ray tracing info

7/29/2019 Ray tracing info

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Ray Tracing II

Lecturer: Erick Fredj

This presentation is largely inspired byPfister and Chan Computer Graphics

Course from MIT 2007.


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Outline

Spatial Acceleration Data Structures

Motivation Distribution Ray Tracing

Monte Carlo Ray Tracing

Importance & Stratified Sampling

Path Tracing & Irradiance Caching

Photon Mapping


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Recap: Ray Tracing


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Recap: Ray Tracingtrace ray

Intersect all objects

color = ambient term

For every light

cast shadow ray

color += local shading termIf mirror

color += colorrefl *

trace reflected ray

If transparent

color += colortrans *

trace transmitted ray


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Ray Tracing Pros and Cons


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Ray Tracing Pros and Cons

Pros:

Easy to implement

Extends well to global illumination

shadows reflections / refractions

multiple light bounces

atmospheric effects

Cons:

Speed! (seconds per frame, not frames per second)


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Speedup Techniques

Why is ray tracing slow? How to improve?


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Speedup Techniques

Why is ray tracing slow? How to improve?

Too many objects, too many rays

Reduce ray-object intersection tests

Many techniques!


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Acceleration of Ray Casting

Goal: Reduce the number of ray/primitive intersections


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Conservative Bounding Region

First check for an intersection

with a conservative bounding region

Early reject


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Bounding Regions

What makes a good bounding region?


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Conservative Bounding Regions

arbitrary convex region

(bounding half-spaces)

non-alignedbounding box

boundingsphere

axis-aligned

bounding box


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Regular Grid


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Create Grid

Find bounding box of scene

Choose grid resolution (nx, ny, nz)

gridx neednot = gridy


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Insert Primitives into Grid

Primitives that

overlap multiple

cells?

Insert into multiple

cells (use pointers)


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For Each Cell Along a Ray

contain an

intersection?

Yes: return closest

intersection

No: continue


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Preventing Repeated Computation

Perform the

computation

once, "mark the

object

Don't re-intersect

marked objects


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Don't Return Distant Intersections

If intersection t

is not within

the cell range,

continue

(there may besomething

closer)


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Adaptive Grids

Subdivide until each cell contains no more than n elements, or

maximum depth d is reached

Nested Grids Octree/(Quadtree)


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Primitives in an Adaptive Grid

Can live at intermediate

levels, or be pushed to

lowest level of grid

Octree/(Quadtree)


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Binary Space Partition (BSP) Tree

Recursively partition space by planes

Every cell is a convex polyhedron


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Simple recursive algorithms

Example: point finding


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Trace rays by recursion on tree

BSP construction enables simple front-to-back traversal


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Grid Discussion

Regular + easy to construct + easy to

traverse - may be only sparsely filled -

geometry may still be clumped

Adaptive + grid complexity matches

geometric density - more expensive

to traverse (especially BSP tree)


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Outline




Importance & Stratified Sampling Path Tracing & Irradiance Caching

Photon Mapping


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Shadows & Light Sources

http://www.davidfay.com/index.php

http://3media.initialized.org/photos/200

0-10-18/index_gall.htm
http://www.davidfay.com/index.phphttp://3media.initialized.org/photos/2000-10-18/index_gall.htmhttp://3media.initialized.org/photos/2000-10-18/index_gall.htmhttp://3media.initialized.org/photos/2000-10-18/index_gall.htmhttp://3media.initialized.org/photos/2000-10-18/index_gall.htmhttp://3media.initialized.org/photos/2000-10-18/index_gall.htmhttp://3media.initialized.org/photos/2000-10-18/index_gall.htmhttp://3media.initialized.org/photos/2000-10-18/index_gall.htmhttp://3media.initialized.org/photos/2000-10-18/index_gall.htmhttp://www.davidfay.com/index.php


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Shadows

One shadow ray per intersection per point light source

no shadow rays

one shadow ray


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Soft Shadows

Multiple shadow rays to

sample area light source

one shadow ray

lots of shadow rays


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Probabilistic Soft Shadows

Multiple shadow rays to

sample area light source

Monte-Carlo ray tracing

generalizes this

lots of shadow rays


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Reflection

One reflection ray per intersection


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Glossy Reflection

Random reflection rays around mirror direction

Justin Legakis


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Glossy Surfaces

Glossy reflections can be handled by Monte Carlo methods


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Motion Blur

Sample objects temporally


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Outline




Importance & Stratified Sampling Path Tracing & Irradiance Caching

Photon Mapping


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Ray Casting

Cast a ray from the eye through each pixel


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Ray Casting


Trace secondary rays (light, reflection, refraction)


