ETNA
Electronic Transactions on Numerical Analysis.
Volume 23, pp. 219-250, 2006.
Copyright  2006, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
A STUDY OF THE FAST SOLUTION OF THE OCCLUDED RADIOSITY
EQUATION
KENDALL ATKINSON AND DAVID CHIEN
Abstract. Consider the numerical solution of the radiosity equation over an occluded surface using a collocation method based on piecewise polynomial interpolation. Hackbusch and Nowak [Numer. Math., 54 (1989)
pp. 463–491] proposed a fast method of solution for boundary integral equations, and we consider the application of
their method to the collocation solution of the radiosity equation. We give a combined analytical and experimental
study of the ‘clustering method’ of Hackbusch and Nowak, yielding further insight into the method.
Key words. Radiosity equation, clustering method, collocation, fast solution method
AMS subject classifications. 65R20, 65F99
1. Introduction. The radiosity equation is a mathematical model for the brightness of
a collection of one or more surfaces when their reflectivity and emissivity are given. The
equation is
(1.1)
with
!#"%$&'(
'
$&
*)+
,)+
the “brightness” or radiosity
at and 3
the
emissivity at
. The function
).
254
gives the reflectivity 6
at )7
, with /10 . In deriving this equation,
the
reflectivity at any point of
is assumed,
to)+be
uniform
in
all
directions
from
;
and
in
addition,
the
diffusion
of
light
from
all
points
is
assumed
to
be
uniform
in
all
directions
from . Such a surface is called a Lambertian diffuse reflector. The radiosity equation is used
in the approximate solution of the ‘global illumination problem’ of computer graphics; see
Cohen and Wallace
[9] and Sillion and Puech [21].
The function is given by
8"
(1.2)
I
"
9(:<;=?>@9A:B;=
C D&C E
F
G(H I
'
DKHLIK
> J F
K
J
C M
N&C O
P
I
Q
1
> and
In
this > is the inner unit normal to at , = > is the angle between
and
I
'R
and = are defined analogously; cf. Figure
1.1.
The
function
is
a
“line
of
sight”
and can “see each
other” along a straight line
function. More precisely, if the points
&"54BS
segment
which
does
not
intersect
at
any
other
point,
then
and otherwise,
&T"
.U.4
on ; otherwise the surface is
/ . An unoccluded surface is one for which
called occluded.
The numerical solution of the unoccluded case by collocation methods has been studied
previously. For example, see [4] for piecewise constant and piecewise linear approximations
over unoccluded piecewise smooth surfaces. For an analysis of the effects of edges and
corners on the solvability of the radiosity equation and its numerical solution, see [14], [15],
and [18]. For a discussion of the numerical
integration needed in implementing collocation
methods, see [6] and [19]. Note that need not be connected; and is usually only piecewise
V
Received September 2, 2005. Accepted for publication May 2, 2006. Recommended by F. Stenger.
Dept of Mathematics, University of Iowa, Iowa City, Iowa 52242 (atkinson@math.uiowa.edu).
Dept of Mathematics, California State University San Marcos, San Marcos, CA 92096 (chien@csusm.edu).
219
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220
K. ATKINSON AND D. CHIEN
P
Q
θP
θQ
nP
n
Q
F IG . 1.1. Illustrative graph for defining radiosity kernel (1.2)
smooth. General introductions to the derivation, numerical solution, and application of the
radiosity equation (1.1) from a computer graphics perspective can be found in the books [9]
and [21].
One of the principal problems in solving (1.2) is the very large operations cost in setting
up and solving the large linear systems that arise when discretizing the integral equation. This
is true with both Galerkin methods and collocation methods. The matrix coefficients with
Galerkin methods are four-fold integrals, and those with collocation methods are two-fold
integrals; and near to edges and corners of , the integrands can be nearly singular. In [5] we
solution of the
proposed and illustrated the use of Hackbusch’s clustering method [12] for the
linear system arising with the collocation method in the case that the surface is unoccluded.
In that case, the operations cost is WQX'Y[Z :]\B^ Y`_ , where Y denotes the number of elements into
which is sub-divided in the discretization of the integral equation. The present paper is an
analytical and experimental study of the clustering method in the case the surface is occluded.
In particular, it is shown that it is not possible to have as good an operations cost as is possible
with unoccluded surfaces.
There has been a great deal of work done by researchers from the computer graphics
community on rapid methods of solution of discretizations of the radiosity equation. For
examples of this, see [10] and [13]. Their perspective is different than that of numerical
analysts, who are usually interested in asymptotic rates of convergence regarded as a function
of the discretization parameter, and this results in a different way of analyzing operation
counts. We comment on this later when our methods are illustrated numerically.
We often write (1.1) in the simpler form
(1.3)
'K
acb
b
'hi"
!#"d$'e
'
*)fg
'hjk'he
or in operator form as
(1.4)
'lh
nm
"%$
P
For a discussion of general stability and convergence results for the collocation solution of
m
this equation, refer to the papers cited
above. We note that in general the integral operator
is not compact for most surfaces of practical interest, and this has made more difficult the
mathematical analysis of collocation methods.
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221
FAST SOLUTION OF THE RADIOSITY EQUATION
We discuss the triangulation of and the definition of the collocation method in o 2.
The clustering method is introduced in o 3, and in o 4 the error analysis from Hackbusch and
Nowak
[12] is specialized to the radiosity equation over polyhedral surfaces. In o 5 the surface
and true solutions used in our experiments are introduced, and a very brief description
is given of the programs and computing equipment used in the experimental study. In o 6
we give our experimental results, discussing the cost as a function of the parameters of the
numerical scheme.
2. Triangulation and collocation. We limit our presentation to polyhedral surfaces.
The treatment of more general surfaces requires certain nuances which we do not want to
consider here; and the problems and methods in which we are interested are illustrated well
with polyhedral surfaces. Moreover, many of the surfaces of practical interest are polyhedral.
Our scheme for the triangulation of is essentially that described in [1, Chap. 5] and
implemented in the boundary element package BIEPACK, described in [2]. We subdivide
the surface into closed triangular elements prqtsvu and we approximate by a low-degree
"
polynomial
over each element. We assume there is a sequence of triangulations of , wyx
C4
pzq@x{ |
0d}&01Yu with some increasing sequence of integer values Y , with Y~€ . In our
codes the values of Y increase by a factor of 4, due to our method for refining a triangulation.
To refine a triangle q x{ | , we connect the midpoints of its sides, creating four new smaller and
congruent triangular elements. There are standard assumptions made on the triangulations.
We describe the triangulation process briefly, and the details are left to [1, Chap. 5], [2].
Associated
with most surfaces are parameterizations of the surface. We assume the sur
face can be written as
N"‚`ƒi„k
with each
s
„…HLHAHL„f‡†
E
a closed polygonal region. Triangulate
s
pzq
C
"Q4]
}
x{ |
, say as
s
Y svu P
PˆPˆP y
This need not be a ‘conforming triangulation’, in contrast
to the situation with finite element
methods for solving partial differential equations. For as a whole, define
†
"
wx
‰
sŠ
ƒK‹ q
s
C
x{ |
"Œ4B
}
Y sŽ
PˆPP y
P
s
Sometimes we dispense with the subscript Y in q x{ | , although it is to be understood implicitly.
We introduce notation and assumptions regarding the mesh size of the triangulations
pLw x u . The mesh size of this triangulation is defined by
(2.1)

U
"‘’?“
ƒR”
”•†
s
 x
ƒe”
’v“
’v›š(œšAz
”
q
|
x?–˜—™
s
x{ |
P
"
Cy4
As noted earlier, the elements of wax are denoted collectively by wax
pzq@x{ | ƒ
0}0%Yu .
With each increase of Y to ž]Y in our programs, the mesh size  decreases to E  .
Let
Ÿ
q
denote the center
of the smallest circle (or smallest ball if it is in 3D) containing
its radius, and ¡ q its area. Introduce
q , q
(2.2)
¢Lx
"%’?“
pr q
¤£
q
)
wxyu
P
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222
K. ATKINSON AND D. CHIEN
We assume there are positive numbers ¥˜¦ , ¥¤§ for which
(2.3)
(2.4)
q
¡
q
¨
¢Ax
¨
¥©¦
E
¢ x
§
¥
)
for all q
wx . These assumptions imply the mesh scheme is uniform over wªx in the manner
with which the elements decrease in size as Y~€ .
For purposes of numerical integration and interpolation over the triangular elements in
w x , we need a parameterization of each such triangular element with respect to a standard
reference triangle in the plane, namely the unit simplex
"
«
p
¬]­j£
¬]­e¬¤®D­
/@0
4
0
)
Let q@x{ |
wx , and let the
ƒR²‡ƒ vertices of q›x{ | be denoted by
£
zation function ±| « ³ ~ x?´ ³ q@x{ | by
¬B­jT"
±+|
"Q4˜
D¬˜
G­
with ®7­
y¯v°
ƒ?
pr¯
ƒ?
¯
P
¯ E
¬]­j)
¯v°?u
. Define a parameteri-
«
. Using this, we can write
",C ·¸
·
·
±f|t¹
§iµK¶
C ·¸
®#¬
¯ E
u
º
C
´±+|
¬Bj­j «
¶
±f|
C
since
±+|@¹
´»±f|
is a constant function, equal to twice the area of qx{ | . This formula
can be used to numerically evaluate the left-hand integral by using numerical integration
formulas developed for the region « ; e.g. see [22].
2.1. Interpolatory projections.
To define collocation methods, we need to consider
interpolation over the surface . The centroid of q x{ | is defined as
Define the operator ¼
)
ƒ
"
±f|
ƒ
X °
° _
ƒ
"
°
¯
ƒi®
®
¯ E
¯v°
P
associated with piecewise constant interpolation over
x
for
|
¼
x
i
•'©"
¶
'
|
(8.)
¶
q
|
"Œ4B
}
by
PˆPP Y
.
