ray tracing

computer graphics • ray tracing

© 2006 fabio pellacini • 1

image formation

computer graphics • ray tracing

© 2006 fabio pellacini • 2

image formation

computer graphics • ray tracing

© 2006 fabio pellacini • 3

rendering computational simulation of image formation

computer graphics • ray tracing

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rendering • given viewer

computer graphics • ray tracing

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rendering • given viewer, geometry

computer graphics • ray tracing

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rendering • given viewer, geometry, materials

computer graphics • ray tracing

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rendering • given viewer, geometry, materials and lights

computer graphics • ray tracing

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rendering • given viewer, geometry, materials and lights • determine visibility

computer graphics • ray tracing

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rendering • given viewer, geometry, materials and lights • determine visibility and simulate lighting

computer graphics • ray tracing

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ray tracing a particular rendering algorithm

computer graphics • ray tracing

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ray tracing algorithm for each pixel { determine viewing direction intersect ray with scene compute illumination store result in pixel }

computer graphics • ray tracing

© 2006 fabio pellacini • 12

vector math review – points and vectors • point – location in 3D space – P = (x,y,z)

• vector – direction and magnitude – V = (u,v,w)

computer graphics • ray tracing

© 2006 fabio pellacini • 13

vector math review – vector operations • dot product – V1•V2=|V1||V2|cos(θ)

• cross product – |V1×V2|=|V1||V2|sin(θ) – V1×V2 is perpendicular to V1 and V2

computer graphics • ray tracing

© 2006 fabio pellacini • 14

vector math review – segments • segment – set of points (line) between two points – P(t) = P0 + t (P1 – P0) with t ∈ [0,1]

computer graphics • ray tracing

© 2006 fabio pellacini • 15

vector math review – rays • ray – infinite line from point in a given direction – P(t) = P0 + t V with t ∈ [0,∞]

computer graphics • ray tracing

© 2006 fabio pellacini • 16

vector math review – planes • plane

– P ∈ plane ↔ (P − P0 ) ⋅ N = 0 ↔ P ⋅ N + d = 0 2 – P = P0 + uxˆ + vyˆ with (u , v) ∈ [−∞, ∞] – N = xˆ × yˆ

computer graphics • ray tracing

© 2006 fabio pellacini • 17

vector math review - triangle • triangle – points in three subspaces defined by triangle edges

• baricentric coordinates P =α A + β B+γ C P ∈ triangle ↔ α + β + γ = 1

( ) ( )

P = C + α CA + β CB with α ≥ 0, β ≥ 0, α + β ≤ 1 – geometric interpretation: relative area γ = AC / Atriangle computer graphics • ray tracing

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ray tracing algorithm for each pixel { determine viewing direction intersect ray with scene compute illumination store result in pixel }

computer graphics • ray tracing

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[Marschner 2004 – original unkwon]

viewer model – window analogy

computer graphics • ray tracing

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viewer model - photography

[Marschner 2004 – original unkwon]

• equiv. to pinhole photography

computer graphics • ray tracing

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viewer model - representation

• position – view point

• orientation – toward, up

• focal length – field of view computer graphics • ray tracing

© 2006 fabio pellacini • 22

viewer model – view frustum

computer graphics • ray tracing

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generate viewing rays

computer graphics • ray tracing

© 2006 fabio pellacini • 24

generate viewing rays

computer graphics • ray tracing

© 2006 fabio pellacini • 25

generate viewing rays

computer graphics • ray tracing

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ray tracing algorithm for each pixel { determine viewing direction intersect ray with scene compute illumination store result in pixel }

computer graphics • ray tracing

© 2006 fabio pellacini • 27

geometry model • simple shapes – spheres – triangles – etc.

