Indirect Illumination

Transcription

1 Indirect Illumination Michael Kazhdan ( /657) HB Ch. 14.1, 14.2 FvDFH 16.1, 16.2

2 Announcements Midterm on October 17 th Office hours: Misha: Monday 2-3 Malone 229 Sing Chun: Friday 9-10 Malone 216 Steve: Tuesday 12-1 Malone 122 Yan: Wednesday 1-2 Malone 122

3 Surface Illumination Calculation Multiple light source: 2 Viewer N 1 V I = I E + K A I A + K D N, + K S V, R n I

4 Overview Direct Illumination Emission at light sources Direct light at surface points Global illumination Shadows Transmissions Inter-object reflections

5 Shadows Shadow term tells if light sources are blocked Cast ray towards each light source i. If the ray is blocked, do not consider the contribution of the light.

6 Shadows Shadow term tells if light sources are blocked Cast ray towards each light source i S i = 0 if ray is blocked, S i = 1 otherwise 1 0 Shadow Term I = I E + K A I A + K D N, + K S V, R n I S

7 Shadows Shadow term tells if light sources are blocked Cast ray towards each light source i S i = 0 if ray is blocked, S i = 1 otherwise 1 S 0 = 1: 0 contributes I = I E + 0 K A I A + K D N, + K S V, R n Shadow Term I S

8 Shadows Shadow term tells if light sources are blocked Cast ray towards each light source i S i = 0 if ray is blocked, S i = 1 otherwise 1 S 0 = 1: 0 contributes S 1 = 0: 1 doesn t contribute I = I E + 0 K A I A + K D N, + K S V, R n Shadow Term I S

9 Ray Casting Trace primary rays from camera Direct illumination from unblocked lights only

10 Recursive Ray Tracing Also trace secondary rays from hit surfaces Consider contributions from: 1. Reflected Rays 2. Refracted Rays

11 Mirror Reflections Also trace secondary rays from hit surfaces Consider contributions from: 1. Reflected Rays 2. Refracted Rays 1 Recall: The specularity of a surface, K S, measures how mirror-like it is. 0 I = I E + K A I A + K D N, + K S V, R n I S

12 Mirror Reflections Also trace secondary rays from hit surfaces Consider contributions from: 1. Reflected Rays 2. Refracted Rays 1 0 Radiance for mirror reflection ray I = I E + K A I A + K D N, + K S V, R n I S + K s I R

13 Mirror Reflections Also trace secondary rays from hit surfaces Consider contributions from: 1. Reflected Rays 2. Refracted Rays 1 0 Radiance for mirror reflection ray I = I E + K A I A + K D N, + K S V, R n I S + K s I R

14 Mirror Reflections Also trace secondary rays from hit surfaces Consider contributions from: 1. Reflected Rays 2. Refracted Rays 1 0 Radiance for mirror reflection ray I = I E + K A I A + K D N, + K S V, R n I S + K s I R

15 Mirror Reflections Also trace secondary rays from hit surfaces Consider contributions from: 1. Reflected Rays 2. Refracted Rays

16 Transparent Refraction Also trace secondary rays from hit surfaces Consider contributions from: 1. Reflected Rays 2. Refracted Rays 1 0 Radiance for refraction ray I = I E + K A I A + K D N, + K S V, R n I S + K s I R + K T I T

17 Transparent Refraction Also trace secondary rays from hit surfaces Consider contributions from: 1. Reflected Rays 2. Refracted Rays 1 0 Radiance for refraction ray I = I E + K A I A + K D N, + K S V, R n I S + K s I R + K T I T

18 Transparent Refraction Also trace secondary rays from hit surfaces Consider contributions from: 1. Reflected Rays 2. Refracted Rays 1 0 Radiance for refraction ray I = I E + K A I A + K D N, + K S V, R n I S + K s I R + K T I T

19 Transparent Refraction Also trace secondary rays from hit surfaces Consider contributions from: 1. Reflected Rays 2. Refracted Rays 1 0 Radiance for refraction ray I = I E + K A I A + K D N, + K S V, R n I S + K s I R + K T I T

20 Transparent Refraction Also trace secondary rays from hit surfaces Consider contributions from: 1. Reflected Rays 2. Refracted Rays 1 0 Radiance for refraction ray I = I E + K A I A + K D N, + K S V, R n I S + K s I R + K T I T

21 Transparent Refraction Also trace secondary rays from hit surfaces Consider contributions from: 1. Reflected Rays 2. Refracted Rays

22 Transparency and Shadow Problem: If a surface is transparent, then rays to the light source may pass through the object Over-shadowing

23 Transparency and Shadow Problem: If a surface is transparent, then rays to the light source may pass through the object Need to modify the shadow term so that instead of representing a binary (0/1) value, it gives the fraction of light passing through. Accumulate transparency values as the ray travels to the light source. I = I E + K A I A + K D N, + K S V, R n I S + K s I R + K T I T

24 Transparency and Shadow Accumulate transparency values as the ray travels to the light source. 1 S 0 = 1: 0 contributes fully 0 I = I E + K A I A + K D N, + K S V, R n I S + K s I R + K T I T