We also wish to consider approximations
based on piecewise linear approximations over
)
i
ƒ
the triangulation wax . Given ½
, we define the interpolating function ¼¾xB½ as follows,
¥
2
basing it on interpolation over the unit simplex « . Let ¿ be a given constant with /@01¿
° ;
and define interpolation nodes in « by
¶
(2.5)
¥
prÀ
ƒz
À E
"
ÀA°]u
p
¿
¿
eL
¿
L4c
DÁ
¿
eLj4˜
7Á
¿
¿
u P
"
«
If ¿
/ , these are the three vertices of ; and otherwise, they are symmetrically placed
«
points in the interior of . Define corresponding Lagrange interpolation basis functions by
 ƒ ¬Bj­jT"
¿
4˜
GÃ
¿
Â
E
¬Bj­ji"
­K
4˜
NÃ
¿
¿
Â
°
¬B­ji"
¬˜
4˜
NÃ
¿
¿
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223
FAST SOLUTION OF THE RADIOSITY EQUATION
¬]­j¤)
«
and for
of (2.5) is given by
".4˜
N¬c
n­
¶
)
¥
i
Æ
°
ÅÆ
¬Bj­jiÄ
For ½
)
. The linear polynomial interpolating
ƒ
¶
« at the nodes
Æ
Â
À
¶
Š
¥
¬Bj­j
P
, define
(2.6)
¼˜x<½
‡
±+|
Æ
°
ÅÆ
¬Bj­jT"
ƒ ½
Š
"Ç4]RÁ
±+|
Æ
Â
À
¬Bj­j(
¬Bj­j¤)
«
'
for }
over each triangular
element
q&|GÈ
, with the
PˆPˆP Y . This interpolates ½
¬
­
interpolating function linear in the parameterization variables and . Let the interpolation
nodes in q| be denoted byÆ
Æ
¯]|r{
"
±+|
À
(
Ég"Œ4]RÁÃa
}
"Œ4]
PPˆP
Y
P
Then (2.6) can be written
(2.7)
¼
x ½
•T"
Š
"Ê4]
Æ
°
ÅÆ
ƒ ½
Æ
Â
¯ |r{
¬B­je
Œ"
±
|
¬Bj­j)
q
|
Æ
ƒz
for }
¯ E PˆPˆP ¯v°x•u ,
PPˆP Y . Collectively, we refer to the interpolation nodes pL¯|r{ u by pr¯
for ¿7ËÌ/ .
Higher degree extensions can be given of the piecewise constant and piecewise linear
interpolation defined above. We refer to [1, Chap. 5] for some of the tools for this, along with
results given in [17]. In our experimental studies, we have considered only piecewise constant
and piecewise linear interpolation. In the numerical experiments discussed in this paper, we
consider collocation methods based on only piecewise linear interpolation. Moreover, we
assume
/
2
¿
4
2
Ã
which forces all interpolation nodes to be interior to each triangle q| .
2.2. Collocation. The collocation method for solving (1.3) or (1.4) can be written abstractly as
(2.8)
'lh
¼˜x
m
yx
"
¼˜x
$
with ¼˜x an interpolatory projection. Our approximations ‡x are
generally not continuous
over the boundaries of the triangular elements of our mesh for , in part because our collocation
node points are chosen interior to the elements of the triangular mesh wªx we impose
on . This is
done, in part, to avoid having node points on edges of the original surface as
I
the normal > in (1.2) is undefined where two edges come together. To render the solution
graphically with standard software, we would need to know values at the corner points of
the triangulation; and that is easily accomplished from the solution we obtain at the interior
points, although we do not consider it here.
Æ "
Let the number of collocation nodes be denoted by x ( x
Y for the centroid method,
"ŒÃ
x
Y for the piecewise linear method), and let p
u denote collectively these nodes. Let
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224Æ
K. ATKINSON AND D. CHIEN
C?ÉK"Q4B
x u denote the Lagrange basis functions for the interpolation scheme being
used. For the centroid rule,
Æ
prÍ
PLPAP
Æ
Æ 4]Ç*)
'i"ÏÎ
Í
q
ÇÑÐ)
/
q
P
For piecewise linear interpolation, use the basis functions implicit in (2.7), which are again
nonzero over only a single triangular element. We also will write
Í
(
|–Ò Ó
"dÃLÕ
Gþ®
} s{ Ô
 ".4]RÁÃa
Â
",4
for the three linear basis functions"5
over
the element qts . Also introduce the parameter Ö
Ã
for the centroid method, and Ö
for the piecewise linear method. These ideas extend
naturally to higher degree collocation functions.
The solution of (2.8)
reduces to the solution of the linear system
Æ
Æ
(2.9)
x
'
'
g
M
Å
ƒ,× ƒ x
sŠ
Ô؊
£a4
x
Å
Â
0
0ÌÖ‡Ú the
with Ù |–Ò Ó
this system symbolically by
'
| – {Ô
b
Þ
x
Æ
Æ
x
'
"
'
x
R
collocation nodes inside
Ö
(2.10)
Æ
"%ß
x
R
'ß
x
q
, for
s
Í
!#"%$'
| –Ò Ó
ÉÛ"Ü4B
PLPAP
x
. We denote
x
Æ
Æ
(
Æ
§–
lÝ
Þ
Æ
"%$&'
e
É"Œ4B
PˆPP
Y P
To set up the linear system requires the evaluation of the integrals in (2.9). These may
or by numerical integration (cf. [6]). Æ Note that
be evaluated
either analytically
(cf. [20])
Rà)á
Õ
"
Æ Æ
b
for
avoids
the need
to Rcalculate
s for some , we have
/ ; and this
Æ
"
U
many of the collocation integrals in (2.9). In particular,
since
x
/
/
{
Æ
over a triangular element containing
in
its
interior;
and
thus
the
system
(2.10)
contains
no
Æ
singular integrals,
although some
are
likely to be
nearly singular when
is near to an edge
'
hTU
/ as
or corner of . In addition, if
varies over q s , the corresponding integral
in (2.9) is zero. For codes evaluating these integrals numerically, see [2] and the discussion
in [6].
For more of a computer graphics perspective on the piecewise linear collocation method,
and for another way to evaluate analytically the needed collocation integrals, see [7], [8].
2.3. Iteration. A standard iterative method of solution of (2.10) is based on a simple
fixed point iteration. Define
(2.11)
Ċ
ÞTâ |(ã
x
for some given initial guess
Þ
x
b
®
x
Þiâ |
x
ä
}
"
/
A4B
PˆPP
. Easily,
x
b
x
ä
âæå
Þ
"‚ß
GÞTâ |eã
x
2*4
Ċ
"
b
x3ç
Þ
x
ä
GÞTâ |
x.è
and the method converges if é x‡é
. In this case, the matrix norm being used is the row
norm, compatible with the maximum norm on êTëRì .
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225
FAST SOLUTION OF THE RADIOSITY EQUATION
b
Examining the elements of x in the case ofÉ piecewise constant interpolation, the ele´'í
ments are all non-negative and for the sum
row, we have
Æ of the
x
Å
sŠ
Æ
b
x
ƒ
"î
{s
'
Æ
&
Æ
h'
a<
It is well-known that
b
é
x‡éÛ0é
m
m
P
é
i
i
when
is regarded as an operator on ïð
to ï¤ð
, and this is known to be bounded
away from 1; cf. [6]. This proves the convergence of (2.11), with a rate that is independent of
the
size of Y . For piecewise linear interpolation, we need to further assume that the reflectivity
is sufficiently small, although this assumption is probably an artifact of our method of
proof.
3. Approximation by clustering. In practical problems x can be quite large, too large
to set up the system for direct solution
by standard Gaussian elimination or standard iteration
methods. For larger values of x , it is impractical to set up this system completely, and socalled ‘fast matrix-vector multiplication methods’ are needed. An example of such is given
in [5] for unoccluded surfaces; and other schemes have been proposed based on the ‘fast
multipole method’ and ‘wavelet compression multi-resolution schemes’ (e.g. [11]). For a
computer graphics perspective in designing such a fast method, see [13].
There are two main practical problems in dealing with the linear system
b
lÝ
(3.1)
x
Þ
x
"dß
b
x
E
when Y is large. First the setup time for the matrix x is proportional to Y , with a large
constant of proportionality. Second, the cost of the iteration method for solving the system
E
is also proportional to Y , even though usually we need only a few iterations. The setup
time seems the greater expense in practice, as some of our timings from [5] suggest. In the
following we propose a method to reduce the cost of both the matrix setup and the iteration
E
procedure to something less than W X Y _ . However, the method will not be as rapid as that
given in [5], as we discuss later. The method is based on the work of Hackbusch and Nowak
[12], who developed such a method for solving boundary integral equation reformulations
of Laplace’s equation in ê ° . Following we describe their work and adapt it to the radiosity
equation. To simplify the presentation and
notation, we consider the collocation method with
"ñ4
only piecewise constant functions ( Ö
in (2.9)), but the central ideas are suitable for
piecewise polynomial approximations of any fixed degree and our numerical examples use
piecewise linear collocation. Based on the results given in [3], it does not seem a worthwhile
b
idea to look at higher degree
collocation methods.
Þ
We do b notÞ compute x explicitly. Rather, for a given vector , we compute an approxb
imation
to x . The cost will be much less than it would be b if based
on first computing
Þ
b
,
each
of which has an
x and then following it with the matrix-vector multiplication
x
Þ
E
operations cost of W X Y _ . This approximation of x is under the control of two parameters ± and ò to control the approximation error, and the
user specifies these parameters. The
)D A4z
parameter ± is a positive integer and the parameter ò
/
; they are introduced below.
Recall
(3.2)
Æ
b
x
ÞK
"
Å
sŠ
x
ƒ s
b
§–
Æ
'
!˜
ÉK".4]
PPˆP Y
P
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226
K. ATKINSON AND D. CHIEN
Æ
q s of the triangulation w x into three distinct subsets,
For each , weÆ subdivide the elements
Æ
called the near field, theÆ far field, and the gray field.
For Æ each , the
gray
field for
contains those elements q s that contain a ‘visibility
'
R
is
neither
identically 0 nor 1 over q s . The line of discontinuity
line’,
meaning
that
'
R
for
, as
varies over q s , is called the visibility line. In the original paper [12,
o 3], there was no need to consider such elements. The larger cost of our adaptation of the
clustering method from [12, o 3] is due toÆ the existence of this gray field.
q s of the triangulation w x that are not in the gray
The near field contains those elements
field and that are ‘sufficiently close’ to . For elements q s in the near field and the gray field,
the corresponding integrals in (3.2) are computed by standard methods; we want to minimize
the number of such integrations. The remaining elements in wªx , those not contained in either
the near field or the gray field, are said to make up the far field. This
separation between
Æ
b
the
near
field
and
the
far
field
is
associated
with
a
parameter
the
user’s control. For
ò under
R
belonging to an element
based on a Taylor
qhs in the far field, we approximate
Ì4
polynomial of degree ±
and then we carry out exactly and efficiently all integrations of
the Taylor polynomial.
