing

Radiosity

Path Tracing

Photon Mapping

Boundaries are not clear! - lots of overlap

A Razdan 5

Overview

Many specific “features”environment mapping

texture mapping

bump mapping

transparency

fog

displacement mapping

shadow mapping

motion blur

…

How to integrate “features” into different rendering algorithms?

A Razdan 6

Light Sources

point light sourcedirectional point light sourcedistributed light source (“area light source”)

A Razdan 7

Surface Lighting Effects

diffuse reflection

specular reflection

reflections fromother surfaces

A Razdan 8

Basic Illumination Models

empirical modelslighting calculations

surface properties (glossy, matte, opaque,…)

background lighting conditionslight-source specificationreflection, absorption

ambient light (background light) Iaapproximation of global diffuse lighting effects

A Razdan 9

Ambient Light Reflection

constant over a surfaceindependent of viewing directiondiffuse-reflection coefficient kd (0 kd 1)

adambdiff IkI

A Razdan 10

Illumination and Shading

shaded surfaces generate a spatial impression

the flatter light falls on a surface,the darker it will appear

therefore:we need the incident light direction

or the position of the (point) light source

A Razdan 11

..

L

L

I = IL · cos

Lambert's Law

when considering the material:

I = kd Il cos

l

l

l

A Razdan 12

Lambertian (Diffuse) Reflection

ideal diffuse reflectors (Lambertian reflectors)brightness depends on orientation of surface

Lambert’s cosine law

cos, lddiffl IkI

A Razdan 13

Diffuse Reflection

angle of incidence [0°,90°] surface illuminated

)(, LNIkI lddiffl

A Razdan 14

Diffuse Reflection Coefficient

varying kd

result for varying values of kd, Ia = 0

0 0.2 0.4 0.6 0.8 1

A Razdan 15

Ambient and Diffuse Reflection

total diffuse reflection )( LNIkIkI ldaal,diff

(sometimeska for ambientlight)

A Razdan 16

Specular Highlights

this area must be lighter than the shading model calculates, because the light source is reflected directly into the viewer's eye

A Razdan 17

Specular Reflection Model

reflection of incident light around specular-reflection angle

empirical Phong model

snll,spec II cos

A Razdan 18

cos 8cos 64cos 128cos 256cos

Specular Reflection Coefficient ns

sncos

A Razdan 19

cos

64cos

256cos

Specular Reflection Coefficient ns

sncos

A Razdan 20

Specular Reflection Coefficient

empirical Phong modelns large shiny surface

ns small dull surface

snll,spec II cos

A Razdan 21

Fresnel Specular Refl. Coefficient

Fresnel’s laws of Reflectionspecular reflection coefficient W()

snll,spec IWI cos)(

specular reflection coefficient as a function of angle of incidence for different materials

A Razdan 22

NLNLR )2(

calculation of R:

Simple Specular Reflection

W() constant for many opaque materials (ks)

snlsl,spec RVIkI )(

LNLNR )2(

A Razdan 23

Specular Reflection Results

snlsl,spec RVIkI )(

A Razdan 24

Simplified Specular Reflection

simplified Phong model with halfway vector H

snlsspec RVIkI )( sn

lsspec HNIkI )(

VLVL

H

A Razdan 25

Diffuse and Specular Reflection

n

l

nlsldlaa

sHNkLNkIIkI1

)()(

ambient

ambient+ diffuse

ambient+ diffuse

+ specular

A Razdan 26

Other Aspects

intensity attenuation with distance

anisotropic light sources (Warn model)

transparency (Snell’s law)

atmospheric effects

shadows

…

A Razdan 27

Polygon-Rendering Methods

application of illumination model to polygon rendering

Flat shading: constant-intensity for polygon

Gouraud Shading: compute light interaction at vertices + interpolate colors

Optimal Solution: compute lighting per pixel

Flat Gouraud

A Razdan 28

Polygon Shading: Interpolation

the shading of a polygon is not constant, because it normally is only an approximation of the real surface interpolation

Gouraud shading: interpolate intensities

Phong shading: interpolate normal vectors

A Razdan 29

Gouraud Shading Overview

intensity-interpolationdetermine average unit normal vector at each polygon vertexapply illumination model to each vertexlinearly interpolate vertex intensities

n

kk

n

kk

V

N

N

N

1

1

A Razdan 30

1. find normal vectors at corners and calculate shading (intensities) there: Ii

Gouraud Shading

I 1

I 2

I 3

A Razdan 31


2. interpolate intensities along the edges linearly: I, I’

