INB382/INN382 Real-Time
Rendering Techniques Lecture 9:
Global Illumination
Ross Brown
CRICOS No. 000213J
Queensland University of Technology
Lecture Contents
•
•
•
•
Global Illumination
Radiosity
Pre-computed Radiosity Transfer
Spherical Harmonics
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Global Illumination
• “There are one-story intellects, two-story
intellects, and three-story intellects with
skylights. All fact collectors with no aim beyond
their facts are one-story men. Two-story men
compare, reason and generalize, using labours
of the fact collectors as well as their own. Threestory men idealize, imagine, and predict. Their
best illuminations come from above through the
skylight.” - Oliver Wendell Holmes
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Lecture Context
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•
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As mentioned before, we have
covered the direct forms of
illumination on object surfaces
Last week we moved further into
more indirect illumination issues by
modelling effects such as reflection
and refraction
We then presented a more
generalised approach known as ray
tracing – simple but costly – not
practical for real-time just yet
We now cover two relevant forms of
global illumination in real-time
rendering – Radiosity and Precomputed Radiance Transfer
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Global Illumination
• In most rendering, the use of a local lighting model
is the norm
• Meaning that only the surface data at the visible
point is used in the lighting calculations
• This is useful for object-precision GPU systems
– you can easily parallelise SIMD instructions on a stream of
vertex and pixel data
• Data is discarded once rendered
• Problematic for global illumination as we showed
last week
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Global Illumination Beginning
• We have covered reflection, refraction and area
lighting, which are global illumination techniques
because the rest of the scene influences the
lighting on the object
• Lighting needs to be thought of as the paths that
the photons take from light sources to the eye
• Local only acounts for one reflection from the
light source to the object, onto the eye
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Global Illumination Beginning
• Global illumination calculates the results of the paths of
photons from any reflecting object in the room –
specular, transparent or diffuse
• The contributions are summed for the object being
rendered
• This process is contained within the rendering equation
developed by Kajiya [2] – more later
• You can thus sum all the possible light paths to the
surface – the more paths, the greater the level of realism
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Radiosity and Ray Tracing
• Thus global illumination is divided into two main
approaches
– Radiosity – calculation of light as radiation transfer
through volume of air
– Ray Tracing – calculation of light as a set of discrete
ray samples through a volume
• We have covered the general idea of ray tracing
• Now we illustrate the main principles of
Radiosity using a series of lectures notes from
Brown University [1]
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Radiosity for Inter-Object Diffuse
Reflection
Color Bleeding
Soft Shadows
No Ambient Term
View Independent
Used in other areas of Engineering
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Pretty Pictures
Reality (actual
photograph)…
Minus Radiosity
Rendering…
Equals the difference
(or error) image
Mostly due to mis-calibration
http://www.graphics.cornell.edu/online/box/compare.html
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The Radiosity Technique:
An Overview
1.
2.
3.
Model scene as “patches”
Each patch has an initial luminance value: all but
luminaires (light sources) are probably zero
Iteratively determine how much luminance travels from
each patch to each other patch until entire system
converges to stable values
We can then render scene from any angle
without recomputing these final patch
luminances – they are viewer independent!
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Overview of Radiosity
• The radiometric term radiosity means rate at which energy
leaves a surface
• Sum of rates at which surface emits energy and reflects (or
transmits) energy received from all other surfaces
• Radiosity simulations are based on thermal engineering model
of emission and reflection of radiation using Finite Element
Analysis (FEA)
• First determine all light interactions in a view-independent way,
then render one or more views
• Consider room with only floor and ceiling:
ceiling
floor
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Overview of Radiosity
• Suppose ceiling is actually a fluorescent droppanel ceiling which emits light…
• Floor gets some of this light and reflects it back
• Ceiling gets some of this reflected light and
sends it back…
• Simulation mimics these successive “bounces”
through progressive refinement – each
iteration contributes less energy so process
converges
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Kajiya’s [2] Rendering Equation
•
Light energy travelling from point i to j is equal to light emitted from i to j,
plus the integral over S (all points on all surfaces) of reflectance from
point k to i to j, times the light from k to i, all attenuated by a geometry
factor
•
L(i → j) is the amount of light travelling along the ray from point i to point
j
•
Le is the amount of light emitted by the surface (luminance)
•
f(k → i → j, λ) is the Bidirectional Reflectance Distribution Function
(BRDF) of the surface. Describes how much of the light incident on the
surface at i from the direction of k leaves the surface in direction of j
– λ is wavelength of light, use a different function for R, G, & B
•
G(i ↔ j) is a geometry term which involves occlusion, distance, and the
angle between the surfaces
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Kajiya’s Rendering Equation
• More complete model of light transport than
either raytracing or polygonal rendering, but
does omit some things, eg., subsurface
scattering
• How do we evaluate this function?
