Today. Part 1: Ray casting Part 2: Lighting Part 3: Recursive Ray tracing Part 4: Acceleration Techniques

Transcription

1 Ray Tracing

2 Today Part 1: Ray casting Part 2: Lighting Part 3: Recursive Ray tracing Part 4: Acceleration Techniques

3 Example 1

4 Example 2

5 What is 3D Rendering? Construct an image from a 3D model Camera Light Rendering 3D Model View Plane

6 n the Physical World How Do We See? Option 1: shoot rays from lights and see how they move in space until they reach the eye. Problem? When tracing rays from light sources to the eye many are wasted since they never reach the eye Solution: reverse the order shoot rays from the eye until they hit the light?

7 Part : Ray Casting How to find the color of a pixel in an image?

8 First Approximation: Ray Casting 1. Shoot Rays from center of projection through pixels Geometry 2. Find intersection with scene objects 3. Calculate Lighting an approximated color at intersection point Rays through view plane Samples on view plane Eye position

9 Ray Casting mage RayCast (Camera camera, Scene scene, int width, int height) { mage image = new mage(width, height); for (int i = 0; i < width; i++) { for (int j = 0; j < height; j++) { Ray ray = ConstructRayThroughPixel(camera, i, j); ntersection hit = Findntersection(ray, scene); image[i][j] = GetColor(scene, ray, hit); } } return image; }

10 Stages Per mage Pixel 1. Shoot Rays from center of projection through pixels: Ray ray = ConstructRayThroughPixel(camera, i, j); 2. Find intersection with scene objects ntersection hit = Findntersection(ray, scene); 3. Calculate an approximated color at intersection point image[i][j] = GetColor(scene, ray, hit);

11 Part : Geometry Ray ntersections

12 Constructing Ray Through a Pixel Up direction =V up View Plane back Towards = V to P 0 Right direction = V right V The Ray is defined by: P 0 + tv P

13 2D Example V up W P0 V to D P 1 H

14 2D Example V right = V to x V up Q = frustum half-angle dist = distance to view plane P Q V to width = 2*dist*tan(Q ) 0 dist P 1 V The Ray is defined by: P 0 + tv To define the ray we must find V V right To find V we find P = pixel intersection point: P width P 1 = P 0 + dist*v to P = P 1 ± i * V right for i (0,width/2) Now we can find V = (P - P 0 ) / P - P 0

15 Ray-Scene ntersection 1. Finding the intersection point with a geometric primitive 2. Find closest intersection point in a group of objects (scene) 3. Acceleration techniques 1. Bounding volume hierarchies 2. Spatial partitions

16 Triangle & Sphere ntersection v p 0 P 2 P 0 V P 1 r O p 3 p 1 p 2

17 Other Primitive ntersections Cone, cylinder, ellipsoid: Similar to sphere Box ntersect 3 front-facing planes, return closest Convex polygon Same as triangle (check point-inpolygon algebraically) Concave polygon Same plane intersection More complex point-in-polygon test Algorithms for 3D object intersection:

18 ntersecting a Group of Objects Must find the nearest intersection E D F C A B

19 ntersecting a Scene A Brute Force Approach: ntersection Findntersection(Ray ray, Scene scene) { min_t = infinity min_primitive = NULL For each primitive in scene { t = ntersect(ray, primitive ); if (t < min_t) then min_primitive = primitive min_t = t } } return ntersection(min_t, min_primitive) }

20 Alisaing?

21 Aliasing? Yet again we are sampling a continuous world and get sampling artifacts. Better results: we shoot more than one ray trough each pixel and average their color! (again: smoothing vs. aliasing) P 0

22 Ray Casting 1. Shoot Rays from center of projection through pixels 2. Find intersection with scene objects 3. Calculate an approximated color at intersection point Rays through view plane Samples on view plane Eye position

23 Part : Lighting Determining Pixel Color

24 From Model to Picture Using just color creates a flat image Real color is created by the interaction of light with surface (normal and material) Wireframe Model Without llumination With llumination

25 What we see is not just color! image[i][j] = GetColor(scene, ray, hit); How do we compute radiance for a sample ray? Depends on: Light properties Surface material properties Geometry (interrelations) Other factors: shadows, atmosphere, smoke, fog

26 Different Types of Lights

27 Different Types of Materials

28 Other Factors

29 Goal Must derive computer models for: Emission at light sources Scattering at surfaces Reception at the camera Global Effects Desirable features: Accurate Concise Efficient to compute

30 rradiance The irradiance is a two dimensional function describing the incoming light at a given point.

