Course Roadmap Rasterization Pipeline Rasterization Core Concepts Transforms & Projection • Sampling Texture Mapping • Antialiasing Visibility, Shading, Overall Pipeline • Transforms Geometric Modeling Intro to Geometry Curves and Surfaces Core Concepts Meshes and Geometry Processing • Splines, Bezier Curves • Topological Mesh Representations Today: Ray-Tracing & Acceleration • Subdivision, Geometry Processing Lighting & Materials Cameras & Imaging CS184/284A Ren Ng Lecture 10: Intro to Ray-Tracing & Accelerating Ray-Scene Intersection Computer Graphics and Imaging UC Berkeley CS184/284A Basic Ray-Tracing Algorithm Ray Casting Appel 1968 - Ray casting 1. Generate an image by casting one ray per pixel 2. Check for shadows by sending a ray to the light CS184/284A Ren Ng Ray Casting - Generating Eye Rays Pinhole Camera Model eye ray closest scene (starts at eye and intersection point goes through pixel) note: more intersection points eye point image plane light source CS184/284A Ren Ng Ray Casting - Shading Pixels (Local Only) Pinhole Camera Model eye ray (starts at eye and goes through pixel) eye point image plane perform shading calculation here to compute color of pixel light source (e.g. Blinn Phong model) CS184/284A Ren Ng Discussion: Ray Casting vs Rasterization Remember: • Rasterization: 4x4 perspective matrix, project triangles to 2D screen, rasterize pixels • Ray Casting: shoot rays through pixels into scene, intersect with triangles Discussion topics: • Will they make the same picture? • Which algorithm is faster, when? • Other pros/cons? When would you use each algorithm, and why? CS184/284A Ren Ng Recursive Ray Tracing “An improved Illumination model for shaded display” T. Whitted, CACM 1980 Time: • VAX 11/780 (1979) 74m • PC (2006) 6s • GPU (2012) 1/30s Spheres and Checkerboard, T. Whitted, 1979 CS184/284A Ren Ng Recursive Ray Tracing eye point image plane light source CS184/284A Ren Ng Recursive Ray Tracing Mirror ray (specular reflection) eye point image plane light source CS184/284A Ren Ng Recursive Ray Tracing Refractive rays (specular transmission) eye point image plane light source CS184/284A Ren Ng Recursive Ray Tracing eye point image plane Shadow rays light source CS184/284A Ren Ng Recursive Ray Tracing secondary rays primary ray eye point image plane shadow rays light source • Trace secondary rays recursively until hit a non-specular surface (or max desired levels of recursion) • At each hit point, trace shadow rays to test light visibility (no contribution if blocked) • Final pixel color is weighted sum of contributions along rays, as shown • Gives more sophisticated effects (e.g. specular reflection, refraction, shadows), but we will go much further to derive a physically-based illumination model CS184/284A Ren Ng Recursive Ray Tracing CS184/284A Ren Ng Ray-Surface Intersection Ray Intersection With Triangle Mesh Why? • Rendering: visibility, shadows, lighting … • Geometry: inside/outside test How to compute? Let’s break this down: • Simple idea: just intersect ray with each triangle • Simple, but