Photon Mapping (PM),
Progressive PM,
Stochastic PPM
Jiří Vorba
What are they used for?
• Methods for computing the Global
Illumination(GI)
• GI = Direct + Indirect Illumination (multiple
bounces)
Caustics
© H.W.Jensen
Chronology
• Photon Mapping – introduced by Henrik Wann
Jensen, 1994
• Progressive PM – Toshiya Hachisuka, H. W.
Jensen, 2008
• Stochastic PPM – T. Hachisuka, H. W. Jensen,
2009
• Both PPM & SPPM slightly improves the
previous one
Photon Mapping - taxonomy
• Hybrid algorithm – combines “the best” of
two worlds:
– Radiosity – computes radiance (radiosity) values
for every mesh element (i.e. the object-space
method)
– Ray-tracing – computes radiance values for every
pixel by generating paths between the pixel and
light sources
Photon Mapping – taxonomy II
• Multipass method – generally computing
various parts of the light transport and then
combining them together
– special care needs to be take about not computing
the same light transport twice (too bright parts of
the scene)
Photon Mapping
• Two passes:
– Approximation of the light transport in the objectspace – photon shooting (particle tracing), storing
photon-surface interactions in the photon map
– Computing the pixel values in the image plane by
exploiting the information stored in the photon
map – final gather (ray-tracing)
Photon Mapping
Final gather
© Kavita Bala
Photon shooting
• Emitting quanta of energy (photons) from light
sources
• Each photon carries the same amount of
energy

p 
Lights
L
W /s
N
• Emission is driven probabilistically according
to the spatial and directional properties of
light sources and according to their brightness
Photon tracing
• Propagating photons in the scene just as rays
are in the ray-tracing
• Reflection, transmission, absorption
• Standard Monte Carlo sampling techniques
• Still, tracing of photons is not the same as
tracing the rays!
– We are “tracing the density”…
Photon tracing - refraction
• Ray-tracing & ray refraction:
– Change of medium density bends the ray
towards/outwards the normal direction
– Results in change of energy
d  ( x)
2
L ( x,  ) 
cos  dA d 
• Compensation factor:
( 2 /  1)
2
Photon tracing - density
• Photon tracing
– The photon carries the same amount of energy
after refraction
• Why?
• Because we want the flux in the scene to be
captured in the distribution of photons rather
than in the various amounts of energy they
bear
Photons approximates the radiosity
© Philippe Bekaert
Photons approximates the radiosity II
© H.W.Jensen
Photon tracing – Russian Roulette
• Allows as to keep the number of traced
photons low as well as keep their energy
approximately the same
• For instance instead of tracing 1000 photons
with half energy after hitting the surface with
albedo = 0.5 we will trace just 500 photons
with the full energy
– 500 photons were simply absorbed by material
Ray-tracing
• Solving the rendering equation by Monte
Carlo distribution ray-tracing


L ( x,  o)  L e( x,  o) 







 
fr ( x ,  o,  i) L ( x ,  i) | n x   i | d  i

nx
o


i
x
• Use the information stored in the photon map
Photon map query
• We can query the photon map about the
estimate of the reflected radiance from any
surface point


L ( x,  o)  L e( x,  o) 




 
fr ( x ,  o,  i) L ( x ,  i) | n x   i | d  i



L ( x,  o)  L e( x,  o) 

L r ( x,  o)
Radiance estimate – derivation I
• Relationship between radiance and flux

d  ( x,  )
L ( x,  )   

| n x   | dA d 
2

• Reflected radiance equation

L r ( x,  o) 




 
fr ( x ,  o,  i) L ( x ,  i) | n x   i | d  i


L r ( x,  o) 



 
d  ( x,  i)
fr ( x ,  o,  i )  
 | n   i | d i
| n x   i | dA d  i


2
Intermezzo – “back to Radiosity”
• We consider the directionality

L r ( x,  o) 



 d  ( x,  i)
fr ( x ,  o,  i )
 d i
dA d  i

2
• If we reduced the dimensionality of the
problem by assuming the Lamertian surfaces
we would get:


L r ( x,  o)  kx
d  ( x,  i)
 d i  kx E ( x )
dA d  i
2


Density estimation – histogram
method
• Stochastic radiosity uses histogram-based density
estimation method (known from statistics):
E ( x) 
 ( x )
A
• Density of flux is captured in the illumination map
or in the geometry mesh itself
– Size of the bin for capturing the flux carried by
photons is fixed (texel or mesh element)
• Positions of photon hits are not stored
– Flux in bins is accumulated incrementally
Density estimation – histogram
method
© Philippe Bekaert
Density estimation – nearest
neighbour
• What if we fixed the number of photons in
density estimation (N/A) instead of fixing the
surface area?
• Lets look for the N nearest photons
• Density estimation is then determined by the
size of an area where we found thouse N
photons
Density estimation – nearest
neighbour vs. histogram
© Philippe Bekaert
Density estimation – nearest
neighbour
• We need to store the positions of incoming
photons
• The density estimation isn’t computed “online”
– Dependency on the whole set of samples => it has
to be computed a posteriori
• Thus we need to store the photon hits
positions
– Why not store the incoming direction as well?
Radiance estimation – derivation II
• Photon map stores information about
1. Photons hit positions
2. Incoming photon direction
3. Flux carried by the photon

L r ( x,  o) 



 d  ( x,  i)
fr ( x ,  o,  i )
 d i
dA d  i

2
Radiance estimation – derivation III
•
•
Apply the nearest neighbour density estimation &
multiply the samples by their flux and BRDF
Assume that each photon in searched space ΔA hits
the surface right at x:


n
L r ( x,  o) 



fr ( x ,  o,  p )
 ( x ,  p )
A
p 1
•
•
Photons look up can be imagined as expanding the
sphere around x until it contains n photons
Lets suppose that the area around the x is locally flat:

L r ( x,  o) 
n
1
r
2

p 1



f r ( x ,  o ,  p )  ( x ,  p )
Radiance estimation – illustration
© H.W.Jensen
Radiance estimation – problems
around corners
• Accounting for wrong photons
• Will be reduced with rising number of photons
(smaller radius is then sufficient in bright places)
• The look up sphere might be flattened in the
direction of surface normal
• We can remember the surface normal in the
photon hit
–
In radiance estimate in x don’t account for those
photons which surface normal differs too much from
the one in x
Radiance estimation – problems
around corners
© H.W.Jensen
Radiance estimation – problems close
to the walls
• obrazek
© H.W.Jensen
• Disk or sphere projection to locally flat surface
may yield the wrong area estimate
• Solutions:
–
–
Computing the convex hull of accounted photons
Or better: use the kernel-density estimation
technique instead of nearest-neighbour (simply just
filtering the radiance estimate by weighting the
samples)
Direct visualization of photon map
(ray- casting on diffuse surfaces)
© H.W.Jensen
Final gather
•
Lets split the BRDF






fr ( x ,  o,  i )  fr , S ( x ,  o,  i )  fr , D ( x ,  o,  i )
•
Lets split the incoming radiance




L i( x,  )  L i, l( x,  )  L i, c( x,  )  L i, d ( x,  )
•
Lets combine it into the expression for reflected radiance

L r ( x,  o) 





 
fr ( x ,  o,  i) L ( x ,  i) | n x   i | d  i
Reflected radiance equation

L r ( x,  o) 






 
fr ( x ,  o,  i) L ( x ,  i) | n x   i | d  i 




 
f r ( x ,  o ,  in ) L i , l ( x ,  in ) | n x   in | d  in 






 
f r , S ( x ,  o ,  in )( L i , c ( x ,  in )  L i , d ( x ,  in )) | n x   in | d  in 


 

 
f r , D ( x ,  o ,  in ) L i , c ( x ,  in ) | n x   in | d  in 






 
f r , D ( x ,  o ,  in ) L i , d ( x ,  in ) | n x   in | d  in
Creation of caustic
•
Caustics appear on diffuse surfaces after light interaction
with specular surfaces (both reflection and transmission)
More photon maps
•
Create the special photon map for caustic photons
–
–
•
•
Trace them separately (store the photon on the first nonspecular surface -> S+D paths)
Cast photons only against specular surfaces – light
source projection maps
Create the special photon map for indirect
illumination
Create the special photon map for participating
media effects (behind the scope of the talk)
Final gather II
•
Direct illumination term