31 Radiance The radiance is a two dimensional function representing the light reflected from a surface at a given point.

32 Direct llumination Two issues: Emission at light sources Scattering at surfaces Later: global illumination Shadows, reflections

33 Modeling Light Sources L(x,y,z,q,f,l)... describes the intensity of energy, leaving a light source, arriving at location (x,y,z),... from direction (q,f),... with wavelength l (x,y,z) Light

34 Empirical Models deally measure irradiant energy for all situations Too much storage Difficult in practice l

35 Light Source Models (OpenGL) Simple mathematical models: Point light Directional light Spot light

36 Point Light Source Models omni-directional point source (e.g., bulb) ntensity ( 0 ), Position (px, py, pz), Factors (k c, k l, k q ) for attenuation with distance (d) Light d (px, py, pz) L k c k l d 0 k q d 2

37 Directional Light Source Models point light source at infinity (e.g., sun) ntensity ( 0 ), Direction (dx,dy,dz) No attenuation with distance Light L 0( D L) (dx, dy, dz)

38 Spot Light Source Models point light source with direction (e.g., Luxo) ntensity ( 0 ), Position (px, py, pz), Direction D=(dx, dy, dz) L Attenuation k c 0( D L) k d k l q d 2 (px, py, pz) D L g d Spotlight

39 Scattering at surfaces Basic approach is energy conservation: Light_out = Light_in + Light_emitted - Light_absorbed Surface

40 Modeling Surface Reflectance Rs(qi,fi,qr,fr,l)... describes the amount of incident energy, arriving from direction (qi,fi),... leaving in direction (qr,fr), with wavelength l Surface

41 Empirical Models deally measure radiant energy for all combinations of incident angles Too much storage Difficult in practice (qr,fr) Surface l (qi,fi)

42 Bidirectional Reflectance Distribution Function (BRDF) 4-dimensional function which defines light reflection at an opaque surface. The ratio of reflected radiance exiting along w 0 to the irradiance incident on the surface from direction w i : f ( w, w ) r i o dlr( w0) dlr( w0) de ( w ) L ( w )cos( q ) dw i i i i i i

43 BRDF: 4 Dimensional Function

44 Phong Reflectance Model Based on model proposed by Phong in his PhD dissertation 1973 Specular Ambient Diffuse Emission

45 Phong Reflectance Model Simpler analytic approximation model Diffuse reflection + Specular reflection + (Emission) + ambient Surface 45

46 Diffuse Reflection Diffuse definition = spread-out, to pass by spreading every way, to extend in all directions Assume surface reflects equally in all directions Examples: chalk, clay, cloth Surface

47 Modeling Diffuse Reflectance How much light is reflected? Depends on angle of incident light (Does not depend on the viewing angle) dl dacosq q dl da Surface

48 Lambertian Model cosine law (dot product) N L N L cosq Nˆ Lˆ cosq K ( Nˆ Lˆ) D D L N q L Surface

49 Specular Reflection Reflection is strongest near mirror angle Examples: mirrors, metals, glass N R q q L Surface

50 How much light is seen? Depends on angle of incident light and angle to viewer Viewer R a q N q L V

51 Specular Reflection Phong model: reflection depends on cos(a) n Where n is the Phong exponent governing the apparent shininess of the surface Viewer V R a q N q L K ( V R) S S n L

52 Behavior of n (V R)

53 Examples of Different Exponent

54 Light Emission Represents light emanating directly from object Emission 0

55 Ambient Term Represents reflection of all indirect illumination This is a total hack (avoids complexity of global illumination)!

56 Single Light llumination Calculation E K A AL K D ( N L) K ( V L S R) n L Viewer R a q N q L V

57 Multiple Light llumination Calculation E K A AMB ( K i D ( N L i ) L i K S ( V R i ) n L i ) N Viewer L 1 L 2 V

58 For Every Light Wavelength? Enough to calculate for the 3 base colors: L i i n i S i D AMB A E R V K L N K K l l l l l l l ) ) ( ) ( ( rl i i n i rs i rd ramb ra re r R V K L N K K ) ) ( ) ( ( gl i i n i gs i gd bamb ga ge g R V K L N K K ) ) ( ) ( ( bl i i n i bs i bd bamb ba be b R V K L N K K ) ) ( ) ( (

59 Get Color for Ray Casting RGB GetColor(Scene scene, Ray in_ray, Point hit) { // Ambient and Emission calculations RGB color = calcemissioncolor(scene) + calcambientcolor(scene); // Diffuse & Specular calculations for (int i = 0; i < getnumlights(scene); i++) { Light light = getlight(i,scene); color += calcdiffusecolor(scene,hit,light) + calcspecularcolor(scene,hit,light); } return color; }

60 Part : Recursive Ray Tracing Shadows Refractions nter-object reflections

61 Shadows

62 Shadows Occlusions from light sources by other objects Point light source creates hard edges Area light source create soft edges 62

63 Shadow Calculation? nstead of calculating the shadow of each object explicitly we do an implicit calcuation For each intersection point with ray we shoot shadow rays and check if a light source is blocked by other objects then we know it is in shadow.

