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nent is
approximated by planar polygons in object space. These de-
fine the scene geometry. Relevant parts of the natural envi-
ronment are those interacting with the planned building with
respect to hiding, shadowing and interreflection.
The world coordinates of a polygon vertex are determined
from its manually identified image coordinates in two or more
input images with different exterior orientation. This is the in-
tersection in space or the second inversion of the camera
model. The geometrical properties of a polygon are defined
by its coplanar vertices, while the non-geometrical properties
of a polygon are given by an image attribute and a material
attribute. The image attribute references one of the input
images. The material attribute is a reference into the material
library (Fig. 4). The referenced material description defines
all material properties except for the diffuse reflectance 0,7
which is retrieved at rendering time from one of the input
images (sections 4.3 and 6). The default input image for re-
trieving o4, is indicated by the image attribute.
The illumination geometry (section 3.3) is determined as fol-
lows: The solid angle £2, of incident sunlight is calculated
from the distance to the sun and the radius of the sun. The
direction d, to the sun is determined by a point on an object
and the corresponding point on the object's shadow by inter-
section in space. The direction d, to the zenith is determined
similarly, e.g. by two points on the same vertical edge of a
house. Since the world coordinate system may be selected
arbitrarily, the zenith direction is not known a priori. m, > 0
is set arbitrarily.
Details about resection and intersection in space are not de-
tailed here. These are basic procedures in photogrammetry
(Wolf, 1983) (Ruger, 1987).
42 Inversion of the Atmospheric Model
The first inversion of the atmospheric model involves
solving the equation of the atmospheric model (2) for the
spectral transmittance r, , of the atmosphere.
Given are the horizon color L, ; and the reference distance
ds = 1. L, is derived from a suitable RGB triplet which is
manually selected in one of the input images (appendix A).
pi; (i — 1..n, n = 2) are known locations in object space.
They are situated on the surfaces of objects of the same
opaque, diffusely reflecting material. The p; are manually
identified on previously determined polygons in object space
(section 4.1). The known apparent object color L;; is the ra-
diance of the light reaching the camera associated with one
of the input images at the known distance d; from p;. L;, is
derived from the RGB triplet of the reconstructed image func-
tion at p,' (appendix A). p;' are the image coordinates of p;
in the selected input image.
By inserting into (2), each sample contributes one equation
Lo; ti + Lei ( 4. — 1,14 ) = Lii = 0 (7)
to a system of n non-linear equations which is redundant for
n » 2. For each wavelength within the visible spectrum, the
system of equations (7) is iteratively solved for the unknowns
which are the true object color Lj; and the spectral transmit-
tance r,, of the atmosphere. L,; is of no interest since it ex-
presses no general property of the scene. The iteration is
started with r,, = 1 and L,, — L;;, such that d; = min (d;,).
A simplifying assumption is that the radiance L,; of the light
leaving the objects towards the observer is the same at all
locations p;. The radiance of the diffusely reflected light is
uniformly distributed in all directions but it depends on the il-
lumination of the surface and the orientation of the surface
relative to the light sources.
Nakamae et al. (Nakamae, 1986) use an equivalent method
based on the same atmospheric model (2). Wavelength de-
pendency is not considered and world coordinates are ob-
tained from topographical maps.
The second inversion of the atmospheric model involves
solving the equation of the atmospheric model (2) for the true
color L,, of an object. This inversion will also be used during
rendering (section 6).
Given are the horizon color L ,;, the spectral transmittance
7, of the atmosphere and the reference distance d, = 1.
L, ; and r,, are known from the first inversion of the atmo-
spheric model (see above). p is a known location in object
space. The known apparent object color L; is the radiance
of the light reaching the camera associated with one of the
input images at the known distance d from p. L, is derived
from the RGB triplet of the reconstructed image function at
p' (appendix A). p' are the image coordinates of p in the se-
lected input image.
Inserting into (2) and solving for L,; yields
Lu = (4; 7bs)fa =d Elan (8)
4.3 Inversion of the Illumination Model
The first inversion of the illumination model involves solv-
ing the equation of the illumination model (6) for the weights
k,, k, and k, of the three illumination components which
make up natural daylight (section 3.3).
Given are the scene geometry (section 4.1) and the illumina-
tion geometry d;, Q,, d, and m, (sections 3.3 and 4.1). L; is
the known relative spectral distribution function of the day-
light components. p; (i = 1..n, n = 3) are known locations
in object space. They are situated on the surfaces of objects
of the same opaque, diffusely reflecting material with materi-
al parameters k, = 1 and 0,7 Where 04; is known up to a
constant factor only. Each p; receives a different amount of il-
lumination by the natural daylight. The p; are manually identi-
fied on previously determined polygons in object space (sec-
tion 4.1). The known apparent object color L;, is the ra-
diance of the light reaching the camera associated with one
of the input images at the known distance d; from p;. L,, is
derived from the RGB triplet of the reconstructed image func-
tion at p;' (appendix A). p;' are the image coordinates of p;
in the selected input image. The true object color L,;; is de-
termined from L;, by the second inversion of the atmospher-
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