"
ƒz
We have a sequence of triangulations wóx Y "
Y å Y
PPˆP Y`ô with wx]õ the most current
¨
Â
Ô
subdivision of . For simplicity, we assume Y‡Ô
ž Y å ,
/ . Let
wx Ó
"
Ô
Ù q
£
|
"Œ4]
}
PPˆP
Y•Ô Ú
Â
"
/
A4B
PˆPP
ö
P
Â
We refer to this triangulation as being at ‘level ’. The collection
"
÷
wxvø
„nHAHLHL„
wx]õ
can be regarded as a tree structure of clusters of the elements from the finest subdivision wyx]õ
of , from the various levels of the
refinement process. For more on this tree structure and its
Æ
properties, see [12, o 3].
) ÷
Given a collocation point , we say a cluster ù
is admissible if
Æ
(3.3)
Æ
'
ù
0úò
C
Ÿ
ù
AC
Æ
)
Æ
and
if
is constant as
varies over ù . If an element q
wóx]õ is not admissible
Æ
Æ
and
is
not
in
the
gray
field
of
,
then
we
say
it
is
in
the
near
field
of
. The far field for
is the closure of the complement in of the union of the near field and gray field for .
As ò decreases,b theÞ size of the near field increases, and this generally increases the cost of
approximating x . The size of )D
the gray
field is independent of ò . For theoretical reasons,
A4z
we assume there is a constant ò å
and we consider the parameters ò satisfying
/
(3.4)
/
2
ò0úò å
.
if ò å is chosen sufficiently small, then
In the later theoretical analysis of o 4, it is shown that
Æ
various estimates will be true for all ò satisfying (3.4).
An admissible covering of with respect to is a collection
Æ
û
with each ù
)
û
"
prù
ƒz
ù üzuÝÈ
PPˆP L
÷
D"
‰,ü
satisfying
ù
)
wx]õ
or
ù
admissible P
ƒ ù
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227
FAST SOLUTION OF THE RADIOSITY EQUATION
Æ
It is shown in [12, Prop. 3.9] that for each , there is a unique minimal admissible covering,
Æ
b of containing a minimum number of clusters ù .
unique in theÆ sense
Þ
To evaluate
x
, begin by writing the unique minimal admissible covering of with
respect to by
Æþý
D"
ô
ç q
Æ
Æþÿ
„…HLHAHL„
„
ô
q
è
Æ
Fù
Æ
ƒg„nHLHAHL„
ù J
Æ
ô
with each ù an admissible cluster and each q – a non-admissible element of the
current
ô
triangulation w x õ (meaning q – belongs to either the near field or the gray field of ). Then
(3.5)
Æ
b
x
Þ
"
Å
sŠ
Æ
ü
ƒ –
Æ
b
a<
õ
–
§
Å
®
sŠ
ƒ
•x
–
Æ
b
'
R
P
The
function •x
is the piecewise constant function over waxBõ with values determined from
Þ
. The first integrals (over the near field and gray field) are evaluated by traditional means,
by analytic or numerical integration, and ) this is discussed at greater length in [6], [20]. For
a gray field integral over an element q
waxBõ , it is important to determine accurately the
location of the visibility line that intersects q , thus allowing an accurate numerical or exact
integral to be computed. For the the remaining integrals over the far field, we proceed as
follows.
Æ
3.1. The far field integration.
For the integral over ù(s in (3.5), we use Taylor poly
R
nomial approximations of
and then perform the remaining integration exactly. We
now explain
how to do this with some efficiency. This follows ideas
in [12]. ) ÷
ú4
. Using a Taylor approximation in of degree ±
about Ÿ ù , write
Let ù
F
(3.6)
G(H I
C M
NC O
> J
Å
Ä
'‡M
Ÿ
ù
*)
ù
with Ÿ ù the center of the circumscribing
circle
for ù , as introduced earlier. The set
",
ƒ?
consists of all multi-integers ¿
with
¿
¿ E ¿ °
¿
á"î
ƒ ¿ E
¨
¿ °
Be
/
¿
"
ƒ
ý
®
®
¿ E
¿ °
2
±
P
As customary, if
ò
then
ò
. We often refer to ±
the Taylor polynomial approximation.
'R
From (3.6), we can obtain an analogous approximation of
:
F
(3.7)
M
NeH I
‚
DHLI
> J F
C M
NC O
I
)
J
Ä
Å
'‡M
l
Ÿ
÷
ù
as the order of
P
Note that'*
1
ishconstant
over any ù
. More important in obtaining (3.7) from
(3.6), the
¤HvI
*
1
quantity
is constant over an element ù . To see
this,
decompose
into
7
components
perpendicular
and
parallel
to
takes
place
parallel
to
ù . All variation in
ù
as I varies over ù , and the part perpendicular to ù remains constant. 'Forming
the
dot
product
Ì
!HRI
with
, the portion parallel to ù is zero, and the result follows that
is constant
over ù .
Next, expand and rearrange the terms in (3.7) into the form
(3.8)
TÄ
Å
' j Ÿ
ù
.)
ù P
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228
K. ATKINSON AND D. CHIEN
Return to (3.5), to the approximation of
Æ
Recalling that
'
–
x
iU.4
x
–
for
Æ
b
Æ
h<‡
P
in a farÆ field element, use (3.8) to write
a<!7Äî
Å
j
Ÿ
ù
–
x
!
P
The integrals
(3.9)
•x
hj
)
ù
÷
)+l
¿
can be evaluated explicitly, and they can be obtained in a preprocessing step before beginning
the iterative solution of (3.1).
What is the operations
cost of calculating these integrals?
The cost of producing (3.9)
is independent of
and
of
the
center
of
expansion
Ÿ
ù , and thus these integrals can be
obtained in W Y steps. For unoccluded surfaces , this will result in there being only W Y
such integrals to be evaluated.
) ÷
In greater detail, let ù
. Then we can
write
Æþý
Æ
for some elements q
"
ù
Æ
ô
)
(3.10)
x
ô
q
µ
. Then
w x õ
–
„…HAHLHr„
ô
q
|
Å
!7"
sŠ
ƒ x
!
õ –
§
h
since •x
is constant over each element in wóx]õ . Produce) the integrals over the elements
÷
of wxBõ , and then
extend those to the remaining clusters ù
using (3.10). Since we are
working with a polyhedral surface,
the
integrals
on
the
right
side
of (3.10) can be evaluated
explicitly. Again the cost is W Y operations.
Following Hackbusch and Nowak [12],
it can be shown that with appropriate
choices of
b
b
"
"
Þ
Þ
ò
ò Y
±
Y , the quantity
and ±
x
x can be approximated, say by x
x with
(3.11)
b
x
Þ
x
b
x
Þ
"
x
W

Þ
é
x é
ð
ð
Þ
x
)
ê ëRì
with the order of the underlying numerical scheme being used. If we assumed sufficient
regularity in the unknown solution , then is the order of the
collocation approximation
"‘4
method being "*
used.
For
piecewise
constant
collocation,
and for piecewise linear
Á
collocation, . The derivation of (3.11) is discussed in detail in o 4.1. The result (3.11)
also assumes the near field integrals in (3.5) are evaluated with an error consistent with the
right side; in some cases they can be evaluated exactly. For notational purposes, we write
b
b
x
Þ
x
b
"
x
Þ
x
with x a square matrix of order x . We never produce this matrix
useful when giving a theoretical analysis of the approximation.
b
x
explicitly, but it is
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FAST SOLUTION OF THE RADIOSITY EQUATION
b
229
Þ
The cost of evaluating
the portion of x x associated with the far field
is shown in [12]
Æ
to be W Y[Z :B\<^ Y operations. Unfortunately, we have the additional cost
of
evaluating theÆ
collocation integrals
in
(2.9)
for
the
elements
.
With
a uniform
qhs in the gray field for
triangulation of , as assumed
in
(2.3)-(2.4),
even
a
single
visibility
line
for
a
field
point
!
cutting across
a face of will lead to W elements intersected by such a line and thus
to
Y
Æ
an additional
operations
cost
of
such
W
Y . When looking at the cumulative effect of W
Y
points , we have an additional operations cost of size
(3.12)
W
Y
Y
P
Thus the total operations cost cannot be brought below W Y Y for most occluded surfaces
of practical interest, even when we retain the efficiency of the approximations
in [12] for the
E near and far fields. This result is an improvement on the cost of W Y
for the standard setup
and iterative solution, but not what we would have liked.
The near-optimal performance of the clustering method in [12] cannot be matched when
applied to the radiosity equation. Moreover, we do not see
how
any fast matrix-vector mul
W
Y
elements
intersected by each
tiplication method can
avoid
this
problem,
due
to
the
visibility line for W Y collocation points.
4. Theoretical error analysis . In this section, we discuss the error caused by the approximation of the far field integrations. The numerical integration of integrals over elements
in the near field and the gray field is discussed at greater length in [6].
As we mentioned in o 2.2, the collocation method for solving (1.3) or (1.4) can be written
abstractly as (2.8),
l
¼
x
m
"
x
¼
$
x
The solution of this equation reduces to the solution of the linear system (2.9), or symbolically
as in (2.10),
b
l
b
b
Þ
x
"‚ß
x
x P
is then approximated by x using the method described in o 3.1. Instead of solving the
above system, we solve the system
x
(4.1)
"
b
l
Þ
x#
"%ß
x
x P
Using standard perturbation theory, we can guarantee the unique solvability of this new
approximation for Y chosen sufficiently large.
²!the
ƒ
b
From
theoretical error analysis for the original collocation method, it is known
that
'lh
¨$
exists and is uniformly bounded for all sufficiently large Y , say for Y
. In
x
addition, if is sufficiently smooth, then
(4.2)
é
Þ+
NÞ
"
x é
W
ð

!
¨%$
Y
with defined following
b
b
(3.11); cf. [6]. From (3.11) (or from Theorem 4.1 given below)
we have that x
x is sufficiently small for all large values of Y . Using standard
b
²‡ƒ
'l3
perturbation theory, we have that
x
also exists and is uniformly bounded in Y .
Then the system in (4.1) is uniquely solvable for all large values of Y .
