International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part Bl. Istanbul 2004 
  
measured, since real measurements involve finite extension 
intervals (resolution) of the concerned geometrical parameters. 
  
  
  
Figure 2. Diagram of the measurement geometry for BRDF. 
The origin of the coordinate system is the point at which the 
central ray of the collimated irradiance E strikes the sample 
surface at angle (9;,9;) with o; «n . The z axis is normal to 
the sample surface, and the y axis lies in the plane defined by z 
and £. The viewing direction is given by (9,,9,) and dO, is 
the solid angle subtended by the receiver. 
A common simplification of the BRDF is to assume the 
concerned reflector as a Lambertian (perfectly diffuser) surface, 
i.c. an infinite ideal surface for which the reflected radiance is 
isotropic with the same value for all directions (9,,6,) 
regardless of how it is irradiated. Therefore, Lambertian 
surfaces constitute a restricted ensemble of reflectors, which are 
included in the more general class of natural surfaces. 
However, it is possible to define another class of reflectors 
whose bi-directional reflectance distribution function 
P sape (.9;,0;,9,,0,) is assumed to be a separable function 
P sapi: 0,9;,0;,9,,0,) & po ()^(9;,0;,9,,0,). If we impose 
according to the definition of reflected irradiance that: 
| 1,.4,.9,.0, )cos9 dO, ox 
(3) 
21, 
then, we obtain an additional definition of albedo p, (A): 
[Paror 0.9,.9,.9, 4.20059, d0, 
27%, (4) 
  
5 pol) = 
A relevant trouble to execute BRDF measurements is the 
necessity to perform a reference measurement over a white 
standard, for instance a sample of Barium Sulphate or a 
Spectralon tile, for any used illumination and viewing geometry. 
The standard experimental procedure would produce a wrong 
or (2.8, ‚6, )= AySOA)LO Op Lor (1.8.6 $, b,)cosS,AQ, (5) 
334 
result since the angular behavior of the investigated sample 
should be normalized to that of the employed reference, which 
has its own angular dependence. 
In order to avoid mixing of angular properties of reference and 
target, the reference measurement has always been executed 
with the same geometry, namely 9, =0° (indicated as 9, ) for 
the illumination angle and 9, = 45° (indicated as S, ) for the 
viewing angle, in the hypothesis to get the spectral 
measurements in the principal plane, ie. $69 —6,-180'. 
Utilising a collimated radiation source emitting the directional 
radiance L,(A), it is easy to demonstrate that the flux 
'^(1.9.,0,) reflected by the reference plate into the 
direction (9,,0,) is: 
with AQ, - [os9,40, . For a measurement executed with a 
Q, 
generic target we obtain a similar expression for the reflected 
flux. The target-to-reference ratio fluxes. p,,, (1,99,0,9,.0,) 
obeys the following expression: 
PI (X) M (9, .00:9,,0,) cos9, 49, 
  
Pa, 0,99.049,.0,) 7 DENT (6) 
pA (X) ^" ($,,0,,9,,0,) cos9, AQ, 
During our measurements the white reference plate always was 
observed at a fixed geometry, and its outcome was employed to 
normalize any target measurement as stated in Eq.6. We point 
out that as an effect of ratio the instrument's sensitivity S(A) is 
cancelled from the retrieved signal p,,,(1,9,,0,9,,0,) from 
which we can deduce the complete target BRDF as explained in 
the following. Under the discussed experimental set-up the 
goniometric head allows a relative measurement of the bi- 
directional reflectance function. The measured reflectance 
Spectrum Pac (4.,95,09,9,,0,) can be expressed as the product 
of a constant «& , which takes into account the directional 
properties of the reference standard, and the target 
Parpr (4,90.00,9,,H,) bi-directional reflectance distribution 
function. 
P4: 0599,0,9,.0,) 2 ap," (07 (84,9¢.9,,0,)cos8, (7) 
In order to determine the unknown coefficient @ we have 
executed a target measurement of directional-hemispherical 
reflectance 077, (4,9,,0,,21,) with a Perkin Elmer Lambda 
hem 
19 double-monochromator, which operates from ultraviolet 
(UV) to short-wave infrared (SWIR), using a deuterium lamp 
(UV range) and a tungsten-halogen lamp (VIS-NIR and SWIR 
ranges) as radiation sources. Then, integrating numerically Eq.7 
Inte 
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