view factor matrix, is the geometric ratio of energy leaving element i that can reach the element j, E the radiance 
due to the direct ill umina tion vector which gives the energy directly incident «1 each element and p fie 
individual reflectance vector which gives the reflectance factor for each element, assumed lambertian in this 
preliminary study. What is unknown in this equation is then die luminance of element i, called radiosity By. The 
so called “radiosity equation”, which balances meaning and outgoing radiation fluxes on discrete surface 
elements, can be written as: 
N 
B. ■ E +p. V F.B. 
1 1 K i L* u 1 
j- i.j*i 
where N is die total number of elements. 
B,: Radiosity on surface element i (total radiation flux density leaving that surface). 
E ( : Radiance due to direct ill uminati on (equivalent to “emission” of the element), 
reflection coefficient 
Fy-:’View facta”: specifies the fraction of radiant flux leaving another surface j that reaches 
surface i. 
As it can be seen on this equation, an element cannot see itself. This equation leads to a rigorous energy balance: 
the total radiation flux leaving any surface element i is equal to the sum of its emitted and reflected flux 
originating from all other surfaces. As said by Gerstl and Borel (1990), the strengh of die radiosity method cones 
from the fact that leaves are treated as individual surfaces that reflect and transmit radiations. This allows the 
analysis of radiative effects due to die discrete nature of leaves and stems and their heterogeneous distribution, 
e.g. mutual shading and clustering of leaves. 
Figure 1: Each element of the canopy is defined by its height h, its length L, its width 1, its orientation 6 and 
optical characteristics. 
Computing view facta allows to account at once fa all possible direct radiative interactions occuring between 
the different elements. The view facta matrix F is computed according to the fish eye method, also described in 
Borel et al. (1991). The basic principle is to compute, fa a given element, a fish eye image of the visible 
hemisphere from this element, then to compute die relative areas occupied by all the surrounding elements on this 
image (figure 2). 
The direct illumination vector, E is computed by rj»lmliitin£ for each eiggnent , with a given 
resolution, what part of its surface is directly lightened by the sun. The individual reflectance vecta p is given as 
input, using field measurements. After the radiosity equation is solved through a Gauss-Jordan algorithm, images 
of the canopy are computed. These images can be generated fa any geometric configuration and allow to 
simulate the bidirectional reflectance of the canopy in all die directions, with any kind of direct illumination.