We next consider how curved surfaces are recovered from apparent contours viewed from uncalibrated cameras.
Differentiating (
8
) with respect to
, we find that the derivative of the depth,
, is computed from the changes in apparent contour and epipole as
follows:
For the exact expression of
, see [
15
]. Note again, the computation of the derivative of the depth,
, is independent of the calibration matrix,
, camera rotation matrix,
, and the initial position of a camera,
.
Substituting (
8
) into (
2
) and using (
1
), the contour generator at time
is reconstructed as follows:
Substituting (
5
), (
8
) and (
9
) into (
3
), we can compute the change in contour generator at time
caused by the camera motion as follows:
From (
5
), it is clear that the the depth to the frontier point,
, at time
is computed from the depth,
, at time
as follows:
For the exact expression of
, see [
15
]. Thus, by computing
iteratively, the curved surface,
, can be reconstructed with respect to the epipolar parameterisation,
, as follows:
where,
is a
column vector whose components are computed from the image measurements
as follows:
In (
13
), the
matrix
represents a 3D affine transformation. This means that even if the
camera is uncalibrated and its translational motion is unknown, the
curved surfaces can be reconstructed up to a 3D affine transformation:
J. Sato
