Shadows & Programmable Shaders

Transcription

1 Advanced Rendering Dr. Alexander G. Gee Shadows & Programmable Shaders Shadows Shadow Mapping Using OpenGL Hardware Soft Shadow Volumes Programmable Pipelines OpenGL Shading Language (GLSL) Coupling GLSL to Applications page 2 1

2 Conceptual Model for Computer Graphics Real Light Real Object Synthetic Light Synthetic Camera Human Eye Synthetic Model Graphics System Display Device page 3 Standard OpenGL Projected Scene page 4 2

3 The Scene of Objects with Shadows page 5 Simple Shadows The shape of the shadow is dependent on The position of the light source The shape of the object The form of the surface on which the plane of the shadow falls View from light source Camera view Top down page 6 3

4 Non-Parallel Point Light Source Shadows Standard light sources are generally those defined at an infinite distance from the objects, as depicted previously The other standard light source is one that represents a light bulb or candle that occupies no space These light sources project rays of light in all directions from the lights spatial position page 7 Eye Linear Texture Coordinate Generation Model Coordinates Modeling Transformation World Coordinates Viewing Transformation View Coordinates Shadows 4D Eye-Linear Projection Transformation (s, t, r, q) Projection Coordinates 3D Normalization Transformation and Clipping against default cube Normalized Coordinates 3D Viewport Transformation + Depth Range Display Coordinates 2D page 8 4

5 Visualizing the Shadow Mapping Technique Shadow Projection Light s point-of-view Depth Map Vertex Distance Kilgard 2002 SIGGRAPH Course 31, NVIDIA Corporation page 9 Hardware Shadow Map Filtering Example GL_NEAREST: blocky GL_LINEAR: antialiased edges Low shadow map resolution used to heightens filtering artifacts Kilgard 2002 SIGGRAPH Course 31, NVIDIA Corporation page 10 5

6 Combining Projective Texturing & Spotlights Use a spotlight-style projected texture to give shadow maps a spotlight falloff Kilgard 2002 SIGGRAPH Course 31, NVIDIA Corporation page 11 Real Soft Shadows Assarsson & Akenine-Möller 2003 Graphics Hardware, Chalmers University of Technology page 12 6

7 Hard vs. Soft Shadows Two different light source types point source area source umbra penumbra umbra Assarsson & Akenine-Möller 2003 Graphics Hardware, Chalmers University of Technology page 13 Area / Volumetric Lights Give Soft Shadows Real lights have area or volume Thus, soft shadows are more realistic HARD SOFT Assarsson & Akenine-Möller 2003 Graphics Hardware, Chalmers University of Technology page 14 7

8 Real-Time Soft Shadow Volume Algorithm Same Silhouette Edge Assarsson & Akenine-Möller 2003 Graphics Hardware, Chalmers University of Technology page 15 Soft Shadow Wedge Rasterization Assarsson & Akenine-Möller 2003 Graphics Hardware, Chalmers University of Technology page 16 8

9 Example Soft Shadows Varying the size of the light source Assarsson & Akenine-Möller th Eurographics Workshop, Chalmers University of Technology page 17 Soft Shadows for Two Light Sources Assarsson & Akenine-Möller th Eurographics Workshop, Chalmers University of Technology page 18 9

10 Better Approximation Single Area Light Source 2 x 2 Area Light Sourcs 1024 Point Light Sources Single Area 2 x 2 Area 3 x 3 Area 1024 Points Assarsson & Akenine-Möller 2003 Graphics Hardware, Chalmers University of Technology page 19 Programmable Vertex Shader glcolor glortho glloadmatrix Application Program OpenGL State Vertex Shader Primitive Assembly gl_vertex gl_modelviewmatrix gl_projectionmatrix gl_frontcolor gl_position page 20 10

11 Programmable Fragment Shader OpenGL State Application Program vertices Rasterizer fragments Fragment Shader pixels Frame Buffer gl_position gl_frontcolor (per vertex) gl_frontcolor (interpolated) gl_fragmentcolor page 21 Advanced Rendering Ray Tracing Radiosity page 22 11

12 Overview OpenGL is based on a pipeline model in which primitives are rendered one at time No shadows No multiple reflections Except by tricks or multiple renderings Global approaches Ray tracing Radiosity page 23 Lighting Effects S. Narasimhan 2005 Appearance Modeling page 24 12

