Recollection. Models Pixels. Model transformation Viewport transformation Clipping Rasterization Texturing + Lights & shadows
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- Charleen Hodges
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1 Recollection Models Pixels Model transformation Viewport transformation Clipping Rasterization Texturing + Lights & shadows Can be computed in different stages 1
2 So far we came to
3 Geometry model 3
4 Surface color 4
5 Now: Shading 5
6 Important recollections
7 Bilinear interpolation A D P X Q B C 7
8 Bilinear interpolation 4 corner points A,B,C,D with known values 1 internal point X with unknown value P = A + u.(b-a), Q = D + u.(c-d) X = P + v.(q-p) Matrix representation X A 1 v v 1 u, u u 0,1, v 0, 1 B D C 8
9 Application: texture mapping Interpolate D A = P, D C = Q, P Q = X P Q P Q 9
10 Application: texture filtering Consider 4 neighboring texels Weighted average 10
11 Lighting and shading
12 General problem For a point in space, calculate lighting conditions and modulate the inherent object color to produce final pixel color 12
13 Lighting and shading What is lighting Computing amount of radiance (per wavelength) reflected from object towards the camera What is shading Creating illusion of space in planar images Usually uses lighting but other options are available too e.g. depth shading 13
14 Lighting
15 Light source types Omnidirectional Spotlight Area Directional Object - what are the differences? 15
16 Elementary theory Light-surface interaction Reflection Refraction Snell s law Surface normal vector Real world is a bit different 16
17 Surface types Reflective Diffuse Lambertian Both 17
18 Light reflection distribution Mirror Matte directional component indirectional component 18
19 Lighting models Empiric e.g. Phong lighting model cheap computation physically incorrect visually plausible Physically-based energy transfer, light propagation closer to real-world physics expensive 19
20 Local illumination models Fast but inaccurate Ignore other objects (i.e. it s not global) Empirical (no physical background) Many physical effects are impossible to achieve Computer games, real-time rendering 20
21 Diffuse light 21
22 Ambient light 22
23 Diffuse + ambient 23
24 Diffuse + ambient + specular 24
25 Phong lighting model Ambient + Diffuse + Specular components without ambient with ambient Simulates global light scattered in the scene and reflected from other objects 25
26 Phong lighting model Ambient + Diffuse + Specular components Lambert law ambient + diffuse I d n l ambient + diffuse + specular 26
27 Phong lighting model Ambient + Diffuse + Specular components Directional view vector I s r v k shine 27
28 Specular component r.v = r. v.cos(rv) absolute parameter k s exponential parameter shininess (gloss) 28
29 Phong lighting model I k a I a k I k I d d s s lights k, I coefficients can depend on wavelength what defines surface lighting properties? k a, k d, k s, k shine 29
30 Other lighting models Blinn-Phong generalization of Phong s model Cook-Torrance microfacets Oren-Nayar rough surfaces Anisotropic microfacet distribution 30
31 Surface normal vector Perpendicular to the surface at the point Computation: Usually from tangent vectors Vector product n u v Depends on the object representation n v u Vector normalization nˆ n n 31
32 Tangent vectors Parametric representation X = x(u,v) Y = y(u,v) Z = z(u,v) Partial derivation by u,v vectors t u, t u Polygonal representation Tangent vectors are edge vectors Mind the orientation! 32
33 Lighting a polygon Scanline rasterization For each pixel evaluate lighting model compute normal vector, view vector, light vector get surface parameters evaluate formula Expensive therefore: shading 33
34 Shading
35 Shading Object color is altered to give impression of light and depth Usually incorporates lighting Often only an approximation of real physics 35
36 Two stages of lighting 1. Evaluate illumination for some points of the object = LIGHTING 2. Use results from (1) to compute illumination of the rest of the object = SHADING 36
37 Flat shading One normal per face Entire face = one color 37
38 Flat (constant) shading 1 normal vector per object face (polygon) 1 lighting value per object face Entire polygon = 1 color 38
39 Gouraud shading Per-vertex lighting Color is interpolated over the face 39
40 Gouraud shading 1 normal vector per 1 surface vertex i.e. 4 lighting values / quad, 3 values / triangle Rest of the polygon lighting value interpolation Bands Chance of missing specular Realtime 40
41 Example A[0, 0] = 80% intensity B[10, 6] = 20% C[10, 9] = 80% D[0, 10] = 40% Interpolate light intensity at S[5, 8] HINT: Bilinear interpolation A D => P B C => Q P Q => S 41
42 Phong shading NOT Phong lighting model Entire surface normal is interpolated instead of interpolating only the lighting value Per pixel lighting Slower 42
43 Towards photorealism
44 Real world effects light refraction mutual object reflection caustics chromatic aberration color bleeding (soft) shadows
45 Refraction, caustics 45
46 Reflections 46
47 Chromatic aberration 47
48 Color bleeding 48
49 Raytracing 49
50 Raytracing Tracing a beam from viewer s eye through each screen pixel. Find first beam intersection with objects Compute local lighting Trace reflected and refracted beams Combine the results with local result recursively 50
51 Raytracing what s inside Line-object intersection expensive computation speed-up by e.g. scene subdivision (octree) or bounding volumes take the nearest intersection Example intersections: Sphere Triangle 51
52 Line-sphere intersection Line A,p : L = P + t * p Sphere C,r : (S C) 2 = R 2 Intersect: ( P + t * p C ) 2 = R 2 Quadratic equation 0,1,2 roots 52
53 Line-triangle intersection Line P,p : L = P + t * p Triangle K,L,M: T = K + u*(l-k) + v*(m-k) Line-plane intersection Plane: ax + by + cx + d = 0 Check if intersection is inside triangle 53
54 Raytracing what s inside Compute reflected and refracted rays evaluate light coming from their direction Combine with the local lighting result C comb( l. C, r. C, t. C L R T ) 54
55 Let s think the combinations C comb( l. C, r. C, t. C L R T ) 55
56 Raytracing pros and cons No need for polygonal representation works with both volume and boundary rep. works with CSG objects, F-reps, meshes... No explicit rasterization takes place Computationally expensive Does not compute soft shadows 56
57 Examples: POVRAY 57
58 Radiosity Object hit by light is a new light source Energy (light) exchange between objects Indirect illumination 58
59 Real world radiosity Light reflectors in photography 59
60 Radiosity Physically based Object hit by light becomes a new light source Not only object-light interaction But also object-object light interaction Energy exchange between objects 60
61 General situation 61
62 The math behind it Energy is either emitted (E) or bounced (B) All reflections are perfectly diffuse Surface A i radiosity: B i E i p i A B Form factors F ij how two surface elements A i and A j affect each other j j F ij 62
63 Elementary example Problem: B 1 =? B 2 =? Let E 1 =E 2 =0 63
64 Energy preservation Bounced = Emitted + p*received B B E p ( B F B F B F ) p B F B F B F ) ( E 1 p B F B 1 p F ) ( p B F E Repeat for all 3 surfaces 64
65 Result = linear system E 1 = E 2 = 0 E 0 = light source parameter In real tens of thousands of surfaces E E E B B B F p F p F p p F p F p F F p F p F p
66 Radiosity pros and cons Physically correct Extreme computational expenses Indirect light soft realistic shadows Area lights and object lights are easy to do Color bleeding possible too Only diffuse light transfer = Problems with reflections and specular light 66
67 Example Indirect light Color bleeding Soft shadows Area light 67
68 Radiosity example direct illumination indirect illumination 68
69 Raytracing vs. radiosity 69
70 Questions?
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