• complex shapes – later in the course

computer graphics • ray tracing

© 2006 fabio pellacini • 28

ray-sphere intersection – algebraic ⎧P (t ) = E + tI 2 ⎨ 2 P O ( t ) − = R ⎩

point on ray

E + tI − O = R 2

by substitution

2

computer graphics • ray tracing

point on sphere

© 2006 fabio pellacini • 29

ray-sphere intersection – algebraic at 2 + bt + c = 0 algebraic equation a= I

2

b = 2I ⋅ (E − O ) c = E−O − R 2

2

computer graphics • ray tracing

© 2006 fabio pellacini • 30

ray-sphere intersection – algebraic d = b − 4ac determinant if d < 0 no solutions otherwise 2

P (t ) ⇔ t − = (− b − d ) / 2a

computer graphics • ray tracing

© 2006 fabio pellacini • 31

ray-sphere intersection – geometric

W = O−E t p = W ⋅ Iˆ if t p < 0 no solution

computer graphics • ray tracing

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ray-sphere intersection – geometric

d = W⋅W −t 2

2 p

if d > R no solution 2

2

computer graphics • ray tracing

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ray-sphere intersection – geometric

tr = R − d 2

2

P (t ) ⇔ t − = t p − t r

computer graphics • ray tracing

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ray-sphere intersection – normal • surface normal need for lighting computation

N = (P − O ) / P − O

computer graphics • ray tracing

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ray-triangle intersection 1 • intersect with a plane • check if the intersection point is in triangle

computer graphics • ray tracing

© 2006 fabio pellacini • 36

ray-plane intersection ⎧P (t ) = E + tI point on ray ⎨ ⎩P ⋅ N + d = 0 point on plane (E + tI ) ⋅ N + d = 0 by substitution E⋅N + d P (t ) ⇔ t = − I⋅N

computer graphics • ray tracing

© 2006 fabio pellacini • 37

ray-triangle intersection 1 • triangle is intersection of 3 half-spaces • check if P is on right side by comparing “normals”

computer graphics • ray tracing

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ray-triangle intersection 1 intersects if

(B − A ) × (P − A ) ⋅ N > 0 (C − B ) × (P − B ) ⋅ N > 0 ( A − C ) × (P − C ) ⋅ N > 0

computer graphics • ray tracing

© 2006 fabio pellacini • 39

ray-triangle intersection 2 • use baricentric coordinates (useful later) • algebraic formulation point on ray ⎧P (t ) = E + tI ⎨ ⎩P (α , β ) = αA + β B + (1 − α − β )C point in the triangle E + tI = αA + β B + (1 − α − β )C

computer graphics • ray tracing

by substitution

© 2006 fabio pellacini • 40

ray-triangle intersection 2 E + tI = α (A − C ) + β (B − C ) + C → α (A − C) + β (B − C) − tI = E − C → α a + β b − tI = e → ⎡t ⎤ [− I a b]⎢⎢α ⎥⎥ = e ⎢⎣ β ⎥⎦

computer graphics • ray tracing

© 2006 fabio pellacini • 41

ray-triangle intersection 2 use Cramer' s rule | e a b | (e × a ) ⋅ b = t= | − I a b | (I × b ) ⋅ a | − I e b | (I × b ) ⋅ e = α= | − I a b | (I × b ) ⋅ a | − I a e | (e × a ) ⋅ I = β= | − I a b | (I × b ) ⋅ a test for α ≥ 0, β ≥ 0,α + β ≤ 1 computer graphics • ray tracing

© 2006 fabio pellacini • 42

images so far

computer graphics • ray tracing

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intersecting many shapes • intersect each primitive • pick closest intersection

computer graphics • ray tracing

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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; } } }

computer graphics • ray tracing

© 2006 fabio pellacini • 45

images so far

computer graphics • ray tracing

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ray tracing algorithm for each pixel { determine viewing direction intersect ray with scene compute illumination store result in pixel }

computer graphics • ray tracing

© 2006 fabio pellacini • 47

shading variation in observed color across a surface

computer graphics • ray tracing

© 2006 fabio pellacini • 48

lighting patterns of illumination in the environment

computer graphics • ray tracing

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shading • compute reflected light • depends on – – – –

viewer position incoming light, i.e. lighting surface geometry surface material

• more on this later

computer graphics • ray tracing

© 2006 fabio pellacini • 50

materials dielectric

[Marschner 2004]

metals

computer graphics • ray tracing

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materials dielectric

[Marschner 2004]

metals

computer graphics • ray tracing

© 2006 fabio pellacini • 52

shading models • empirical shading models – produce believable images – simple and efficient – only for simple materials

