B-Splines and Its Applications - Bezier and Splines in Image Processing and Machine Vision - page 119

axes. The curl component measures the change in orientation of patches in
the image, while the divergence term indicates scale or change in size. The
deformation term indicates the distortion of the image shape as a shear. Use
of differential invariants of the image velocity field is significant in the sense
that the deformation component provides information about the orientation
of surface and the divergence component can provide an estimate of the time
to contact or collision.
We shall now check the conditions under which the image velocity field
can be well approximated by its first order terms. The requirement is trans-
formation that should be locally equivalent to an ane transformation, i.e.,
parallel lines remain parallel. In other words, transformation from a plane
in the world to the image must be described by an ane mapping. This is
what we call weak perspective. One can establish after an examination of the
quadratic terms in the equation of image velocity about the vicinity of a point
in the image,

λ
λ

<< 1 , (5.39)

and

Ω . δ
Ω . Q << 1 .

(5.40)

We note that δ , the difference between two rays, defines the field of view
in radians and λ is the depth of relief in the field of view. An empirical

result says that if the distance to the object is greater than the depth of relief
by an order of magnitude [161], then the assumption of weak perspective
is a good approximation to perspective projection. It is true that at close
distances “looming” or “fanning” effects will become prominent and the ane
transformation is not sucient to describe the changes in the image. In many
practical cases, it is possible to restrict our attention to small fields of view in
which the weak perspective model is valid.

5.7.2 3D Shape and Viewer Ego-motion

In the above section, we have seen the differential invariants expressed
in terms of viewer's translation ( U 1 /λ, U 2 /λ, U 3 /λ ) and surface orientation
( λ x /λ, λ y /λ ). From the previous equations through some algebraic manipula-
tions one can write,

2 Ω 3 + ( − U 1 λ y + U 2 λ x )
λ 2

(5.41)

curl v = −

div v =2 U λ + ( U 1 λ x + U 2 λ y )

λ 2

(5.42)

( U 1 λ x − U 2 λ y )
λ 2

(5.43)

( def v )cos2 μ =