Write
Þ
Þ
x
"QÞ+
NÞ
x
®%Þ
x
Þ
x
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230
K. ATKINSON AND D. CHIEN
Þ+
é
Þ
x é
Þ
GÞ
0Qé
ð
Using the identity
Þ
(4.3)
x
Þ
we have
Þ
é
x
Þ
x`é
b
".l
x
ð
²‡ƒ
H
b
x
é
x
b
²!ƒ
x
Þ
é
ð
x
b
lÝ
&
0Ύ
®
x é
b
é
Þ
x é
jÞ
x
x
b
&
x
P
ð
x
Þ
x‡é P
When combined with (3.11) and (4.2), we have
Þ
é
Þ
"
x é
W
ð
•

¨%$
Y
P
We note an implicit assumption tob this work and to that of Hackbusch and Nowak [12].
We want
to choose the approximation x in such a way that the operations count for calcub
Þ
lating x x is minimized while simultaneously maintaining the rate of convergence of the
original collocation solution in (4.2). This is the reason for desiring the result (3.11), as it
leads via (4.3) to the desired rate of convergence and to the operation counts that we obtain.
In contrast,
many
fast methods used in the computer graphics community ask only that the
b
b
Þ
Þ
x x be smaller than something related to the pixel size of the display deerror x x
vice; e.g., see [13]. Both of these perspectives are reasonable, but they can lead to markedly
different interpretations of the phrase ‘optimal operation counts’.
4.1. The error in clustering. We follow
the development in [12] to bound
b
b
é
x
Þ
x
x é
and to bound the numbers of elements in the near and far fields of the triangulation.
We do so
m
in the context of all our sub-surfaces being polygonal and planar, and of being the radiosity
integral operator. In this case, more precise results can be obtained and some
steps in the
U
¢ x õ , with Y ô
proof are simplified. As notation, recall the definition of ¢ x in (2.2) and let ¢
b
the index for the current finest level mesh w x õ .
b
T
HEOREM 4.1. Let ± be the order of the Taylor expansion of the kernel. Let
x and
x be described as above. Then
(4.4)
é
b
b
x
x
jÞ
x`é
0d¥
¾²‡ƒ?Á
where
¥
'`)(ai"
'
(
Þ
é
x‡é
¡
ð
¢
i
E j4¤®
E
ò å ¥
ò å
E
¦
E O Á °
ò å
4c
Proof. Define
}
ò
4©®
²!ƒ "
ò å
ã O P
*(h
+'ygH
C '
.(‡C O
Y-, ²!ƒv*' /(
°
where
, , and
. Let }
be its Taylor expansion of degree
Y , are vectors in ê
14
(
)
­¤)G ˜4z
°
around å
. There exists
such that the remainder
±
ê
/
²!ƒv*'&
+(
"
"
*' 0(aK
}
4
±.162
]­4
)(h
+(
å
}
å
4
²!ƒ *' 0(ai"
*'&
+(
Cæ'&
+(
*(
]­54
å
±.132
K
G­(*(@
+(
å
G­(*(@
+(
H
å
å
LC O
®D­((h
+( g
.'•H
å
å
Y-,
Cæ'&
+( G­(*(t
.( AC O
å
å
Y
,
P
ETNA
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231
FAST SOLUTION OF THE RADIOSITY EQUATION
Define
7
­j©"
G­0]H
The analytic continuation
, Y
G­0óC O
C
7
Cy
9 C2M4?Ð
ä
"
¿
C[
+9;óC O
P
P
9A9
9
n­j
ƒ
ã
CC
C
9(:B;a=
Y ,
C [
:9AªC O
0
0
C óC ° C 4˜
+B
4
¿
4
CC ó
C?
dC •
9 C!C ó
CþC °
0
C•
9 C]"%B
for
C°
P
¿
C [
:9AªC °
4
"
C óC °FE 4˜
+B > G> E °
> HI> E
E
E
E
4
C ­AC2CBt2
for
C[
+9;óCˆC
C
,
Y
2M4
0úò å
7
Á É=<
4
"
C óC
> ?> Š-@
C*[
:9ABKH
D9<LC]"
C
Cª
C
*6BT"
4
­ji"
±.1
Since 7
°
ê
,
Y
CÛ
:9;óC O
Cauchy’s integral formula
yields
7
â
)
,
Y
with
¿
¿
4
8
ê
Û
:9;]KH
9T"
is holomorphic in
­)
and
CÝ
9 n­AC¨JBÛ
14
for /
21­©2M4Ý2CBÝ",C y
9 C
, we have
7
4
E
â
E
ä
E ±.1
E
E
E
E
4
E
E
ƒ
ã
E
E
Á E
4
BÛ
14r
C°
¿
B
"
ƒ
ã
D9<LC
C
<
> ?> Š-@
Á 0
E
C óC ° Cæ4˜
+B
Á 9Ý
G­j
E 4
0
E
99
E Á ÉK<
E E
> ?> Š-@
E
E
7
4
E
E "
­j
7
C
9 n­AC
ã
C 9•C
ƒ
B
C óC ° Cæ4˜
.B
¿
C ° *B[
14r
ƒ
ã
P
For
4
BÝ"
4
4©®
Á
2
"
4
¿
the above equation
becomes
7
4
E
â
E
ä
E ±+1
E
E 0
Á
E
"
¿
2
0
E
'­j
2
®%4
E
4˜
4©®
4˜
¿
ò å
ò å
®d4z
¿
Có
C
¿
Á
°
4
4
Có
C°
2
4˜
¿
2
4˜
E
¿
Á
°
ò å
4
2
4˜
¿
ò å
¿
4˜
¿
ã
4
ƒ
4
¿
4
4
4B
2
Á
4
¿
)
¿
2
E
4˜
2
Á
E
Á
°
4
Á
°
E
Á
E
E
²
®%4
¿
E
°
4
Có
C
4
E 4˜
®%4
¿
4©®
"
2
4˜
ò å
ò å
4
Á
2
4˜
ò å
4
ㇰ
¿
C óC ° P
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232
K. ATKINSON AND D. CHIEN
The estimate
C [
LóC
E
"
Có
C
E C óC
E
E 4˜
E
E
E 0
Có
C
E
E
Có
C
E
E
E
4©®
E 0
Có
C
E
4©®
¿70
E
ò å
E
implies that
4©®
4
ò å
C [
LóC P
Có
C 0
Thus,
7
C
¾²‡ƒ?]LC]"
4
E
â
E
(4.5)
0
Ý"M'&
+(
,
å
h"%(t
.(
2
å
,
4©®
E
'­j
E 0
E ±+1
E
E
E
4©®
For
ä
ò å
4˜
ò å
Á
4
4
2
Û
Lh"N'
+(
4˜
2
ò å
4
4˜
2
¿
Ch
( +(
, with
C
Có
C
C '&
+(
01ò
²
°
¿
°
ò å
C[
:ªC4
2
å
ㇰ
4
ò å
4©®
ㇰ
4
ò å
Á
ò å
P
C
å
, where
òk0úò å
and define
"
ò
ò
4©®
O
E P
ò
Now, (4.5) becomes
C
'`P(aK
}
4©®
0
"
¥
2
4˜
ò å
ò å
²‡ƒ?Á
2
ò
4
4
ò å
¿
Á
4
ò å
4©®
4˜
2
ò å
4˜
2
4
ò å
4©®
"
0
ò å
4˜
2
²!ƒv*' P(aAC]"*C !
² ƒv*'&
+( Q(h
+( LC
å
å
Á
4T®
ㇰ
°
}
2
4˜
4
ò å
2
ㇰ
4
4˜
C (t
.(
ㇰ
Á
2
C&
' +(
ò
ò å ²
C '
+(‡C °
ò å
C[
:ªC*4
C
å
å
4T®
4
C
2
4©®
2
Cæ'&
.(
°
ò å
C '&
+(‡C
j4©®
"
4
å
ò å
Ì*(t
.(
4˜
å
°
AC
4
ò
O Á ° Á
ò å ò
C
&
'
+
(‡C °
ã O
å
P
For the radiosity kernel,
}
Let }
²!ƒ
'ªhi"ÇF
M
NHLI
'
DKHAI
> J F
C ‚
D&C O
ú4
be the Taylor expansion of degree ±
F
NKHLI
about
> J
C D&C O
J
P
for
å
P
The error is
C
}
ª
}
¾²‡ƒ?yLC]".C NgHAI
0
F
C N&C ¥
J
C
²‡ƒvÁ
M
GgHAI
E F
E
E E
ò
C ‚
D&C
C M
NC O
°
0
¥
> J
²!ƒ?Á
ò
C NC E
²!ƒ]'T E
E
}
E
P
E
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etna@mcs.kent.edu
233
FAST SOLUTION OF THE RADIOSITY EQUATION
*)f
Given any point
R
R
)
For every ù
'
û
‰
T"
.
)
pLù
, ù
¤£
û
H
0
dist
ò
R
Ð)
, ù
Ë
H
ò
dist
—™
R
Ð)
'
˜¨
ù
ù
Ð
ù
˜2
ù
Ð
ù
)
For every ù
R
ù
¨
q
P
ò P
u
ò P
¢
Ë
¦
¥
)
for some q
R
)
Note, ù
means that ù is a panel in near field. For every ù
panel in w x õ or a cluster which contains several panels. Therefore,
, which implies
œz'
;
. Let
, which implies
ù
œz'
;
w.r.t.
is admissible with respect to
ù
'
—™
For every ù
is an admissible covering of
w x õ P
,
œr
;
—™
¨
ù
ù
¢
Ë
ò
P
¦ ò
¥
Thus,
R
'
b
LS
È
'
@
where
b
'i"
@
p
&CLC ‚
N&C
B
0
u
and
BÝ"
¢
Since
'T
4T®J' E
is increasing for
"
ò
'+)
ò
4©®
F/
O
ò
4
,
J
ò å
4©®
O
0
E
P
¦ ò
¥
ò å
E P
Then,
BÝ"
"
¥
¢
¦ ò
¨
¥
¢
¦
ƒ
U
ã
¢
¦
¥
U
where
¥
"
O
ƒ ø
U
ã
4©®
ò å ¥¤¦
"
U
ò å
ø
E
P
¢
O
4i®
ò å ¥
¦
ò å
E
"
¥
¢
, ù is either a
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234
K. ATKINSON AND D. CHIEN
The additional error caused by the Taylor expansion of the kernel is
b
E
Þ
E
E
E E
E
0
E
EV
E
²!ƒvÁ
0Ì¥
E
E
E
ò
Þ
}
Þ
é
W8XZY
Þ
é
i
x é
ð
E 4©®
¢
E
E
E
E
E
E
E
E !