64 Shadows Rays Shadow terms tell which light sources are blocked Cast ray towards each light source L i S i = 0 if ray is blocked, S i = 1 otherwise i i i n S D A A E S R V K L N K K ) ) ( ) ( ( Shadow Term

65 Get Color for Ray Casting with Shadows RGB GetColor(Scene scene, Ray in_ray, Point hit) { // Ambient and Emission calculations RGB color = calcemissioncolor(scene) + calcambientcolor(scene); // Diffuse and Specular calculations for (int i = 0; i < getnumlights(scene); i++) { Light light = getlight(i,scene); Ray light_ray = ConstructRaytoLight(hit, light) // Add color only if light is not occluded if!occluded(light_ray, scene, light) { color += calcdiffusecolor(scene,hit,light) + calcspecularcolor(scene,hit,light); } } return color; }

66 Recursive Ray Tracing Ray casting cannot account for global effects of interactions between objects such as reflections and refractions of light. Ray tracing models these by recursively following the (reverse) rays from the intersection points

67 Ray Casting vs. Tracing

68 Casting vs. Tracing Fromulas Trace primary rays from camera: direct illumination from unblocked lights only L L L n S D A A E S R V K L N K K ) ) ( ) ( ( T T R R L L L n S D A A E K K S R V K L N K K ) ) ( ) ( ( Also trace secondary rays from hit surfaces: global illumination from mirror reflection and transparency

69 Mirror reflections Trace secondary ray in direction of mirror reflection Evaluate radiance along secondary ray and include it into illumination model Radiance for mirror reflection ray n E K A A ( K L D( N L) KS ( V R) ) S L L K S R K T T

70 Transparency Trace secondary ray in direction of refraction and include it into illumination model Transparency coefficient is fraction transmitted K T = 1 if object is translucent, K T = 0 if object is opaque 0 < K T < 1if object is semitranslucent n E K A A ( K L D( N L) KS ( V R) ) S L L K S R Radiance for refraction ray K T T

71 Refractive Transparency For thin surfaces, can ignore change in direction Assume light travels straight through surface T L N Q i h i h r L T Qr T Q i

72 Refractive Transparency For solid objects, apply Snell s law: h sin Q h sin Q r r i i T hi ( cos Qi cos Qr ) N h r h i h r hi h r L T Qr N Q i L

73 Recursion Ray Tree Ray tree represents illumination computation Ray traced through scene Ray tree

74 Gather Components Ray traced through scene Ray tree T T R S L L L n S D A A E K K S R V K L N K K ) ) ( ) ( (

75 Ray Tracing mage RayTrace(Camera camera, Scene scene, int width, int height) { mage image = new mage(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] = calccolor(scene, ray, hit,0); } } return image; }

76 Recursive Calculation of Color for Ray Tracing Regular part like ray casting (Note: without shadows!) Recursive part (Note: only reflection) RGB CalcColor(Scene scene, Ray in_ray, int level) { Point hit = Findntersection(in_ray, scene); RGB color = calcemissioncolor(scene) + calcambientcolor(scene); for (int i = 0; i < getnumlights(scene); i++) { Light light = getlight(i,scene); color += calcdiffusecolor(scene,hit,light) + calcspecularcolor(scene,hit,light); } if (level == MAX_LEVEL) return rgb(0,0,0); } Vector normal = getnormalatpoint(hit); Ray out_ray = ConstructOutRay (in_ray, normal); color += K_s * CalcColor(scene, out_ray, level+1); return color; // Here should come a similar part for // a Refractive Ray

77 Effect of Recursion Depth Depth=1 Depth=1 Depth=2 Depth=3 Depth=5

78 Difference Between Levels 1 st Diff 5 st Diff (amplified)

79 Direct Diffuse + ndirect Specular and transmission 79

80 + Soft Shadows 80

81 + Caustics = the envelope of light rays reflected or refracted by a curved surface or object 81

82 + ndirect Diffuse llumination 82

83 Example

84 Example "Balanza" Jaime Vives Piqueres (2002)

85 Part V: Acceleration Techniques Bounding volume hierarchies Spatial partitions: Uniform grids Octrees BSP trees

86 Bounding Volumes Key idea: check for intersection with simple shape first Use bounding volume for shapes f ray doesn t intersect bounding volume, then it doesn t intersect its contents!

