11_Radiosity
COMP3421
Radiosity, Splines, Colour
The lighting equation we looked at earlier only
handled direct lighting from sources:
We added an ambient fudge term to account
for all other light in the scene.
Without this term, surfaces not facing a light
source are black.
Recap: Global Lighting
Global lighting
In reality, the light falling on a surface comes
from everywhere. Light from one surface is
reflected onto another surface and then
another, and another, and…
Methods that take this kind of multi-bounce
lighting into account are called global lighting
methods.
Raytracing and Radiosity
There are two main methods for global lighting:
• Raytracing models specular reflection and
refraction.
• Radiosity models diffuse reflection.
Both methods are computationally expensive
and are rarely suitable for real-time rendering.
Radiosity
Radiosity is a global illumination technique
which performs indirect diffuse lighting.
direct lighting +
ambient
global
illumination
Radiosity
Direct lighting techniques only take into
account light coming directly from a source.
Raytracing takes into account specular
reflections of other objects.
Radiosity takes into account diffuse reflections
of everything else in the scene.
Ray tracing vs Radiosity
reflected light
n
incoming light incoming
light
reflected light
Specular
reflection
Diffuse
reflection
Ray tracing vs Radiosity
reflected light
n
incoming light incoming
light
Specular
reflection
Diffuse
reflection
view
Finite elements
We can solve the radiosity problem using a
finite element method.
We divide the scene up into small patches.
We then calculate the
energy transfer from each
patch to every other patch.
Energy transfer
The basic equation for energy transfer is:
where ρ is the diffuse reflection coefficient.
Energy transfer
The light input to a patch is a weighted sum of
the light output by every other patch.
Bi is the radiosity of patch i
Ei is the energy emitted by patch i
ρi is the reflectivity of patch i
Fij is a form factor which encodes what
fraction of light from patch j reaches patch i.
Form factors
The form factors Fij depend on
• the shapes of patches i and j
• the distance between the patches
• the relative orientation of the patches
ni nj
Aj
Ai
θi
θj r
Form factors
Mathematically:
Calculating form factors in this way is difficult
and does not take into account occlusion.
ni nj
Aj
Ai
θi
θj r
Nusselt Analog
An easier equivalent approach:
1.render the scene onto a unit
hemisphere from the patch’s point
of view.
2.project the hemisphere
orthographically on a unit circle.
3.divide by the area of the circle
Nusselt Analog
Aj
Fij = A/B
The hemicube method
A simpler method is to render the
scene onto a hemicube and weight the
pixels to account for the distortion.
Solving
We can view the radiosity equation as a system
of linear equations
B1 = E1 + ρ1(B1F11 + B2F12 +…+ BNF1N )
B2 = E2 + ρ2 (B1F21 + B2F22 +…+ BNF2N )
!
BN = EN + ρN (B1FN1 + B2FN 2 +…+ BNFNN )
Solving
This system of equations can be expressed as a
matrix equation:
In practice n is very large making exact
solutions impossible.
1− ρ1F11 −ρ1F12 ! −ρ1F1N
−ρ2F21 1− ρ2F22 ! −ρ2F2N
! ! ” !
−ρNFN1 −ρNFN 2 ! 1− ρFNN
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
B1
B2
!
BN
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
=
E1
E2
!
EN
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
Iterative approximation
One simple solution is merely to update the
radiosity values in multiple passes:
for each iteration:
for each patch i:
Bnew[i] = E[i]
for each patch j:
Bnew[i] +=
rho[i] * F[i,j] * Bold[j];
swap Bold and Bnew
F 0 1
0 0 0.185
1 0.185 0
0
1
30°
60°
B0 B1
1 0 0.8
2 0.072 0.8
3 0.074 0.807
4 0.075 0.807
F01 = F10 = (cos(30)− cos(60)) / 2 ≈ 0.185
Iterative approximation
Using direct rendering
for each iteration:
for each patch i:
Bnew[i] = E[i]
S = RenderScene(i,Bold)
B = Sum of pixels in S
Bnew[i] += rho[i]*B
swap Bold and Bnew
Iterative approximation
first pass
(direct lighting)
second pass
(one bounce)
third pass
(two bounces)
16th Pass
Progressive refinement
The iterative approach is inefficient as it spends
a lot of time computing inputs from patches
that make minimal or no contribution.