slow (accelerate next time) • Note: can have 0, 1 or multiple intersections CS184/284A Ren Ng Ray Equation Ray is defined by its origin and a direction vector Example: Ray equation: 0 t< 1 point along ray “time” origin unit direction CS184/284A Ren Ng Plane Equation Plane is defined by normal vector and a point on plane Example: r(t)=o + t d, 0 t< 1 p :(p p0) N =0 r(t)=o + t d, 0 t< − · 1 Plane Equation:ax +r(tby)=+o +czt d,+0 d pt<=0:(p p0) N =0 1− · p :(p p0) N =0 ax + by + cz + d =0 − · Set p = r(t) and solve for t ax + by + cz + d =0 Set p = r(t) and solve for t all points on plane any pointSet(normalpp= vectorr(tp)0 and) N solve=( for(pot +p0)t dN =(po +0)t dNp=00) N =0 − · − · − · − · CS184/284A (p p0) N =(o + t d p0) N =0 Ren Ngp0 o) N − · − · p0 o)t =N − · p0 o) N d N t = − · t = − · · d N d N · · Ray Intersection With Plane Ray equation: r(t)=o + t d, 0 t< 1 p :(p p0) N =0 −Plane· equation:r(t)=o + t d, 0 t< r(t)=o + t d, 0 t< ax + by + cz + d =0 1 r(t)=p :(op+ t dp,)0 N t<=0 1 0 1 p :(p p0) N =0 p :(p p0) NSet=0p =−r(t) and· solve for t − · ax + by + cz + d =0 − · ax + by +(pczSolve+p0d) =0 Nfor=( intersectiono + t d p0) N =0 − · Set p−= r(t·) and solveax for t+ by + cz + d =0 Set p =pr0 (t)o and) N solve for t t(p= p−0) N· =(o + t d p0) N =0 Set p = r(t) and solve for t (p p−0) Nd ·=(N o + t d p−0) N·=0 · p0 o) N − · t = −− · · (p0 o) N d N (p p0) N =(o + t d p0) N =0 t = − · r(t)=oCheck:+ t d, 0 t< d N · 1 − · − · p :(p· p0) N =0 p0 o) N CS184/284A − · Ren Ng t = − · ax + by + cz + d =0 d N Set p = r(t) and solve for t · (p p0) N =(o + t d p0) N =0 − · − · p0 o) N t = − · d N · Ray Intersection With Triangle Triangle is in a plane • Ray-plane intersection • Test if hit point is inside triangle (Assignment 1!) Many ways to optimize… CS184/284A Ren Ng Can Optimize: e.g. Möller Trumbore Algorithm CS184/284A Ren Ng p :(p c)2 R2 =0 − − (o + t d c)2 R2 =0 Ray Intersection With Sphere − − 2 Ray: r(t)=o + t d, 0 t< at + bt+ c =0, where 1 2 2 pp:(:(pp cc))2 RR2 =0=0 p :(p p0) N =0 2 2 a = d d− − Sphere:−p :(·p pc:()2p Rc2)=0R =0 p :(·−p c−)2 R2 =0 ax + by +pcz:(+p d−=0c)2 −−R2 =0− b = 2(o −c) d − − − − · 2 2 Set p = r(t) and solve for t ((oo++ttdd cc))2 RR2 =0=02 Solve for intersection:2 2 c =(o c−) (o − c) R (o + t d(o +c)t2d Rc2)=0R =0 (o +−−t d· c−)−2 R−2 =0 (p (po0)+ tNd=(−co)2+−t−Rd2 =0p−0) N =0 − − − · − · 2 − − at2 + bt+ c =0, where p0 o) N at +bbt+pbc2=04,acwhere t = 2 − · 2 at2 + btat+ +c =0dbt+N, wherec =0, where t a==at−d ±+dbt+−c =0, where at2 + bt+ c =0· , where a = d ·d 2a a = d ad= d d b =a 2(=·od dc) d a = d ·d · b = 2(o −·c) ·d b = 2(·ob =c 2() od c) d b = 2(−o ·c) d 2 − · cc=(=(oo cc)) ((oo cc)) RR2 b = 2(o −c) ·d 2 − −· ·− − c =(o c−=(c) o·(o c)c)(o R2c) R c =(−o ·c) −(o −c) R2 c =(o −c) ·(o−−c)· −R−2 − − · − − 2 − · − − b pb2 4ac CS184/284A t = −b ±pb −4Renac Ng 2b pb2 4ac t = − ± −2 b pb 4ac b 22aapb 4ac t = −b t±=pb−2 −±4ac − t = − ± − t = − ± 2a − 2a 2a 2a Ray Intersection With Implicit Surface Ray: r(t)=o + t d, 0 t< 1 p :(p p0) N =0 p : f(p)=0 − · p : f(p)=0 ax + byGeneral+ cz + d