25 Transparency and Shadow Accumulate transparency values as the ray travels to the light source. 1 S 0 = 1: 0 contributes fully S 1 =?: 0 I = I E + K A I A + K D N, + K S V, R n I S + K s I R + K T I T

26 Transparency and Shadow Accumulate transparency values as the ray travels to the light source. 1 S 0 = 1: 0 contributes fully S 1 = 1 K T?: K T 0 I = I E + K A I A + K D N, + K S V, R n I S + K s I R + K T I T

27 Transparency and Shadow Accumulate transparency values as the ray travels to the light source. 1 S 0 = 1: 0 contributes fully S 1 = 1 K T K T?: K T 0 I = I E + K A I A + K D N, + K S V, R n I S + K s I R + K T I T

28 Transparency and Shadow Accumulate transparency values as the ray travels to the light source. 1 K T S 0 = 1: 0 contributes fully S 1 = 1 K T K T : 1 contributes partially 0 I = I E + K A I A + K D N, + K S V, R n I S + K s I R + K T I T

29 Transparency and Shadow Accumulate transparency values as the ray travels to the light source. 1 K T S 0 = 1: 0 contributes fully S 1 = 1 K T K T : 1 contributes partially For solid models can use the distance travelled through the surface instead: S = e d K T 0 I = I E + K A I A + K D N, + K S V, R n I S + K s I R + K T I T

30 Transparency and Shadow Accumulate transparency values as the ray travels to the light source.

31 Transparent Refraction When a light of light passes through a transparent object, the ray of light can bend, (θ i θ r ). N θ i T θ r

32 Snell s aw The way that light bends is determined by the indices of refraction of the internal and external materials η i and η r : N η r sin θ r = η i sin θ i θ i T θ r The index of refraction of air is η = 1.

33 Snell s aw The way that light bends is determined by the indices of refraction of the internal and external materials η i and η r : N η r sin θ r = η i sin θ i θ i T θ r T = η i η r cos θ i cos θ r N η i η r

34 Snell s aw The way that light bends is determined by the indices of refraction of the internal and external materials η i and η r : η r sin θ r = η i sin θ i 1 0

35 Snell s aw The way that light bends is determined by the indices of refraction of the internal and external materials η i and η r :

36 Snell s aw and Shadows Problem (Caustics): If a surface is transparent, then rays to the light source may not travel in a straight line This is difficult to address with ray-tracing

37 Termination Criteria How do we determine when to stop recursing? Depth of iteration» Bounds the number of times a ray will bounce around the scene Cut-off value» Ignores contribution from bounces that contribute very little

38 Termination Criteria Pixel GetColor( scene, ray, ir, depth, cutoff ) { Pixel p(0,0,0) Ray reflect, refract Intersection hit = FindIntersection( ray, scene ) if ( hit ) { p += GetSurfaceColor( hit.position ) reflect.direction = Reflect( ray.direction, hit.normal ) reflect.position = hit.position + reflect.direction* if( depth >0 && hit.kspec>cutoff ) p += GetColor( scene, reflect, ir, depth-1, cutoff/hit.kspec )*hit.kspec } refract.direction = Refract( ray.direction, hit.normal, ir, hit.ir ) refract.position = hit.position + refract.direction* if( depth >0 && hit.ktran>cutoff ) p += GetColor( scene, refract, hit.ir, depth-1, cutoff/hit.ktran ) *hit.ktran } return p

39 Termination Criteria Pixel GetColor( scene, ray, ir, depth, cutoff ) { Pixel p(0,0,0) Ray reflect, refract Intersection hit = FindIntersection( ray, scene ) if ( hit ) { p += GetSurfaceColor( hit.position ) reflect.direction = Reflect( ray.direction, hit.normal ) reflect.position = hit.position + reflect.direction* if( depth >0 && hit.kspec>cutoff ) p += GetColor( scene, reflect, ir, depth-1, cutoff/hit.kspec )*hit.kspec } refract.direction = Refract( ray.direction, hit.normal, ir, hit.ir ) refract.position = hit.position + refract.direction* if( depth >0 && hit.ktran>cutoff ) p += GetColor( scene, refract, hit.ir, depth-1, cutoff/hit.ktran ) *hit.ktran } return p I = I E + K A I A + K D N, + K S V, R n I S + K s I R + K T I T

40 Termination Criteria Pixel GetColor( scene, ray, ir, depth, cutoff ) { Pixel p(0,0,0) Ray reflect, refract Intersection hit = FindIntersection( ray, scene ) if ( hit ) { p += GetSurfaceColor( hit.position ) reflect.direction = Reflect( ray.direction, hit.normal ) reflect.position = hit.position + reflect.direction* if( depth >0 && hit.kspec>cutoff ) p += GetColor( scene, reflect, ir, depth-1, cutoff/hit.kspec )*hit.kspec } refract.direction = Refract( ray.direction, hit.normal, ir, hit.ir ) refract.position = hit.position + refract.direction* if( depth >0 && hit.ktran>cutoff ) p += GetColor( scene, refract, hit.ir, depth-1, cutoff/hit.ktran ) *hit.ktran } return p I = I E + K A I A + K D N, + K S V, R n I S + K s I R + K T I T