– very difficult to solve complicated integral equations
analytically
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Rendering a Scene
• A scene has:
– geometry
– luminaires (light sources)
– observation point
• Light transport to camera must be computed
for every incoming angle to observation point,
bouncing off all geometry (in all directions
combined)
• This is far too hard
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Rendering a Scene
• Both raytracing and radiosity are crude approximations to
rendering equation
• Raytracing: consider only a very small, finite number of rays
and ignore diffuse inter-object reflections, perhaps using
ambient hack to approximate that lost component. Can use
either a BRDF or a simpler (e.g., Phong) model for luminaires
• Radiosity: approximate integral over differential source areas
dA(i) with finite sum over finite areas; consider light transport
not on basis of individual rays but on basis of energy transport
between finite patches (e.g., quads or triangles that result
from (adaptive) meshing) – remember Lecture 5???
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Adaptive Meshes for Radiosity
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What Can We Do?
•
•
Design an alternative simulation of
real transfer of light energy
With any luck, will be more
accurate, but accuracy is relative
– hall of mirrors is specular 
raytracing
– museum with latex-painted walls
is diffuse  radiosity
•
Best solutions are hybrid
techniques: use raytracing for
specular components and
radiosity for diffuse components
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What Can We Do?
• Radiosity approximates global diffuse interobject reflection by considering how each pair of
surface elements (patches) in scene send and
receive light energy, an O(n2) operation best
accomplished by progressive refinement
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Some Important Symbols
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Energy = light energy = radiosity for our purposes – it should really be rate, ie., energy/unit
time…
Ei: initial amount of energy radiating from i’th patch
Bi: final amount of energy radiating from i’th patch
Bj: final amount of energy radiating from j’th patch
Fj-i: fraction of energy Bi emitted by j’th patch that is gathered by i’th patch (relationship
between i’th and j’th patches based on geometry: distance, relative angles)
Fj-i Bj: total amount of patch j's energy sent to patch i
ρi: fraction of incoming energy to a patch that is then exported in next iteration
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Let’s Arrange Those Symbols
• Amount of light/radiosity/energy a patch finally emits is initial
emission plus sum of emissions due to other n-1 patches in scene
emitting to this patch - recursive definition. Note: F1-1 is zero
only for planar patches!
• Why is it zero for only planar patches?
Bi  Ei  i
Sender1
Sender2
…
Sendern
or
F
j i B j
1 j  n
Receiver patchi
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j i B j
ij n
Ei  Bi  i
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F
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Arranging Those Symbols
B1 - r1(F1-1B1 + F2-1B2 + F3-1B3 + ...) = E1
• Thus:
• Rewrite as a vector
product:
• And the whole system:
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Arranging Those Symbols Some More
Decompose the first matrix as:
Can be rewritten:
• (I – D()F)B = E
– where D() is diagonal matrix with i as its ith
diagonal entry, and F is called a Form Factor
Matrix and is based on geometry between
patches
• If we know E, D() , and F, we can determine B
• If we let
A = I – D()F
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Arranging Those Symbols Even More
• Then we are solving (for B) the equation
AB = E
• This is a linear system, and methods for
solving these are well-known, e.g. Gaussian
elimination or Gauss-Seidel iteration
(although which method is best depends on
nature of matrix A) – part of introductory
linear algebra courses
• Typically want B, knowing E and A
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Progressive Refinement
• Let us look at the floor ceiling problem again
• Both ceiling and floor act as lights emitting
and reflecting light uniformly over areas (all
surfaces considered such in radiosity)
C emits 12
F gets 1/3 C
