Precomputed Radiance Transfer and Relighting
[CSE 190 Assignment 3]
Sean Wenzel
Brittany Factura
University of California, San Diego
University of California, San Diego
[email protected]
[email protected]
ABSTRACT
Recently, a body of work known as precomputed radiance
transfer, or precomputation-based relighting, has become a
popular technique for real-time rendering in varying illumination, and recently also viewpoint for dynamic scenes.
Some variant of this technique is now widely used in movies
and games. Good references are the seminal papers by
Sloan, Kautz and Snyder in SIGGRAPH 02, and Ng, Ramamoorthi, Hanrahan in SIGGRAPH 03. In this assignment, we aim to understand and implement Precomputed
Radiance Transfer.
Precomputed Radiance Transfer
Precomputed Radiance Transfer splits the problem of realtime global illumination into two parts: a precomputed
step, and one solved in real time. Functions ρ, V, and
G can be grouped into a single function T, known as the
transfer function
Direct lighting: light reflected/scattered at x in the direction ωo after arriving directly from light sources or other
emitting surfaces along all possible directions.
Z
Le (wi )ρ(x, wi , wo )V (x, x0 )G(x, x0 )dωi
Lo (x, wo ) =
ω
1.
INTRODUCTION
The high cost of global illumination algorithms has limited
their use in real-time applications. Precomputed radiance
transfer is an attempt to surpass the challenge of real-time
global illumination by splitting the problem into two parts:
one that is solved in a precomputed step and another that
is solved in real time.
The Rendering Equation
All rendering algorithms can be seen as solutions to particular formulations of the rendering equation.
L(x, wo ) = Le (x, wo )+
Z
L(x0 , wi )ρ(x, wi , wo )V (x, x0 )G(x, x0 )dωi
ω
where,
ω: the domain of all possible directions
L(x’,wi ): the amount of light arriving at x from
another point x’ along ωi
ρ(x,wi ,wo ): the function that tells how much of the
incoming light arriving at x along ωo
V(x,x’): binary visibility function involving x and x’
G(x,x’): geometric relationship between x and x’
The light leaving any point x in a given direction ωo is
computed as the amount of light that x emits in direction
ωo , plus the reflected/scattered light from x in the outgoing direction ωo after it has reached x from all incoming
directions.
Local geometric aspects are implicitly defined by x. Material properties are expressed as p. The incoming light is
represented by L. Occlusions are obtained as the result of
the visibility function V.
and after grouping,
Z
Le (wi )T (x, wi , wo , x0 )dωi
Lo (x, wo ) =
(1)
ω
Where Le is the lighting function that describes the arriving light intensity at point x and T is the transfer function
that expresses how x responds to the incoming illumination.
To determine the illumination of a point over a surface, it
is necessary to integrate, along with its material properties (ρ), the lighting function (Le ), the visibility function
(V) and the geometric factor (G). The lighting and transfer
functions are estimated during a pre-processing step, and
their values are used to approximate the integral in real
time. We then use basis functions to project and reconstruct arbitrary functions.
Projection and Reconstruction of Functions
Given a function space, a basis functions β is an infinite
set of functions used to project and reconstruct arbitrary
functions. The projection of a function f into any function
βi gives a measure ci of how similar these two functions
are, and is accomplished by integrated the product f and βi
over the entire domain of f:
Z
ci = f (x)βi (x)dx
The process of reconstruction allows for the original function f to be recovered as a linear combination of the basis
functions βi , each modulated by its corresponding coefficient ci . Its accuracy increases the more basis functions are
used; with an infinite number of basis functions (and an
infinite number of coefficients), the original function can be
recovered.
PRT uses orthonormal basis functions:
(
Z
0, if i 6= j
βi (x)βj (x)dx =
1, if i = j
Orthonormality guarantees that the integral of the product
of two basis functions will be either zero if the functions
are different, or one if they are the same. Thus, let ck
and dk be the coefficients associated with the projections
of any two functions f(x) and g(x), respectively, into the
basis functions βk , for all k. The integral of the product
f(x)g(x) is obtained as the sum of the products ck dk .
n
X
c k dk
(2)
Spherical Harmonics
Spherical harmonics are orthonormal functions defined over
the unit sphere, where the 2D domain can be seen as the
set of all possible directions. In their general formulation,
spherical harmonic functions are defined over complex numbers, but Precomputed Radiance Transfer is mainly concerned in approximating real functions.
Legendre polynomials are the core of the spherical harmonic
functions. The associated Legendre polynomials return real
numbers. They are characterized by two parameters, l,
which is usually called band, and m, that varies according to l. They are defined over real numbers in the interval
between [-l, l ].
k=1
This provides a powerful mechanism for factoring the evaluation of Equation (1) in two steps:
(i) the projection of both Le and T into an orthonormal
basis, which can be done during pre-processing, and
(ii) the evaluation of the integral as a dot product
(Equation 2), which can be efficiently done in real time.
Precomputed Radiance Transfer Eq Derivation
Let both lighting and transfer functions from Equation (1)
be projected into the same set basis functions βk . The
original functions can be approximated as:
Le (wi ) ≈
n
X
lk βk (wi ),
where
R
Le (wi )βk (wi )dwi
k=1
lk =
T (x, wi , wo , x0 ) ≈
n
X
tk βk (wi ),
where
k=1
tk =
R
ω
For each term k of the sum, the associated lighting coefficient will be multiplied by the entire integral. But this
integral represents a projection operation. So, each term
of the sum will produce a new coefficient that is the result
of projecting the transfer function into the associated basis
function, resulting in the following equation:
Lo (x, wo ) =
n
X
lk t k
k=1
which is a dot product of the lighting and the transfer coefficients. While this only handles direct lighting, it handles
occlusions with respect to the light source. To perform extra bounces, a similar derivation can be done with higher
order terms of the Neumann Expansion, which leads to the
final PRT equation:
L(x, wo ) =
2.
IMPLEMENTATION
The PRT algorithm consists of two steps:
(i) preprocessing - projection of the lighting and transfer
functions into some orthonormal basis functions
(ii) real time rendering - the evaluation of Equation (3).
Lighting functions can be represented as environment maps
stored as spherical maps or cube maps (light probes).
For transfer functions, two types will be considered:
(i) unshadowed transfer functions, which ignore the
visibility function (V(x,x’)) and
(ii) shadowed transfer functions, which take the occlusion
term into account.
We can also add indirect lighting, or interreflected transfer
function, to the shadowed case. The transfer function will
be evaluated for each vertex present in the scene.
T (x, wi , wo , x0 )βk (wi )dwi
Substituting into Equation (1),
Z
n
X
lk
βk (wi )T (x, wi , wo , x0 )dwi
Lo (x, wo ) =
k=1
Spherical harmonic functions are parametrized using θ and
φ, with scaling factors Klm for normalizing the functions.
A spherical harmonics approximation of a function using l
bands requires l2 coefficients.
n
X
k=0
lk (t0k + t1k + t2k + ...)
(3)
3.
LITERATURE
4.
RESULTS AND DISCUSSIONS
5.
FUTURE WORKS
6.
REFERENCES