implicit=0 surface: Set p = r(t) and solve forf(ot+ t d)=0 Substitute ray equation: f(o + t d)=0 (p p0) N =(o + t d p0) N =0 Solve− ·for real, positive− roots· p0 o) N t = − · d N · CS184/284A Ren Ng Accelerating Ray-Surface Intersection Ray Tracing – Performance Challenges Simple ray-scene intersection • Exhaustively test ray-intersection with every object Problem: • Exhaustive algorithm = #pixels ⨉ #objects • Very slow! CS184/284A Ren Ng Ray Tracing – Performance Challenges Jun Yan, Tracy Renderer San Miguel Scene, 10.7M triangles Ray Tracing – Performance Challenges Deussen et al; Pharr & Humphreys, PBRT Plant Ecosystem, 20M triangles Pre-Class Discussion: Accelerating Ray-Scene Intersection ~1 million pixels, ~20 million triangles Deussen et al; Pharr & Humphreys, PBRT In pairs, brainstorm accelerations, small or big ideas. Write down 3-4 ideas. Pre-Class Discussion: Accelerating Ray-Scene Intersection Brainstorm 3 or 4 accelerations, small or big ideas. • Subdivide the world • If recursive, truncate the perhaps using an octtree number of recursions, and estimate the tail of • Raycasting for a group computation of pixels at a time • Parallelize raycasting • Something like • Do every second ray, and mipmapping, LOD -- interpolate further away, simpler • Sort triangles from front to geometry back • Bounding box • Ray-marching until in proximity CS184/284A Ren Ng Bounding Volumes Bounding Volumes Quick way to avoid intersections: bound complex object with a simple volume • Object is fully contained in the volume • If it doesn’t hit the volume, it doesn’t hit the object • So test bvol first, then test object if it hits CS184/284A Ren Ng Ray-Intersection With Box Could intersect with 6 faces individually Better way: box is the intersection of 3 slabs CS184/284A Ren Ng Ray Intersection with Axis-Aligned Box 2D example; 3D is the same! Compute intersections with slabs and take intersection of tmin/tmax intervals tmax x0 x1 y0 y1 x0 x1 y0 y1 x x y y − − − − − − tmax0 − 1 − 0 − 1 tmax tmin tmin o, d o, d o, d o, d o, d o, d tmin x0 x1 y0 y1 x x y y x x y y − − − 0 − 1 − 0 − 1 0 − 1 − 0 − 1 Note: tmin < 0 x0 x1 x0y0 x1y1 y0 y1 x0 x1 x0y0 x1y1 y0 y1 x0 x1 x y0 x y1 y y − − − − − − − − − − − − − −0 − −1 − 0 − 1 Intersections with x planes Intersections with y planes Final intersection result How do we know when the ray misses the box? r(t)=o + t d, 0 t< 1 p :(p p0) N =0 Optimize Ray-Plane Intersection− For· Axis-Aligned Planes? ax + by + cz + d =0 Set p = r(t) and solve for t (p p0) N =(o + t d p0) N =0 − · − · r(t)=o + t d,(p00 t<o) N t = − 1· p :(p p0) N =0 General − · d N ax + by + cz + d =0 · Set p =3 subtractions,r(t) and solve 6 multiplies, for t 1 division (p r(tp)=0) oN+=(t d,o0+ tt<d p0) N =0 − · −1 · p0 o) N p :(p p0) N =0 t = − · − · d N Perpendicular (p0ax +oby) +Ncz + d =0 · p0x ox t = − · Set p ==r(t) andt solve= for t − to x-axis d N ) dx · (p p0) N =(o + t d p0) N =0 − · − · p0 o) N1 subtraction, 1 division t = − · d N · CS184/284A Ren Ng Uniform Spatial Partitions (Grids) Preprocess – Build Acceleration Grid 1.