13 The Cornell Box Physically based rendering: a simple physical environment for which the properties of lighting, geometry, and material reflectance have been measured and their simulation via computers Measured S. Narasimhan 2005 Appearance Modeling Simulated page 25 The Cornell Box Difference image as a simple pixel-by-pixel subtraction of one image from the other Edge effects stem from misregistration between the actual positions and rendered coordinates Measured S. Narasimhan 2005 Appearance Modeling Difference Simulated page 26 13

14 Ray Tracing Ray tracing handles Global reflections and refractions (transmissions) Hidden surface removal Shadows and other special effects Whitted presented the first general recursive ray tracing model in 1980 (Others had used some of the ideas before) page 27 Overview Ray tracing follows specularly reflected and refracted rays through a scene The rays are infinitely thin, so aliasing is a problem Ray tracing can be used as a basic technique for volume rendering Ray tracing is recursive; secondary rays are followed recursively from primary rays page 28 14

15 Tracing Light Rays Follow rays of light from a point source Can account for reflection and refraction translucent surface shadow Angel 2006 Interactive Computer Graphics 4 th Ed. page 29 Computation Should be able to handle all physical interactions Ray tracing paradigm is not computational Most rays do not affect what we see Scattering produces many (infinite) additional rays Alternative: ray casting page 30 15

16 Ray Casting Only rays that reach the eye matter Reverse direction and cast rays Need at least one ray per pixel Center-of-Projection Projection Plane page 31 Ray Casting Quadrics Ray casting has become the standard way to visualize quadrics which are implicit surfaces in CSG systems Constructive Solid Geometry (CSG) Primitives are solids Build objects with set operations Union, intersection, set difference page 32 16

17 Ray Casting a Sphere Ray is parametric P(α) = p + α v Sphere is quadric (x x c ) 2 + (y y c ) 2 + (z z c ) 2 r 2 = 0 Resulting equation is a scalar quadratic equation which gives entry and exit points of ray (or no solution if ray misses) P(α 2 ) v p P(α 1 ) page 33 Basic Ray Tracing Algorithm Fire an eye ray r through the center of a pixel to find the intersection with the closest object A (specular) reflected ray is followed from the intersection A (specular) refracted ray is followed from the intersection To calculate shadows, fire a shadow ray (or shadow feeler) s from the intersection point to each light source Reflection, refraction, and shadow are handled by secondary rays from the point of intersection page 34 17

18 Reflection (Specular) reflection ray is given by r = l 2 n (n l) where n and l are the unit surface normal and unit incident vector hitting the surface Note that l is first an eye ray, then for secondary rays it can be a reflected or refracted ray; its role changes as rays are cast from ray-object intersection points l r page 35 Refraction (Specular) refraction (transmission) ray is given η1 η1 t = [ ( )] η l n cosθ + n l 2 t η2 where η 1 and η 2 are the indices of refraction in the incident and transmitted media (e.g., air and water), and θ t is the angle between the transmitted ray and the negative normal, and cos θ t = 1 η2 η ( ) ( 1 ( n l) ) l r t page 36 18

19 Shadow Rays Even if a point is visible, it will not be lit unless we can see a light source from that point Shadow rays are fired from the intersection point to each light source l s page 37 Ray Trees Each reflection and refraction ray may spawn their own shadow, reflection and refraction rays, recursively generating a ray tree The ray tree can be built to any depth subject to The ray does not hit an object The amount of work (time) we are willing to spend The amount of contributed illumination from the rays l 1 l 2 r r 1 t 1 page 38 19

20 Ray Trees eye r l 1 l 2 r 1 obj 1 t 1 obj 2 r 1 t 1 l 1 obj 3 r r 2 l 2 l 3 obj 4 l 4 page 39 Light Intensity Equation The ray tree is evaluated bottom-up with each node s intensity a function of its children s intensity At each intersection point I = Ilocal + krg I reflected + ktg Itransmitted where I local is the intensity at the intersection k rg is the global reflectivity (between 0 and 1) I reflected is the reflected ray s returned intensity k tg is the global transmissivity (between 0 and 1) I transmitted is the refracted ray s returned intensity The local intensity can be computed from any illumination model, e.g. I local [ k ( n l) + k ( n h) ] α = I aka + I p d s page 40 20

