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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004 
2.1. Lambertian Reflectance Model 
Lambertian surfaces are surfaces having only diffuse 
reflectance, i.e. surfaces which reflect light in all directions The 
brightness of a Lambertian surface is proportional to the energy 
of the incident light The amount of light energy falling on a 
surface element is proportional to the arca of the surface 
clement as seen from the light source position that is cosine 
function of the angle between the surface orientation and the 
light source direction (7) . Therefore the Lambertian surface can 
be modeled as the product of the strength of the light 
source E, , the albedo of the surface A and the foreshortened 
area cosi as follows: 
I, 2 E, Acosi (1) 
Where /, is the reflectance map (figurel). If the surface 
normal and the light source direction both are unit vector the 
above formula can be rewritten as: 
‘=F ANS (2) 
Where “ - ’represents dot product. 
Recent work by Wolff has demonstrated that the Lambertian 
model only really applies when the angle of incidence and the 
angle of reflection is small (relative to the surface normal). 
Importantly, Wolff has developed a simple modification of 
Lambert's law, which accurately accounts for all illumination 
and viewing directions. 
2.2. Specular Reflectance Model 
Specularity only occurs when the incident angle of the light 
source is equal to the reflected angle. It is formed by two 
components: the specular spike and the specular lobe. 
The specular spike is zero in all directions except for a very 
narrow range around the direction of specular reflection. 
The specular lobe spreads around the direction of specular 
reflection. The simplest model for specular reflection is 
described by the following delta function: 
1 =E 610 20) (3) 
Where / is the specular brightness, Æ, is the strength of the 
Ü 
specular component, 0. is the angle between the light source 
direction and the viewing direction and 0. is the angle 
between the surface normal and the viewing direction. This 
model assumes that the highlight caused by specular reflection 
is only a single point, but in real life this assumption is not true. 
  
  
  
  
  
Figure 2. Sprcular reflection 
2.3. Hybrid Reflectance Model 
Most surfaces in the real world are neither purely Lambertian 
nor purely specular, they are a combination of both. That is, 
they are hybrid surfaces. One straightforward equation for a 
hybrid surface is: 
1=(1-a)l, +al, (4) 
Where / is the total brightness for the hybrid surface, 
I A ; are the specular brightness and Lambertian brightness, 
respectively, and @ is the weight of the specular component. 
One of the hybrid models that are used in photogrammetry is 
the Lommel-Seeliger that assumes the radiance observed at a 
sensor comes from light scattered by all particles in the 
medium lying within the field of view of the sensor. Therefore 
the Lommel-Seeliger law contains not only the incidence angle 
I but also the emittance angle €: 
COSI 
cos: +cose 
Where /, is the Lommel seeliger brightness. This law is a 
good description of the light scattering behaviour of low albedo 
surface. In the figure 3a ,3b the two reflectance models are 
depicted graphically with respect to the incidence and 
emittance angle. According to the figures and the equations, for 
the lommel-seeliger módel for a vertical image of a horizontal