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C reflects 75%
F reflects 50%
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C gets 1/3 F
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Progressive Refinement
• Let ceiling emit 12 units of light per second
• Let floor reflect get 1/3 of light from ceiling
(based on geometry) reflect 50% of what it gets
• Let ceiling get 1/3 of floor’s light (based on
geometry), and reflect 75% of what it gets
• Writing B1 for ceiling’s total light, and B2 for
floor’s, and E1 and E2 for light generated by
each:
3 1
E
=
12,
B
=
E
+
r
(F
B
)
=
12
+
× B2
1
1
1
2-1 2
• Ceiling: 1
4 3
• Floor: E = 0, B = E + r (F B ) = 0 + 1 × 1 B
2
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2
2
1- 2
1
2
3
1
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Progressive Refinement
• thus:
B1 = E1 + ρ1(F2-1(E2 + ρ2(F1-2B1)))
• becoming: B1 = E1 + ρ1F2-1E2 + ρ1 ρ2 F1-2F1-2B1
• which simplifies to:
F E
B1 
1   1 2 F 2  1F 1  2
1
2 1
2
F E
B2 
1   2  1F 1  2 F 2  1
2
1  2
1
• In general, this algebra is too complex, but we can find the solution
iteratively using progressive refinement
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Progressive Refinement
• Iterative method 1, gathering energy: send out light from
emitters everywhere, accumulate it, resend from all
patches… Each iteration uses radiosity values from
previous iteration as estimates for recursive form. Iterate by
rows.
• Bk = E + D(r)FBk-1
• B1 = E
• B2 = E + D(r)FB1
• B3 = E + D(r)FB2
• Where Bk is your kth guess at radiosity values B1, B2…
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Progressive Refinement
Results for our example (notice what happens to values):
{12, 0} = {B1, B2}= {E1, E2}
ì
î
ü
þ
{12, 2} ={ E1 + r1F2-1B2, E 2 + r2 F1-2 B1} = í12 + 0, 0 + 1× 1 ×12ý
{12.5, 2} =
2 3
3 1
1 1 

12    2, 0   12
4 3
2 3 

{12.5, 2.08373} =
3 1
1 1


12    2, 0   12.5
4 3
2 3


{12.5208, 2.08373}
{12.5208, 2.08681}
{12.5217, 2.08681}
{12.5217, 2.08695}
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General Radiosity Equation
Sender2
…
Sendern
Sender1
Receiver
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General Radiosity Equation
The radiosity equation for normalized unit areas of Lambertian diffuse patches is:
(B j A j ) × F j-i
Bi = E i + ri å
Ai
1£ j£n
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•
•
Bi is total radiosity in watts/m2 (i.e. energy/unit-time / unit-area) radiating from
patch i
– Note that we are now calculating Bi (and Ei) per unit area
Ei is light emitted in watts/m2
ρi is fraction of incident energy reflected by patch i (related to diffuse reflection
coefficient kd in simple lighting model)
Aj is area of jth patch
(Bj Aj) is total energy radiated by patch j with area Aj (i.e., unit radiosity x area)
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General Radiosity Equation
(B j A j )× F j-i
Bi = E i + ri å
Ai
1£ j£n
• Fj-i is fraction of energy leaving (“exported by”) patch j arriving at
patch i
• Fj-i is dimensionless form factor that takes into account shape and
relative orientation of each patch and occlusion by other patches
• It is a function of (r, θi, and θj)
• Geometrically, Fj-i is relative area receiver patch i subtends in
sender patch j’s “view”, a hemisphere centred over patch j
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General Radiosity Equation
Receiveri
Senderj
θi
θj
– patches may be concave and self-reflect; Fi-i 0
•
å
n
i=1
F j-i = 1
for all i, i.e., convex linear combination (conservation
of energy)
• (B j A j ) × F j-i is total amount of energy leaving patch j arriving at
patch i
• ( B j A j )  F j i Ai is total amount of energy leaving patch j arriving
at unit area of patch i
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General Radiosity Equation
• Reciprocity relationship between Fi-j and Fj-i:
Fi- j
Aj
=
F j-i
Ai
or Ai × Fi- j = F j-i × A j
• Which means form factors scaled for unit area of
receiver patch are equal
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General Radiosity Equation
• Therefore,
Bi  Ei  i
B F
j
i j
1 j  n
– which is easier to deal with, if less intuitive
– says that radiosity of receiver patch i is energy emitted by that
patch + attenuated sum of each sender j’s radiosity times form
factor from i to j (not from j to i)
– for each receiver row i iterate across all sender columns j to
gather energy those senders emit, attenuated by Fi-j.