21 Diffuse Surfaces Theoretically the scattering at each point of intersection generates an infinite number of new rays that should be traced In practice, we only trace the transmitted and reflected rays but use the Phong model to compute shade at the point of intersection Consequently, ray tracing works best for specular surfaces Radiosity works best for perfectly diffuse (Lambertian) surfaces page 41 Building a Ray Tracer Best expressed recursively Image based approach For each ray. Find intersection with closest surface Need whole object database available Complexity of calculation limits object types Compute lighting at surface Trace reflected and transmitted rays page 42 21

22 Recursive Ray Tracer Color Trace(Point p, Vector d, int step) { if(step > max) return(background_color) Point q = Intersect(p, d, status) if(status==light) return(light_source_color) if(status==no_inter) return(background_color) Normal n = Normal(q) Vector re = Reflection(q, n) Vector ra = Refraction(q, n) } Color local = Phong(q, n, r) Color reflected = Trace(q, re, step+1) Color refracted = Trace(q, ra, step+1) return(local + reflected + refracted) page 43 Ray Casting For Intersections A basic ray tracer intersects the eye ray with every object in the scene The nearest object hit spawns secondary rays that are also intersected with every object in the scene It has been estimated that 95% of the time in ray tracing is spend on intersection tests Quadric surfaces are often used in ray tracing because ray-quadric surface intersections are easy to compute Spheres (a quadric surface) is often used as a bounding volume Ray-polygon intersections are easy (but expensive for a fine polygon mesh) Ray-box intersections are also fairly easy, and (generalized) boxes make good bounding volumes Intersection tests for bicubic patches, fractals, surfaces of revolutions, and other objects have been developed page 44 22

23 Computing Intersections Sphere Intersection Algebraic Solution Geometric Solution Polygon Intersection page 45 Sphere Intersection - Algebraic Solution 1 Ray-sphere intersection calculations are easy Define a sphere (x x s ) 2 + (y y s ) 2 + (z z s ) 2 = r 2 where c = (x s, y s, z s ) is the center of the sphere and r is the radius Define a ray from the point of projection x = x 0 + x d t y = y 0 + y d t z = z 0 + z d t where p 0 = (x 0, y 0, z 0 ) is the ray s origin and r d = x d, y d, z d is the unit direction vector for the ray page 46 23

24 Sphere Intersection - Algebraic Solution 2 Plug in the parametric line into the sphere equation yields the quadratic equation At 2 + Bt + C = 0 where A = x d2 + y d2 + z 2 d B = 2[x d (x 0 x s ) + y d (y 0 y s ) + z d (z 0 z s )] C = (x 0 x s ) 2 + (y 0 y s ) 2 + (z 0 z s ) 2 r 2 If B 2 4AC < 0, the ray and sphere do not intersect If B 2 4AC < 0, ray grazes sphere If B 2 4AC < 0, the smallest positive t corresponds to the intersection B ± B 2 4AC If the intersection is t± = 2A then the surface normal is n =,, xi xs r yi ys r zi zs r page 47 Sphere Intersection - Geometric Solution 1 A more efficient solution can be determined geometrically Find if the ray s intersection is outside the sphere v 2 (r 0 c r 0 c) = (x s x 0 ) 2 + (y s y 0 ) 2 + (z s z 0 ) 2 r 2 Find the closest approach of the ray to the sphere s center t ca = v cosθ ( r0c r0 c) = v r0c rd = x x, y y, z s 0 s (this assumes x d, y d, z d is a unit vector) 0 s z 0 x, y, z d d d page 48 24

25 Sphere Intersection - Geometric Solution 2 If the ray is outside and points away from the sphere it misses the sphere, that is, if v 2 r 2 and t ca < 0 the sphere is missed If the origin is inside the sphere or t ca 0, find the squared distance from t ca to the sphere s surface This is the half chord distance squared If the ray misses the sphere, it can be negative t hc2 = r 2 (v 2 t ca2 ) If t hc < 0, the ray misses the sphere If t hc 0, then 2 t = t if v 2 r 2 ca t hc and 2 t = t if v 2 < r 2 ca + t hc page 49 Polygon Intersection If p is the plane of the polygon, then n p + d = 0 Let the ray be given by r(t) = r 0 + r d t The intersection of the plane and the ray is at parameter value d + n r t = 0 n r d This calculation requires 12 floating point operations and 3 tests If n r d = 0, the intersection is rejected If t < 0, the intersection is rejected If a closer intersection has been found the intersection is rejected page 50 25