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Ray Casting


Trace secondary rays (light, reflection, refraction)

Accumulate radiance contribution


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Monte-Carlo computation of

Take a square

Take a random point (x,y) in the square

Test if it is inside the disc (x2+y2 < 1)

The probability is /4


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Monte-Carlo computation of

The probability is /4

Count the inside ratio n = # inside / total # trials

n * 4

The error depends on the number or trials


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Review of probability (discrete)

Random variable can take discrete values xi

Probability pi for each xi

0 < pi


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Example: Fair Dice

5.3

6543216

1

6

16

1

6

1

i

i

i

iidice

x

xpxE


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Variance & Standard Deviation

Variance 2:

Measure of deviation from expected value

Expected value of square difference (MSE)

Also

Standard deviation :

Square root of variance (notion of error, RMS)

222

ii xExxExE

222 xExE


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Monte Carlo integration

Functionf(x) of x[a b]

We want to compute

Consider a random variablex

Ifx has uniform distribution, I=E[f(x)]/(b-a) By definition of the expected value

Expected value of f(x) is the integral times (b-a)

Accurate on average

dxxfIb

a


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Sum of Random Variables

N independent identically-distributed (IID) variables xi

Share same probability (uniform here)

Define

n

j

iN xfN

F1

1


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Variance of FN

After some math:

i.e. stddev (error) decreases by

n

i

iN

N

xfF

1

22

N

xfFN

2

2

N


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Example

1

0

45 dxxI


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Example

1

0

45 dxxI

We know itshould be 1.0

In practice with

uniform

samples:


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Monte-Carlo Recap

Expected value is the integrand

Accurate on average

Variance decrease in 1/N

Error decreases inn1


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Monte-Carlo Ray Tracing


Cast random rays from the visible point

Recurse


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Recurse


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Recurse


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1 sample per pixel


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256 sample per pixel


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Results


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Outline






Photon Mapping


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Importance Sampling

Consider N random samples drawn with probabilityp(x)

Define estimator as:

Probabilityp allows us to sample the domain more smartly

N

i

xp

xf

ii

NI 1

1


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Importance Sampling

Choosep wisely to reduce variance

p that resembles f

Does not change convergence rate (still sqrt)

But decreases the constant

N

i

xp

xf

i

i

NI

1

1

bad uniform good


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Monte-Carlo

Take reflectance into account

Multiply incoming radiance by BRDF value


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Sampling a BRDF


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Sampling a BRDF


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Sampling a BRDF


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Sampling a BRDF


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Stratified sampling

With uniform sampling, we can get unlucky

E.g. all samples in a corner

To prevent it, subdivide domain

into non-overlapping regions i

Each region is called a stratum

Take one random samples per i


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Stratified Sampling

Cheap and effective

Typical example: jittering for antialiasing

Signal processing perspective: better than uniform

because less aliasing (spatial patterns)

Monte-Carlo perspective: better than random

because lower variance (error for a given pixel)


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Outline





Path Tracing & Irradiance Cache

Photon Mapping


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Monte Carlo Path Tracing

Trace only one secondary ray per recursion

But send many primary rays per pixel


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Results

10 paths/pixel


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Results

100 paths/pixel


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Path Tracing is costly

Needs tons of rays per pixel


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Direct illumination


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Global Illumination


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Indirect Illumination


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Irradiance cache

The indirect illumination is smooth


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Irradiance cache



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Irradiance cache


Interpolate nearby values


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Irradiance cache


Interpolate nearby values

But do full calculation for direct lighting


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Irradiance caching


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Outline






Photon Mapping

h


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Photon mapping

Preprocess: cast rays from light sources

Store photons

h i


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Photon mapping

Preprocess: cast rays from light sources

Store photons (position + light power + incoming direction)

h


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Photon map

Efficiently store photons for fast access

Use hierarchical spatial structure (kd-tree)

Ph i d i


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Photon mapping - rendering

For secondary rays

reconstruct irradiance using adjacent stored photon

Take the k closest photons

Ph l


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Photon map results

Ph i i


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Photon mapping - caustics

Special photon map for specular reflection and refraction


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1000 paths/pixel


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Photon mapping

Ph t M i R


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Photon Mapping Recap

Preprocess: Path tracing

Store photons

Special map for caustics (because not as uniform)

Rendering

Primary rays

Direct lighting

Indirect lighting

Reconstruct irradiance using k nearest photons Irradiance caching

Reconstruct caustics from caustic map

E d


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End

Good Luck

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