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ISPRS Commission III, Vol.34, Part 3A »Photogrammetric Computer Vision“, Graz, 2002 
L(x,0,,9,)= L(x.0,.9,) 
+ [1,(<.0,0,6,.,)L,(x,0,p)cosedæ (1) 
Q 
Notations and definitions used here are those given by Sillion 
and Puech (1994). Additional information on transfer in global 
illumination can be found in Arvo (1993). 
e  L(x,0,0), L(x06,¢,) and L;(x, 6,@) are respectively the 
radiance leaving point x in the viewing direction 
(0,,,), the emitted radiance by point x in the same 
direction, and the incident radiance impinging the 
point x from direction (0, q). 
e CQ is the set of directions (0,9) in the hemisphere 
covering the surface at point x. 
e 1,090,090) is the bidirectional reflectance 
distribution function (BDRF) describing the reflective 
properties of point x (Nicodemus et al., 1977). 
Both emitted and incident radiance, the BRDF, and the 
response g of the instrument depend on the wavelength A. 
BRDF models exist in the visible range (Rusinkiewicz, 1997). 
They usually depend on about five parameters that can be found 
in databases such as Columbia-Utrecht (2001). Both models and 
databases are nor assessed in the infrared range. For the sake of 
concision, we assume lambertian reflectors and emitters, and the 
isotropy of the BRDF. Considering fluxes G, E, and B; instead 
of radiance L, L, and L;, and expressing the environment of the 
object i, the previous equation becomes: 
G,,,, 7 Ja(0E AA Mr, edo, 008,04A — 
F;; is the form-factor and represents the rate of energy exchange 
between object i and object j. It can be computed (Schröder ef 
al., 1993) or estimated (Wallace et al., 1989). Due to spectral 
quantities and integral equations, the radiosity method (Foley, 
1996; Watt, 2000) cannot be used easily, except in considering 
many systems of equations: one system per spectral sample in a 
given spectral range. Nevertheless, under useful simulation 
conditions, the number of neighbours N is generally small as 
well as the form-factor. The previous equation is solved with an 
iterative method, considering emission and reflection at order k. 
Equation (2) will be used to compute spectral radiance leaving 
objects. Radiosity method can be applied when equation (2) is 
integrated from zero to infinity, to compute energy flux balance 
impinging objects. 
2.2 The prediction of emitted flux and of the temperature 
The main difficulty related to equation (2) is to compute the E; 
terms that correspond to the self-emission of the object. They 
can be expressed as the product of the emissivity & of the object 
by the blackbody function LP at temperature 7,. For lambertian 
emitters, it comes: 
E(A)e n e, A)" (r..A) G) 
The surface temperature of the object has to be known to make 
the simulation. This temperature is governed by the heat 
equation, which can be written under thermodynamical 
conditions usually encountered in landscapes: 
oT 
—=K-AT 4 
eek (4) 
where K is the thermal diffusivity, and A is the Laplacian 
operator. This equation may be solved using e.g. the finite 
difference method or the method proposed by Deardorff (1978). 
Because of thermal inertial, the knowledge of this temperature 
at t requests the temperature at the previous moment (#-dt). The 
in-depth temperature of the object, and the energy flux balance 
at the surface of the object are the two boundary conditions 
required to solve a second-order differential equation. Given an 
initial state, an iterative process is harnessed to compute surface 
temperature at the instant of simulation. The interactions 
between physical parameters, and their variations in time are 
modelled. Methods exist to compute these interactions and their 
changes in time (Johnson, 1995; Jaloustre-Audouin et al, 
1997). Because it is iterative, and because of the complexity of 
models used for the energy flux balance determination, this 
process is by far the more demanding in computation time of all 
the simulation processes. 
3. METHODOLOGY USED FOR THE SYNTHESIS 
3.1 Specifications 
In the infrared range, the spectral flux coming from an object 
depends upon the meteorological conditions existing in the 
scene: surface temperature, air temperature, humidity of the 
object, humidity of the atmosphere... and the optical properties 
of the object itself in the considered spectral range. To perform 
a simulation, four types of inputs are provided to the simulator: 
e the 3-D landscape geometrical description. The 
landscape is expressed as a set of located and oriented 
objects, having their own surrounding. Each object is 
made of several facets, each of them made of the same 
material, 
e the set of the conditions of the simulation for the 
present time and past hours: day, time, place, sky 
cloudiness..., 
e a database of thermal and optical characteristics of all 
primary materials existing in the landscape (albedo, 
spectral reflectance (ASTER, 2000), specific heat, 
thermal conductivity, roughness, leaf area index for 
vegetation,...), 
e the spectral response of the sensor g. 
3.2 Main difficulties related to infrared range 
In short-wave radiation, computation is made on facets, which 
are flat external parts of an object. In thermal infrared range, the 
synthesis process requests that each point of a facet exhibits the 
same energy flux balance at instant t: this facet is homogeneous 
with respect to energy flux balance. Considering e.g. the 
shadow variations, this same facet will not be necessary 
homogeneous at £--dt, and so on. This introduces difficulties to 
compute temperatures. The smallest entities representing the 
scene may be considered, voxels for instance. With voxels, 
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