Parallelization. Memory coherency helps when distributing rays to various threads/processors
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2 Ra Tracing OpenGL projects triangles onto the image plane and rasteries them to determine which piels the cover Scanline rendering is a per triangle operation Ra Tracing instead works as a per piel operation For each piel, a ra is created to represent the light coming inwards from the world that hits that piel That ra is traced backwards out of the camera into the world to find out where the light came from and thus determine what color to shade that piel /
3 Paralleliation Ra tracing is inherentl parallel, since the ras for one piel are independent of the ras for other piels Can take advantage of modern parallel CPU/GPU/Clusters to significantl accelerate a ra tracer Threading (e.g., Pthread, OpenMP) distributes ras among cores Message Passing Interface (MPI) distributes ras among processors across different machines OptiX/CUA distributes ras on the GPU Memor coherenc helps when distributing ras to various threads/processors Assign the spatiall neighboring ras (on the image plane) to the same core/processor These ras tend to intersect with the same objects in the scene, and thus need access to the same memor /
4 Tpicall not used in games Real Time Ra-Tracing Could be used in games, but tpicall considered too epensive However, GPUs are ver good at tasks that are easil paralleliable For eample, NVIIA Opti real time ra tracer 4/
5 Basic Ra Tracing 5/
6 Ra Tracing Generate an image b backwards tracing the path of light through piels on the image plane Simulate the interaction of light with objects 6/
7 Aperture and Image Plane Start with an ee pupil or aperture (a focal point), along with an image plane with piels 7/
8 Constructing Ras Ra equation: Rt () = E+ tp ( E) t [0, + ) Ee pupil located at t=0 Piel center located at t= 8/
9 Object Intersection Shoot the ra from the ee through the piel, and find the first object it hits 9/
10 Shadow Ra Shoot a shadow ra toward the light, and see if the intersection point is shadowed 0/
11 Shading If the light is visible from the intersection point, compute the shading using the light source /
12 Shading the Intersection Point Similar to the OpenGL shading model The shading on each intersection point is the sum of contributions from all light sources Cast ras from the intersection point to all light sources Light tpes: Ambient light, point light, directional light, spot light, area light, volume light, etc. Material properties: iffusion, specular, shininess, emission, etc. Shading models: iffusive shading, Phong shading, BRFs, etc. /
13 Piel Color Use the results of the shading to record a color for that piel /
14 Pseudocode Image Ratrace (Ee ee, Scene scene, int width, int height) { Image image = new Image (width, height) ; for (int i = 0 ; i < height ; i++) for (int j = 0 ; j < width ; j++) { Ra ra = RaThruPiel (ee, i, j) ; Intersection hit = Intersect (ra, scene) ; image[i][j] = FindColor (hit) ; } return image ; } 4/
15 Lights 5/
16 Light Tpes Point Light irectional Light Spot Light Area Light Area Light from a light tube Volume light 6/
17 Area Lights Treat the area light like a bunch of point lights Shoot a number of ras from the intersection point to different points on the area light Take the average of the results Creates soft shadows 7/
18 Question If I break an area light into 00 point lights, how do I set the strength of each point light? 8/
19 Shadows 9/
20 Shadow Ras etect shadows b casting ras to the light source R( t) = S + t( L S) t [ ε,) Test for occluder intersection No occluder intersection, shade normall (e.g. Phong, BRF) Yes occluder intersection, skip light (don t skip ambient) 0/
21 Spurious Self-Occlusion Incorrect self-shadowing Correct /
22 Spurious Self-Occlusion Add ε to the starting point of shadow ras to avoid accidental re-intersection with the original surface: t [ε,) This can often fail for graing shadow ras near the objects silhouette Better to offset the intersection point in the normal direction from the surface The direction of the shadow ra shot from the perturbed point to the light ma be slightl different from the direction of the original shadow ra Ma also need to avoid placing the new starting point too close to or inside other nearb objects! The perturbed point ma still be inside the object offset in the ra direction offset in the normal direction /