Gouraud Shading

I 1

I 2

I 3

I= t·I 1+(1-t)·I 2 I'=u·I 2+(1-u)·I 3

A Razdan 32


2. interpolate intensities along the edges linearly: I, I’

3. interpolate intensities along scanlines linearly: Ip

Gouraud Shading

I 1

I 2

I 3

I= t·I 1+(1-t)·I 2 I'=u·I 2+(1-u)·I 3

v·I +(1–v)·I '

A Razdan 33

Gouraud Shading

interpolating intensities

221

411

21

244 I

yyyy

Iyyyy

I

545

44

45

5I

xx

xxI

xx

xxI

ppp

A Razdan 34

Gouraud Shading

incremental update

221

11

21

2I

yyyy

Iyyyy

I

21

12

yyII

II

A Razdan 35

Problems of Gouraud Shading

darklight

dark

darkdark

dark

highlights can get lost or growcorners on silhouette remainMachband effect is visible at some edges

A Razdan 36

Gouraud Shading Results

no intensity discontinuities

Mach bands due to linear intensity interpolation

problems with highlights

flat Gouraud

A Razdan 37

Phong Shading

instead of intensities the normal vectors are interpolated, and for every point the shading calculation is performed separately

darklight

dark

A Razdan 38

Phong Shading Principle

normal-vector interpolationdetermine average unit normal vector at each polygon vertex

linearly interpolate vertex normals

apply illumination model along each scan line

A Razdan 39

Phong Shading Overview

1. normal vectors at corner points2. interpolate normal vectors along the edges3. interpolate normal vectors along scanlines

& calculate shading (intensities) for every pixel

P

A Razdan 40

Phong Shading Normal Vectors

normal-vector interpolation

221

1Nyy

yy

121

2Nyy

yyN

A Razdan 41

Phong Shading

incremental normal vector update along and between scan lines

comparison to Gouraud shadingbetter highlights

less Mach banding

more costly

A Razdan 42

Ray Tracing Concepts

visibility calculation

L1

1

shading

L2 2nd light source

2

light source

A Razdan 43


L1

L2

shadows

reflection

shading of the reflected object

A Razdan 44


for whole image:ray through every pixel

shadows

reflection

visibility calculation

shading

for perspectiveprojection: eye point

A Razdan 45

Ray-Tracing Principle

tracing a ray from the projection reference point through a pixel position with multiple reflections and transmissions

A Razdan 46

Ray-Tracing Properties

highly realistic imagesvery time consumingglobal reflection, transmissionvisible-surface

detectionshadowstransparencymultiple light

sources

A Razdan 47

Ray-Tracing

principles of geometric optics

ray-tracing coordinate reference frame

A Razdan 48

Shading: Diffuse Shading

Id … illumination caused by diffuse shadingxxx … any shading model

(Phong, Blinn, Cook/Torrance,…)

Id = xxx

A Razdan 49

Ray-Tracing: Shadows

P

L0

a light source influences the result only if there is no intersection with 0 < s < 1

ray = intersection point + s vector to light source

ray = P + s (L0 – P)

P …intersection pointL0…light source position

A Razdan 50

Ray-Tracing: Shadows and Shading

shadow ray along Lambient light

diffuse reflectionspecular reflection

kaIa

kd(N.L)

ks(H.N)ns

unit vectors at the surface of an object

intersected by an incoming ray along

direction u

Id = kdIa + kd(N.L)

+ ks(H.N)ns

A Razdan 51

Ray-Tracing: Reflection

Ir = kr · Xr

=

Hr

Ir … illumination caused by reflexionkr … reflection coefficient of the materialXr … shading in the reflected direction

Xr

A Razdan 52

Ray-Tracing: Transparency

A

B

It = kt · Xt

It … illumination caused by transparencykt … transparency coefficient of the materialXt … shading in the transparency direction

Xt

A

B

sinA::sinB = B:A

A Razdan 53

Ray-Tr.: Reflection & Transparency

reflection ray

transparency raySnell’s law

ir

ir

sinsin

R = u (2u.N)N

A Razdan 54

Ray-Tracing: Rays & Ray Tree

primary, secondary rays

corresponding binary ray-tracing tree

reflection and refraction ray paths for one pixel

A Razdan 55

Ray-Tracing: Complete Shading Method

I = Id + Ir + It

additional requirement: kd + kr + kt 1

A Razdan 56

4.IF surface of S is transparent THEN trace secondary ray; shading+=influence of transp.

