B-Splines and Its Applications - Bezier and Splines in Image Processing and Machine Vision - page 120

( U 1 λ y + U 2 λ x )
λ 2

(5.44)

( def v )sin2 μ = .

The average image translation ( u 0 ,v 0 ) can always be canceled out by appro-
priate camera rotations, while divergence and deformation remain unaffected
by viewer rotation, such as panning or tilting of the camera or eye move-
ments, whereas these rotations could lead to considerable changes in image
point velocities or disparities.
Differential invariants depend on the viewer motion, depth, and surface
orientation. When the translations are scaled by depth, λ , we get a 2-D vector,
say A , given by

A = U λ , U λ
=

(5.45)

U − ( U.Q ) Q
λ

.

Similarly, when the depth gradient is scaled by depth, λ , we get a 2-D vector,
say F to represent the surface orientation, given by
F = λ λ , λ λ
=

(5.46)

gradλ
λ

.

| provides the tangent of the slant of the surface, i.e., tangent of the angle

between the surface normal and visual direction. It is zero for a frontal view
and infinite when the viewer is in the tangent plane of the surface. Direction of
F provides the direction in the image of increasing distance and this is equal
to the tilt, τ , of the surface tangent plane. Hence,

F |

| F | = tan σ, F = τ.

Relation between the differential invariants, motion parameters, and surface
orientation can, therefore, be shown as

curl v = − 2 Ω . Q + | F ∧ A | . (5.47)

div v = 2 U . Q
λ

+ F . A . (5.48)

def v = | F || A | , (5.49)

and

A + F

μ =

. (5.50)

2

Note that μ bisects the sum of the angles of F and A .

5.7.3 Geometric Significance

Formulation in the preceding section clearly shows the speed-scale ambiguity
and the bas-relief ambiguity. Translational velocities appear scaled by depth.
So, we note that a nearby object moving slowly or a far-away object moving