• physically-based shading models – can reproduce accurate effects – more complex

• will concentrate empirical plastic-like model – more on this later in the course computer graphics • ray tracing

© 2006 fabio pellacini • 53

Phong 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)

computer graphics • ray tracing

© 2006 fabio pellacini • 54

diffuse reflection • light reflects equally in all directions – i.e. surface looks the same from all view points – view-independent

• but incident light depends on angle – beam of light is more spread on an oblique surface

computer graphics • ray tracing

© 2006 fabio pellacini • 55

lambertian shading model diffuse coefficient

Cd = k d Cl max(0, N ⋅ L) reflected diffuse illumination illumination from light

computer graphics • ray tracing

© 2006 fabio pellacini • 56

lambertian shading model • produces matte appearance

kd

computer graphics • ray tracing

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images so far

computer graphics • ray tracing

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specular shading • light reflects mostly around mirror direction – i.e. surface looks different from view points – view-dependent

• Phong specular model – empirical, but looks good enough – use cosine of mirror and view direction

• Blinn-Phong specular model – slightly better than Phong – use cosine of bisector and normal direction computer graphics • ray tracing

© 2006 fabio pellacini • 59

Phong specular model mirror direction

R = −L + 2(N ⋅ L )N Cs = k s Cl max(0, V ⋅ R )

n

reflected specular illumination specular coefficient

computer graphics • ray tracing

specular exponent

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Phong specular model • produces highlights, shiny appearance n

ks

computer graphics • ray tracing

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Blinn-Phong variation bisector direction

H = (L + V ) / L + V Cs = k s Cl max(0, N ⋅ H )

n

reflected specular illumination specular coefficient

computer graphics • ray tracing

specular exponent

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shading model • putting the pieces together

C = Cd + C s =

= Cl [k d max(0, N ⋅ L) + k s max(0, V ⋅ R ) n ]

computer graphics • ray tracing

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images so far

computer graphics • ray tracing

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lighting • determines how much light reaches a point • depends on – light geometry – light emission – scene geometry

computer graphics • ray tracing

© 2006 fabio pellacini • 65

light source models • describe how light is emitted from light sources • empirical light source models – point, directional, spot

• physically-based light source models – area lights, sky model – will cover later in the course

computer graphics • ray tracing

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point lights • light emitted equally from point in all directions – simulates local lights – sometimes r^2 falloff for a bit more realism

L =|| S − P || Cl = C / r 2

computer graphics • ray tracing

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directional lights • light emitted from infinity in one direction – simulates distant lights, e.g. sun

L=D Cl = C

computer graphics • ray tracing

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spot lights • same as point light, but only emit in a cone – simulate theatrical lights – cone falloff model arbitrary

L =|| S − P || Cl = C ⋅ att / r 2

computer graphics • ray tracing

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multiple lights • add contribution for each light

C = ∑ (Cd )i + (Cs ) i = i

= ∑ (Cl ) i [k d max(0, N ⋅ (L)i ) + k s max(0, V ⋅ (R ) i ) n ] i

computer graphics • ray tracing

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image so far

computer graphics • ray tracing

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illumination models • describe how light spreads in the environments • direct illumination – incoming lights comes directly from sources • shadows

• indirect illumination – incoming lights comes from other objects • specular reflections (mirrors), diffuse inter-reflections

computer graphics • ray tracing

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[PCG]

illumination models

computer graphics • ray tracing

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ray tracing lighting model • captures – – – –

point/directional/spot light source models sharp shadows sharp reflections/refractions hacked diffuse inter-reflections: ambient term