87 Bounding Volume Hierarchies Build hierarchy of bounding volumes Bounding volume of interior node contains all children 1 3 E 1 D F 2 C 3 A B D E F A 2 C B

88 Ray Cast Using BVH Use hierarchy to accelerate ray intersections ntersect node contents only if hit bounding volume 1 3 E 1 D F 2 C 3 A B D E F A 2 C B

89 Sort Hits & Detect Early Termination Findntersection(Ray ray, Node node) { // Find intersections with bounding volumes // of child node... // Sort intersections front to back... // Process intersections // checking for early termination min_t = infinity; for each intersected child i { if (min_t < bv_t[i]) break; shape_t = Findntersection(ray, child); if (shape_t < min_t) { min_t = shape_t;} } return min_t; }

90 Uniform Grid Construct uniform grid over scene ndex primitives according to overlaps with grid cells D E F C A B

91 Uniform Grid Trace rays through grid cells E Fast ncremental D F Only check primitives in intersected grid cells Given an entry point into a cell and a vector, its easy to calculate exit point A B C

92 Uniform Grid Potential problem: How choose suitable grid resolution? Too little benefit if grid is too coarse D E F Too much cost if grid is too fine A C B

93 Octree (Quadtree in 2D) An octree is a tree data structure that represents a recursive, hierarchical subdivision of 3-dimensional space into cubes. Each internal node represents a cube that is divided by 3 axis aligned planes into 8 equal size sub-cubes.

94 Octree Storage

95 Octree Scene Construction Constructs adaptive grid over scene: Recursively subdivide box-shaped cells into 8 octants ndex primitives by overlaps with cells D A E F C Fewer cells than a grid (why?) B Quadtree

96 Ray Cast with Octree Trace rays through neighbor cells Fewer cells but More complex neighbor finding than a grid A trade-off between fewer cells and more expensive traversal D A E B F C

97 Other Uses of Octree Very useful in computer graphics for: ntersections Collisions Color quantization Surface reconstruction (meshing) 97

98 Binary Space Partition (BSP) Tree Recursively partition space by planes Every cell is a convex polyhedron 1 3 E D F C A 4 B 2

99 Example: BSP Point Finding Simple recursive algorithms 1 P 3 E D F C A 4 B 2

100 Ray Cast with BSP Recursion on BSP tree enables simple frontto-back traversal 1 P 3 E D F C A 4 B 2

101 Ray Cast with BSP Tree RayTreentersect(Ray ray, Node node, double min, double max) { if (Node is a leaf) return intersection of closest primitive in cell, or NULL if none else dist = distance of the ray point to split plane of node near_child = child of node that contains the origin of Ray far_child = other child of node if the interval to look is on near side return RayTreentersect(ray, near_child, min, max) else if the interval to look is on far side return RayTreentersect(ray, far_child, min, max) else if the interval to look is on both side if (RayTreentersect(ray, near_child, min, dist)) return ; else return RayTreentersect(ray, far_child, dist, max) }

102 Other Accelerations Screen space coherence Check last hit first Beam tracing Pencil tracing Cone tracing Memory coherence Large scenes Parallelism Ray casting is embarrassingly parallelizable More...

103 Simple Model for Exercise Use only reflections and shadow rays

104 Summary Ray casting (direct llumination) Usually use simple analytic approximations for light source emission and surface reflectance Recursive ray tracing (global illumination) ncorporate shadows, mirror reflections, and pure refractions Note: remember these are approximation so that it will be practical to compute

105 Appendix: llumination Terminology Radiant power [flux] (F) Rate at which light energy is transmitted (in Watts). Radiant ntensity () Power radiated onto a unit solid angle in direction (in Watts/sr) e.g.: energy distribution of a light source (inverse square law) Radiance (L) Radiant intensity per unit projected surface area (in Watts/m 2 sr) e.g.: light carried by a single ray (no inverse square law) rradiance (E) ncident flux density on a locally planar area (in Watts/m 2 ) e.g.: light hitting a surface along a Radiosity (B) Exitant flux density from a locally planar area (in Watts/ m 2 )