26 Polygon Inside/Outside Testing 1 If the intersection is not rejected, determine if it is inside the polygon The solution presented here is based on triangles, and works for convex polygons A polygon with n vertices can be broken up into n 2 triangles Given a point p, v 0 p = αv 0 v 1 + βv 0 v 2 A point p is inside the triangle (v 0, v 1, v 2 ) if α 0, β 0, and α + β 1 Rewritten as x p x 0 = α(x 1 x 0 ) + β(x 2 x 0 ) y p y 0 = α(y 1 y 0 ) + β(y 2 y 0 ) z p z 0 = α(z 1 z 0 ) + β(z 2 z 0 ) page 51 Polygon Inside/Outside Testing 2 A more general algorithm Reduce dimensionality project polygon and intersection point orthographically onto coordinate plane which yields largest projection Translate polygon so intersection point is at origin Draw ray from origin along positive u axis Count how many times it intersects polygon Even number of intersections outside Odd number of intersections inside page 52 26

27 Efficient Considerations in Ray Tracing Three basic approaches to making ray tracing more efficient are To compute fewer intersections Adaptive depth control and adaptive sampling To compute intersections faster Bounding volume, light buffers, space subdivision To use generalized rays Beams, cones, pencils Adaptive depth control allows the depth of the ray tree to change, getting deeper when the returned intensity is large and more shallow when the returned intensity is small Bounding volumes can be used to quickly discard objects that can not be intersected page 53 Adaptive Depth Control The percentage of a scene that consists of highly reflective and transparent surfaces is, in general, small It is inefficient to trace all rays to a maximum depth The ray tree can be pruned when little intensity will result from the pruned sub-tree Rays are attenuated as they pass through a scene Adaptive depth control uses properties of the materials the ray hits to determine this attenuation Reflected rays are attenuated by the global reflection coefficient k rg Refracted rays are attenuated by the global refraction coefficient k tg page 54 27

28 Bounding Volumes in Ray Tracing Bounding volumes, or extents, or enclosures are the most basic tool for ray tracing acceleration A bounding volume encloses a given object (or objects) It should be easier to compute the ray-bounding volume intersection than the ray-object intersection Using simple bounding volumes substitutes simple intersection checks for more costly ones, but does not decrease the number of intersections The simplicity of the intersection test is not the only criterion; the void area of bounding volume should also be considered page 55 Distributed Ray Tracing Cook, Porter, and Carpenter introduced this concept in 1984 Distributed ray tracing is a Monte Carlo technique that stochastically distributes the direction of rays It supersamples the image, firing more than one type Sampling the reflected ray according to a specular distribution function produces gloss (blurred reflection) Sampling the transmitted ray produces translucency (blurred transparency) Sampling the solid angle of the light source produces penumbras (soft shadows) Sampling the camera lens area produces depth of field (focusing) Sampling in time produces motion blur page 56 28

29 Example Ray Traced Image Turner Whitted 1980 page 57 Radisoity page 58 29

30 Introduction Ray tracing is best with many highly specular sufaces Not characteristic of real scenes Rendering equation describes general shading problem Radiosity solves rendering equation for perfectly diffuse surfaces page 59 Phong Shading Plastic Looking Scene No shadows No object interactions S. Narasimhan 2005 Appearance Modeling page 60 30

31 Ray Tracing Scene does not look realistic enough Is window flush with wall? Where is the corner of the room? Is the carpet and wood supposed to be this dark? S. Narasimhan 2005 Appearance Modeling page 61 Radiosity Indirect lighting affects realism Window has depth Room has a corner Carpet and wood on table is lighter Wall has a shade of pink S. Narasimhan 2005 Appearance Modeling page 62 31

32 Introduction The radiosity method and ray tracing are the two major post-phong approaches to rendering objects Radiosity was introduced to computer graphics in 1984 by a group from Cornell (Goral, Torrance, and Greenberg) Radiosity is a theoretically rigorous method to the diffuse interaction problem in a closed environment Specular reflection is difficult to include Assumes conservation of light energy in a closed environment Assumes all emission and reflection processes are ideal (Lambertian) diffuse page 63 Terminology Energy ~ light (incident, transmitted) Must be conserved Energy flux = luminous flux = power = energy/unit time Measured in lumens Depends on wavelength so we can integrate over spectrum using luminous efficiency curve of sensor Energy density (Φ) = energy flux/unit area page 64 32

