GR2 Advanced Computer Graphics AGR

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GR2Advanced Computer GraphicsAGR

• Lecture 15
• Radiosity

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Review

• First a review of the two rendering approaches we
  have studied
• Phong reflection model
• ray tracing

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Phong Reflection Model

• This is the most common approach to rendering
• objects represented as polygonal faces
• intensity of faces calculated by Phong local
  illumination model
• polygons projected to viewplane
• Gouraud shading applied with Z buffer to
  determine visibility

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Phong Reflection Model

• Strengths
• simple and efficient
• models ambient, diffuse and specular reflection
• Limitations
• only considers light incident from a light
  source, and not inter-object reflections - ie it
  is a local illumination method (ambient term is
  approximation to global illumination
• empirical rather than theoretical base
• objects typically have plastic appearance

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Ray Tracing

• Ray traced from viewpoint through pixel until
  first object intersected
• Colour calculated as summation of
• local Phong reflection at that point
• specularly reflected light from direction of
  reflection
• transmitted light from refraction direction if
  transparent

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Ray Tracing

• This is done recursively
• Colour of light incoming along reflection
  direction found by
• tracing ray back until it hits an object
• colour of light emitted by object is itself
  summation of local component, reflected component
  and transmitted component
• and so on

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Ray Tracing - Strengths and Weaknesses

• Advantages
• increased realism through ability to handle
  inter-object reflection
• Disadvantages
• much more expensive than local reflection
• still empirical
• only handles specular inter-object reflection
• entire calculation is view-dependent

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Radiosity

• Based on the physics of heat transfer between
  surfaces
• Developed in 1980s at Cornell University in US
  (Cohen, Greenberg)
• Determine energy balance of light transfer
  between all surfaces in an enclosed space
• equilibrium reached between emission of light and
  partial absorption of light
• Assume surfaces are opaque, are perfect diffuse
  reflectors and are represented as sets of
  rectangular patches Ai

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Radiosity Examples
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Radiosity - Definition

• Radiosity defined as
• energy (Bi ) per unit area leaving a surface
  patch (Ai ) per unit time

Bi Ai Ei Ai Ri ? ( Fj-i Bj Aj) i1,2,..N
j
light emitted
light reflected
light leaving
Ei is the light energy emitted by Ai per unit
area Ri is the fraction of incident light
reflected in all directions Fj-i is the fraction
of energy leaving Aj that reaches Ai
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Radiosity - Pictorial Definition
Form factor Fj-i is fraction of energy leaving Aj
that reaches Ai. It is determined by
the relative orientation of the patches and
distance r between them
Aj
?j
r
Nj
Ni
Form factors are hard to calculate! Lets assume
for now we can do it.
?i
Ai
Bi Ai Ei Ai Ri ? (Fj-i Bj Aj)
j
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Simplifying the Equation
Bi Ai Ei Ai Ri ? (Fj-i Bj Aj) ie Bi Ai Ei
Ai Ri ? (Bj Fj-i Aj)
j
j
There is a reciprocity relationship Fj-i Aj
Fi-jAi
Hence
Bi Ai Ei Ai Ri ? (Bj Fi-j Ai)
j
So the radiosity of patch Ai is given by
Bi Ei Ri ? ( Bj Fi-j )
j
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Creating a System of Equations

• We get one equation for each patch
• assume we can calculate form factors
• then N equations for N unknowns B1,B2,..BN
• First equation is
• (1-R1F1-1)B1 - (R1F1-2)B2 - (R1F1-3)B3 - .. -
  (R1F1-N)BN E1
• ..and we get in all N equations like this.
• Generally Fi-i will be zero - why?
• Most of the Ei will be zero - why?

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Solving the Equations

• Equations are solved iteratively - ie a first
  guess chosen, then repeatedly refined until
  deemed to converge
• Suppose we guess solution as
• B1(0), B2(0), .. BN(0)
• Rewrite first equation as
• B1 E1 (R1F1-2)B2 .. (R1F1-N)BN
• Then we can improve estimate of B1 by
• B1(1) E1 (R1F1-2)B2(0) .. (R1F1-N)BN(0)
• .. and so on for other Bi

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Solving the Equations

• This gives an improved estimate
• B1(1), B2(1), .. BN(1)
• and we can continue until the iteration
  converges.
• This is known as the Jacobi iterative method
• An improved method is Gauss-Seidel iteration
• this always uses best available values
• eg B1(1) (rather than B1(0) ) used to calculate
  B2(1), etc

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Rendering

• What have we calculated?
• The Bi are the intensities of light emanating
  from each patch
• the form factors do not depend on wavelength ?,
  but the Ri do
• thus Bi depend on ?, so we need to calculate
  BiRED, BiGREEN, BiBLUE
• We get vertex intensities by averaging the
  intensities of surrounding faces
• Then we can pass to Gouraud renderer code for
  interpolated shading

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Pause at this stage

• Number of equations can be very large
• Calculation is not view dependent
• Only diffuse reflection
• We still have not seen how to calculate the form
  factors!
• Their calculation dominates

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Form Factors

• The calculation of the form factors is
  unfortunately quite hard
• We begin by looking at the form factor between
  two infinitesimal areas on the patches

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Form Factors - Notation
Aj
Form factor Fdi-dj gives the fraction of energy
reaching dAj from dAi.
dAj
Nj
?j
Ni
r
?i
Ai
r is distance between elements Ni, Nj are the
normals ?i, ?j are angles made with normals by
line joining elements
dAi
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Form Factor Calculation 2D Cross Section View
Draw in 2D but imagine this in 3D create unit
hemisphere with dAi at centre light
emits/reflects equally in all directions from dAi
form factor Fdi-dj is fraction of
energy reaching dAj from dAi
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Form Factor Calculation 2D Cross Section View
We begin calculation by projecting dAj onto
the surface of the hemisphere (blue) this
resulting area is (cos ?j / r2 ) dAj
Nj
?j
r
This gives us the relative area of light energy
reaching dAj from dAi
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Form Factor Calculation 2D Cross Section View
But we need a measure per unit area of Ai so need
to adjust for orientation of Ai This gives a
corrected area as ( cos ?i cos ?j / r2 ) dAj
?i
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Form Factor Calculation 2D Cross Section View
Total energy comes from integrating over whole
hemisphere - this comes to ? Hence form factor
is given by Fdi-dj ( cos ?i cos ?j ) / ( ?
r2 )dAj
dAi
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Form Factor Calculation 2D Cross Section View
Finally we need to sum up for ALL dAj. This
means integrating over the whole of the
patch Fdi-j ? (cos ?i cos ?j ) /( ? r2 ) dAj
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Form Factor Calculation
Strictly speaking, we should now integrate over
all dAi. In practice, we take Fdi-j as
representative of Fi-j and this assumption Fi-j
Fdi-j works OK in practice.
However the calculation of the integral ? (cos ?i
cos ?j ) /( ? r2 ) dAj is extremely difficult.
Next lecture will discuss an approximation.