A better approach is to prioritise patches by
how much light they output, as these patches
will have the greatest contribution to the scene.
Progressive refinement
for each patch i:
B[i] = dB[i] = E[i]
iterate:
select patch i with max dB[i]:
calculate F[i][j] for all j
for each patch j:
dRad = rho[j] * B[i] *
F[i][j] * A[j] / A[i]
B[j] += dRad
dB[j] += dRad
dB[i] = 0
In practice
Radiosity is computationally expensive, so
rarely suitable for real-time rendering.
However, it can be used in conjunction with
light mapping.
The payoff
Geometric light
sources
Real-time Global
Illumination
http://www.youtube.com/watch?
v=Pq39Xb7OdH8
https://www.youtube.com/watch?
v=BmOppCTh1nA
https://www.youtube.com/watch?
v=h2lbZC3ihUU
Sources
http://freespace.virgin.net/hugo.elias/radiosity/
radiosity.htm
http://www.cs.uu.nl/docs/vakken/gr/2011/
gr_lectures.html
http://www.siggraph.org/education/materials/
HyperGraph/radiosity/overview_2.htm
http://http.developer.nvidia.com/GPUGems2/
gpugems2_chapter39.html
http://freespace.virgin.net/hugo.elias/radiosity/radiosity.htm
http://freespace.virgin.net/hugo.elias/radiosity/radiosity.htm
http://www.cs.uu.nl/docs/vakken/gr/2011/gr_lectures.html
http://www.cs.uu.nl/docs/vakken/gr/2011/gr_lectures.html
http://www.siggraph.org/education/materials/HyperGraph/radiosity/overview_2.htm
http://www.siggraph.org/education/materials/HyperGraph/radiosity/overview_2.htm
http://http.developer.nvidia.com/GPUGems2/gpugems2_chapter39.html
http://http.developer.nvidia.com/GPUGems2/gpugems2_chapter39.html
COMP3421
B-Splines
Quick Recap: Curves
We want a general purpose solution for
drawing curved lines and surfaces. It should:
• Be easy and intuitive to draw curves
• Support a wide variety of shapes,
including both standard circles, ellipses,
etc and “freehand” curves.
• Be computationally cheap.
Bézier curves
Have the general form:
where m is the degree of the curve
and P0…Pm are the control points.
Bernstein polynomials
where:
is the binomial function.
Bernstein polynomials
For the most common case, m = 3:
Problems
Local control – Moving one control point
affects the entire curve.
Incomplete – No circles, elipses, conic
sections, etc.
Problem: Local control
These curves suffer from non-local control.
Moving one control point affects the entire
curve.
Each Bernstein polynomial is active (non-zero)
over the entire interval [0,1]. The curve is a
blend of these functions so every control point
has an effect on the curve for all t from [0,1]
Splines
A spline is a smooth piecewise-polynomial
function (for some measurement of
smoothness).
The places where the polynomials join are
called knots.
A joined sequence of Bézier curves is an
example of a spline.
Local control
A spline provides local control.
A control point only affects the curve within a
limited neighbourhood.
Bézier splines
We can draw longer curves as sequences of
Bézier sections with common endpoints:
Parametric Continuity
A curve is said to have Cn continuity if the nth
derivative is continuous for all t:
C0: the curve is connected.
C1: a point travelling along the curve doesn’t
have any instantaneous changes in velocity.
C2: no instantaneous changes in acceleration
Geometric Continuity
A curve is said to have Gn continuity if the
normalised derivative is continuous for all t.
G1 means tangents to the curve are continuous
G2 means the curve has continuous curvature.
Continuity
Geometric continuity is important if we are
drawing a curve.
Parametric continuity is important if we are
using a curve as a guide for motion.
