Path Integral Formulation II & Light Path Expressions CS295, - - PowerPoint PPT Presentation

Path Integral Formulation II & Light Path Expressions

CS295, Spring 2017 Shuang Zhao

Computer Science Department University of California, Irvine

CS295, Spring 2017 Shuang Zhao 1

Last Lecture

• Path integral formulation

CS295, Spring 2017 Shuang Zhao 2

Today’s Lecture

• Path integral formulation II
• Light path expressions
• Hints for implementing path tracing

CS295, Spring 2017 Shuang Zhao 3

Recap: Path Integral Formulation

where Ω is the path space and for any , and

CS295, Spring 2017 Shuang Zhao 4

Recap: Local Path Sampling

• The initial vertex is sampled on the light source

(based on emitted radiance Le) or the sensor (based on emitted importance (aka. the measurement function We)

• From the initial vertex, the path is grown one

vertex at a time via a probability density conditioned on the current sub-path

CS295, Spring 2017 Shuang Zhao 5

Local Path Sampling

• Path tracing
• Adjoint particle tracing

CS295, Spring 2017 Shuang Zhao 6

Limitations of Local Path Sampling

• Certain types of transport paths cannot be

sampled

• E.g., caustics caused by point lights and curved

specular surfaces

• In general, the rendering problem is undecidable

(using Turing machines)

CS295, Spring 2017 Shuang Zhao 7

Regular Expression Notation for Paths

• Paths are described using regular expressions
• f the form

L (S|D)* E where

• L denotes the endpoint on the light source
• E denotes the endpoint on the sensor (or “eye”)
• S denotes vertices on “specular” surfaces
• D denotes vertices on “diffuse” surfaces

CS295, Spring 2017 Shuang Zhao 8

Full-Path Regular Expressions

• Extending the notice of S vs. D to the light

source and the sensor

• Light: L (S|D) (S|D)
• LDD: a diffusely emitting sphere
• LDS: a collimated beam
• LSD: a point light
• LSS: a laser beam

CS295, Spring 2017 Shuang Zhao 9

Position Direction

Full-Path Regular Expressions

• Extending the notice of S vs. D to the light

source and the sensor

• Sensor: (S|D) (S|D) E
• DDE: a finite-aperture lens
• SDE: an orthographic camera
• DSE: a pinhole camera
• SSE: an idealized spot meter (measuring the radiance

along a single ray)

CS295, Spring 2017 Shuang Zhao 10

Direction Position

Why using Regular Expression?

• S vs. D suggests how certain quantities can be

sampled

• “Specular” events have to be followed

deterministically

• “Diffuse” events can be sampled continuously
• Remark
• Here “diffuse” does NOT imply uniform distributions.

Instead, it just means the corresponding functions contain no delta functions

CS295, Spring 2017 Shuang Zhao 11

Simulating Caustics

L S D S D D S E

CS295, Spring 2017 Shuang Zhao 12

Point light Pinhole camera Specular surface Diffuse surface D LSD S

Cannot be easily

• btained!

Possible “Fixes”

• Possibility 1: disallowing “specular” events
• LDD (diffusely emitting area sources)
• DDE (finite-aperture lens)
• D (no perfectly specular surfaces)

CS295, Spring 2017 Shuang Zhao 13

Possible “Fixes”

• Possibility 2: introducing path chains
• Example: L S D S D D S E
• Each chain has its endpoints being D/L/E and

intermediate vertices being S

• New methods, called connectors, can be developed

to generate individual chains given its endpoints

CS295, Spring 2017 Shuang Zhao 14

Connectors

• Example 1: planar mirrors (DSD)

CS295, Spring 2017 Shuang Zhao 15

x y x'

Connectors

• Example 2: curved mirrors (DSD)

CS295, Spring 2017 Shuang Zhao 16

[Mitchell & Hanrahan 1992]