ray tracing - fabio...

$: ray tracing - Fabio Pellacinipellacini.di.uniroma1.it/teaching/graphics13a/slides/03_raytracing.pdf · raytracing algorithm for each pixel {determine viewing direction intersect ray$
ray tracing

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image formation

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image formation

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rendering

computational simulation of image formation

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rendering

given viewer, geometry, materials, lights

determine visibility and compute colors

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raytracing

a specific rendering algorithm

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raytracing algorithm

for each pixel {

determine viewing direction

intersect ray with scene

compute illumination

store result in pixel

}

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vector math review

point: location in 3D space

vector: direction and magnitude

P = ( , , )Px Py Pz

v = ( , , )vx vy vz

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vector math review

dot product

cross product

is orthogonal to and

a ⋅ b = |a||b| cos θ

|a × b| = |a||b| sin θ

a × b a b

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vector math review

segment: set of points (line) between two points

with P(t) = A + t(B − A) t ∈ [0, 1]

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vector math review

ray: infinite line from point in a given direction

with P(t) = E + td t ∈ [0, ∞]

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vector math review

coordinate system aka frame

frame : position and orthonormal

axes

default (or world) frame: origin and three major axes

f = { , , , }fO fx fy fz

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vector math review

point coords are defined wrt a frame

wrt (world if not

specified)

P = ( , , )Px Py Pz { , , , }fO fx fy fz

P = ((P − ) ⋅ , (P − ) ⋅ , (P − ) ⋅ )fO fx fO fy fO fz

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vector math review

change of coordinate system

is w.r.t

f → f ′

= ( , , )P′ P ′x P ′

y P ′z P { , , , }f ′

O f ′x f ′

y f ′z

= ((P − ) ⋅ , (P − ) ⋅ , (P − ) ⋅ )P′ f ′O f ′

x f ′O f ′

y f ′O f ′

z

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vector math review

change of coordinate system

is w.r.t

→ ff ′

= ( , , )P′ P ′x P ′

y P ′z P { , , , }f ′

O f ′x f ′

y f ′z

P = + + +f ′O P ′

xf ′x P ′

y f ′y P ′

z f ′z

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vector math review

vector coords are defined wrt a frame

to change coord system, ignore origin

v = + +v′xf ′

x v′y f ′

y v′zf ′

z

= (v ⋅ , v ⋅ , v ⋅ )v′ f ′x f ′

y f ′z

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vector math review

construct a frame from two non-orthonormal vectors ,

assume that is not parallel to

construct a frame from a vector

pick arbitrary and continue as above

z′ y′

z′ y′

z = /| |z′ z′

x = × z/| × z|y′ y′

y = z × xz′

y′

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vector math review

infinite plane

with

normal:

P ∈ plane ; (P − C) ⋅ n = 0 ; P ⋅ n = d

P(u, v) = C + u ⋅ u + v ⋅ v (u, v) ∈ (−∞, ∞)2

n = u × v

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vector math review

triangle baricentric coordinates

with

, ...

P(α, β, γ) = αA + βB + γC α + β + γ = 1P(α, β) = α(A − C) + β(B − C) + Cα = area(BCP)/area(ABC)

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vector math review

sphere

P ∈ sphere ; |P − C| = R

P(u, v) = C + R ⋅ (cos ϕ sin θ, sin ϕ sin θ, cos θ)

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viewing

for each pixel {

-> determine viewing direction


compute illumination


}

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viewer model

a painter tracing objects on a canvas in front

[Marschner 2004 – original unknown]22

viewer model

equivalent to pinhole photography

[Marschner 2004 – original unknown] 23

viewer model -- parameters

camera frame: position and orientation , ,

image plane: distance and size ,

O x y zd w h

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view frustum

all visible points within a truncated pyramid

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generating view rays

for each pixel, ray from camera center to the pixel center

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ray:

point on image plane

r = O + t(Q − O)/|Q − O|Q

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image plane params: , origin at bottom

Q(u, v) = (u − 0.5)wx + (v − 0.5)hy − dz(u, v) ∈ [0, 1]2

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geometry model

simple shapes

spheres, quads, traingles

complex shapes

handled as collections of simple shapes later in the course

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ray-shape intersection

determine visible surface by finding closest intersection

along a ray

ray with

keep explicit bounds on

e.g. used in shadows and to improve numerical precision

if not specified otherwise: ,

mitigate numerical precision issues ("shadow acne")

value is scene depedent: start with

r : E + td t ∈ ( , )tmin tmax

t

= ϵtmin = ∞tmax

ϵ

10−5

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ray-sphere intersection

point on a ray:

point on a sphere:

by substitution:

P(t) = E + td|P(t) − C| = R

|E + td − C| = R

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algebraic equation:

with: , ,

determinant:

no solution for

a + bt + c = 0t2

a = |d|2 b = 2d ⋅ (E − C) c = |E − C −|2 R2

d = − 4acb2

d < 0

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two solutions:

pick smallest such that

= (−b ± )/(2a)t± d√

t t ∈ ( , )tmin tmax

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shading frame at with normal with

and

and

, where ,

P = fO n = fz

P = E + td = (P − C)/RPl

θ = arccos P lz ϕ = arctan( , )P l

y P lx

f = {P, x, y, }Pl x = (sin ϕ, cos ϕ, 0)y = (cos θcos ϕ, cos θsin ϕ, sin θ)

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ray-plane intersection

point on a ray:

point on a plane:

by substitution:

P(t) = E + td(P(t) − C) ⋅ n = 0

(E + td − C) ⋅ n = 0

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one solution for , no/infinite solutions otherwise

check that

d ⋅ n ≠ 0

t =(C − E) ⋅ n

d ⋅ n

t ∈ ( , )tmin tmax

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shading frame: f = {e + td, u, v, n}

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ray-triangle intersection

point on ray:

point on triangle:

by substitution:

P(t) = E + tdP(α, β) = α(A − C) + β(B − C) + C

E + td = α(A − C) + β(B − C) + C

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E + td = α(A − C) + β(B − C) + C →

α(A − C) + β(B − C) − td = E − C →

αa + βb − td = e →

[ ] = e−d a b⎡⎣

t

α

β

⎤⎦

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use Cramer's rule

test for

t = =| |e a b

| |−d a b(e × a) ⋅ b(d × b) ⋅ a

α = =| |−d e b| |−d a b

(d × b) ⋅ e(d × b) ⋅ a

β = =| |−d a e| |−d a b

(e × a) ⋅ d(d × b) ⋅ a

t ∈ ( , ), α ≥ 0, β ≥ 0, α + β ≤ 1tmin tmax 40


shading frame:

create frame by orthonomalization with

,

f = {e + td, u, v, n}

= (B − A) × (C − A)z′ = (B − A)x′

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intersection and coord systems

transform the object

simple for triangles, since they transforms to triangles

but objects may require more complex intersection tests

transform the ray

much more elegant

works on any surface

allow for much simpler intersection tests

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intersection and coord systems

ray wrt (e.g. world)

object defined wrt (in turn defined wrt )

transform rays

transform origin/direction as point/vector

intersect object with transformed ray

use standard intersection tests

transform intersection frame back to

transform origin/axes as point/vectors

r = {E, d} fo′ f ′ f

= { , }r′ E′ d′

o′ r′

f

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image so far

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intersecting many shapes

intersect each primitive

pick closest intersection

essentially a line search

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intersecting many shapes --pseudocode

minDistance = infinity

hit = false

foreach surface s {

if(s.intersect(ray,intersection)) {

if(intersection.distance < minDistance) {

hit = true;

minDistance = intersection.distance;

}

}

}

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image so far

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shading

for each pixel {

determine viewing direction


-> compute illumination


}

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shading

variation in observed color across a surface

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shading

compute reflected light

depends on:

view position

incoming light, i.e. lighting

surface geometry

surface material

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real-world materials

Metals Dielectric

[Marschner 2004] [Marschner 2004]

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real-world materials

Metals Dielectric

[Marschner 2004] [Marschner 2004] 52

shading models

empirical models

produce believable images

simple and efficient

only for simple materials

physically-based shading models

can reproduce accurate effects

more complex and expensive

will concentrate on empirical models first

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shading model

shading model: diffuse + specular reflection

diffuse reflection

light is reflected in every direction equally

colored by surface color

specular reflection

light is reflected only around the mirror direction

white for plastic-like surfaces (glossy paints)

colored for metals (brass, copper, gold)

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incident light

beam of light is more spread on oblique surfaces

incident light depends on angle

light fraction: f = |n ⋅ l|

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surface reflectance

surface reflectance is described by the BRDF, bidirectional

surface distribution functions

BRDF is simple for simple shading models

in general, the BRDF is a function of incoming and outgoing

angles

is the direction from the point to the light

is the direction from the point to the viewer

is the local shading frame that describes surface

orientation (normal and tangent)

ρ(l, v; f)lvf

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lambert diffuse model

simple and efficient diffuse model

light is scattered uniformly in all directions

brdf:

surface color:

(l, v; f) =ρd kd

= (l, v; f) ⋅ |n ⋅ l| = |n ⋅ l|Cd ρd kd

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lambert diffuse model

produce matte appearance

left-to-right: increasing kd

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image so far

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phong specular model

empirical, used to look good enough

cosine of mirror and view direction

reflected direction:

brdf:

r vr = −l + 2(n ⋅ l)n

(l, v; f) = max(0, v ⋅ rρs ks )n

= (l, v; f) ⋅ |n ⋅ l| = max(0, v ⋅ r ⋅ |n ⋅ l|Cs ρs ks )n

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phong specular model

produces highlight, shiny appearance

left-to-right: increasing , top-to-bottom: increasing n ks

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blinn specular model

slightly better than Phong

cosine of bisector and normal

bisector:

brdf:

h nh = (l + v)/|l + v|

(l, v; f) = max(0, n ⋅ hρs ks )n

= (l, v; f) ⋅ |n ⋅ l| = max(0, n ⋅ h ⋅ |n ⋅ l|Cs ρs ks )n

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image so far

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lighting

patterns of illumination in the environment

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lighting

determines how much light reaches a point

depends on:

light geometry

light emission

scene geometry

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light source models

describe how light is emitted from light sources

empirical light source models

point, directional, spot

physically-based light source models

area light, sky model

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point lights

light is emitted equally from a point in all directions

simulate local lighting, different at each surface point

light direction:

light color:

SP

l = (S − P)/|S − P|L = /|S − Pkl |2

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directional lights

light is emitted from infinity in one direction

simulate distant lighting, e.g. sun, same at all surface points

light direction:

light color:

d

Pl = d

L = kl

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spot lights

same as points lights, but only emits in a cone around

simulate theatrical lights

cone falloff model arbitrary

light direction:

light color:

d

l = (S − P)/|S − P|L = ⋅ attenutation/|S − Pkl |2

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shading model with multiple lights

add contribution of all lights for diffuse and specular

for Lambert and Phong

for Lambert and Blinn

i

C = ⋅ ( ( , v; f) + ( , v; f)) ⋅ |n ⋅ |∑iLi ρd li ρs li li

C = ⋅ ( + max(0, v ⋅ ) ⋅ |n ⋅ |∑iLi kd ks ri)

n li

C = ⋅ ( + max(0, n ⋅ ) ⋅ |n ⋅ |∑iLi kd ks hi)

n li

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image so far

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illumination models

describe how light spreads in the environment

direct illumination

incoming light comes directly from light sources

shadows

indirect illumination

incoming light comes from other objects

specular reflections (mirrors)

diffuse inter-reflections

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illumination models

[PCG]

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ray tracing lighting model

point/directional/spot light sources

sharp shadows

sharp reflection/refractions

hacked diffuse inter-reflection: ambient term

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ray traced shadows

light contributes only if visible at surface point

no shadow shadow

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ray traced shadows

send a shadow ray to check if light is visible

visible if no hits or if more than light distancet

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ray traced shadows

shadow ray with

spot/point lights at :

directional lights:

scale lighting by visibility term which is 0 or 1

implementation detail: numerical precision

shadow acne: ray hits the visible point

solution: only intersect if , i.e.

r = P + tl t ∈ ( , )tmin tmax

S = length(S − P)tmax

= ∞tmax

(P)Vi

C = ⋅ (P)( + )|n ⋅ |∑iLi Vi ρd ρs li

t > ϵ = ϵtmin

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image so far

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ambient term hack

light bounces even in diffuse environment

ceiling are not black

shadows are not perfectly black

very expensive to compute

approximate (poorly) with a constant term

C = + ⋅ (P)( + )|n ⋅ |kd La ∑iLi Vi ρd ρs li

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ray traced reflections and refractions

perfectly shiny surfaces reflects objects

recursively trace a ray if material is reflective or refractive

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ray traced reflections and refractions

reflections: along mirror direction ,

scaled by

refractions: along refraction direction scaled by

implementation detail: recursion

avoid hitting visible point:

make sure you do not recurse indefinitely

r = −l + 2(l ⋅ n)nkr

kt

C = + ⋅ (P)( + )|n ⋅ | +kd La ∑iLi Vi ρd ρs li

+ raytrace(P, r) + raytrace(P, t)kr kt

> ϵtmin

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image so far

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antialiasing

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antialiasing: removing jaggies

1 sample/pixel

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1 sample/pixel

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1 sample/pixel

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1 sample/pixel

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1 sample/pixel 9 sample/pixel

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poor-man antialiasing:

for each pixel

take multiple samples

compute average

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ray tracing pseudocode

for(i = 0; i < imageWidth; i ++) {

for(j = 0; j < imageHeight; j ++) {

u = (i + 0.5)/imageWidth;

v = (j + 0.5)/imageHeight;

ray = camera.generateRay(u,v);

c = computeColor(ray);

image[i][j] = c;

}

}

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anti-aliased ray tracing pseudocode

for(i = 0; i < imageWidth; i ++) {

for(j = 0; j < imageHeight; j ++) {

color c = 0;

for(ii = 0; ii < numberOfSamples; ii ++) {

for(jj = 0; jj < numberofSamples; jj ++) {

u = (i+(ii+0.5)/numberOfSamples)/imageWidth;

v = (j+(jj+0.5)/numberofSamples)/imageHeight;

ray = camera.generateRay(u,v);

c += computeColor(ray);

}

}

image[i][j] = c / (numberOfSamples^2);

}

}94

image so far

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ray tracing - fabio...

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