23 Question Wh not just ignore the object completel when casting a shadow ra? /
24 Ra-Object Intersections 4/
25 Ra-Object Intersections Given a ra R(t)=A+t, find the first intersection with an object where t t min and t t ma The object geometr can be a polgon mesh, an implicit surface, a parametric surface, etc. oesn t have to be decomposed into triangles (as was required b OpenGL) Goal: write intersection code for all kinds of different geometries 5/
26 Ra-Sphere Intersections 6/
27 Ra-Sphere Intersection R ( t) = A + t Ra equation: Implicit equation for a sphere: ( X C) = r Combine them together: ( A + t C) = r Quadratic equation in t : t + ( A C) t + ( A C) r With discriminant: = 4[( A C) ] 4( A C) + r = 0 7/ /
28 Ra-Sphere Intersection > 0 = 0 < 0 For the case with two solutions, choose the first intersection (smaller t) 8/
29 Ra-Sphere Intersection Intersection Point: P = A + t int Intersection Normal: N = P P C C P 9/
30 Implicit Surfaces 0/
31 Ra Implicit Surface Intersection The Ra-Sphere intersection was an eample of this! /
32 Ra-Plane Intersections /
33 Ra-Plane Intersection Ra equation: R ( t) = A + t Implicit equation for a plane: a + b + c + d = 0 Combine them together and solve for t to find the point of intersection: a( + t ) + b( + t ) + c( + t ) + d = 0 A A A Just another implicit surface intersection! /
34 Ra-Triangle Intersections 4/
35 Ra-Triangle Intersection First, find the ra-plane intersection with the plane containing the triangle Then, project the triangle vertices and the ra-plane intersection point to a trul plane Finall, run the point-inside-triangle test in as we did for rasteriation in Week 5/
36 Or avoiding projection Compute the normal direction n orthogonal to edge e, and pointing towards the edge s opposite verte in the plane of the triangle: n= e e e e e Given an endpoint P of e, test whether ( P P ) n 0 to see if the intersection point is inside that edge 6/
37 Recall Barcentric Coordinates 7/
38 Ra-Triangle Intersection Ra equation: R ( t) = A + t Parametric equation for triangle: X = P + α P P ) + β ( P ( P Combine: A + t = P + α P P ) + β ( P ( P ) ) P A P P 8/
39 9/ Ra-Triangle Intersection equations with unknowns ) ( ) ( P P P P P t A + + = + β α + + = = = + ) ( ) ( ) ( ) ( ) ( ) ( t t t A A A β α β α β α ),, ( ),,, ( ),,, ( P P P = = =
40 40/ Ra-Triangle Intersection Matri form: Satisfing = t A A A β α α β α 0 0 min t ma t t
41 4/ Cramer s rule The 4 unique matrices have some common columns, so the number of operations can be reduced b reusing numbers when computing determinants Ra-Triangle Intersection A A A = α A A A = β A A A t =
42 Pseudocode bool RaTriangle (Ra R, Vec V, Vec V, Vec V, Interval [t min, t ma ]) { compute t; if(t < t min or t > t ma ) return false; compute α; if(α < 0 or α > ) return false; compute β; if(β < 0 or β > - α) return false; return true; } // Notice the conditions for earl termination. // Recall: for the point inside triangle test, one can return earl as well when testing the edges 4/
43 Question Wh might we need more than one wa of computing ra-triangle intersections? 4/
44 Triangle Normals 44/
45 Barcentric interpolation Normals As usual interpolate from the normals at the triangle s vertices: N = ( α β ) N + αn + βn Bump mapping, normal mapping, displacement mapping all done similarl as in opengl (tetures too ) 45/
46 Transformed Objects 46/
47 Ra Tracing Transformed Objects A triangle is still a triangle after transformation But a sphere can become an ellipsoid: Write another intersection routine? Better et, can reuse ra-sphere intersection code. 47/
48 Ra Tracing Transformed Objects Intersect the untransformed object in object space with the inverse-transformed ra (put into object space) Transform the results back into world coordinates p = Mp and N = M T N Be careful with the normal! 48/
49 Final Project 49/
50 Question 4 Brainstorm some on our final project 50/
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