 

 
f r ( x ,  o ,  in ) L i , l ( x ,  in ) | n x   in | d  in

–
•
Solving directly by casting shadow rays (sampling the light sources)
Specular and glossy term
 
f
(
x
,

,  )( L

r, S
–
–
•
o
in
i, c


 
( x ,  in )  L i , d ( x ,  in )) | n x   in | d  in

Standard MC ray-tracing with importance sampling in favour of
BRDF narrow peak
Could take too much time if we waited until enough photons hit the
BRDF narrow peak
Caustics



 
f r , D ( x ,  o ,  in ) L i , c ( x ,  in ) | n x   in | d  in

–
Direct query into the caustic photon map

Final gather III
• Multiple diffuse reflections
– Resulting illumination is very soft (caustics are in
the other map – in MC methods are caustics the
biggest source of noice)
– Thus we can use MC methods or classical
Irradiance cashing methods





 
f r , D ( x ,  o ,  in ) L i , d ( x ,  in ) | n x   in | d  in
Different components of rendered
solution
In what are the photon maps
especially good at?
• Generating caustics
• Fast convergence of SDS paths
–
© Hachisuka et
al.
Here the classical MC algorithms fails (path tracing,
bidirectional path tracing, metropolis light transport)
SDS - Bottom of the pool
© Wojciech Jarosz
Main disadvantages of PM
•
Biased
–
•
Still, consistent
Cannot get arbitrarily precise results
–
–
We are strictly limited by the physical memory boundary
Thus not suitable in the areas of predictive rendering
Progressive PM
Jiří Vorba
Progressive PM
• How to improve PM so that we sharpen the
details of image function?
– Especially visible on the edges of caustic
– In other words: We want to be able to compute
the image with arbitrarily small bias
• Obviously we need to cope with the physical
memory boundary
First idea
• Use the memory more times than one and
combine the results from N photon maps
• Averaging radiance estimates
– In the point x where we want to query PM we have N
radiance estimates
– For averaging we need to use histogram-density
estimate method = fixing the radius; (remember
histogram vs. nearest neighbour)
• It is not consistent
– Getting smoother results but still blurry
Question
• How to combine illumination approximation
from more photon maps so that we get in the
final solution the results which are not present
in any single photon map we have?
– i.e. to have the consistent radiance estimate
combination
Second idea
lim
N 
N  
1
r
2

 


f r ( x ,  o ,  p )  ( x ,  p )  L r ( x ,  o )
   0 ,1
p 1
• Says: “The algorithm of PM is consistent.”
• So what?! Did it help us? We’ve already heard
that…
• It gives us a clue how to do it
– We have to progressively decrease the radius while
merging the results from multiple photon maps
– More work has to be done to achieve the goal…
Rough schema of an algorithm
• Ray tracing pass finds the hit points (HPs)
• In the HPs we combine the radiance estimates
from every single photon tracing pass
© Hachisuka et al.
Radius reduction
• Main idea:
– We suppose that the density d(x)=n/(πr2) around x is
invariant
– Thus radius can be reduced if we reduce the number
of photons in the estimate
• On the other hand to satisfy the condition of
consistency we need have gain of photons in the
estimate
• Solution:
– Lets keep just fraction of new photons after each
iteration
Radius reduction - derivation
• Use parameter α in range (0,1) to control the fraction
of new photons
• new density
– N(x) photons after i iterations
– M(x) in (i+1)-th iteration
• New total number of photons when keeping the density the
same, just reducing radius
• Put it all together
Radius reduction - derivation
• Thus new radius after (i+1)-th iteration
• Radius is updated in every HP after photon tracing
pass
– At worst it stays the same – when no photon is
caught – we will show later it is still consistent
– When everything goes well the radius decreases
while the photon count rises
Flux correction
• Cannot just average the illumination
– We need to account for those lost photons disposed by α
© Hachisuka et al.
• Unnormalized flux pre-multiplied by BRDF
Flux correction
• We could keep the list of photons in every HP and subtract
flux of those which fall out
– Slow, memory demanding
• Better: assume both photon density and therefore the
illumination are the same. Then
radiance doesn’t change
• Results in:
Resulting radiance estimate
• Normalization by total number of emitted
photons