Bézier splines
If the control points are collinear, the the curve
has G1 continuity:
Drawing
Bézier splines
If the control points are collinear and equally
spaced, the curve has C1 continuity:
Motion
B-splines
We can generalise Bézier splines into a larger
class called basis splines or B-splines.
A B-spline of degree m has equation:
where L is the number of control points, with
B-splines
The function is defined recursively:
(Note: this formulation differs slightly from the
one in the textbook)
Knot vector
The sequence is called the knot
vector.
The knots are ordered so
Knots mark the limits of the influence of each
control point.
Control point Pk affects the curve between
knots tk and tk+m+1.
Number of Knots
The number of knots in the knot vector is
always equal to the number of control points
plus the order of the curve. E.g., a cubic (m=3)
with five control points has 9 items in the knot
vector. For example:
(0,0.125,0.25,0.375,0.5,0.625,0.75,0.875,1)
Uniform /
Non-uniform
Uniform B-splines have equally spaced knots.
Non-uniform B-splines allow knots to be
positioned arbitrarily and even repeat.
A multiple knot is a knot value that is repeated
several times.
Multiple knots create discontinuities in the
derivatives.
Continuity
A polynomial of degree m has Cm continuity.
A knot of multiplicity k reduces the continuity
by k.
So, a uniform B-spline of degree m has Cm-1
continuity.
Interpolation
A uniform B-spline approximates all of its
control points.
A common modification is to have knots of
multiplicity m+1 at the beginning and end in
order to interpolate the endpoints. This is
called clamping.
Moving Controls and
Knots
Moving Controls: Adjacent control points on
top of one another causes the curve to pass
closer to that point. With m adjacent control
points the curve passes through that point.
Moving Knots: Across a normal knot the
continuity for and degree curve is Cm-1. Each
extra knot with the same value reduces
continuity at that value by one.
Quadratic and Cubic
The most commonly used B-splines are
quadratic (m=2) and cubic (m=3).
Uniform quadratic splines have C1 (and G1)
continuity.
Uniform cubic splines have C2 (and G2)
continuity.
Bezier and B-Spline
A Bézier curve of degree m is a clamped uniform
B-spline of degree m with L=m+1 control points.
A Bézier spline of degree m is a sequence of
bezier curves connected at knots of multiplicity
m.
A quadratic piecewise Bézier knot vector with 9
control points will look like this
(0,0,0,0.25,0.25,0.5,0.5,0.75,0.75,1,1,1).
Incomplete
Conic sections are the intersection between
cones and planes.
Rational Bézier Curves
We can create a greater variety of curve shapes
if we weight the control points:
A higher weight draws the curve closer to that
point.
This is called a rational Bézier curve.
Rational Bézier Curves
Rational Bézier curves can exactly represent all
conic sections (circles, ellipses, parabolas,
hyperbolas).
This is not possible with normal Bézier curves.
If all weights are the same, it is the same as a
Bezier curve
Rational B-splines
We can also weight control points in B-splines
to get rational B-splines:
NURBS
Non-uniform rational B-splines are known as
NURBS.
NURBS provide a power yet efficient and
designer-friendly class of curves.
Closed curves
A unclamped uniform B-spline of degree m is a
closed loop if the first m control points match
the last m control points.
http://nurbscalculator.in/
http://nurbscalculator.in/
Surfaces
Surfaces
We can create 2D surfaces by parameterising
over two variables:
Where is any particular spline function
we choose (Bezier, B-spline, NURBS)
and denote an LxM array of control points.
COMP3421
Colour Theory
What is colour?
The experience of colour is complex, involving:
• Physics of light,
• Electromagnetic radiation
• Biology of the eye,
• Neuropsychology of the visual system.
Physics of light
Light is an electromagnetic wave, the same as
radio waves, microwaves, X-rays, etc.
The visible spectrum (for humans) consists of
waves with wavelength between 400 and 700
nanometers.
Non-spectral colours
Some light sources, such as lasers, emit
light of essentially a single wavelength or
“pure spectral” light (red,violet and colors of the
rainbow).
Other colours (e.g. white, purple, pink,brown) are
non-spectral.
There is no single wavelength for these colours, rather
they are mixtures of light of different wavelengths.