41 Termination Criteria if( depth >0 && hit.kspec>cutoff ) p += GetColor( scene, reflect, ir, depth-1, cutoff/hit.kspec )*hit.kspec; The first reflected ray should be cast if it can contribute an intensity greater than the cut-off. K S 1 > cut-off 1 K S 1 0

42 Termination Criteria if( depth >0 && hit.kspec>cutoff ) p += GetColor( scene, reflect, ir, depth-1, cutoff/hit.kspec )*hit.kspec; The second reflected ray should be cast if it can contribute an intensity greater than the cut-off. K S 2 K S 1 > cut-off 1 K S 2 > cut-off K S 1 K S 2 K S 1 0

43 Termination Criteria if( depth >0 && hit.kspec>cutoff ) p += GetColor( scene, reflect, ir, depth-1, cutoff/hit.kspec )*hit.kspec; The second reflected ray should be cast if it can contribute a global intensity greater than the cut-off. K S 2 K S 1 > cut-off 1 K S 2 > cut-off K S 1 The second reflected ray should be cast if it can reflect 0 at least cut-off K1 of the local intensity S K S 2 K S 1

44 Termination Criteria Pixel GetColor( scene, ray, ir, depth, cutoff ) { Pixel p(0,0,0) Ray reflect, refract Intersection hit = FindIntersection( ray, scene ) if ( hit ) { p += GetSurfaceColor( hit.position ) reflect.direction = Reflect( ray.direction, hit.normal ) reflect.position = hit.position + reflect.direction* if( depth >0 && hit.kspec>cutoff ) p += GetColor( scene, reflect, ir, depth-1, cutoff/hit.kspec )*hit.kspec } refract.direction = Refract( ray.direction, hit.normal, ir, hit.ir ) refract.position = hit.position + refract.direction* if( depth >0 && hit.ktran>cutoff ) p += GetColor( scene, refract, hit.ir, depth-1, cutoff/hit.ktran ) *hit.ktran } return p I = I E + K A I A + K D N, + K S V, R n I S + K s I R + K T I T

45 Termination Criteria Pixel GetColor( scene, ray, ir, depth, cutoff ) { Pixel p(0,0,0) Ray reflect, refract Intersection hit = FindIntersection( ray, scene ) if ( hit ) { p += GetSurfaceColor( hit.position ) Why do we need the terms? To ensure that the new ray does not hit its starting location! reflect.direction = Reflect( ray.direction, hit.normal ) reflect.position = hit.position + reflect.direction* if( depth >0 && hit.kspec>cutoff ) p += GetColor( scene, reflect, ir, depth-1, cutoff/hit.kspec )*hit.kspec } refract.direction = Refract( ray.direction, hit.normal, ir, hit.ir ) refract.position = hit.position + refract.direction* if( depth >0 && hit.ktran>cutoff ) p += GetColor( scene, refract, hit.ir, depth-1, cutoff/hit.ktran ) *hit.ktran } return p

46 Termination Criteria Pixel GetColor( scene, ray, ir, depth, cutoff ) { Pixel p(0,0,0) Ray reflect, refract Intersection hit = FindIntersection( ray, scene ) if ( hit ) { p += GetSurfaceColor( hit.position ) reflect.direction = Reflect( ray.direction, hit.normal ) reflect.position = hit.position + reflect.direction* if( depth >0 && hit.kspec>cutoff ) p += GetColor( scene, reflect, ir, depth-1, cutoff/hit.kspec )*hit.kspec } } return p refract.direction = Refract( ray.direction, hit.normal, ir, hit.ir ) refract.position = hit.position + refract.direction* if( depth >0 && hit.ktran>cutoff ) p += GetColor( scene, refract, hit.ir, depth-1, cutoff/hit.ktran ) *hit.ktran Warning: In practice, cutoff is a scalar while hit.ktran/kspec are RGB values.

47 Illumination Examples Ray tracing (rays to point light source) Courtesy Henrik Wann Jensen

48 Illumination Examples Soft shadows (rays to area light source) Courtesy Henrik Wann Jensen

49 Illumination Examples Caustics (rays from light source) Courtesy Henrik Wann Jensen

50 Illumination Examples Full Global Illumination Courtesy Henrik Wann Jensen

51 Recursive Ray Tracing GetColor calls RayTrace recursively Image RayTrace( Camera camera, Scene scene, int width, int height, int depth, float cutoff ) { Image image = new Image( width, height ); for( int i = 0 ; i<width ; i++ ) for( int j = 0 ; j<height ; j++ ) { Ray ray = ConstructRayThroughPixel( camera, i, j ); image[i][j] = GetColor( scene, ray, 1, depth, cutoff ); } return image; }

52 Summary Ray casting (direct Illumination) Usually use simple analytic approximations for light source emission and surface reflectance Recursive ray tracing (global illumination) Incorporate shadows, mirror reflections, and pure refractions All of this is an approximation so that it is practical to compute More on global illumination later!