TURNER WHITTED’S RAY-TRACING ALGORITHM: PRACTICE

PROGRAMAÇÃO 3D

MEIC/IST

Ray Tracing History

Ray Tracing History

Whitted ray tracing

It combines in a single model:

Hidden surface removal

Shading due to direct illumination

Shading due to indirect illumination (reflection and refraction effects due to reflective/translucid objects)

Shadow computation (hard shadows)

Image Rendering

xo

zo

Objects

Pixel(RGB)

Light

Illumination

yo

Camera

xe

ye

ze eye

Ray Casting

For each pixel in the viewport;

shoot a ray;

for each object in the scene

compute intersection ray-object;

store the closest intersection;

if there is an intersection

shade the pixel using color, lights, materials;

else /* ray misses all objects */

shade the pixel with background color

Shadows Only shades the intersection if not in shadow

Ray Casting

How does RT differ fromRasterization (slide from CS348b: image synthesis by Matt Pharr)

Global Illumination

Global Illumination or Light Transport

Light Transport and Shading The appearance of objects, only depends on

the way light interacts with matter and travels trough space.

Shading: Interaction light-matter

Light transport: determine and follow pathlight rays due to inter-reflections

Forward tracing aka Light Tracing

Backward Tracing

Backward Tracing

Ray-Tracing de Turner Whitted

Define the viewpoint, the view window and the viewport resolutionfor each pixel in the viewport{

compute a ray in World space from the eye towards the pixel;pixel_color = trace ( scene, eye, primary ray direction, 1);

}

• Backward Ray Tracer

• We trace light rays from the eye through a pixel in the viewport - primary rays

• ie we follow light beams in the reverse direction of the light propagation

• Find the intersection with the first object during the backwards trace of the ray

• The color of the ray (hence at the required pixel) is made up of 3 contributions:

-local color due to direct illumination (it can be in shadow) and ambient light ;

-color from a ray coming from the reflection direction – reflected ray;

- color from a ray coming from the refraction direction – transmitted or refratced ray;

• Shadow feelers, reflected and refracted rays are called secondary rays

Recursive nature

Algorithm’s recursive nature

R1

T1

R2

T2

R3

L1

L3

L2

R1

R2

L1

L3L2

T1

T2R3

Result

Mirror Maze

Mirror Maze

Color trace (Scene scene, Vector3d origin, Vector3d ray direction, int depth){

intersect ray with all objects and find a hit point (if any) closest to the start of the rayif (!intersection point) return BACKGROUND;else {

color = object material’s ambient color;compute normal at the hit point;for (each source light) {

L = unit light vector from hit point to light source;if (L • normal>0)

if (!point in shadow); //trace shadow raycolor += diffuse color + specular color;

}if (depth >= maxDepth) return color;

if (reflective object) {rRay = calculate ray in the reflected direction;rColor = trace(scene, point, rRay direction, depth+1);reduce rColor by the specular reflection coefficient and add to color; }

if (translucid object) {tRay = calculate ray in the refracted direction;tColor = trace(scene, point, tRay direction, depth+1);reduce tColor by the transmittance coefficient and add to color; }

return color;}

Rays

Parametric formulation: p = o + td

p: a point on the ray

o: origin of the ray

t: scalar parameter

d: unit vector giving the direction of the ray

The axes in the Camera Frame

Viewing direction

v

u

eyexe

ye

ze

ResX (pixels)width

Res

Y(p

ixel

s)h

eig

ht

viewing window-viewporttransformation

o xe

ye

t (top)

b (bottom)

l (le

ft)

r (r

igh

t)

Camera Position and Orientation

eye = viewerat = target point in the center of viewing window (near plane in OpenGL), up = up direction

World Coordinates

eyeat

fovy

upeye

at

up

x0

z0

y0

near

far

view = at - eye

Camera Frame - xe ye ze (uvn)

at

zeeye

up

z0

y0

x0

view

up

at

eye

z0

y0

x0

data:eye, at, up

view = at - eye

ze 1

eye ateye at

Camera Frame - xe ye ze (u v -n)

at

eye

up

z0

y0

x0

view

zexe

at

eye

up

z0

y0

x0

view

zexe

ye

e

e

e zupzup

x

1

eee xzy eye

up ze

xeye

at

Primary Rays

zw

xw eyeo

d pxy eye

yw

dop ttraio )(:

d

Computing Primary Rays

vuop ˆ)(ˆ)(1 yvxuxy

wsX

xxu

Re)(

hsY

yyv

Re)(

v

h

eexysY

yh

sX

xw yxop ˆ

Reˆ

Re1

(w/2, h/2)

uResX-1o1

pxy

10 2 x3

ResY-1

1

2

y

w

(-w/2, -h/2)