33 Terminology Intensity ~ brightness Brightness is perceptual = flux/area-solid angle = power/unit projected area per solid angle Measured in candela Φ = I da dω page 65 Rendering Equation Outgoing light is from two sources Emission Reflection of incoming light Must integrate over all incoming light Integrate over hemisphere Must account for foreshortening of incoming light n l r θ i θ r φ v page 66 33

34 Rendering Equation Rendering equation is an energy balance Energy in = energy out Integrate over hemisphere Fredholm integral equation Cannot be solved analytically in general Various approximations of R bd give standard rendering models Should also add an occlusion term in front of right side to account for other objects blocking light from reaching surface page 67 Conservation of Energy = + Emitted power = self-emitted power + received & reflected power S. Narasimhan 2005 Appearance Modeling page 68 34

35 Rendered Image Angel 2006 Interactive Computer Graphics 4 th Ed. page 69 Patches Angel 2006 Interactive Computer Graphics 4 th Ed. page 70 35

36 Diffuse Interreflections - Radiosity Consider Lambertian surfaces and sources Radiance independent of viewing direction Consider total power leaving per unit area of a surface Can simulate soft shadows and color bleeding from diffuse surfaces Used abundantly in heat transfer literature page 71 Irradiance and Radiosity Irradiance E Power received per unit surface area Units: W/m 2 Radiosity Power per unit area leaving the surface (like irradiance) S. Narasimhan 2005 Appearance Modeling page 72 36

37 Radiosity Radiosity is a global illumination model that assumes all surfaces are diffuse and computes view independent illumination. Radiosity computes the intensity reflected from each small surface region (differential area) equally in all directions (including the eye...). This intensity includes energy emitted by the surface itself and reflected energy from other objects. Following an (expensive) preprocessing to compute radiosity, the model is shaded and can be rendered interactively. J. Huang 2002 Computer Graphics, Ohio State page 73 Basic Definitions Radiosity: (B) Energy per unit area per unit time. Emission: (E) Energy per unit area per unit time that the surface emits itself (e. g., light source). Reflectivity: (ρ) The fraction of light which is reflected from a surface. (0 ρ 1) Form-Factor: (F) The fraction of the light leaving one surface which arrives to another. (0 F 1) J. Huang 2002 Computer Graphics, Ohio State page 74 37

38 The Basic Radiosity Equation We will compute the light emitted from a single differential surface area da i It consists of: 1. Light emitted by da i 2. Light reflected by da i depends on light emitted by other da j, fraction of it reaches da i The fraction depends on the geometric relationship between da i and da j : the form-factor J. Huang 2002 Computer Graphics, Ohio State page 75 Form-Factor F j i = the fraction of power emitted by j, which is received by i Area If i is smaller, it receives less power Orientation If i faces j, it receives more power Distance If i is further away, it receives less power j i S. Narasimhan 2005 Appearance Modeling page 76 38

39 The Basic Radiosity Equation The relationship between a single differential area s radiosity and the radiosities of the rest of the environment: J. Huang 2002 Computer Graphics, Ohio State page 77 The Discrete Radiosity Equation We can t operate in continuous space. We need a finite problem! We divide the surfaces into small discrete areas called patches. We assume that radiosity and emission do no vary across the patch area The radiosity of a patch is then: Or, more conveniently: J. Huang 2002 Computer Graphics, Ohio State page 78 39

40 The Reciprocity Relationship If we had equal sized emitters and receivers, the fraction of energy emitted by one and received by the other would be identical to the fraction of energy going the other way Thus, the form-factors from A i to A j and from A j to A i are related by the ratios of their areas: Thus: The radiosity equation is now: J. Huang 2002 Computer Graphics, Ohio State page 79 Matrix Formulation For an environment of N patches: Because the sum of form-factor along a lines is less then 1, this matrix is diagonally dominant and therefore iterative Gauss-Siedel is guaranteed to (quickly) converge to a solution! J. Huang 2002 Computer Graphics, Ohio State page 80 40

41 The Form-factor The form-factor is purely a function of geometric relationship between patches and thus does not depend on viewer position or surface reflectivity attributes Between differential areas: Between patches: J. Huang 2002 Computer Graphics, Ohio State page 81 Computation Hurdles Computation of the form-factors Solutions: hemicube, elements Solution of the matrix in time O(N 2 ) Solution: progressive shooting J. Huang 2002 Computer Graphics, Ohio State page 82 41