3.IF surface of S is reflective THEN trace secondary ray; shading+=influence of refl.

2.FOR all light sources L DO trace shadow feeler S -> L IF no inters. between S, L THEN shading+=influence of L

1.trace primary ray eye -> p find closest intersection S

FOR all pixels p DO

Ray-Tracing: Basic Algorithm

A Razdan 57

Ray-Tracing Examples

A Razdan 58


A Razdan 59

True Global Illumination Example

A Razdan 60

„Professional“ Ray tracing

www.oyonale.com

A Razdan 61

Requirements for Object Data(to use them for ray-tracing)

intersection calculation ray - object possible

surface normal calculation possibleB-Rep: simple

CSG: recursive evaluation

A Razdan 62

Ray-Surface Intersection

ray equation

for primary rays

for secondaryrays

s.uPP 0

prppix

prppix

PP

PPu

u = R

u = T

describing a ray with an initial-position vector P0 and unit direction vector u

A Razdan 63

r

P0

Pc

Pu

Ray-Sphere Intersection

parametric ray equation inserted into sphere equation

022 rPP c

022 0 rPsu P c

0 (u² = 1))(2 222 rPsPus

222)( rPPuPus

0PPP c

( )

( )

A Razdan 64

Ray-Sphere Intersection

discriminant negative no intersections

222)( rPPuPus

“sphereflake”:7381 spheres

3 light sources

22 )( uPuPrPus because u2=1

(to avoid roundoff errors when r2 << |P|2)

A Razdan 65

Sphereflake

A Razdan 66

Ray-Polyhedron Intersection

use bounding sphere to eliminate easy caseslocate front facessolving plane equation

uNPND

s 0

u.N < 0

Ax + By + Cz + D = 0

N.(P0 + su) = DP0

N = (A, B, C)N.P = D

P

P

A Razdan 67

Ray-Polyhedron Intersection

intersection point inside polygon boundaries?

inside-outside test

smallest s to insidepoint is first intersectionpoint of polyhedron

A Razdan 68

Ray-Surface Intersection

quadric, spline surfaces:parametric ray equation inserted into surface definition

methods like numerical root-finding, incremental calculations

ray-traced scene with global

reflection of texture

A Razdan 69

Reducing Object-Intersection Calc.

bounding volumesbounding volume hierarchies

A Razdan 70


space-subdivision methodsregular grid

octree

A Razdan 71


space-subdivision methodsincremental grid traversal

3D Bresenham processing of potential exit faces

ray traversal through a subregion of a cube enclosing a scene

A Razdan 72

Incremental Grid Traversal

ray direction u / ray entry position Pinpotential exit facesnormal vectors

check signs of components of u

0 kNu

)1,0,0(

)0,1,0(

)0,0,1(

kN

A Razdan 73

calculation of exit positions, select smallest sk

example:


usPP kinkout

,

DkPN koutk

,

uN

PNDksk

inkk

)0,0,1(kN

x

kk u

xxs 0

A Razdan 74


variation: trial exit planeperpendicular to largest component of uexit point in 0=> done{1, 2, 3, 4} => side clear{5, 6, 7, 8}=> extra calc.

sectors of the trial exit plane

A Razdan 75


1298 polygons, 4 spheres, 76 cylinders, 35 quadrics5 light sources

A Razdan 76

http://www.eleves.ens.fr:8080/home/massimin/ray/povray.html.en


A Razdan 77

Ray-Tracing Example

A Razdan 78

1 Billion Raytraced Triangles

www.openrt.de

A Razdan 79

Radiosity Method

describes the physical process of light distribution in a diffuse reflecting environment

areas that are not illuminated directly are also not completely dark

every object acts as a secondary light source

A Razdan 80

Radiosity

Radiosity B is the „radiant flux per unit area“ that is leaving a surface

A Razdan 81

Radiosity Equation

hemi

k. d B

incoming light from the environment

self emission(only for light sources)