• more accurate lighting later in the course

computer graphics • ray tracing

© 2006 fabio pellacini • 74

ray traced shadows • light contributes only if visible at surface point no shadow

computer graphics • ray tracing

shadow

© 2006 fabio pellacini • 75

ray traced shadows • cast a “shadow-ray” to check if light is visible • visible if no-hits or if distance > light distance

computer graphics • ray tracing

© 2006 fabio pellacini • 76

ray traced shadows • scale light by a visibility term

C = ∑ (Cd ) i + (C s ) i = i

= ∑ (Cl ) i Vi (P )[k d max(0, N ⋅ (L) i ) + k s max(0, V ⋅ (R ) i ) n ] i

light source visibility {0,1}

computer graphics • ray tracing

© 2006 fabio pellacini • 77

images so far

computer graphics • ray tracing

© 2006 fabio pellacini • 78

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 = Ca + ∑ [(Cd )i + (Cs )i ] = i

= k a Camb + ∑ [(Cd ) i + (Cs ) i ] i

ambient coefficient computer graphics • ray tracing

ambient illumination © 2006 fabio pellacini • 79

ray traced reflections • perfectly shiny surfaces reflects objects – recursively trace a ray if material is reflective • along mirror direction, scaled by reflection coeff. • mirror direction calculated for Phong

computer graphics • ray tracing

© 2006 fabio pellacini • 80

ray traced reflections • recursively ray trace, scale by reflection coeff. • refraction is the same along the transmission dir.

C = Ca + ∑ [(Cd ) i + (Cs ) i ] + Cr + Ct = i

= Ca + ∑ [(Cd ) i + (Cs ) i ] + i

+ k r raytrace(P, R ) + kt raytrace(P, T ) reflection coefficient

computer graphics • ray tracing

refraction coefficient

© 2006 fabio pellacini • 81

images so far

computer graphics • ray tracing

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antialiasing - removing jaggies 1 sample/pixel

computer graphics • ray tracing

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antialiasing - removing jaggies 1 sample/pixel

computer graphics • ray tracing

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antialiasing - removing jaggies 1 sample/pixel

computer graphics • ray tracing

© 2006 fabio pellacini • 85

antialiasing - removing jaggies 1 sample/pixel

computer graphics • ray tracing

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antialiasing - removing jaggies 1 sample/pixel

computer graphics • ray tracing

9 samples/pixel

© 2006 fabio pellacini • 87

antialiasing - removing jaggies 1 sample/pixel

computer graphics • ray tracing

9 samples/pixel

© 2006 fabio pellacini • 88

antialiasing - removing jaggies 1 sample/pixel

computer graphics • ray tracing

9 samples/pixel

© 2006 fabio pellacini • 89

antialiasing - removing jaggies 1 sample/pixel

computer graphics • ray tracing

9 samples/pixel

© 2006 fabio pellacini • 90

antialiasing - removing jaggies 1 sample/pixel

computer graphics • ray tracing

9 samples/pixel

© 2006 fabio pellacini • 91

antialiasing - removing jaggies • for each pixel – take multiple samples – compute average

• poor-man antialiasing – more on this later in the course

computer graphics • ray tracing

© 2006 fabio pellacini • 92

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; } }

computer graphics • ray tracing

© 2006 fabio pellacini • 93

antialiased 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 ++) { x = (i+(ii+0.5)/numberOfSamples)/imageWidth; y = (j+(jj+0.5)/numberofSamples)/imageHeight; ray = camera.generateRay(x,y); c += computeColor(ray); } } image[i][j] = c / (numberOfSamples^2); } }

computer graphics • ray tracing

© 2006 fabio pellacini • 94

images so far

computer graphics • ray tracing

© 2006 fabio pellacini • 95

ray tracing details • numerical precision issues come up – shadow acne: ray hits the visible point – solution: only intersect if distance > epsilon

• make sure you do not recurse infinitely when computing reflections or refractions – solution: simply stop after a given number

computer graphics • ray tracing

© 2006 fabio pellacini • 96

ray tracing wrap-up • simple and general algorithm – – – –

ray cast to evaluate visibility simplified shading model ray cast to evaluate shadows recursive execution for reflections/refractions

• inefficient – linear with number of primitives – sub-linear ray cast later in the course

• not photorealistic – too clean – missing soft shadows, realistic materials, realistic camera model, diffuse inter-reflection

• base of most general algorithms computer graphics • ray tracing

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