thetic
e pla-
lep, a
mina-
viron-
three-
adow-
it ren-
'ound
in the
nanu-
ected
of the
e the
1d re-
eters
rated
| onto
man-
back-
allows
)heric
topo-
1988)
d into
Janar
sional
ture
al ob-
jae et
d and
ough
ar po-
of the
s, are
or the
sional
rically
del is
sional
of the
raphs
n illu-
Jows.
al pa-
3. MODELS
Models serve to simplify the description of some aspects of
the physical world. This section introduces important radio-
metric quantities and models for the representation of light
sources, materials, geometrical objects and the camera as
well as models for the simulation of atmospheric effects and
illumination effects.
3.1 Radiometric Quantities
Irradiance E — dó / dF [Wm ?] at a point on a surface
element is the radiant power dd [W] incident on the surface
element, divided by the area dF [m 2] of the surface ele-
ment. Radiance L — d?4 / (cos0 dF dw) [Wm ?sr ^] at
a point on a surface element and in a given direction lis the
radiant power dó incident on the surface element or leaving
the surface element in a cone containing I, divided by the
solid angle dw [sr] of that cone and the projected area
cos dF = À - À dF ot the surface element. à is the unit sur-
face normal (Fig. 1). The corresponding spectral distribution
functions are given by E, = dE / dà [Wm ?] and
L, — dL / dA [Wm sr !]. E, and f, denote relative spec-
tral distribution functions (spectral distribution functions in an
arbitrary unit) (Judd, 1975). From the above definitions: :
dE = L cosOdw = Lii-ldo. (1)
3.2 The Atmospheric Model
Due to the scattering and absorption of light within the atmo-
sphere, an object which moves away from the observer
changes its color depending on the distance from the observ-
er, the current weather conditions and the pollution of the air.
The atmospheric effect is simulated by the atmospheric mod-
el (Schachter, 1983):
L; = Ly Tat Lab A Takt )- (2)
Ly, represents the true object color. This is the light which
leaves the surface of an object towards the observer. L; rep-
resents the apparent object color. This is the light which
reaches the observer at a distance d from the object. ai
represents the horizon color which is L; for d — o. t,,is
the spectral transmittance of the atmosphere. This is the
fraction of light energy which is neither absorbed nor scat-
tered away from the straight direction of light propagation
within the reference distance d,. The atmosphere described
by (2) is homogeneous and isotropic. Note that the term col-
or is used for simplicity though L,;, L, and L,, are color
stimuli in terms of spectral radiance.
3.3 The Light Source Model
Natural daylight is assumed to be composed of direct sun-
light, diffuse ambient light and diffuse skylight.
The direct sunlight (Fig. 2) is defined by the direction d, to
the light source (the sun) and by the solid angle $2, «€ 2z
and the radiance L,, — k, L; of the incident light. k, is a
weighting factor. By (1), the irradiance E,; of the sunlight in-
cident at a point p on a surface F is
Fig. 1 Definition of radiometric quantities.
2;
En = | Luk ldo =kiiyi 0. (3)
0
The diffuse ambient light (Fig. 2) is a uniform illumination
term. It is defined by the radiance L, = k, L, of the incident
light. k, is a weighting factor. By (1), the irradiance E, of
the ambient light incident at a point p on a surface F is
2x
Ea m | arido m o (4)
0
The diffuse skylight (Fig. 3) is sunlight scattered by the at-
mosphere. It is emitted by the sky hemisphere above the ho-
rizon. Because of its large radius, all objects are approxi-
mately in the center of the sky hemisphere. The skylight is
defined by the direction d, to the zenith which is the highest
point of the sky hemisphere, the parameter m, > 0 (see be-
low) and the radiance L; — k, L, of the incident light. kj is
a weighting factor. By (1), the irradiance E, of the skylight
incident at a point p on a surface F is
Q 4m2
Bu o | iidem kılı 300 kan (5)
0
i=1
Q x 2x represents the part of the sky hemisphere with
ñ -1 > 0 and d, 1 > 0 which is visible from p. The solid
angle 2 is subdivided into 4m? sky facets (m, facets in po-
lar direction x 4m, facets in azimuth direction) of equal size
Aw = 2x/(4m?) and the integral is approximated by a sum.
í, is the direction to the center of the i-th sky facet. The sum-
mation is done for all facets the centers of which are visible
from p. ó; — 1 if the center of the i-th sky facet is visible
from p, and ó; = 0 else.
The illumination geometry is defined by di, Q;, d, and m;.
Kaneda et al. (Kaneda, 1991) model natural daylight as be-
ing composed of direct sunlight and diffuse, non-uniformly in-
cident skylight. The contribution of the skylight is determined
by integrating over the visible parts of the sky dome which is
subdivided into band sources. Wavelength dependency as
well as absorption and scattering in a non-homogeneous at-
mosphere are taken into account. The discretization of distri-
buted light sources is described by Verbeck & Greenberg
(Verbeck, 1984).
435