Ray at the left-bottom corner of the unit square pixel

Ray at the center of the square pixel

eexysY

yh

sX

xw yxop ˆ

Re

5.0ˆ

Re

5.01

for unit square pixels, ie. s=1;

Computing Primary Rays

zo

xo

eyeo yo

dop ttray )(:

d

eeefsX

xw

sY

yhd xyzd ˆ

2

1

Reˆ

2

1

Reˆ

d f eye at

Ray at the left-bottom corner of the square pixel

Camera Data in C

struct _Camera {

/* Camera definition*/

Vector eye, at, up;

float fovy;

float near,far; //hither and yon planes

int ResX,ResY;

float w,h;

Vector xe,ye,ze; //uvn frame

};

typedef struct _Camera Camera;

Camera* camCreate( Vector eye, Vector at, Vector up,

double fovy, double near, double far, int ResX, int ResY );

Ray camGetPrimaryRay( Camera camera, double x, double y );

The Camera object

ze 1

eye ateye at e

e

e zupzup

x

1

eee xzy

Initialization:

Ray in parametic form : o + td (normalize d; why?)

eyeo

d f eye at

h 2d f tanfov

2

w ResX

ResYh

Data input: fov, ResX, ResY, near, far, eye, at, up

Given: x, y

eeefsX

xw

sY

yhd xyzd ˆ

2

1

Reˆ

2

1

Reˆ

Ray at the left-bottom corner of the square pixel

Shadow Feeleers

No light at that point

ip

sr̂

sitt rpp ˆ)(:ray Shadow

Intersections computation

• Read the slides “Geometry Intersections”• Self-intersections problem due to floating-point precision (secondary rays and shadow-feelers) •Solution: Slightly offset intersections • Read the article : Andrew Woo et al., “It’s Really Not a Rendering Bug, You See…”, IEEE CG&A, September 1996, vol. 21

Reflection Model for Lambert Surfaces

luz incidente luz

incidenteluz incidente

1. Reflete igualmente em todas as direções2. Diffuse Intensity: linear variation with angle cos

Diffuse Reflection Component

cos

cos

cos

dbb

dgg

drr

b

g

r

kl

kl

kl

I

I

I

- /2 0 /2

cos

1

n̂

L̂

ip

Diffuse Reflection Component

I, l, k [0 ,1 ]

Eq. 1

LnLn

Ln

Ln

Ln

ˆˆˆˆ

ˆˆ

ˆˆ

ˆˆ

db

dg

dr

b

g

r

dbb

dgg

drr

dbb

dgg

drr

b

g

r

k

k

k

l

l

l

kl

kl

kl

kl

kl

kl

I

I

I

cos

cos

cos

dbb

dgg

drr

b

g

r

kl

kl

kl

I

I

I

Ambient plus Diffuse:

Ln ˆˆ

db

dg

dr

b

g

r

db

dg

dr

ab

ag

ar

b

g

r

k

k

k

l

l

l

k

k

k

I

I

I

I

I

I

Local Specular Reflection Component

I, l, k [0 ,1 ]

