Luminance: A New Visual Feature for Visual Servoing - Visual Servoing via Advanced Numerical Methods - page 72

k

n

i

L

R

θ

θ

V

α

P

Fig. 5.1 The Phong illumination model [20]

where we have introduced the interaction matrix L x associated to x . Consequently,
from (5.15) and (5.16), we obtain

I L x v + I t = kK s u k − 1
1

∇ u 1 + K d u 2 (5.17)

that is a general formulation of the OFCE considering the Phong illumination model.
Thereafter, by explicitly computing the total time derivative of u 1 and u 2 and
writing

u 1 = L 1 v and u 2 = L 2 v , (5.18)

we obtain the interaction matrix related to the intensity at pixel x in the general case

I L x + kK s u 1 k − 1 L 1 + K d L 2 . L I = − ∇ (5.19)

I L x associated to the intensity
under temporal constancy (see (5.9)), i.e. in the Lambertian case ( K s = 0) and when
u 2 = 0( i.e. the lighting direction is motionless with respect to the point P ).
To compute the vectors L 1 and L 2 involved in (5.19) we have to explicitly ex-
press u 1 and u 2 . However, to do that, we have to assume some hypothesis about
how n and L move with respect to the observer. Various cases have been studied in
[6]. Nevertheless, to make this chapter more readable, we report here only the case
where the light source is mounted on the camera and only give the final equation.
However, all the details can be found in [6].
In this case, considering a directional light source, we simply have L =

Note that we recover the interaction matrix − ∇

− k .After

tedious computations, it can be shown that

L 2 = n z L x + L 4 − ∇ (5.20)

n z ) and L 4 = 000 n y − n x 0 . where n =( n x , n y ,

L 1 is expressed as follows:

L 1 = V J R + R J V L x + L 3

(5.21)

where J R and J V are respectively the Jacobian matrices related to R and V (see [6])

with respect to x , while L 3 = 000 L 3 x L 3 y L 3 z with