Luminance: A New Visual Feature for Visual Servoing - Visual Servoing via Advanced Numerical Methods - page 70

where I k • is nothing but the k -th line of the image. I ( r ) is then a vector of size
N

× ×

M is the size of the image. As mentioned in Section 5.1, an
estimation of the interaction matrix is at the center of the development of any visual
servoing scheme. In our case, we have to derive the interaction matrix related to the
luminance of a pixel in the image, that is

M where N

I ( x , t + dt ) − I ( x , t )

lim
dt

= L I ( x ) v (5.4)

dt

→ 0

x =( x y ) being the normalized coordinates of the projection p of a point physical P

belonging to the scene.
Before computing the interaction matrix L I ( x ) in the general case, lets first con-
sider the simpler case where the temporal luminance constancy hypothesis is as-
sumed, as it is done in most of computer vision applications. Let us also assume
that p has a small displacement dx in the time interval dt

,

I ( x + dx , t + dt )= I ( x , t ) . (5.5)

If dx is small enough, a first order Taylor series expansion of (5.5) around x can be
performed yielding the so-called optical flow constraint equation (OFCE) [13]

I x + I t = 0 ∇ (5.6)

t ) 1

with ∇ I the spatial gradient of I ( x , and I t = ∂ I ( x , t ) / ∂ t . Moreover, considering

the interaction matrix L x related to x ( i.e. x = L x v )
L x = −

(1 + x 2 ) y 1 / Z 0 x / Z y −

(5.7)

Z 1 + y 2 0 − 1 / Zy / − xy − x

(5.6) gives

I L x v I t = − ∇ . (5.8)

However, note that I t is nothing but the left part of (5.4). Consequently, from (5.4)
and (5.8), we obtain the interaction matrix L I ( x ) related to I at pixel x

I L x . L I ( x )= − ∇ (5.9)

Of course, because of the hypothesis required to derive (5.5), (5.9) can only be
valid for Lambertian scenes, that is for surfaces reflecting the light with the same
intensity in each direction. Besides, (5.9) is also only valid for a motionless lighting
source with respect to the scene.
Indeed, to derive the interaction matrix in the general case, we have to consider
a more realistic reflection model than the Lambert's one. The Lambert's model can
only explain the behavior of non homogeneous opaque dielectric material [22]. It
only describes a diffuse reflection component and does not take into account the

1

Let us point out that the computation of ∇ I is the only image processing step necessary to
implement our method.