• Note: we should calculate this for all wavelengths --approximate with BiR, BiG, BiB.
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Computing Form Factors
Receiverj
Senderi
•
Form factor from differential sending area dAi to differential receiving area dAj is:
dFdi-dj =
cosqi cosq j
H ij dA j
2
p×r
for ray of length r between patches, at angles i, j to normals of the areas.
Hij is 1 if dAj is visible from dAi and 0 otherwise
•
We will motivate this equation…
Applet: http://www.cs.brown.edu/exploratories/freeSoftware/catalogs/lighting_and_shading.html
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Computing Form Factors
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When two patches directly face
each other, maximum energy is
transmitted from Ai to Aj
Their normal vectors are parallel,
cosj = 1, cosi =1 since i = j =
0°
Rotate Aj so that it is perpendicular
to Ai
Now cosi is still 1, but cosj = 0
since j = 90
In general, we calculate energy
fraction by multiplying by cosj
Tilting Ai means multiplying by cosi
Same as Lambertian diffuse
reflection from direct lighting
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Computing Form Factors
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•
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•
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From where does the r2 term
arise?
The inverse-square law of light
propagation
Consider a patch A1, at a distance
R = 1 from light source L
If P photons hit area A1, their
density is P/A1
These same P photons pass
through A2
Since A2 is twice as far from L, by
similar triangles, it has four times
the area of A1
Therefore each similar patch on
A2 receives 1/4 of the photons
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Computing Form Factors
• The  in the formula is a normalizing factor
matching area of circle
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Computing Form Factors
•
Now consider a differential patch
dAi radiating to finite patch Aj
• Fdi-j can be computed by
projecting those parts of Aj visible
from dAi onto the unit hemisphere
centred about dAi.
•
•
•
Form factor is effectively the ratio
of curved patch area to the total
surface area of the hemisphere.
Total surface area encompasses
all energy emitted by dAi
But is costly to computer in this
form
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Approximating Form Factors
•
Rather than projecting Aj onto a
hemisphere, Cohen and Greenberg
proposed projecting onto the upper half
of a cube centred about dAi, with its top
face parallel to the surface
•
Each face of the hemicube is divided into
equal sized square cells
•
Think of each face of the cube as a film
plane which records what a patch, dAi ,
“sees” in each of the five directions;
centre of dAi acts as Centre Of Projection
•
In other words, think of the cells on the
faces as pixels and use the frustum
formed by the vertices of Aj onto each
face.
•
What does this remind you of?
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Approximating Form Factors
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•
•
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Identity of closest intersecting
patch is recorded at each cell (the
survivor of the z buffer algorithm).
Each hemicube cell p is
associated with a precomputed
delta form factor value,
Where θp is angle between p’s
surface normal and vector of
length r between dAi and p, and
where ΔA is area of cell
Can approximate Fdi-j for any
patch j by summing Fp
associated with each cell p
covered by patch Aj’s hemicube
projection
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• Provides occlusion
determination of patches
through z buffer (albeit
approximately, limited by the
granularity/resolution of the
cells on each plane)
• This eliminates the need to
compute Hi-j
Fp 
cos  i cos  p
 r
2
A
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Faster Progressive Refinement:
Shooting
•
Gathering: must process consecutive rows of receivers,
for each receiver looping through each column/sender; all
rows must be processed for each iteration of the algorithm
–
•
all form factors must be calculated before first iteration of GaussSeidel occurs
Shooting: shoot energy in order from brightest to least
bright patch (i.e., most significant light sources first)
–
–
accumulate at receivers
iteratively shoot from patch that has largest amount of “unshot”
radiosity (e.g., for a single light source, the patch which has the
largest form factor with that source will be next patch to shoot
from)
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Faster Progressive Refinement:
Shooting
• Computing form factor can be done by using a hemicube for each
shooter that can be computed and discarded for each receiver
(shown next)
– solves O(n2) processing / storage problem for each gathering iteration
– shooting converges faster than gathering
• In shooting, iterate by column not from left to right but in order of
patch with most “unshot” radiosity
– can form an estimate of light on all destination patches with only first
column shot, while gathering by row from all senders lights up only that
row’s patch.