ò å ¥ ¦
j
4
©
®
E E
¢
ò å
[ \
ð
¡
x
C D&C E
ð
x‡é
E
'
W8XZY[\
x‡é
²!ƒv'©Þ
é
ò
²!ƒ Á
¥
'
x
ò
²!ƒvÁ
0Ì¥
Þ
'¤K
}
ä
E â>
"
b
'K
x
E
ò å ¥
ò å
E
¦
E P
Thus the proof is completed.
*'•
4.2. The number of clusters.û We
give an estimation
of the number «
of clusters in
'y
'
the minimum
admissible
covering
with
respect
to
and
the
number
of
non-admissible
û *'•
triangles in
.
Recall the definitions of ¢ and ¥¦ in (2.2) and (2.3), respectively. Since is the union of
planes,
b
(4.6)
¡
9^]fi
@
B E
0
We begin with the following lemma.
L EMMA 4.2. The condition (4.6) implies
(4.7)
¨
E
Y‡¢
B@¨
for all
_9›)
/
°
ê
P
T
¡
Ë1/
and
Proof. Since qÏÈ
Then,
¡
Ti"
ä â §
¡
¡
ba`
q
¡
ì
Ÿ
0
q
01¡
Å
§ d
q
jb]f
bc`
â §
q
Ÿ
b]fi
q
Å
§ d
õ
)
for all q
w x õ P
,
ä 0
E
q
ì
E
q
0
0
Å
§ d
õ
E
q
ì
¢
E
P
"
Y‡¢
E
P
õ
Before we prove the next theorem,
we assume the
following condition. This condition
û_e
describes how a limited part of û_ise covered by a part of an admissible covering. An upper
bound of number of elements
in B is given by (4.9), provided the clusters in the admissible
covering all have the size ù 0 . 9+)D
¨fB
C ONDITION
4.3. For any point
andûb e
Ë/ there is an admissible covering
9
÷
with respect to for which there exists
a
subset
È
with
b `
g]
(4.8)
(4.9)
h
û
ù
9
e
0
0
B
Áa
È
„
pAù
)
for all ù
Ð5B? E
P
e
û
)
u
û
e S
wx]õ
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235
FAST SOLUTION OF THE RADIOSITY EQUATION
*'•
and ò ' be the ratio. The number «
T HEOREM 4.4. Let Y be the number of elements
û 'y
of cluster in the minimum admissible covering
w.r.t.
is bounded by the following
inequality:
º
*'y
«
(4.10)
C '&
÷
"
®
i
C '
E
ò
Y
for all
')+
P
. First we assume that
`
b
implying
0
Á
Z :B\
is iAC
Ÿ
4
2 ò
ú)
Proof. Let the size of
E
4
0Ì¥
'y
È
"‚Á
for P
iLC
Ÿ
Ë .
In later step, we consider the case of
ƒR
²
In this step, we calculate how 9t
many
clusters
each
annulus
contains. For all i
"N'
"
"‚Á
j
, and
kj
we apply the condition (4.3) with
, B"NB
kj ò
"
j
"
/
‡4]
PAPAP
P
ž
û
There are coverings j with
`
b
g]
'y
h
û
„
È
Á
jK0
j )
Ùvù
2
"‚Á
4
j
Let
û
e
û
È
j
Note that
it satisfies
"MÁ
4
ž
4
2 ò
lon
ba`
*'•mS
E
ž
Ct
( +'KC¨
©"
ù
j
H
tj
Á
*'`
Ð]Á
j
e "
j )
Ù ù
for any
C&
' +óC¨
H ™rq
s
Ë
"
i
/
û
£
j
]
ù
j6" p
Ú
P
b
tj
tj
ž
B
0
()
H ™rq
s p
"
ù
"
ž
ù
"
j
ž
j "p
B
j ¨
ù
ò
. If ù
¨
u
j ]k
j
S
wx]õ
È
_
„
pLù
is not a panel,
j
kj
Á
+B
j
"
kj
Á
P
)
û
'
for any ù
e j u
'
û_e
is admissible w.r.t. . The number of elements in the set j is
h
û
e
j 0
h
û
j 0
kj ò
ž
X
û-e
)
,
C '&
+(‡Cz
dC (t
.óC
ò
b
)D û-e
PLPAP(P
jò 7
b
]
j
b
Thus, û-e ù 0 ò dist
ù . We can conclude
that ù' is admissible w.r.t.
j ]f
Hence, j is an admissible covering of
w.r.t. . We have
and ù
‡4B
7
b
is an annulus, and j contains clusters which covers
*' *'y
E
ûbe
by (4.8). Since
`-l
b
j "
dist
Ð
by
j
û
b
E
j
2 jò
b
and define j
j Ú
E
j
B
û
Á
ž
2 ò
4
E
P
)
û
j
e
.
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236
K. ATKINSON AND D. CHIEN
Let ï
¨
/
be the first integer with
Bvu"
(¢
ù
Ð
)
¥¤¦
û
u
ƒe²
Á
"
ò
ò
ž
B
¢
0
ž
P
¦
¥
j ’s are getting smaller when i ’s are getting larger.) Then no
is fixed from (2.3)
and
BIu
) û u
û u
È1w x õ . Let
can satisfy ù 0
; hence by (4.8), all ù
are panels, i.e.,
u
e "
û
e
û
Since the clusters of
satisfies
‰
)
pAù
û
"
‰
"
„
jŠ å
"%g]
å
e „
`
û ƒ e „nHAHLHL„
e
û
e
û
)
u
û
P
to
e
û
ebw
with ù
u
u ²!ƒ
j u
u e ²!ƒ „
û
there is no ù
e ‡„G„
û
)
pLù
u
û
3xÌg]Ky ‰
u
ù•u
b
P
j
P
)
h
û
û
û_e
)
û
0
h
Å
e
û
0
ï
®%4rjÁ
ï
h
e ®
j
jŠ å
",
*'•~}
E
2
)
ù
ï or ù
Thus, is a covering of . If ù
, then
j for some i
)
ù is admissible. In the latter case ù
w x õ holds. Therefore, all ù
'
û
w.r.t. . The number of clusters u in²‡ƒ is estimated in u
u
h
ý!z{|
b
„
jŠ å
*'•T"M
ø
û
)
prù
)
e C
û
‰
prù
b
e )
pLù
"
u
u ²‡ƒ
û
are not disjoint, we restrict
û
û
u
0Ì¥
Å
j 0
jŠ å
4
2 ò
. In the former case
w x õ are admissible
S
û
Á
E
ž
4
2 ò
jŠ å
E
4
4
2 ò
û
0
E
ž
h
Å
u
û
u
û
)
P
is the first integer
such that the following relation is true,
ƒe² u
Á
ò
ž
"-
¢
0
Á
u
²
0
¦
¥
Á
Then
ïÌ01Z :B\ E
"
Z :B\ E
¥
¦ ò
2
€ ¥
¡
Now we proof the case of
¦
i
C '&
ý
®
‚
Ÿ
iAC
¦
ý
#
'&
C&
' Z :B\ E
2
ý
"
W
¥¤¦ ò
Á
¢
4
P
(Use (4.7))
X Z :B\
X
Ác®
ò
E
Y _r_ P
. Let
®
¨
ï
ý
¦ ò
Y
T
j
ƒ
‚
ý
¡
" òY
Z :]\ E
iAC
Ë
k
„' "M9
C3
„' ¥
4 ý 01Z :]\ E€
¢
"-
¢
ò¥
i
Ÿ
Ÿ
iAC P
„'
û
. Let
Ÿ
0
Note that *'` @
be the) admissible covering w.r.t. as
constructed before.
¨
K
'
„' ÷
ù
ù
ù
Since dist
dist
for
all
,
is
also
admissible
w.r.t.
.
Thus, the estimate
'+)
(4.10) is true for all
.
T HEOREM
4.5. Let Y be the numberû of
panel and ò be the ratio. Assume (4.6). The
*'•
e *'y
number «
of non-admissible panelsº5… in
is bounded by the inequality:
(4.11)
«
e *'•
E
4
01¥
2 ò
4
for all
')+
P
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etna@mcs.kent.edu
237
FAST SOLUTION OF THE RADIOSITY EQUATION
'
coverings
w.r.t. e have
the
Proof. Proposition
3.9(b) of [12] states that allû admissible
'y
e *'•
û *'•
« *'•
w x õ . Then,
same number «
of non-admissible
panels.
Let
be
has
non'
b
admissible
panels w.r.t. .
BN"ÇÁ Ð
)
*'y
'
p
¢ ò . For any q
w x õ and q
È
, it is admissible w.r.t. . This is
Let
@
because that
C&
' for q
È
p
b
@
*'•
ACa¨CBÛ
q
¢
and ¢ is defined in (2.2). Now,
C†'&
ò
Thus, such q
Ÿ
Ÿ
ACa¨
q
*B[
ò
i"‚Á
¢
e "
)
pzq
h
*'y
@
qîÈ
¨
¢
û-e
0
b
C
wx]õ
¨
ò¢
« e *'y
is admissible by (3.3). Therefore,
û
¢
q
P
u
where
P
With (4.6) and (2.4), we have
«
E
e *'yiH ¢
¥
Å
§
0
ˆ‡
§
¡
T"
q
„
¡
p
)
e û
u
0#¡
b
Thus,
«
e 'y
0
B E
¢
E
¥¤§
"
¢
where
º
Á
H
¥¤§
E
2
º
¢
E
"
ò
4
*'•-]fi
@
¥
B E
P
E
4
2 ò
0
4
"
¥
ž
¥¤§
P
5. The experimental problems. For , we use the
following
5-piece
surface. It consists
ƒ E , ° , O ,
of four squares
and
one
rectangular
piece,
labeled
as
,
.
In
particular,
'Q(
^
‰ `ƒ˜" F J
/ Ÿ
¹ F / Ÿ J in the
-plane,
facing
upward;
Š
Š
9t"Œ4
‰ E
J ¹ F/
J in the plane
and
;
° are the bottom and top, respectively, of F /
"
@
9
%
"
Á
‰
O
F / ¥ J ¹ F / ¥ J in the plane
,
facing
downward;
"
' ‹9¤£
'
9
4
ƒv
‰ E
using the side that is visible from
p
Ÿ
/@0
0̟
/›0
0
u
^
and O .
‹Š&
We choose the parameters Ÿ
¥ to satisfy
(5.1)
¥
2%Š&
ÁˆŠá2
Ÿ
P
Š&
G"
This
surface is illustrated in Figure 5.1. In all of our examples, we use Ÿ
¥
DŒaÁL4r
.
A surprising number of phenomena can be studied with the use of this quite simple
surface. ƒ
‰
contains ‘shadow lines’.
