CS180: Introduction to Computer GraphicsSpring 2019, Lingqi Yan, UC Santa Barbara

Ray Tracing 2 (Acceleration) Lecture 13:

http://www.cs.ucsb.edu/~lingqi/teaching/cs180.html

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Last Lecture• Shadow mapping for rasterizers

• Introduction to ray tracing

• Ray generation

• Ray object intersection- Ray sphere intersection

- Ray implicit surface intersection

- Ray triangle intersection = Ray plane intersection + inside test

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Accelerating Ray-Surface Intersection

Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Ray Tracing – Performance Challenges

Simple ray-scene intersection

• Exhaustively test ray-intersection with every triangle

• Find the closest hit (with minimum t)

Problem:

• Naive algorithm = #pixels ⨉ # traingles (⨉ #bounces)

• Very slow!

For generality, we use the term objects instead of triangles later (but doesn’t necessarily mean entire objects)

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Ray Tracing – Performance Challenges

San Miguel Scene, 10.7M triangles

Jun Yan, Tracy Renderer

Ray Tracing – Performance Challenges

Plant Ecosystem, 20M triangles

Deussen et al; Pharr & Humphreys, PBRT

Bounding Volumes

Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

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

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Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Ray-Intersection With Box

Could intersect with 6 faces individually Better way: box is the intersection of 3 pairs of slabs

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Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Ray Intersection with Axis-Aligned Box2D example; 3D is the same! Compute intersections with slabs and take intersection of tmin/tmax intervals

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o,do,d

x0 � x1 � y0 � y1x0 � x1 � y0 � y1

x0 � x1 � y0 � y1

x0 � x1 � y0 � y1

tmin

tmax

o,do,d

x0 � x1 � y0 � y1x0 � x1 � y0 � y1

x0 � x1 � y0 � y1

x0 � x1 � y0 � y1

tmin

tmax

Note: tmin < 0

Intersections with y planes Intersections with x planes

o,do,d

x0 � x1 � y0 � y1x0 � x1 � y0 � y1

x0 � x1 � y0 � y1

x0 � x1 � y0 � y1

tmin

tmax

Final intersection resultHow do we know when the ray misses the box?

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

• Recall: a box (3D) = three pairs of infinitely large slabs

• Key ideas

- The ray enters the box only when it enters all pairs of slabs

- The ray exits the box as long as it exits any pair of slabs

• For each pair, calculate the tmin and tmax (negative is fine)

• For the 3D box, tenter = max{tmin}, texit = min{tmax}

• If tenter < texit, we know the ray stays a while in the box(so they must intersect!) (not done yet, see the next slide)

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Ray Intersection with Axis-Aligned Box

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

• However, ray is not a line

- Time t should always be positive for physical correctness!

• What if texit < 0?

- The box is “behind” the ray — no intersection!

• What if tenter < 0?

- The ray’s origin is inside the box — still intersection!

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Ray Intersection with Axis-Aligned Box

Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Optimize Ray-Plane Intersection For Axis-Aligned Planes?

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r(t) = o+ td, 0 t < 1p : (p� p0) ·N = 0

ax+ by + cz + d = 0

Set p = r(t) and solve for t

(p� p0) ·N = (o+ td� p0) ·N = 0

t =p0 � o) ·N

d ·N

r(t) = o+ td, 0 t < 1p : (p� p0) ·N = 0

ax+ by + cz + d = 0

Set p = r(t) and solve for t

(p� p0) ·N = (o+ td� p0) ·N = 0

t =(p0 � o) ·N

d ·N

3 subtractions, 6 multiplies, 1 division

t =(p0 � o) ·N

d ·N =) t =p0

x � ox

dx

r(t) = o+ td, 0 t < 1p : (p� p0) ·N = 0

ax+ by + cz + d = 0

Set p = r(t) and solve for t

(p� p0) ·N = (o+ td� p0) ·N = 0

t =p0 � o) ·N

d ·N1 subtraction, 1 division

General

Perpendicularto x-axis

Uniform Spatial Partitions (Grids)

Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Preprocess – Build Acceleration Grid

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1. Find bounding box

Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Preprocess – Build Acceleration Grid

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1. Find bounding box 2. Create grid

Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Preprocess – Build Acceleration Grid

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1. Find bounding box 2. Create grid 3. Store each object

in overlapping cells

Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Ray-Scene Intersection

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Step through grid in ray traversal order

For each grid cell Test intersection with all objects stored at that cell

Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Grid Resolution?

One cell

• No speedup

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Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Grid Resolution?

Too many cells

• Inefficiency due to extraneous grid traversal

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Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Grid Resolution?

Heuristic:

• #cells = C * #objs

• C ≈ 27 in 3D

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Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Uniform Grids – When They Work Well

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Deussen et al; Pharr & Humphreys, PBRT

Grids work well on large collections of objectsthat are distributed evenly in size and space

Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Jun Yan, Tracy Renderer

Uniform Grids – When They Fail

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“Teapot in a stadium” problem

Spatial Partitions

Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Spatial Hierarchies

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A

A

Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

B

A

Spatial Hierarchies

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A

B

Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

4 5

D 3

2 C

Spatial Hierarchies

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A

BC

D

1

2

34

5

1 B

A

Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Spatial Partitioning Examples

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BSP-TreeKD-Tree

Note: you could have these in both 2D and 3D. In lecture we will illustrate principles in 2D.

Oct-Tree

Object Partitions &

Bounding Volume Hierarchy (BVH)

Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

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

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Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Bounding Volume Hierarchy (BVH)

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Root

Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Bounding Volume Hierarchy (BVH)

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Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

Bounding Volume Hierarchy (BVH)

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Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

C D

B

Bounding Volume Hierarchy (BVH)

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A

AB

C

D

Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

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

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Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

BVH Pre-Processing

• Find bounding box

• Recursively split set of objects in two subsets

• Recompute the bounding box of the subsets

• Stop when there are just a few objects in each set

• Store obj reference(s) in each leaf node

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Ren NgCS184/284A

BVH Pre-Processing

Choosing the set partition

• Choose a dimension to split

• Heuristic #1: Always choose the longest axis in node

• Heuristic #2: Split node at location of median object

Termination criteria?

• Heuristic: stop when node contains few elements (e.g. 5)

Slide courtesy of Prof. Ren Ng, UC Berkeley

CS180, Spring 2019 Lingqi Yan, UC Santa Barbara

BVH Recursive Traversal

Intersect(Ray ray, BVH 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 the closer of hit1, hit2;

}

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node

child1 child2

Thank you!