Eq.2

n̂

L̂

r̂

v̂

-1 -0,5 0 0,5 1

n=4

n=1

n=8

nvr ˆˆ

n

sbb

n

sgg

n

srr

especularb

g

r

kl

kl

kl

I

I

I

cos

cos

cos

Specular area

Local Specular Reflection Component

n

sb

sg

sr

b

g

r

db

dg

dr

b

g

r

db

dg

dr

ab

ag

ar

b

g

r

k

k

k

l

l

l

k

k

k

l

l

l

k

k

k

I

I

I

I

I

I

vrLn ˆˆˆˆ

Local Specular Reflection Component

n̂

L̂

r̂ v̂

ip

luzrefletida

Mirror Reflection Vector

LnnLr ˆˆˆˆ2ˆ

LLh n

h h

Ln

nnLL ˆ)ˆˆ( n

hLr nˆ

n̂

L̂ r̂

Phong Model: Local Reflection Distribution

nsluzdifluzamb kCkCCC vrLn ˆˆˆˆ

n̂

L̂

r̂v̂

ip

Ambient Diffuse Specular

Multiple Light Sources

luzes

n

sb

sg

sr

b

g

r

db

dg

dr

b

g

r

db

dg

dr

ab

ag

ar

b

g

r

k

k

k

l

l

l

k

k

k

l

l

l

k

k

k

I

I

I

I

I

I

vrLn ˆˆˆˆ

vnnvr ˆˆ)ˆˆ(2ˆ r

luzes

n

r

sb

sg

sr

b

g

r

db

dg

dr

b

g

r

db

dg

dr

ab

ag

ar

b

g

r

k

k

k

l

l

l

k

k

k

l

l

l

k

k

k

I

I

I

I

I

I

LrLn ˆˆˆˆ

LnnLr ˆˆ)ˆˆ(2ˆ

First Lab Exercise

Model with multiple lights and shadow

otherwise1

shadowin if0s

f

luzes

n

r

sb

sg

sr

b

g

r

db

dg

dr

b

g

r

s

db

dg

dr

ab

ag

ar

b

g

r

k

k

k

l

l

l

k

k

k

l

l

l

f

k

k

k

I

I

I

I

I

I

LrLn ˆˆˆˆ

Indirect Illumination: Mirror reflectionsfrom other objects

Reflective surface

n̂

v̂

iprr̂

ritt rpp ˆ)(:ray reflected

Indirect illumination: Mirror Reflection Vector

VVr nnr ˆˆ2ˆ

VVn h

h h

Vn

nn ˆ)ˆ( VVn

hr nVrˆ

n̂

Vrr̂

Translucid objects

1

2

h1

h2

n̂

v̂

ip

tr̂

ti ttrefratadoRaio rpp ˆ)(:

1

2

2

1

sin

sin

h

h

Snell law

n̂

t̂

t

t

vv

t1ˆ

Indirect Illumination: Refracted Ray

i

n̂ vnnvv ˆˆ)ˆˆ( t

v̂

ip

tr̂

nv

tvti vsin

i

t

it

h

h sinsin

tt 2sin1cos

)ˆ(cosˆsin ntr ttt

titt rpp ˆ)(:ray refracted

t

Advertência: Refração não é simples!

Objectos refletores e transparentes

redução da

transparência

redução da

reflexão

)(

)(

)(

)1(

)(

)(

)(

ˆˆˆˆ

tb

tg

tr

rb

rg

rr

luzes

n

r

sb

sg

sr

b

g

r

db

dg

dr

b

g

r

s

db

dg

dr

ab

ag

ar

b

g

r

I

I

I

o

I

I

I

k

k

k

k

l

l

l

k

k

k

l

l

l

f

k

k

k

I

I

I

I

I

I

r

r

r

r

r

r

LrLn

Texturas

u

v

1.0

1.0

0 u1.0

1.0

0

v

u1.0

1.0

0

v

Texturas 2D = Imagens onde o domínio é u, v [0,1]×[0,1] R2

Sistemas de coordenada de textura na caixa

u

v

u

v

x

y

z

u

v

(1,1)(0,0)

(0,0)

(0,0)

face x=xmax

minmax

min

yy

yyu

minmax

min

zz

zzv

Sistema de coordenadas de textura na esfera

x

y

z

sinsin

cossin

cos

y

x

z

z

yx

xy

22

tan

/tan

1

1

v

u

)1(

)1(

wuj

hvi

i

j

Sistema de coordenadas de textura no triângulo

u

v

(u1,v1)

(u2,v2)

(u3,v3)

u1= u3 u2

v1= v2

v3

1

2

3

Textura no triângulo e coordenadas baricêntricas

p1

v12

v23

v31

2/)(ˆ

2/)(ˆ

2/)(ˆ

1123

3312

2231

ppvn

ppvn

ppvn

i

i

i

A

A

A

A3

A1A2

321 AAAAT

T

T

T

AAL

AAL

AAL

/

/

/

33

22

11

n̂

p2

p3

int

ip

3

3

3

2

2

2

1

1

1v

uL

v

uL

v

uL

v

u

i

i

)(1 213 LLL

Textura

Texturas de rugosidade (bump textures) eTexturas de deslocamentos (displacement mapping)

Pertubar aleatoriamente

as normais dos objetos

n̂

),(ˆ'ˆ nn

Pertubar aleatoriamente

as posições dos pontos

Textura ambiente(environment maps)

xy

z

u

v

(0,0)

(1,1)

face dos raios x>y e x>z

x

xyu

2

x

xzv

2

y=-x

y=x

z=-x

z=x

Processamento anti-aliasing

•Lance um raio para cada sub-pixel•Faça uma média dos valores obtidos

dx, dy = variáveis aleatórias

dxdya sub-pixels

pixel

Anti-aliasing

(a) original (b) uniforme (c) “jittered”

pixel pixel pixel

Refinamento Progresivo

subdivisão finalsegunda subdivisão

primeira subdivisãoamostragem inicial

pixels sendo visitados pixels já visitados

Radiosidade e Ray Tracing

Klaus Mueller, Stony Brook University, Computer Science (CSE 564)

global illuminationstandard raytracing

Todays State of the Art- Some Snapshots

RT Acceleration

More efficient computations

Hierarchical data structures: Kd-Tree, BVH o Grids