L ( x,  ) 

 ( x,  )
1
R ( x )
2
N emitted
• Consistency is kept even in those places not
receiving any photons (division by N)
Algorithm review
•
Hp structure:
1. Ray-tracing pass
–
Trace ray across specular
bounces until first
non-specular surface is hit
–
Remember hp, its corresponding pixel, intersection details, weight
2. Photon tracing passes
–
–
–
–
Shoot photons into the new map
In each hp query the photon map
Update hp’s statistics
Possibly write the image for preview
Main disadvantages of PPM
• Biased
– Still, consistent
– In theory we can get arbitrarily precise solution (no
memory restrictions)
• Bad behaviour on glossy reflections
– Could be solved by casting more samples => more hps =>
memory restrictions
Examples of glossy reflections with
PPM
Stochastic PPM
Jiří Vorba
Distribution ray-tracing effects
•
What does it actually mean to compute glossy reflections,
depth of field, motion blur, anti-aliasing etc.?
–
•
Integration over some (additional) domain
Glossy reflection (usual hemisphere integration)

L r ( x,  o) 




 
fr ( x ,  o,  i) L ( x ,  i) | n x   i | d  i

•
Anti-aliasing and motion blur together
–
Pixel value Ij , importance function Wj of pixel j, time range T, all
surface points in the scene A including the image plane
Ij 
 W ( x , p ) L ( x , p , t ) dx dp dt
j
TAA
How would have we done it by PPM?
• For brevity suppose that we want to know the average
value of radiance over some region S (e.g. anti-aliasing)
and we have n uniformly distributed samples xk over the
region S
• This needs keeping the track of n HP’s statistics for all
sets of HPs
How to overcome the memory
boundary?
• We can keep the shared statistics for the region S
– The region S in implementations is usually the pixel
– Thus we are fine with one HP for every pixel
• For region S it is important to share the radius and α
– It can vary between regions even at the beginning of the
algorithm if we want to
– This is the only condition for consistency
How SPPM actually works?
1. Ray-tracing pass
–
Generating of hit points
2. Photon mapping pass (n times)
–
progressive update of shared statistics
3. Ray-tracing pass
–
loop
Regenerating of hit points
© Hachisuka et al.
PPM vs. SPPM
© Hachisuka et al.
Optimal light paths connection
Jiří Vorba
Which disadvantages of PM was not
mentioned?
•
•
•
Final gather is just a heuristic!
In general it doesn’t work well for every scene
It is not problem in PPM
– It lacks the final gather…
– Well, on the other hand it looses robustness due to
glossy scenes
•
SPPM more robust in comparison with PPM
– But due to “partial” final gather for glossy reflections
the problem is the same as with PM
Examples
© Jaroslav Křivánek
What went wrong?
• Distribution samples in the corners are too
correlated
– We can see the reflection of photon map
(illumination approximation)
• Distribution rays traveling the longer way
compensate the error by querying
uncorrelated parts of the photon map
Examples II – znova s mensim poctem
fotonu (neni videt problem)
What went wrong II
•
Direct photon map visualization is good for
caustic from highly glossy surface
– It is not that good for overall appearance (direct
visualization of illumination approximation)
•
Indirect visualization (final gather) is good for
overall appearance
– Turning the whole path into the ray-tracing is bad for
caustic
•
So question is: “When should we query the
photon map?”
What approach to try?
•
Inspiration by Erich Veach
–
•
In bi-directional path-tracing he connects paths from light
and from camera by weighting the whole path
contribution
–
•
Technique of Multiple Importance Sampling
This prevents the same paths generated from different
sampling techniques to contribute twice to the overall
illumination
Radiance estimate (photon map query) is connection of
paths as well
–
–
Could we somehow apply Veache’s approach?
It means more photon map queries on “the same final gather
path” and weight their contributions
Thank you
http://ms.mff.cuni.cz/~vorbj4am/pm
_ppm_sppm.pptx