42 Nusselt s Analog The inner integral in the previous double integral represents the form-factor from a differential area to a finite patch. This quantity can also be computed by the fraction of the base of the hemisphere covered by the projection: J. Huang 2002 Computer Graphics, Ohio State page 83 The Hemi-cube Any patch which covers the same projected area on the hemisphere has the same form-factor Any surface can be used to project the patches onto, without changing the form-factor J. Huang 2002 Computer Graphics, Ohio State page 84 42

43 The Hemi-cube Compute the delta form-factor of each grid cells F and store in a table (next slide...) Project all patches onto the hemi-cube screen (use Z- buffer rendering hardware?), drawing a patch-id instead of color Sum the delta form factors of all grid cells covered by the patch s id J. Huang 2002 Computer Graphics, Ohio State page 85 Computing the Delta Form-Factor J. Huang 2002 Computer Graphics, Ohio State page 86 43

44 Problems with the Hemi-cube Because we compute only the inner integral serious inaccuracies can occur if the size of the patch is large relative to the distance Because the hemisphere is divided into discrete solid angles, a number of aliasing problems may occur J. Huang 2002 Computer Graphics, Ohio State page 87 Patches and Elements Patches are used for emitting light Some patches are divided into elements, which are used to more accurately compute the received light after the patch solution have been computed J. Huang 2002 Computer Graphics, Ohio State page 88 44

45 Progressive Radiosity The problem: we need to solve a set of N equations O(N 2 ) time In the conventional radiosity we gather the contribution of other patches to patch i J. Huang 2002 Computer Graphics, Ohio State page 89 Shooting Radiosity Shoot the radiosity of patch i and update the radiosity of all other patches J. Huang 2002 Computer Graphics, Ohio State page 90 45

46 Visual Comparison Assume the scene is displayed after each update: In the row solution (gather), every image will show another element being fully illuminated In the column solution (shoot), every image will show the whole image change due to the contribution by one element. The magnitude of this change depends on the amount of energy B i A i sent out from the shooting patch Therefore, the image will converge faster to the final solution if the patches with highest added energy B i A i shoot first! J. Huang 2002 Computer Graphics, Ohio State page 91 The Total Process Form-Factor Calculation Solution to the System of Equations Input Scene Geometry Reflectance Properties Radiosity Solution Visualization Viewing Conditions Radiosity Image page 92 46

47 Planar Piecewise Constancy Assumption Subdivide scene into small uniform polygons S. Narasimhan 2005 Appearance Modeling page 93 Uniform vs. Adaptive Problem: Texture mapping is equivalent to a uniform mesh subdivision Resolution is too low in some areas, too high in others uniform adaptive Solution: Coarse geometric hierarchical subdivision, followed by finer texture map subdivision G. Coombe, M. Harris, A. Lastra 2004 Radiosity on Graphics Hardware page 94 47

48 Adaptive Subdivision Subdivide geometry using quad-tree Store energy at leaf texture (32x32 resolution) Data structure managed on CPU Calculate subdivision criteria in fragment programs Check for gradient or value discontinuities Occlusion queries get result G. Coombe, M. Harris, A. Lastra 2004 Radiosity on Graphics Hardware page 95 Sample Scene S. Narasimhan 2005 Appearance Modeling page 96 48

49 Sample Scene S. Narasimhan 2005 Appearance Modeling page 97 Sample Scene S. Narasimhan 2005 Appearance Modeling page 98 49

50 Sample Scene S. Narasimhan 2005 Appearance Modeling page 99 Summary Classic radiosity = finite element method Assumptions Diffuse reflectance Usually polygonal surfaces Advantages Soft shadows and indirect lighting View-independent solution Precompute for a set of light sources Useful for walkthroughs page

51 Image vs. Object Space Image space: Ray Tracing Trace backwards from viewer View-dependent calculation Result: rasterized image (pixel by pixel) Object space: Radiosity Assume only diffuse-diffuse interactions View-independent calculation Result 3D model, color for each surface patch Can render with OpenGL page 101 Two Pass Solution First Pass: Diffuse Interreflections View independent, global diffuse illumination computed with radiosity. Second Pass: Specular Interreflections View dependent, global specular illumination computed with ray-tracing. Combine strengths of radiosity and ray-tracing. page

52 Questions? page