More precisely,
has shadow
lines Ralong
the boundRÁˆŠÊ
ÁŠ*
ÁŠ
ÁˆŠ
F/
F/
J ¹
J ; cf. Figure.
aries of the squares F /
¥ J ¹
¥ J and F /
5.2. Therefore one can study the effects of ‘discontinuity meshing’ (cf. [16]). For
an analysis of this with examples for both piecewise constant and piecewise linear
collocation; see [6].
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etna@mcs.kent.edu
238
K. ATKINSON AND D. CHIEN
z
(0,C,2)
(0,B,1)
(A,A,1)
(A,A,0)
x
F IG . 5.1. The 5-piece surface z
(0,C,2)
(0,B,1)
(A,A,1)
(A,A,0)
x
‰
F IG . 5.2. The 5-piece surface with shadow lines showing
is an occluded surface. For example,
° cannot be seen from any portion of either
ƒ
or , and only portions of
can be seen from O . Therefore, there will be
^
visibility lines cutting through a number of elements.
‰
There will
usually
be singular behaviour in the radiosity solution along the common
ƒ
edge of
and . See [18] for an analysis of this
singular behaviour. There
is also
ƒZ]
^
singular behaviour associated with vertices of , as with the endpoints of
;
^
cf. [14].
ƒ
‰
For collocation points near to the common edge of
and , the collocation in^
tegrals are nearly singular for those triangular elements near to the collocation point,
ƒ
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239
FAST SOLUTION OF THE RADIOSITY EQUATION
but on the opposite sub-surface. Examples are given in [6] along with a numerical
scheme for the accurate and efficient evaluation of such integrals. Analytic methods
for evaluating these integrals are discussed at length in [19], [20].
We could use more complicated and physically realistic surfaces, but our surface is sufficient for studying the principal mathematical questions we have in regard to the cost of the
clustering method. To use a physically realistic surface would have required constructing a
visibility complex for the surface, a nontrivial task that would have greatly complicated our
programming without adding any additional insight on the questions we are studying.
5.1. Test solutions. We have $ carried out tests for a variety of true solutions . In these
cases, we calculate the emissivity using highly accurate numerical
integration. Then the
collocation procedure is applied to find the approximate solution x which is then compared
to the known true solution . In this paper we report
on onlyŒtwo cases.
Õ3"64B
‰
is constant over each sub-surface s ,
. The collocation error is zero
PAPLP
and thus we have the exact solution of the discretized linear system (2.9). We can
compare it to the solution obtained using the clustering method, thus measuring the
effect of various approximations involvedKƒ in clustering.
‰
The significance of the shadow lines in
is that the first partial derivative of is
often discontinuous when the derivative is in a direction perpendicular
to the shadow
Ä
line. This affects the accuracy of the approximation ¼¾x
. To study this phenomena, we use the following true solution :
(5.2)
'
(
’”“ '`‹(–•

'`‹(y9<g"Ž
¹š™


E
"
®
#
Ÿ
ÁŠŒ
.'•
"
ã
#
°˜—
Ÿ
ÁˆŠŒ
+(
'`0(•9<©) ƒ
㜛˜
*' 0(•‹9¤)+
4]


/

’”“
„f
°
O
^

’ “ *'`‹(aT"Mž
„f
*' 0(•‹9¤)f
‘
E
²
“
â EŸ
²
ä
, â E‹Ÿ
² Aä
P
'`‹(y
The function
is used toRÁˆdecrease
the size of away from the shadow
/
Š
RÁˆŠ
¨
F/
J ¹ F/
J . The quantity
line
on
the
boundary
of
to
if
ã is equal
2
¨
‘
"
4
¶
¶
and it equals zero if
, ¶ we have
/ . The exponent ¡
/
/ . With ¡
¶
ƒ first
a continuous nonlinear function which has bounded,
but some
discontinuous
RÁˆŠ
RÁˆŠ
"
J ¹ F/
J . With ¡
order partial derivatives along the boundary of F /
E , we
have an algebraic singularity along this boundary, and we can study the effects
of
U
/ on
different
types
of
triangulations
for
such
a
solution
function.
We
choose
ƒ
RÁˆŠ
ÁŠ
F/
F/
J ¹
J . We have
both
and on the subset of
outside of the square
ƒ
^
also forced the function to be non-symmetric over , to lessen any special effects
associated with
the¬A­ symmetry
of the various
sub-surfaces.
U%ö£¢
4
"Q4
­
In all cases
we
use
Y
Y
0
.
Choosing
does not present a problem, as we still
m
2,4
have é é
due to not being a closed surface. The cases we present here are sufficient
for illustrating the fast matrix-vector multiplication method of this paper. Additional studies
on the accuracy of piecewise constant and piecewise linear collocation are given in [6].
5.2. Computing. We use the basic framework of [1, o 5.1]; and to implement it we use
the software given in [2], with the addition of codes to implement the fast matrix-vector
multiplication described in o 3.1.
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240
K. ATKINSON AND D. CHIEN
The timings were done on a Dell PowerEdge 6650 workstation. This computer has four
2.0Ghz/2MB Cache Xeon processors with 12 GB RAM. The computer was networked, but
otherwise was restricted to only the given program being timed, except that we might run the
program for up to three data sets simultaneously. Based on the clustering method described
in o 3, three lists of triangles are created: far field, gray field, and near field. These lists are
very large, along with the lists of integrals for the gray and near fields. Our computer RAM
memory cannot hold all of them, so a cache scheme is used to store the lists for rapid retrieval
during the iteration phase.
6. Experimental results. The central questions
involve the cost of our method for the
Æ
radiosity equation on an occluded
surface.
We
begin
by
looking at the number of quantities
Æ
« '
in the near field, gray field, and
far
field.
Let
denote
the number of admissable clusters
in the far field for the point , and
Æ
"̐’?“
V
«
>
« '
P
Æ
Define «m¤ and «m¥ similarly, for the near field and the gray field as
cation nodes. From Theorems 4.4 and 4.5,
«
(6.1)
V
V ò
0Ì¥
«m¤
(6.2)
²m¦
0d¥
Ác®
F
Z :B\
ranges over the collo-
¦
ò
Y J
²¦
¤
ò
with Á constants ¥ V , ¥ ¤ that are independent of ò and Y . From (4.10) and (4.11) we know
§
; below we examine its size empirically. Note that these
formulas depend on ò , not on
0
the degree of the Taylor expansion used in approximating in the far field. In addition for a
uniform triangulation scheme (as we are using), it is straightforward to show that
«m¥
(6.3)
0Ì¥
¥
Y
P
The constant ¥ ¥ is independent of ò .
We give empirical results for (6.1)-(6.3) as both ò and Y vary, beginning with the gray
field. Table 6.1 contains results for «m¥ , and as we noted the results do not depend on ò .
TABLE 6.1
Values of ¨© for varying ª
Y
«
¥
10
2
40
6
160
14
640
28
2560
60
10240
118
40960
248
163840
508
In Figure 6.1, we give a log-log graph of Y vs. «¥ , together with a linear least squares fit to
determine experimentally the value of = in
«Q¥
4v¬
"%ö
Yb« P
4v¬Bí
"
Œa4v®
P
The value obtained with a fit to the values /@01Yn0
, which is consistent
ž</ is =
P
with the theoretically expected value of 0.5 from (6.3).
For the near field, we give values of «¤ for varying Y and ò in Table 6.2, and a log-log
graph of «m¤ vs. Y is given in Figure 6.2. The upper bound in (6.2) is independent of Y ; and
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241
FAST SOLUTION OF THE RADIOSITY EQUATION
7
6
G
log(σ )
5
4
3
2
1
0
0
2
4
6
log(n)
8
10
12
14
F IG . 6.1. Gray field values of ¨ˆ© and linear fit to log-log graph of values.
TABLE 6.2
Values of ¨ˆ¯ for varying ª and °
Y
"
ò
10
40
160
640
2560
10240
40960
163840
P
4rÁŒ
ò
«m¤
"
6
16
70
128
232
262
262
262
P/
¬BÁŒ
"
¬<ÁŒ
"
ò
6
24
78
300
508
922
1044
1044
"
P/
Ãó4rÁŒ
6
24
96
334
1206
2022
3700
4184
Ãa4zÁŒ
for the larger values of Y (with ò
and ò
), we have that «¤ is independent
P/
P/
of Y as well.
A table of values « V for the far field are given in Table 6.3. To look at the values
¦
graphically,
we consider the theoretical formula of (6.1). We graph the values of Z :]\ Y vs.
« V
ò
, as then the theory would say
¦
ò
(6.4)
«
Ä
V
Ä
Ác®
V Z]
: \ F
ò
®
§F
V
¥
Z :B\ ò
¥
¦
Y J
Z :]\
Y J
"*4
ŒŒ
and the graph should be linear. The graphical
results are shown in Figure 6.3 with §
P
§ "Á
rather than the theoretical value of
. The results do show straight lines, approximately,
for larger values of Y , and they are parallel for varying values of ò , as would be predicted by
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242
K. ATKINSON AND D. CHIEN
4
10
3
σ
N
10
2
10
η =.125
η =.0625
η =.03125
1
10
0
10
2
4
6
8
log(n)
10
F IG . 6.2. The near field values of ¨ ¯
12
14
for varying ª and °
TABLE 6.3
Values of ¨± for varying ª and °
Y
ò
"
10
40
160
640
2560
10240
40960
163840
P
V
«
4rÁŒ
ò
"
2
8
60
248
670
1116
1824
2598
P/
¬BÁŒ
0
8
32
256
972
2686
4464
7242
ò
"
P/
Ãó4rÁŒ
0
0
32
128
1034
3860
10878
17754
ŒŒ
",4
(6.4). To" determine
the experimental value of §
, we observed the variation of « V vs.
P
4I¬]í
ò for Y
žB/ and we attempted to fit it based on the Hackbush model of (6.1) with a
",4 ŒŒ
suitable value of § . The results, using a log-log scale, are shown in Figure 6.4 for §
.
P
and there is close agreement for smaller values of ò .
6.1. Testing
the program. By using a solution that is constant or linear over each
sub-surface s , the true collocation solution !x equals , and the true error in the numerical
method based on o 3.1 satisfies
(6.5)
•x
"
yx
yx P
This allows testing the effects of the other sources of error in the numerical method. In this
case the remaining error in solving the radiosity equation using the collocation method with
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243
FAST SOLUTION OF THE RADIOSITY EQUATION
120
η =.125
η =.0625
η =.03125
100
γ
ησ
F
80
60
40
20
0
2
4
6
8
log(n)
10
12
14
F IG . 6.3. The number of elements in the far field, normalized by °I² .
b
b method of o 3 is due to the following.
the fast matrix-vector multiplication
Ä
‰
The approximation x
x .