– iterating over column of brightest emitter lights up all patches reachable
by emitter as a zeroth order approximation, improved by each
successive emitter (and then brightest unshot patch)
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Details on Shooting
•
Each row of matrix used in “gathering” (I – D()F) represents estimate of patch i’s
radiosity Bi based on estimates of other patch radiosities
–
each term in summation gathers light from patch j for all j:
k
k-1
k-1
Bi (due to B j ) = ri B j Fi- j , for all j
–
therefore,
B
•
= ri å
n
k-1
B
j Fi- j
j=1
For shooting, shoot from patch i to each patch j in turn; again
k
Bj
–
•
k
i
(due to
F j-i , for all j
) = r j Bk-1
i
k-1
Bi
for each receiver j, keep adding radiosity from successive sources i in order of decreasing
radiosity
So given an estimate of Bi from shooter patch i, we can estimate its impact on all
receiving patches j, at the cost of computing Fj-i for each receiver patch j, i.e., via n
hemicubes. But that is still too much work.
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Details on Shooting
•
•
•
•
•
•
Using reciprocity again, or just the
original formulation, which requires
only the hemicube over patch i!
Thus only a single hemicube and its
n form factors need be computed
each pass!
Note that for a given shooter i, we
loop through all receiving patches j.
Given our notation for the form
factor matrix, holding i constant and
looping through all j corresponds to
traversing a column
Java applet on the right available
from link below
NB: need Java 1.6 to run the app!!
B (due to B
k
j
k-1
i
)=r B
j
k-1
i
i- j
F
Ai
Aj
Applet: http://www.cs.brown.edu/exploratories/freeSoftware/catalogs/lighting_and_shading.html
Video: https://youtu.be/AiMBG8-rYE4
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Radiosity Pseudocode
•
Algorithm for fast progressive refinement through shooting:
U0 = e; // Set Matrix to initial emission values U = unshot energy
B0 = e;
t = 1;
do {
i = index_of(MAX(ut-1)); // Max row
precalculate hemicubei;
// Shoot the radiance to each row
for (j = 1; j < n; ++) {
bjt = uit-1 Fi-j Ai/Aj + bjt-1;
ujt = uit-1 Fi-j Ai/Aj + ujt-1;
}
uit = uit-1 Fj-j;
++t;
} while Bt-Bt-1 > tolerance; // ie. Matrix has not changed
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Limitations of Radiosity
• Assumption that radiation is uniform in all
directions
• Assumption that radiosity is piecewise
constant
– usual renderings make this assumption, but then interpolate
cheaply to fake a nice-looking answer
– this introduces quantifiable errors
• Computation of form factors Fi-j can be tough
– especially with intervening surfaces, etc.
• Assumption that reflectivity is independent of
directions to source and destination
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Limitations of Radiosity
• No volumetric objects (though there are equations and
algorithms for calculating surface-to-volume form factors)
• No transparency or translucency
• Independence from wavelength – no fluorescence or
phosphorescence
• Independence from phase – no diffraction
• Enormity of matrices! For large scenes, 10K x 10K
matrices are not uncommon (shooting reduces need to
have it all memory resident)
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More Comments
• Even with these limitations, it
produces great pictures
• For n surface patches, we
have to build an n x n matrix
and solve Ax = b, which takes
O(n2), this gets rather
expensive for large scenes
• Could we do it in O(n) instead?
