ray tracing 1 image formation 2 image formation 3 rendering - - PowerPoint PPT Presentation

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|sinθ 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

→ f f ′ = ( , , ) P′ P ′

x P ′ y P ′ z

P { , , , } f ′

O f ′ x f ′ y f ′ z

P = + + + f ′

O

P ′

xf ′ x

P ′

yf ′ y

P ′

zf ′ 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′

yf ′ 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 × x z′ 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 = 1 P(α,β) = α(A − C) + β(B − C) + C α = area(BCP)/area(ABC)

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vector math review

sphere

P ∈ sphere ; |P − C| = R P(u,v) = C + R ⋅ (cos ϕ sinθ,sinϕ sinθ,cos θ)

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viewing

for each pixel {

• > determine viewing direction

intersect ray with scene compute illumination store result in pixel }

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viewer model

a painter tracing objects on a canvas in front

[Marschner 2004 – original unknown]

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viewer model

equivalent to pinhole photography

[Marschner 2004 – original unknown]

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viewer model -- parameters

camera frame: position and orientation , , image plane: distance and size ,

O x y z d 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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generating view rays

ray: point on image plane

r = O + t(Q − O)/|Q − O| Q

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generating view rays

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 = ϵ tmin = ∞ 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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ray-sphere intersection

algebraic equation: with: , , determinant: no solution for

a + bt + c = 0 t2 a = |d|2 b = 2d ⋅ (E − C) c = |E − C − |2 R2 d = − 4ac b2 d < 0

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ray-sphere intersection

two solutions: pick smallest such that

= (−b ± )/(2a) t± d √ t t ∈ ( , ) tmin tmax

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ray-sphere intersection

shading frame at with normal with and and , where ,

P = fO n = fz P = E + td = (P − C)/R Pl θ = arccos P l

z

ϕ = arctan( , ) P l

y P l x

f = {P,x,y, } Pl x = (sinϕ,cos ϕ,0) y = (cos θcos ϕ,cos θsinϕ,sinθ)

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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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ray-plane intersection

• ne solution for

, no/infinite solutions otherwise check that

d ⋅ n ≠ 0 t = (C − E) ⋅ n d ⋅ n t ∈ ( , ) tmin tmax

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ray-plane intersection

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

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ray-triangle intersection

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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ray-triangle intersection

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 ≤ 1 tmin tmax

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ray-triangle intersection

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)

• bject

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} f

• ′

f ′ f = { , } r′ E′ d′

• ′

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 intersect ray with scene

• > compute illumination

store result in pixel }

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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]

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shading models

empirical models produce believable images simple and efficient

• nly 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

• rientation (normal and tangent)

ρ(l,v;f) l v f

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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 v r = −l + 2(n ⋅ l)n (l,v;f) = max(0,v ⋅ r ρs 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 n h = (l + v)/|l + v| (l,v;f) = max(0,n ⋅ h ρs 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:

S P l = (S − P)/|S − P| L = /|S − P kl |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 P l = 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 − P kl |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 ⋅ | ∑i Li ρd li ρs li li C = ⋅ ( + max(0,v ⋅ ) ⋅ |n ⋅ | ∑i Li kd ks ri)n li C = ⋅ ( + max(0,n ⋅ ) ⋅ |n ⋅ | ∑i Li 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 distance

t

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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 ⋅ | ∑i Li Vi ρd ρs li t > ϵ = ϵ tmin

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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 ⋅ | kdLa ∑i Li 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)n kr kt C = + ⋅ (P)( + )|n ⋅ | + kdLa ∑i Li Vi ρd ρs li + raytrace(P,r) + raytrace(P,t) kr kt > ϵ tmin

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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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antialiasing: removing jaggies

1 sample/pixel

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antialiasing: removing jaggies

1 sample/pixel

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antialiasing: removing jaggies

1 sample/pixel

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antialiasing: removing jaggies

1 sample/pixel 9 sample/pixel

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antialiasing: removing jaggies

1 sample/pixel 9 sample/pixel

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antialiasing: removing jaggies

1 sample/pixel 9 sample/pixel

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antialiasing: removing jaggies

1 sample/pixel 9 sample/pixel

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antialiasing: removing jaggies

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); } }

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image so far

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