reflected lightfrom environment

B = E + k. d B

hemi

radiosityof the point

E

hemi

I(x) dx =

hemi

d B

A Razdan 82

Radiosity Equation

to calculate the light influence between surfaces

B = E + k. d B

hemi

Radiosity = total light leaving a surface point

B…radiosity hemi…half space over pointE…self emission k…reflection coefficient

A Razdan 83

Radiosity Properties

diffuse interreflections in a sceneradiant energy transfersconservation of energy, closed environmentssubdivision of sceneinto patches withconstant radiosity Bk

A Razdan 84

Radiosity: Subdivision into Patches

Bk = Ek + k . Fjk.Bj

j=1

n

the scene is discretized into n "patches"

(plane polygons) Pk, for each of these

patches a constant radiosity Bk is assumed:

k diffuse reflection coefficient of patch k

Fjk “formfactor”: describes how much % of the

influence on patch k comes from patch j; geometric size

B = E + k. d B

hemi

A Razdan 85

Radiosity Model

Bk radiosity of patch k

Ek self-emission of patch k

BjFjk contribution of other patches

Fjk form factor, contribution of Bj to Bk

k reflectivity factor of patch k

kj

j jkkkk B FEB

A Razdan 86

Radiosity Equation

solving the radiosity equation

kj

j jkkkk B FEB

kkj

j jkkk EB FB

nn2n1n

2

1

E

EE

B

BB

FF

FFFF

...

...

...

...

2

1

2

1

nn

n2122

n1211

11

......

......

A Razdan 87

Projection of a Polygon

aA

. .

a·cos

A·cos

a . cos

A . cos

A Razdan 88

Radiosity: Form Factors

form factor Fjk: contribution of patch j to patch k

2r

Ak

cos kcos j

Ak

Ak'

Pj

.

j k

.

r

FP Aj k

Fjk = 1

,F dAdA kj d

A Razdan 89


2,

coscos

r

dAF kkj

dAdA kj

form factor Fjk: contribution of patch j to patch k

kA k

kj dAr 2

coscos

,F AdA kj

emitted energyfrom dAj on Ak

j kA A jk

kj

jjk dAdA

rAF 2

coscos1

form factor isaverage overarea Aj

A Razdan 90


form factor properties

conservation of energy

uniform light reflection

no self-incidence

11

n

kjkF

0jjF

AjFjk AkFkj

A Razdan 91


form factor calculationmost expensive step in radiosity calculation

numerical integration (Monte Carlo methods)

hemicube approach (hemisphere replaced by hemicube)

Pk

Pj

A Razdan 92

Radiosity Equation

solving the radiosity equationGaussian elimination

Gauss-Seidel iteration

very time and storage intensive

nn2n1n

2

1

E

EE

B

BB

FF

FFFF

...

...

...

...

2

1

2

1

nn

n2122

n1211

11

......

......

A Razdan 93

Radiosity Equation

solving the radiosity equationGauss-Seidel iteration

kj

ij jkkk

ik B FEB 1

Pk

Bj

“gathering”

A Razdan 94

Radiosity Equation

“gathering” vs.“shooting”

kj

ij jkkk

ik B FEB 1

Pk

B j

Pk

B k

x

x

x

x

x

xxxxxxx

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

A Razdan 95

Progressive Refinement Radiosity (1)

“shooting”select brightest patch k and distribute its radiosity Bk

kj

j jkkkk B FEB j jkkk B FB j B todue

k kjjj B FB k B todue

kk

jjkjj B

AA

FB k B todue jkjkjk FAFA

A Razdan 96

Progressive Refinement Radiosity (2)[one refinement step]

for selected patch k/* set up hemicube, calculate form factors Fjk */

for each patch j {rad := j*Bk*Fjk*Aj/AkBj := Bj + radBj := Bj + rad}

Bk := 0

A Razdan 97

Progressive Refinement Radiosity (3)

initially Bk=Bk=Ek,

select patch withhighest BkAk

cathedral rendered with progressive

refinement radiosity

form factors computed with ray-

tracing methods

A Razdan 98

Radiosity Example Images (1)

image of a constructivist museum rendered with progressive refinement radiosity

A Razdan 99


stair tower of a building at Cornell University rendered with progressive refinement radiosity