• The answer, for lots of nice
scenes, is “Yes”
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• The Google search engine
uses an system much like
radiosity to rank its pages
– Site rankings are determined
not only by the number of links
from various sources, but by
the number of links coming
into those sources (and so on)
– After multiple iterations
through the link network, site
rankings stabilize
– Site importance is like
luminance, and every site is
initially considered an
“emitter”
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Making Radiosity Fast
• One approach is importance driven radiosity: if I turn on a bright
light in the graphics lab with the door open, it’ll lighten my office
a little…
• …but not much
• By taking each light source and asking “what’s illuminated by
this, really?” we can follow a “shooting” strategy in which
unshot radiosity is weighted by its importance, i.e., how likely it
is to affect the scene from my point of view
• No longer a view-independent solution…but much faster
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CRICOS No. 000213J
Precomputed Radiance Transfer PRT
• Imagine that time is stationary. Picture a volume of
space filled with photons – each cube of space can be
said to have a constant photon density. Picture this field
of photons in linear motion
• We need to find out how many photons collide with a
stationary surface for each unit of time, a value called
flux which measured in joules/second or watts
• Divide the flux by the differential area, we get a value
called the irradiance, measured in watts/meter2
• We thus model this using Spherical Harmonic functions
– rest of PRT based on paper - Spherical Harmonic
Lighting: The Gritty Details, Robin Green[4]
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CRICOS No. 000213J
Spherical Harmonics (SH)
•
•
•
•
•
SH lighting paper assumes knowledge
of the use of Basis Functions.
Basis Functions are small pieces of
signal that can be scaled and
combined to produce an approximation
to original function
Process of working out how much of
each basis function to sum is called
Projection.
To approximate a function using basis
functions we must work out a scalar
value that represents how much the
original function f(x) is like the each
Basis Function Bi(x).
We do this by integrating the product
f(x)Bi(x) over the full domain of f – use
summation for an approximation
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CRICOS No. 000213J
Legendre Polynomials
•
•
•
The associated Legendre
polynomials are at the heart of the
Spherical Harmonics, a
mathematical system analogous
to the Fourier transform but
defined across the surface of a
sphere
SH functions in general are
defined on Imaginary Numbers but
we are only interested in
approximating real functions over
the sphere (i.e. light intensity
fields)
Lecture will be working only with
the Real Spherical Harmonics
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R
CRICOS No. 000213J
SH series for varying values of l and m
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R
CRICOS No. 000213J
Projecting into SH Space
• Note how the first band is just a constant positive value
• If you render a self-shadowing model using just the 0band coefficients the resulting looks just like an
accessibility shader with points deep in crevices (high
curvature) shaded darker than points on flat surfaces –
See Lecture 8
• The l = 1 band coefficients cover signals that have only
one cycle per sphere and each one points along the x, y,
or z-axis
• Linear combinations of just these functions give us very
good approximations to the cosine term in the diffuse
surface reflectance model
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CRICOS No. 000213J
Projection Process
• Process for projecting a
spherical function into SH
coefficients is simple
• Calculate a single coefficient
for a specific band you just
integrate the product of your
function f and the SH function
y
• Working out how much your
function is “like” the basis
function – Slide 58
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CRICOS No. 000213J
Example Complex SH Function
Approximations
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CRICOS No. 000213J
Approximating Lighting using SH
• To project a function into SH
coefficients we want to
integrate the product of the
function and an SH function
• We must evaluate this integral
using Monte Carlo integration
An example lighting function displayed as
a color (left) and a spherical plot (right)
• where xj is our array of precalculated samples (Slide 59)
and the function f is the
product
•
f(xj) = light(xj)yi(xj).
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Example (left) of
a physically
sampled scene
using a
spherical silver
ball and a
camera
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CRICOS No. 000213J
Spherical Harmonic Sampling Basis
•
•
•
•
•
Uses a set of ortho-normal basis
functions on the surface of a sphere –
just like X,Y,Z axes
Rendering equation that we want to
integrate over the surface of a sphere
So all we need to do is generate
evenly distributed points (more
technically called unbiased random
samples) over the surface of a sphere.