Colour perception
The retina (back of the eye) has two different
kinds of photoreceptor cells: rods and cones.
Rods are good at handling low-level lighting (e.g.
moonlight). They do not detect different
colours and are poor at distinguishing detail.
Cones respond better in brighter light levels.
They are better at discerning detail and colour.
The Eye
http://open.umich.edu/education/med/
resources/second-look-series/materials
http://open.umich.edu/education/med/resources/second-look-series/materials
http://open.umich.edu/education/med/resources/second-look-series/materials
Tristimulus Theory
Most people have three different kinds of cones
which are sensitive to different wavelengths.
Colour blending
As a result of this, different mixtures of light
will appear to have the same colour, because
they stimulate the cones in the same way.
For example, a mixture of red and green light
will appear to be yellow.
Colour blending
We can take advantage of this in a computer by
having monitors with only red, blue and green
phosphors in pixels.
Other colours are made by mixing these lights
together.
Colour blending
Can we make all colours this way?
No. Some colours require a negative amount of
one of the primaries (typically red).
Colour blending
What does this mean?
Algebraically, we write:
to indicate that colour C is equivalent (appears
the same as) r units of red, g units of green and
b units of blue.
Colour blending
A colour with wavelength 500nm (cyan/teal)
has:
We can rearrange this as:
So if we add 0.3 units of red to colour C, it will
look the same as the given combination of
green and blue.
C
0.3
R
0.49
G
0.11
BX XData source:
http://www.cvrl.org/
http://www.cvrl.org/
Colour blending
A colour with wavelength 500nm (cyan/teal)
has:
We can rearrange this as:
So if we add 0.3 units of red to colour C, it will
look the same as the given combination of
green and blue.
C
0.3
R
0.49
G
0.11
BX XData source:
http://www.cvrl.org/
http://www.cvrl.org/
Tristimulus Theory and
Colour Blending
https://graphics.stanford.edu/courses/cs178/
applets/locus.html
https://graphics.stanford.edu/courses/cs178/
applets/colormatching.html
https://graphics.stanford.edu/courses/cs178/applets/locus.html
https://graphics.stanford.edu/courses/cs178/applets/locus.html
https://graphics.stanford.edu/courses/cs178/applets/colormatching.html
https://graphics.stanford.edu/courses/cs178/applets/colormatching.html
Complementary Colors
Colours that add to give white (or at least grey)
are called complementary colours
eg red and cyan
Retinal fatigue causes complementary colours to
be seen in after-images
http://www.animations.physics.unsw.edu.au/jw/
light/complementary-colours.htm
http://www.animations.physics.unsw.edu.au/jw/light/complementary-colours.htm
http://www.animations.physics.unsw.edu.au/jw/light/complementary-colours.htm
Describing colour
We can describe a colour in terms of its:
• Hue – the colour of the dominant
wavelength such as red
• Luminance – the total power of the light
(related to brightness)
• Saturation – the purity of the light
i.e the percentage of the luminance given
by the dominant hue (the more grey it is
the more unsaturated)
Spectral Density
Hue is the peak or dominant wavelength
Luminance is related to the intensity or area under the entire spectrum
Saturation is the percentage of intensity in the dominant area
Physics vs Perception
We need to be careful with our language.
Physical and perceptual descriptions of light
differ.
A red light and a blue light of the same physical
intensity will not have the same perceived
brightness (the red will appear brighter).
Intensity, Power = physical properties
Luminance, Brightness = perceptual properties
Describing colour
“Computer science offers a few poorer cousins to these perceptual spaces that may
also turn up in your software interface, such as HSV and HLS. They are easy
mathematical transformations of RGB, and they seem to be perceptual systems because
they make use of the hue-lightness/value-saturation terminology. But take a close look;
don’t be fooled. Perceptual color dimensions are poorly scaled by the color
specifications that are provided in these and some other systems. For example,
saturation and lightness are confounded, so a saturation scale may also contain a wide
range of lightnesses (for example, it may progress from white to green which is a
combination of both lightness and saturation). Likewise, hue and lightness are
confounded so, for example, a saturated yellow and saturated blue may be designated
as the same ‘lightness’ but have wide differences in perceived lightness. These flaws
make the systems difficult to use to control the look of a color scheme in a systematic
manner. If much tweaking is required to achieve the desired effect, the system offers
little benefit over grappling with raw specifications in RGB or CMY. “
http://www.personal.psu.edu/cab38/ColorSch/ASApaper.html
http://www.personal.psu.edu/cab38/ColorSch/ASApaper.html
Standardisation
A problem with describing colours as RGB
values is that depends on what wavelengths we
define as red, green and blue.
