Bidirectional Path Tracingshz/courses/cs295/slides/12_bdpt.pdf · Bidirectional Path Tracing •Build light transport paths by connecting two sub-paths starting from the light source

Bidirectional Path Tracing

CS295, Spring 2017

Shuang Zhao

Computer Science Department

University of California, Irvine

CS295, Spring 2017 Shuang Zhao 1

Last Lecture

• Path integral formulation II

• Light path expressions

• Hints for implementing path tracing


Today’s Lecture

• Bidirectional path tracing


Recap: Path Integral Formulation

where Ω is the path space and for any ,and


Recap: Local Path Sampling

• Path tracing

• Adjoint particle tracing


Example: Modified Cornell Box

• Area light facing up• The scene is lit largely

indirectly

• Camera facing a mirrored ball• The scene is observed

indirectly

• Difficult to render for both path tracing and adjoint particle tracing


Cornell Box

Mirrored

ball


• Difficult to render for both path tracing and adjoint particle tracing


Path tracing Adjoint particle tracing

Unrenderable!


• Path tracing

• Most shadow rays give zero contribution by hitting the wrong side of the light source

• Largely disabling next-event estimation!


Cornell Box

Mirrored

ball

Unrenderable!


• Adjoint particle tracing

• Most shadow rays give zero contribution

• Cannot get back to the pinhole camera by randomly hitting the mirrored ball


Cornell Box

Mirrored

ball

X


• Is there a better way?

• Yes: bidirectional path tracing (BDPT)• Trace two sub-paths from both ends

• Connect in between


Cornell Box

Mirrored

ball

Light

sub-path

Sensor

sub-path



Path tracing Bidirectional path tracing

(Rendered in equal-time)

Bidirectional Path Tracing

• Build light transport paths by connecting two sub-paths starting from the light source and the sensor, respectively

• Use multiple importance sampling (MIS) to properly weight each path


[Veach 1997]

Building Transport Paths

• For any , create light sub-path and sensor sub-path (using local path sampling)

• The full path is obtained by concatenating these two pieces:



• Remark 1: can be zero because:

• andmay be zero (e.g., when zt-1 does not belong to the hemisphere around the normal at ys-1)

• may be zero (e.g., when are

mutually invisible)



• Remark 2: there is more than one sampling technique for each path length• (k + 1) techniques for paths with k vertices


[Veach

1997]


• For each s and t, the construction of gives a probability density

• Similar to the unidirectional case,equals the product of densities of sampling both sub-paths

• More details on this soon



• Using multiple importance sampling, we can combine all these path sampling schemes, resulting in:

where ws,t is the weighting function

• Balance heuristic

• Power heuristic


Building Paths in Batches

• Instead of generating each path sample independently (i.e., one light and one sensor sub-path for

each ), path samples can be built in batches

• First, generate both sub-paths


• Then, connecting each pair of vertices ys, zt

yields a full path sample

• (nL nE) full paths can be constructed from one pair of light and sensor sub-paths



Connecting

edge



• (nL nE) full paths can be constructed from one pair of light and sensor sub-paths






• Remark 1: to ensure unbiasedness, nL, nE

have to be picked randomly (e.g., via Russian

roulette) per batch



• Remark 2: after obtaining nL, nE for batch i,

becomes

• nL, nE vary between batches


Sub-Path Generation

• Light sub-path

• Sample y0 on the surface of a light source with density

• After creating each vertex yi, terminate the path with probability 1 – q(yi), causing nL = i + 1

• Otherwise, draw a direction ωi based on Le(y0, ω) when i = 0 and when i > 0

• Let yi + 1 = RayTrace(yi, ωi). Then,


Computing Probability Densities

• Given nL, nE and the two corresponding sub-paths and , for any , we have

where


Capturing the

change of measure

Survival

probability

Evaluating Path Contribution

• To compute the contribution of a single path sample , we first consider the unweighted contribution



• Recall that


Light sub-path Sensor sub-path

Connecting

edge


• Then,

where



• It follows that


Evaluating Weighting Function

• When using the balance heuristic, we have

where for all j

• Most terms in pi cancels with those in ps when evaluating pi /ps


Implementation Details

• Survival probability q can be picked adaptively

• For each yi on the light sub-path, it is desirable to have

• If , q(yi) then becomes the surface albedo at yi


Bidirectional Path Tracingshz/courses/cs295/slides/12_bdpt.pdf · Bidirectional Path Tracing •Build light transport paths by connecting two sub-paths starting from the light source

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