A Razdan 100


2 lighting schemes for an opera production:(left) day view (right) night view

A Razdan 101

Environment Mapping (1)

reflection mappingdefined over the surface of an enclosing universe (sphere, cube, cylinder)

A Razdan 102

Blinn, CACM 76

Haeberli/Segal


A Razdan 103


information in the environment mapintensity values for light sourcesskybackgroundobjects

pixel areaprojected ontosurfacereflected onto environmentmap

A Razdan 104


A Razdan 105

Adding Surface Detail (1)

most objects do not have smooth surfacesbrick wallsgravel roadsshag carpets

surface texture required

A Razdan 106

Adding Surface Detail (2)

modeling surface detail with polygonssmall polygon facets(e.g., checkerboard squares)

facets overlaid on surface polygon (parent)

parent surface used for visibility calculations

facets used for illumination calculations

impractical for intricate surface structure

A Razdan 107

Texture Mapping (1)

texture patternsmapped ontosurfaces

texture pattern:raster imageor procedure (modifies surface intensities)

A Razdan 108

Texture Mapping (2)

texture mappingtexture scanning (s,t)(x,y)inverse scanning (x,y)(s,t)

texture-surface transformation

vvv

uuu

ctbsatsvv

ctbsatsuu

),(

),(

A Razdan 109

Texture Mapping (3)

projecting pixel areas to texture space =

= inverse scanning (x,y)(s,t)

calculation ofanti-aliasing with filter operations

VPM 1-TM 1-

A Razdan 110

Texture-Map.: Cylindrical Surface

u , with 0 /2

v z with 0 z

x rcos u,y rsin u,z v

MVP

A Razdan 111

M-1VP

pixel surface point (x,y,z)(x,y,z) (u,v):

Texture-Map.: Cylindrical Surface

M-1T

tvsu ,2/

zvxyu ),/(tan 1

vtus ,/2

MT

A Razdan 112

Texture Mapping: Anti-aliasing

anti-aliasing with filter operationsproject largerpixel area intotexture space

mip-mapping

summed-areatable method

A Razdan 113

Procedural Texturing (1)

procedural texture definitiontexture_function (x,y,z) returns intensity

avoid MT

2D (surface texturing) or 3D (solid texturing)

stochastic variations (noise function)

exampleswood grains

marble

leaves

Copyright Alias|Wavefront

A Razdan 114


a scene with surface characteristics generated using solid-texture methods

A Razdan 115


gem faces:polygonal facets

rest: quadric & bicubic patches

surface texturing

procedural texturingfor steamy jungle atmosphere

A Razdan 116

Bump Mapping (1)

bumps are visible because of shading

modeling of bumps is very costly.trick: insert a detail structure T:

T

v

u

+ =

A Razdan 117

Bump Mapping Examples

Blinn, SIGGRAPH78 Blinn, SIGGRAPH78 Blinn, SIGGRAPH78

A Razdan 118

Perlin, SIGGRAPH85

Bump Mapping (2)

surface roughness simulatedperturbation function variessurface normal locally

bump map b(u,v)

),( vuP

vu PPN NNn /

surface point

surface normal

nvubvuPvuP ),(),(),( modified surface point

A Razdan 119

Bump Mapping (3)

nvubvuPvuP ),(),(),(

vu PPN

uuuu bnnbPbnPu

P

)(

, , nbPPnbPP vvvuuu

)()()( nnbbPnbnPbPPN vuvuuvvu

0nn

)()( vuuv PnbnPbNN


A Razdan 120

Bump Mapping (4)

bump map b(u,v) defined as raster imagebu, bv: approximated with finite differences


A Razdan 121

Displacement Mapping

“Correct version of bump mapping”Surface points are moved from their original positionOutline of object changesMuch harder to implement than bump mapping

Rare in practice

Latest hardware partially supports it

A Razdan 122

Combination of Mapping Methods

armor surface =bump mapping +

environment mapping +

texture mapping

CST 494/598 Intro to Applied Computer Graphics Anshuman Razdan [email protected].

Documents

ambient light slide

point light source slide

surfaces slide

flatter light

spec iicos slide

lots of overlap slide

light sources models

lniki lddiffl slide