Different displays emit different frequencies,
which means the same RGB value will result in
slightly different colours.
We need a standard that is independent of the
particular display.
The CIE standard
The CIE standard, also known as the XYZ
model, is a way of describing colours as a three
dimensional vector (x,y,z) with:
X, Y and Z are called imaginary colours.
It is impossible to create pure X, it is just a
useful mathematical representation.
The CIE standard
E = (1/3, 1/3, 1/3)
Non-spectral
colours
Spectral
colours
Gamut
Any output device has a certain range of
colours it can represent, which we call its
gamut.
We can depict this as an area on the CIE chart.
If a monitor has red, green and blue phosphors
then the gamut is the interior of the triangle
joining those points.
RGB Gamut
RGB Gamut
What is wrong with
this picture?
Gamut mapping
How do we map an (x, y, z) colour from outside
the gamut to a colour we can display?
We want to maintain:
• Approximately the same hue
• Relative saturation to other colours in
the image.
Rendering intents
C
There are four
standard rendering
intents which
describe approaches
to gamut mapping.
The definitions are
informal.
Implementations
vary.
Absolute colormetric
C
C’
Map C to the
nearest point within
the gamut.
Distorts hues.
Does not preserve
relative saturation.
Relative colormetric
C
C’
Desaturate C until it
lies in the gamut.
Maintains hues more
closely.
Does not preserve
relative saturation.
Perceptual
C1
C1′
Desaturate all
colours until they all
lie in the gamut.
Maintains hues.
Preserves relative
saturation.
Removes a lot of
saturation.
C2
C2′
C3
C3′
Saturation
Attempt to maintain
saturated colours.
There appears to be
no standard
algorithmic
implementation.
C1
C1′
C2 C2′
C3
C3′
Demo
http://graphics.stanford.edu/courses/cs178/
applets/gamutmapping.html
http://graphics.stanford.edu/courses/cs178/applets/gamutmapping.html
http://graphics.stanford.edu/courses/cs178/applets/gamutmapping.html
Colour space
Standard colour representations:
• RGB = Red, Green, Blue
• CMYK = Cyan, Magenta, Yellow, Black
• HSV = Hue, Saturation, Value (Brightness)
• HSL = Hue, Saturation, Lightness
RGB
Colour is expressed as the addition of red,
green and blue components.
This is called additive colour
mixing. It is the most common
model for computer displays.
CMY
CMY is a subtractive colour model, typically
used in describing printed media.
Cyan, magenta and yellow are the contrasting
colours to red, green and blue
respectively. I.e.:
Cyan pigment/ink
absorbs red light.
CMYK
Real coloured inks do not absorb light perfectly,
so darker colours are achieved by adding black
ink to lower the overall brightness.
The K in CMYK stands for “key” and refers to
black ink.
HSV
HSV (aka HSB) is an attempt to describe
colours in terms that have more perceptual
meaning (but see earlier proviso).
H represents the hue as an angle
from 0° (red) to 360° (red)
S represents the saturation
from 0 (grey) to 1 (full colour)
V represents the value/brightness
form 0 (black) to 1 (bright colour).
HSL
HSL (aka HLS) replaces the brightness
parameter with a (perhaps) more intuitive
lightness value.
H represents the hue as an angle
from 0° (red) to 360° (red)
S represents the saturation
from 0 (grey) to 1 (full colour)
L represents the
lightness form 0 (black) to 1 (white).
Video
https://www.youtube.com/watch?
v=z9Sen1HTu5o