Recollection. Models Pixels. Model transformation Viewport transformation Clipping Rasterization Texturing + Lights & shadows

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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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