‰
Numerical integration errors for the near field and the gray field collocation integrals
(cf. (2.9))
Æ
(6.6)
‰
&
Æ
R
§–
Íg| –Ò Ó
a<
P
Integration errors in computing the right hand emissivity function
(6.7)
$&'T"
'
&
*)
for the given known $solution . Of course, in actual graphics computations this is
not a difficulty since is given explicitly.
The numerical integration errors for (6.6)
and (6.7) can be quite
difficult "%
to Õcontrol, especially
6)D
p
when integrating over a sub-surface
, and near to a
s with the field point
| , }
common edge joining s and | . The numerical integration methods used here are discussed
in [6].
To illustrate the effects of these possible errors for our numerical method, we used the
true solution
(6.8)
'`0(•9<¤)+ ƒ 4]
'`0(•9<g"&³
/
'`0(•9<¤)+
E
„3
°
„f
O
„f
^
P
In order to study the effect of the fast matrix-vector multiplication method of o 3.1, the integrations of (6.6) and (6.7) were carried out to very high accuracy. Thus the only source of
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244
K. ATKINSON AND D. CHIEN
5
10
Far field σF
Hackbusch model
4
σ
F
10
3
10
2
10 −2
10
−1
0
10
η
10
F IG . 6.4. The model (6.1) with ´µ.¶˜· ¸˜¸I¹ and its comparison with the graph of ° vs. ¨± .
The errors ºœ¯
¬
žB/
Á Œˆ¬
4
/
Á
$ ¤
¢B?žIBh"
¢B?AžvBh"MÁ
Y
žB/
4I¬
TABLE 6.4
for (6.8) with °»µ½¼I· ¸
¬ Œ
P ž
à î
P
Á Ì
/
Á
/
Á
žB/
® ¬
žB/
/
4v¬]í
Á
Á
ž</
E
E
E
P
PÀ¾¾ E
­a4 E
P
­a4 E
P
­BÁ E
P
Ã
Ã
Ã
Ã
Ã
Ã
Ã
Ã
Ã
Ã
Ã
Ã
/
/
/
/
/
/
4v¬
4
/vž
P
P
Á]Á
P
4
P
4LÃ
P
4v­
P
P
4v®
4v®
¾
E
E
E
E
E
E
E
¢BzAžIBh"M¬
ž
Á
/vž
/vž
Ã
/vž
/vž
4
Pž
Á
Œ
P¿¾
Ψ
Œ P Á­
Œ PÁ
Œ P Á­¾
Œ P Á­
/vž
/vž
/vž
P
b
Œ
E
E
E
E
E
E
E
/
Œ
/
Œ
/
Œ
/
Œ
/
Œ
/
Œ
Ä
/
b
error in the quantity !x
•x of (6.5) is the approximation
x
x , which depends on
both
the Taylor order ± and the far-field parameter ò . Tables 6.4-6.6 give
the maximum for
"MÁ ¬
. Because the
•x
•x at the collocation nodes for three values of ò and for orders ±
ž
'`0(a
region is symmetric in "á4]and
the
solution
is
also
symmetric
(being
constant),
the numerÃaŒ
"QÁa ‹¬
ical results for orders ±
are the same as those for orders ±
, respectively.
ž
In the tables,
Æ
Æ
(6.9)
$
"
Æ
Š
Æ
¤
C4
É
Ã
’?“
ƒ
{¿Á¿Á¿Á { °x
C
yx
yx
'
AC
with p
0
0
Yu the interpolation nodes for the triangulation and ‡x the true collocation solution of (6.8). This type of example gives some idea of the error to be expected when
dealing with an unknown function .
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FAST SOLUTION OF THE RADIOSITY EQUATION
The errors º_¯
žB/
4I¬
/
¬
žB/
Á Œ¬
4
Á
/
/
žB/
®¬
žB/
4I¬]í
P
4
E
E
E
Á
¾
4 Áˆ®
/
žB/
P
P
/vž
/vž
žB/
4I¬
/
Á
/
/
žB/
®¬
žB/
4I¬]í
/
žB/
¢B?AžIB"Ĭ
® ­­ Ý4I¬
E Ý4Œ
P
Á ­
P ¾ E
4
­ ®
P¿¾ E /
Á ®
®
P / E /
à Ãó4
®
E
/ ®
P
ž P žBž E /
Œ Ãó4 ®
E
P
/
TABLE 6.6
for (6.8) with °3µg¼I· ¼˜ÅI¶0ظ
$ ¤
¢B?žIBh"
ž
® ­­ Ý4v¬
E
P
Á ­
Ý4IŒ
P ¾ E
Á ¬­
Ý4IŒ
¢B?žIBh"MÁ
® ­­ Ý4v¬
E Ý4IŒ
P
Á ­
P ¾ E
Á ¬­
Ý4IŒ
E
®
ΠP
PÀ¾ / E /
à 4
¬
P ž E /
4 4]4
Œ
E
/ ¬
P
Œ
¾ P ¾ E /
Y
4
$ ¤
¢B?žIBh"
ž
® ­­ Ý4v¬
E
P
Á ­
Ý4IŒ
P ¾ E
4 ®­
E / ¾
­ P ¬ P/ E / ¾
Œˆ­ ¾ P Œ E
/ ¾
¾ P ¬ ¾ E
/ ¾
¾P / E / ¾
/vž
The errors º ¯
¬
žB/
Á Œ¬
TABLE 6.5
for (6.8) with °»µÂ¼I·¿¶0ظ
¢B?žIBh"MÁ
® ­­ Ý4v¬
E Ý4IŒ
P
Á ­
P ¾ E
Π4
Œ
P ¾ E /
4 ]
­ à E /vž
P
4 ÁŒ
Y
245
¢B?AžIB"Ĭ
® ­­ Ý4I¬
E Ý4Œ
P
Á ­
P ¾ E
Á ¬­
Ý4Œ
E
¬ P ¬­ Ý4Œ
E Ý4rÁ
E
¬¾P ¬ ® Ý4
E /®
P
à 4
E / ®
/
P
Á ­ ® E /
P
P
Ã]Ã
E Ý4rÃ
¾ Ã E Ý4rÃ
Œ P¿Œ]
E
¬ P ­¬ Ý4rÃ
E
P
Ã
P
4
b
We want b to illustrate
and confirm experimentally the result (4.4) in Theorem 4.1, looking
jÞ
at the error
x
x
x when it is regarded as a function of the parameter ò for various
"
4I¬]í
"
Á
± P Let Y
žB/ , and consider using the orders ±
ž
values of the
%Taylor
order
4›"
4]Ã<
(degrees ±
for our Taylor expansion in the far field. The numerical results are
given in Table 6.7, and empirically they imply
(6.10)
é
Þ
x
Þ
From Theorem 4.1, fixing both Y
expansion,
(6.11)
é
"
x é
W
ð
•
ò
"‚Áa
±
4
the number of elements, and ±
b
x
b
Æ
x
jÞ
x`é
"
jÁ
W
ð
ò
ž P
, the degree of the Taylor
P
When we combine (4.4) and (6.10), we have that the theoretical rates of convergence to zero
x agree with the experimental results given in Table 6.7 and (6.10).
of x
In order forÁ (6.11)
to be
consistent
with the earlier bound (3.11), the value of ò must be
"
chosen so that ò
W
 . This results in a relation between ò , Y , , and ± ,
ò
"
n
²
W"LY
E
#
S
cf. [12, Lemma 5.2]. Our experimental results imply that it is probably better to experiment
with various choices of ± and ò for cases in which the solution is known. Then choose values
of ± and ò so as to give sufficient accuracy while also minimizing the operations cost. We
further illustrate this in the following subsection.
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246
K. ATKINSON AND D. CHIEN
TABLE 6.7
The error (6.5) as a function of °
¢B?AžvBh"MÁ
ò
/ P
/ P
/ P
/ P/
/ P/
é
Þ
Œ
x
Þ
Ratio
x é
Ã
c
­<Á
Œ ­®
P
4 Á®
Á
¢B?AžIBÝ"
E
E
ž
E tžŒ
P
Ã
Ã
P/ E
Œ
8¬
¾ P ¾ E
ÁŒ
4zÁŒ
¬BÁŒ
Ãa4rÁŒ
P
ž P
Œ
Áˆ¬
Ã
x
4I®
Þ
x é
E tžŒ
4 Ã
P ž E
¬
¾ P ¬/® E 8­¾
ž P
E 8®
Á ­®
E
P
®
ž PÀ¾
ž P
é
Þ
¾
ž P /]/
P
ž
Ratio
­
ÁvÃ
4
4v¬
4v¬
P
¬
¾P Á
P
P
Á
−2
10
−3
10
−4
E
n
10
−5
10
−6
10
−7
10
1
2
10
3
10
4
10
5
10
n
10
6
10
F IG . 6.5. Maximum errors in linear collocation solution.
6.2. Convergence results. We consider the numerical
solution
of the radiosity
equation
"
4BˆÇ#"
4 Œ
"
4
when the true solution is chosen as in (5.2) with ¡
/ P . We have
P , and chosen a smaller value of in order to accelerate the convergence of the iteration in (2.11);
but the average cost of the individual iterates is not affected by this choice of . We begin
with the uniform triangulation of the surface. leading to the number Y of triangles
as listed in
ƒ
the preceding Tables 6.1-6.3. With this triangulation of the sub-surface , the shadow lines
of Figure 5.2 will intersect with some of the triangular elements. As in (6.9), we evaluate the
error at the collocation nodes; define
Æ
Æ
$
¤
"
Æ
Š
›’v“
ƒ
{¿Á¿Á¿Á { °x
C
'
•x
'
AC
P
These maximum errors are given in Table 6.8, and they are illustrated graphically in Figure
6.5. Note
that for a given number Y of elements, the order of the approximating linear system
Ã
equals Y .
