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 (p r(tp)=0) oN+=(t d,o0+ tt 1. Find bounding box CS184/284A Ren Ng Preprocess – Build Acceleration Grid 1. Find bounding box 2. Create grid CS184/284A Ren Ng Preprocess – Build Acceleration Grid 1. Find bounding box 2. Create grid 3. Store each object in overlapping cells CS184/284A Ren Ng Ray-Scene Intersection Step through grid in ray traversal order (3D line - 3D DDA) For each grid cell Test intersection with all objects stored at that cell CS184/284A Ren Ng Grid Resolution? One cell • No speedup CS184/284A Ren Ng Grid Resolution? Too many cells • Inefficiency due to extraneous grid traversal CS184/284A Ren Ng Grid Resolution? Heuristic: • #cells = C * #objs • C ≈ 27 in 3D CS184/284A Ren Ng Careful! Objects Overlapping Multiple Cells What goes wrong here? • First intersection found (red) is not the nearest! Solution? • Check intersection point is inside cell Optimize • Cache intersection to avoid re-testing (mailboxing) CS184/284A Ren Ng Uniform Grids – When They Work Well Deussen et al; Pharr & Humphreys, PBRT Grids work well on large collections of objects that are distributed evenly in size and space CS184/284A Ren Ng Uniform Grids – When They Fail Jun Yan, Tracy Renderer “Teapot in a stadium” problem CS184/284A Ren Ng Non-Uniform Spatial Partitions: Spatial Hierarchies Spatial Hierarchies A A CS184/284A Ren Ng Spatial Hierarchies A B B A CS184/284A Ren Ng Spatial Hierarchies A B C B C A CS184/284A Ren Ng Spatial Hierarchies A B C B C D D A CS184/284A Ren Ng Spatial Hierarchies A 2 1 B C 1 B 2 C 4 D 3 D 3 4 5 5 A CS184/284A Ren Ng Attendance If you are seated in class, go to this form and sign in: • https://tinyurl.com/184lecture Notes: • Time-stamp will be taken when you submit form. Do it now, won’t count later. • Don’t tell friends outside class to fill it out now, because we will audit at some point in semester. • Failing audit will have large negative consequence. You don’t need to, because you have an alternative! CS184/284A Ren Ng Spatial Partitioning Variants Oct-Tree KD-Tree BSP-Tree Note: you could have these in both 2D and 3D. In lecture we will illustrate principles in 2D, but for assignment you will implement 3D versions. CS184/284A Ren Ng KD-Trees Internal nodes store • split axis: x-, y-, or z-axis • split position: coordinate of split plane along axis • children: reference to child nodes Leaf nodes store • list of objects • mailbox information CS184/284A Ren Ng KD-Tree Pre-Processing • Find bounding box • Recursively split cells, axis-aligned planes • Until termination C criteria met (e.g. max B #splits or min #objs) • Store obj references with each leaf node D A CS184/284A Ren Ng KD-Tree Pre-Processing A Root 2 1 B Internal Nodes 2 C C 1 B D 3 4 4 5 D 3 Leaf Nodes 5 Only leaf nodes store references to geometry A CS184/284A Ren Ng KD-Tree Pre-Processing Choosing the split plane • Simple: midpoint, median split • Ideal: split to minimize expected cost of ray intersection Termination criteria? • Simple: common to prescribe maximum tree depth (empirical 8 + 1.3 log N, N = #objs) [PBRT] • Ideal: stop when splitting does not reduce expected cost of ray intersection CS184/284A Ren Ng Top-Down Recursive In-Order Traversal A 1 B C B 2 C D 3 D 4 5 A CS184/284A Ren Ng Top-Down Recursive In-Order Traversal tmin tsplit A 1 B C B 2 C D 3 tmax D 4 5 Internal node: split A CS184/284A Ren Ng Top-Down Recursive In-Order Traversal tmin tmax A 1 B C B 2 C D 3 D 4 5 Leaf node: intersect A all objects CS184/284A Ren Ng Top-Down Recursive In-Order Traversal tmin A 1 B t C split B 2 C D 3 tmax D 4 5 Internal node: split A CS184/284A Ren Ng Top-Down Recursive In-Order Traversal tmin A 1 B t C max B 2 C D 3 D 4 5 Leaf node: intersect A all objects CS184/284A Ren Ng Top-Down Recursive In-Order Traversal A 1 B t split C tmin B 2 C D 3 tmax D 4 5 Internal node: split A CS184/284A Ren Ng Top-Down Recursive In-Order Traversal A 1 B C tmin B 2 C D 3 tmax D 4 5 Leaf node: intersect A all objects CS184/284A Ren Ng Top-Down Recursive In-Order Traversal A 1 B C B 2 C thit D 3 D 4 5 Intersection found A CS184/284A Ren Ng KD-Trees Traversal – Recursive Step W.L.O.G. consider x-axis split with ray moving right t =(x o )/d split split x x t =(x o )/d split split x x t =(x o )/d split split x x tsplit =(xsplit ox)/dx t =(x o )/d split split x x CS184/284A Ren Ng Object Partitions & Bounding Volume Hierarchy (BVH) Spatial vs Object Partitions Spatial partition (e.g.KD-tree) • Partition space into non- overlapping regions • Objects can be contained in multiple regions Object partition (e.g. BVH) • Partition set of objects into disjoint subsets • Bounding boxes for each set may overlap in space CS184/284A Ren Ng Bounding Volume Hierarchy (BVH) Root CS184/284A Ren Ng Bounding Volume Hierarchy (BVH) CS184/284A Ren Ng Bounding Volume Hierarchy (BVH) CS184/284A Ren Ng Bounding Volume Hierarchy (BVH) A C B C D D B A CS184/284A Ren Ng Bounding Volume Hierarchy (BVH) Internal nodes store • Bounding box • Children: reference to child nodes Leaf nodes store • Bounding box • List of objects Nodes represent subset of primitives in scene • All objects in subtree CS184/284A Ren Ng BVH Pre-Processing • Find bounding box • Recursively split set of objects in two subsets • Stop when there are just a few objects in each set • Store obj reference(s) in each leaf node CS184/284A Ren Ng BVH Pre-Processing Choosing the set partition • Choose a dimension to split or optimize over x,y,z • Simple #1: Split objects around spatial midpoint • Simple #2: Split at location of median object • Ideal: split to minimize expected cost of ray intersection Termination criteria? • Simple: stop when node contains few elements (e.g. 5) • Ideal: stop when splitting does not reduce expected cost of ray intersection CS184/284A Ren Ng BVH Recursive Traversal Intersect (Ray ray, BVH node) node if (ray misses node.bbox) return; if (node is a leaf node) test intersection with all objs; return closest intersection; hit1 = Intersect (ray, node.child1); hit2 = Intersect (ray, node.child2); return closer of hit1, hit2; child1 child2 CS184/284A Ren Ng Optimizing Hierarchical Partitions (How to Split?) How to Split into Two Sets? (BVH) CS184/284A Ren Ng How to Split into Two Sets? (BVH) CS184/284A Ren Ng How to Split into Two Sets? (BVH) Split at median element? Child nodes have equal numbers of elements CS184/284A Ren Ng How to Split into Two Sets? (BVH) A better split? Smaller bounding boxes, avoid overlap and empty space CS184/284A Ren Ng Which Hierarchy Is Fastest? Key insight: a good partition minimizes the average cost of tracing a ray CS184/284A Ren Ng Which Hierarchy Is Fastest? What is the average cost of tracing a ray? For leaf node: Cost(node) = cost of intersecting all triangles = C_isect * TriCount(node) C_isect = cost of intersecting a triangle TriCount(node) = number of triangles in node CS184/284A Ren Ng Which Hierarchy Is Fastest? What is the average cost of tracing a ray? For internal node: Cost(node) = C_trav + Prob(hit L)*Cost(L) + Prob(hit R)*Cost(R) C_trav = cost of traversing a cell Cost(L) = cost of traversing left child Cost(R) = cost of traversing right child CS184/284A Ren Ng Optimizing Hierarchical Partitions Example: Surface Area Heuristic Algorithm Ray Intersection Probability The probability of a random ray hitting a convex shape A enclosed by another convex shape B is the ratio of their surface areas, SA / SB. SB S P (hitA hitB)= A | SB SA CS184/284A Ren Ng Estimating Cost with Surface Area Heuristic (SAH) Probabilities of ray intersecting a node • If assume uniform ray distribution, no occlusions, then probability is proportional to node’s surface area Cost of processing a node • Common approximation is #triangles in node’s subtree Cost(cell) = C_trav + SA(L)*TriCount(L) + SA(R)*TriCount(R) SA(node) = surface area of bbox of node C_trav = ratio of cost to traverse vs. cost to intersect tri C_trav = 1:8 in PBRT [Pharr & Humphreys] C_trav = 1:1.5 in a highly optimized version CS184/284A Ren Ng Partition Implementation Constrain search to axis-aligned spatial partitions • Choose an axis • Choose a split plane on that axis • Partition objects into two halves by centroid • 2N–2 candidate split planes for node with N primitives. (Why?) Partition Implementation (Efficient) Efficient modern approximation: split spatial extent of primitives into B buckets (B is typically small: B < 32) b0 b1 b2 b3 b4 b5 b6 b7 For each axis: x,y,z: initialize buckets For each object p in node: b = compute_bucket(p.centroid) b.bbox.union(p.bbox); b.prim_count++; For each of the B–1 possible partitioning planes evaluate SAH Execute lowest cost partitioning found (or make node a leaf) Cost-Optimization Applies to Spatial Partitions Too • Discussed optimization of BVH construction • But principles are general and apply to spatial partitions as well • E.g. to optimize KD-Tree construction • Goal is to minimize average cost of intersecting ray with tree • Can still apply Surface Area Heuristic • Note that surface areas and number of nodes in children differ from BVH CS184/284A Ren Ng Things to Remember Linear vs logarithmic ray-intersection techniques Many techniques for accelerating ray-intersection • Spatial partitions: Grids and KD-Trees • Object partitions: Bounding Volume Hierarchies Optimize hierarchy construction based on minimizing cost of intersecting ray against hierarchy • Leads to Surface Area Heuristic for best partition CS184/284A Ren Ng Acknowledgments Thanks to Pat Hanrahan, Kayvon Fatahalian, Mark Pauly and Steve Marschner for lecture resources. CS184/284A Ren Ng