B-Splines and Its Applications - Bezier and Splines in Image Processing and Machine Vision - page 121

quickly have the same effects and, therefore, introduces an ambiguity known
as speed-scale ambiguity. Similarly, increasing the slant of the surface F while
scaling the movement by the same amount will leave the local image velocity
field unchanged. As a result, the ambiguity, viz. the bas-relief ambiguity, arises.
Therefore, we conclude that from two weak perspective views and with no
knowledge of the viewer translation, it is impossible to determine whether the
deformation in the image is due to a large and a small slant or due to a

small rotation and a large slant. So, a nearby “shallow” object will produce
the same effect as a far-away “deep” structure. As a consequence, we can only
recover the depth gradient F up to an unknown scale.
it is interesting to note the similarity between motion parallax [109, 140,
38] which relate the relative image velocity between two nearby points Q (1)
t

| A |

and Q (2)
t

to their relative inverse depths,

Q ] 1

,

1
λ (1)

Q (2)
t

Q (1)
t

− =[( U ∧ Q ) ∧

λ (2) −

(5.51)

and the equation relating image deformation to surface orientation

grad ( 1

λ ) .

def Q t = | ( U ∧ Q ) ∧ Q | (5.52)

The results are essentially the same, relating local measurements of relative
image velocities to scene structure in a simple way which is uncorrupted by
the rotational image velocity component. In the first case, the depths are dis-
continuous and differences of discrete velocities are related to the difference
of inverse depths. In the latter case, the surface is assumed smooth and con-
tinuous and derivatives of image velocities are related to derivatives of inverse
depth.

5.7.4 Constraints

It is dicult to completely solve for the structure and motion due to insuf-
ficient information. We have six equations in eight unknowns of the scene
structure and motion. For a complete solution in a single neighborhood we
need to compute second order derivatives to get more equations [109, 171].

Case: Known Translation and Arbitrary Rotation

In this case, we can use equations (5.48), (5.49) and (5.50) to unambiguously
recover the surface orientation and the distance to the object in temporal
units. For the speed-scale ambiguity, we can express the latter as a time to
contact. The axis of expansion ( μ ) of the deformation component and the
projection in the image of the direction of translation ( A ) allow the recovery
of the tilt of the surface equation (5.50). Now subtract the contribution due