Although the ratios by which the errors decrease show a great deal of variation, the
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247
FAST SOLUTION OF THE RADIOSITY EQUATION
TABLE 6.8
Maximum errors in linear collocation solution
$
Y
4
žB/
4I¬
¬
žB/
Á Œ¬
Á
/
P
4
­
P/
Ã
/
P/
­
Á­
žB/
ž P
4 B
à Ã
P
žB/
ž P
® ¬
žB/
/
4I¬]í
­
­ØÉ
¢
cÃ
E tŒ
E
E tžŒ
E 8¬
E 8¬
E
E ¾
/
® P Œ]Ã
/
4
¤
Ã
ž
¬
­P ­
/ P
à Œ 4
4LÃ
P
Á
¾P /
à Á]Á
P
Á
Œ
PÀ¾
TABLE 6.9
Errors for °Fµ½¼I· ¸
¢B?žIBh"Œ4
Y
4
ž</
4I¬
¬
žB/
Á Œ¬
4
/
Á
/
4
4
žB/
®¬
žB/
4I¬]í
4
/
/
žB/
4
4
í
E
P ¾ E
4
Pž E
P žB/ E
4
Pž E
4
Pž E
4
Pž E
P
4
$ ¤
¢B?žIB"%Ã
Ã
4
/
4
Á
Ã
Á
Ã
Á
Ã
Á
Ã
Á
PÀ¾
Ã
Á
®
/
/
/
/
/
/
Ãa4
P
Ã
E
E
E
E
E
E
E
4
P/
¨
P
¬
P
P
P
®
®¾
/
ž
Ã
/
Iteration
time (seconds)
¢B?žIBh"dÃ
¢B?AžIB"fŒ
¢B?AžIB"Č
4
Ã
E
E
P ž E
ÃBÃ
E
P
4
P¿¾ E
­
P / E
­¬ E
P
/
® P¬
P ¾
4 ¬
/vž
/vž
Á
/vž
Á
/vž
Á
/vž
Á
/vž
¢B?AžvB".4
Ã
/
/
Á
/
4rÁŒ
P /B/
Œ
P/
/]ž
P/
/]ž
P
/]ž
¬¬
Œ
Ã
4v®
/]ž
4]4
/]ž
P /]/
P/
žB/
P/
P/
Á
Pž
/ P
P /]/
ŒŒ
¬ ¾ ­ Œ
P /
Á a
Œ 4v¬
P
4I¬ ­
/
¾ Œ/
Pž
ÁŒ
4
Á
¬
4zÁ4
ž
P
P
Œ
P
ÃAŒ
Á]Á
/
4LÃa4Œ
ÀP ¾ / v
¾ /
4 ¬­
ž P
Á
­AŒ
¾ P
4LÃBÃ ­
P
overall
trend
is more uniform, as is indicated in Figure 6.5. If we use the values of
¨š¬
Y
ž</ we obtain an empirical rate of convergence of
$
with È
".
P 4
P /]/
¾
"
. From (2.1)-(2.4), Y
$
W
W
x
x
for
²
"
x
²!ƒ
$
X Y
E
X 
Ä
W
_
, and this leads to
_
E
X 
_
P
This is the expected rate of convergence
for a linear collocation method, even though is not
ƒ
continuously differentiable
over
.
For
a further discussion, see [6].
Þ
Þ
The error é
x é varies with Y , ò , and ± . To give some insight, we give values in
Æ
Æ
Tables 6.9-6.11 for
Æ
$
¤
"
Š
’?“
ƒ
{¿Á¿Á¿Á { °x
C
'
x
AC
for selected values of the far field parameter ò and the Taylor order ± . We also give the
average time (in seconds) for each iteration" (as Ãó
described
in o 2.3). "îÃ
4rÁŒ
%4"ÏÁB
The case with parameter values of ò
and
order
(degree ±
/P
±
"
4v¬Bí
Ã
give completely sufficient accuracy up through Y
gave
ž</ . Using an order ±
Ë
essentially no additional4 accuracy.
For
smaller
values
of
Y , we can use larger values of ò .
Á
"
¬BÁŒ
For example, with Yd0
,
/
žB/ , we can have essentially the same accuracy with ò
P/
although we do not give those values here for reasons of space.
To look at the cost, we show in Figure 6.6 the average iteration time
as a function of ò ,
".4v¬Bí
žB/ , and there
for varying values of Taylor approximation order ± . This is done for Y U
S
É ò ±
Y
are similar graphs for smaller values of Y . Experimentally, the time É
satisfies
(6.12)
É
ÄdövÊ
´Ë
±
Y
²
ò
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248
K. ATKINSON AND D. CHIEN
TABLE 6.10
Errors for °FµÂ¼I·¿¶0ظ
¢B?AžvBh".4
Y
4
žB/
4I¬
/
¬
žB/
Á Œˆ¬
4
/
Á
4]4
¾P 4 ž
®¬
¾P 4 ž
/
¾P ž
ž</
Œ
Á
P/
¾P 4
/
/
P/
4
žB/
žB/
4v¬]í
Ã
P
4
$ ¤
¢B?žIB"%Ã
Ã
E
E
E
E
E
E
E
4
/
Ã
/
® P ŒvÃ
/vž
P
4
/vž
Œ
Ã
/
­
P/
4]4
­ P Œˆ­
Œ
4
Œ
4
/
Œ
/
/
P
®
P/
Œ
P/
Ã
E
E
E
E
E
E
E
4
/
/vž
Œ
¬
Œ
/
Œ
/
¢B?AžIBÝ".4
Ã
E
E
P
4
­ P/ E
à 4
/ E
­ P
¬ ¬ E
P
4 4
P / E
4
P/ ¾ E
/
/
Ã
/
® P Œ]Ã
Œ
/
Iteration
time (seconds)
¢B?žIBh"dÃ
¢B?žIB"Č
¢BzAžIBh"Č
/
P /B/
Œ
P
/
Œ
Á
¬
P
4
/
¾
4
P /]ž
Ã
/]ž
/
Ã
¾
ž]ž</
4]4
Œ
Œ
¬AŒ¾
/
ÁŒ®
¾
Á Ã
P
/
Π4 P]
Á Ã
P
/
P /]/]žB/
Á
/
P /vž
Ã
Ã]Ã
P
P
¾
¾
4v®
Œ­ ˆ
Œ ¬
P
Á ®
Œ
®
P
4zÁ
ž P
Ã
P /]/
P /vž
¾
Ã
¾
Á
¾ /Œ
Á ®
P /
4Œ Œ
P /
¬
¾Ãž¬ P / ¬
Pž
ž P
TABLE 6.11
Errors for °3µg¼I· ¼˜Ì˜Ã˜¸
¢B?AžvBh".4
Y
4
žB/
4I¬
/
¬
žB/
Á Œˆ¬
4
/
Á
4
Ã
/
žB/
®¬
žB/
4v¬]í
Ã
ž
/
ž
ž</
ž
E
E
P
­ P/ E
4]4
E
P
Ã
P ¾ E
P ž]ž E
Œ Pž E
/
® P ŒvÃ
$ ¤
¢B?žIB"%Ã
Ã
4
/
/
Œ
/
¬
/
¬
/
¬
P
4
/vž
/
Ã
/
® P ŒvÃ
Œ
­
Ã
P/
­
P/
Áˆ­
ž P
4 ]
à Ã
P
ž P
­
ž
E
E
E
E
E
E
E
"*4
Ã
4
/
/
Œ
­
Á ­
P/
ž P
4 B
à Ã
P
¬
/
­
P/
Ã
¬
/
P
4
/
/
Ã
/
® P Œ]Ã
Œ
/vž
Iteration
time (seconds)
¢B?žIBh"dÃ
¢B?žIB"Č
¢BzAžIBh"Č
¾
ž P
­
ž
E
E
E
E
E
E
E
¢B?AžIBÝ".4
Ã
/
P /B/vž</
Œˆ®BÃ
P/
­ó4
ž</
P
Œ
/
/]ž
Œ
/
4
¬
/
­
ÃBÃ
¾
/
¬BÃ
¾Á P Œ
¬BÁB
¬
/
Œ
/ P/
/
P
4
P /]/]žB/
¬ Ã
/
P/
­ ®]Ã
P /
4
4 Œ
I
/
®® P Ã
P ž
­
¾ Þ ®<ÁvP¿Ã ¾
Ã
P /]/]ž
¬ 4
ó
/
P/
­B
Á ŒvÃ
P
4
/ PÀ¾
4B4rÁ Á
­®a4 P Á
P
ž
­®
4
ž
­­
with the experimental value È P P . This is consistent with the sizes for « V and «m¤ given
in (6.1) and (6.2), and of course, «m¥ depends on only Y and is independent of both ± and
increases, which is to be
ò . In the graph we also see that the time increases as the order ±
expected. The large increases in calculation time are associated with decreasing the size of ò
rather than with increasing the order ± .
In Figure 6.7 we show the growth of time as a function of Y for varying values of ò . From
this log-log graph it is clear
thatÃathat
the rate of increase is a power law in Y for larger values
"
4rÁŒ
of Y . Using the case of ò
/ P/
, we obtain that
(6.13)
É
"Q4
4I®
Ăö
x ³
ʸ ë
ò
±
Y
ü
with À P P ƒ . This is better than the worst case scenario of (3.12) which predicted an overall
cost of W X Y Á ^ _ . For our values of Y and ò , it appears that the cost arising from treating the
far field and near field is much larger and more significant than is that for the gray field.
Summary. We have examined the cost of the clustering method of Hackbusch and Nowak
[12], both analytically and empirically. The empirical results agree with the form of the
theoretical growth formulas given in o 4, while experimentally these formulas have somewhat
smaller exponents as regards the cost; this is illustrated in ( 6.12) and (6.13). The experimental
timings also show that it is important to determine adequate values of ± and ò , for example,
by proceeding as we did in subsections 6.1 and 6.2 with known true solutions. Too small a
value of ò or too large a value of ± will lead to unnecessary calculation and greatly increased
computation times, as is illustrated in Tables 6.9-6.11.
The clustering method allows the setup and solution of the discretized radiosity equation
with a very large number of elements. In our examples, the number of equations to be set up
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FAST SOLUTION OF THE RADIOSITY EQUATION
4
time
10
order=1
order=2
order=3
order=4
order=5
order=6
3
10
2
10 −2
10
−1
0
10
η
10
F IG . 6.6. Time as a function of ° , for varying values of Taylor approximation order.
4v¬Bí
Ã
žB/ in our examples,
and solved
is Y , with Y the number of elements being as large as
®ó4IŒBÁ
/ equations. We believe this illustrates the potential importance of this method.
i.e. ž
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ETNA
Kent State University
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250
K. ATKINSON AND D. CHIEN
4
10
3
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2
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1
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10
0
10
η =.5
η =.25
η =.125
η =.0625
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asymptotic
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n
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