Taking a pair of independent canonical
random numbers ξx and ξy we can
map this “square” of random values
into spherical coordinates using the
transform
This forms a light sample basis for the
later calculations
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CRICOS No. 000213J
Generating SH Coefficients
•
Applying this process to the light source we defined
earlier with 10,000 samples over 4 bands gives us
this vector of coefficients:
[ 0.39925,
- 0.21075, 0.28687, 0.28277,
- 0.31530, - 0.00040, 0.13159, 0.00098, - 0.09359,
- 0.00072, 0.12290, 0.30458, - 0.16427, - 0.00062, - 0.09126 ]
•
Reconstructing the SH functions for checking
purposes from these 15 coefficients is simply a case
of calculating a weighted sum of the basis functions:
An example approximated lighting function displayed as a color and a spherical plot
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CRICOS No. 000213J
Can also model shadows with SH
•
•
•
The transfer function (occlusion factor) in the lighting model can also be
stored as a 16 coefficient SH function
As you can see this is a different way of recording shadowing at a point on a
model without forcing us to do the final integral
Can rotate object and still get correct values for shadows on surface without
recasting the rays – an improvement on last lecture approach
Light
Distribution
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Transfer
(Occlusion)
real world
Final Light
Distribution
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CRICOS No. 000213J
Indirect Lighting Steps
•
•
•
•
Geometrically, the idea of
interreflected light is simple
Each point on the model already
knows how much direct
illumination it has, encoded in
the form of an SH transfer
function
Fire rays to find sample points
that can reflect light back onto
our position and add a cosine
weighted copy of that transfer
function back into our own
For example, point A in the
illustration above has fired a ray
and hit point B
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CRICOS No. 000213J
Shadowed Indirect Lighting
• A rendering using 5th order
diffuse shadowed SH transfer
functions
• Note the soft shadowing from
the constant hemisphere light
source
• NB: This is how light probes
work in Unity 5
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CRICOS No. 000213J
Real-Time Rendering using SH
• Now we have a set of SH coefficients for each vertex, how do we
build a renderer using current graphics hardware that will give us
real-time frame rates?
• Basic calculation for SH lighting is the dot product between an SH
projected light source and the SH transfer function vertex
• Approximate the complete solution over an object by filling in the
gaps between vertices using Gouraud shading
• Typically only need 4 coefficients per vertex – one extra 4 value
colour
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CRICOS No. 000213J
Real-Time Rendering using SH
• Can rotate the SH
coefficients so that the
object is able to have its first
pass lighting updated in real
time
• Now have a cheap
rendering technique for
generalised area light
sources in a scene
• Can add a specular term
and/or environment maps to
give required appearance
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CRICOS No. 000213J
Direct X Browser PRT Demonstration
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•
Within the DirectX
SDK is a demo
browser
•
Run the PRT demo
to see similar to the
movie on the left
•
Choose scene 4 to
obtain the head
figure
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CRICOS No. 000213J
Radiosity and Unity 5
• For years this has been a
grand challenge of CG, to
run radiosity in real time
• Enlighten’s
implementation [3]
performs a partial
version in real time on a
tablet in Unity 5!!!
• See here
– https://www.youtube.com/w
atch?v=Wrt5aLHI8ME
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CRICOS No. 000213J
Unity 5 GI Basics [5]
• Unity GI performed in
background and baked
on for static objects
• Precomputed Realtime GI
– encodes all possible
bounces into lightmap
data structure texture [6]
• Directional Light is
bounced into the scene
via this data structure in
real time due to no
position requirements
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CRICOS No. 000213J
Dynamic Objects and GI
• Dynamic objects cannot
cast light into the scene
• But they can use light
probes to sample the
bouncing light in the
scene to be lit
themselves
• Light probes are placed
where dynamic objects
are moving and are a
spherical panoramic
view of the environment
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CRICOS No. 000213J
Dynamic Objects and GI
• Light probes are stored
as Spherical Harmonic
approximations
• Interpolated internally
to light dynamic object
at that point
• Indirect, probe and
direct light blended to
provide final lighting
model in scene
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CRICOS No. 000213J
Unity 5 Demonstration Video
Video: https://youtu.be/G-dpRCdfFc8
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References
1.
John F. Hughes, Andries van Dam, Introduction to Radiosity,
www.cs.brown.edu/courses/cs123/lectures.shtml 29/04/2007
2.
J Kajiya. “The Rendering Equation.” SIGGRAPH 1984, pp.
143-150
3.
4.
www.geomerics.com 29/04/2007
Robin Green, Spherical Harmonic Lighting: The Gritty Details www1.cs.columbia.edu/~cs4162/slides/spherical-harmoniclighting.pdf
5.
http://docs.unity3d.com/Manual/GIIntro.html 04/05/2015
6.
http://www.geomerics.com/wpcontent/uploads/2014/03/radiosity_architecture.pdf
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