B-Splines and Its Applications - Bezier and Splines in Image Processing and Machine Vision - page 122

to the surface orientation and viewer translation parallel to the image axis
from the image divergence equation (5.48). This equals − A ).

The remaining component of divergence is due to movement towards or away
from the object. This can be used to recover the time to contact t c as

| def v | cos( τ

λ
U . Q .

t c =

(5.53)

The time to contact fixes the viewer translation in temporal units. It allows
the specification of the magnitude of translation parallel to the image plane
A , up to the same speed-scale ambiguity. The magnitude of deformation can
be used to recover the slant σ of the surface from equation (5.49).
The advantage of this formulation is that camera rotations do not affect
the estimation of shape and distance. Effects of errors in the direction of
translation are evident as scalings in depth or by a relief transformation [92].
If the cameras or eye rotate to keep the object of interest in the middle
of the image, the eight unknowns reduce to six. The magnitude of rotations
needed to bring the object back to the center of the image determines A and
hence allows us to solve for these unknowns. The major effect of any error in
the estimate of rotation is to scale depth and orientations.
Even without any additional assumptions, we can get useful information
from the first order differential invariants. Inspection of equations (5.48) and
(5.49) shows that the time to contact must lie in an interval given by

1
t c

= div v

def v
2 2 ±

. (5.54)

The upper bound on time to contact occurs when the component of viewer
translation parallel to the image plane is in the opposite direction to the
depth gradient. The lower bound occurs when the translation is parallel to
the depth gradient. The upper and lower estimates of time to contact are equal
when there is no deformation component. This is the case in which the viewer
translation is along the ray. The estimate of time to contact is then exact.
A similar equation has been described by [157]. Subbarao's result suggests
the curl and deformation components can be used to estimate bounds on the
rotational component about the ray,

curl v
2 ±

def v
2

Ω . Q = − . (5.55)

Koenderink and Van Doorn [95] showed that when weak perspective is a
valid approximation, the deformation component alone in a small field of view
can provide surface shape information. As a result, recovery of a 3D shape
can be made up to a scale and relief transformation.
Two different cases are described next.