increases, but the accuracy of data is not improved. If sonar
images in the same area are available and the resolution of the
sonar images is higher than that of the grid, slopes of seafloor
surface points between bathymetric grid points can be
calculated by the relaxation procedure (4). The combination of
depths of grid points and slopes makes it possible to obtain
depth information between grid points. Because image
information is used to derive depth information within the
boundaries of existing grid points, bathymetric data densified
with this technique describe the seafloor surface more exactly.
To simplify the experiment presented in this paper, the
Lambertian model is applied for the reflectance map. More
complex backscattering models which are better suited for
sonar images will be studied and used for further
development. Under the Lambertian assumption, image
intensity is proportional to the albedo (intrinsic reflectivity)
and the cosine of the angle between the local normal vector of
the surface and the vector of illumination (Kober and Leberl,
1991, Fronkot and Chellappa, 1987, Grimson, 1983). Since it
is difficult to use one constant albedo parameter for the whole
area covered by the image in the implementation, albedo
parameters are determined in small subareas with the
assumption that the albedo parameter varies very slowly in
such small sub-areas and can be treated as constants.
The transformation from object space to image space is very
important in the relaxation procedure. Control points with
known coordinates in both the image system and the object
coordinate system have to be utilized to determine the
transformation parameters. For images (e.g. GLORIA images)
which do not have depth information acquired in swaths
during cruises, these control points are only available along
track lines. Clearly, a linear distribution of control points in an
image is not a good configuration to determine transformation
parameters for the whole image area. It is unlike the
processing of satellite images for land remote sensing or
aerophotographs; we usually don't have known marked points
on the sea floor which can be identified on sonar images.
Therefore, control points across track lines are interpolated in
order to achieve a stable transformation between image and
object spaces.
Figure 1. The grid is densified by a factor of two
782
The relaxation model (4) is implemented in a hierarchical
manner. For instance, figure 1 shows a grid being densified by
a factor of two by three passes for the shape from shading
procedure. Actually, the densification factor can be other than
two. This hierarchical process captures surface relief with
different frequencies and reduces computational time (Li,
1990, Terzopoulos, 1986b).
2.3 Boundary Constraints
Known grid points are used as boundary points for deriving
depths from surface slopes (Z,, Z,). Thus, depths of all points
with slopes can also be computed. On the other hand, these
known grid points are treated as tie points for the
reconstructed surface. As shown by figure 2 for a 4x4 grid of a
subarea, for instance, we have four known grid points and 12
unknown points. At every point (both known and unknown)
we have (Z,, Z,) obtained by relaxation procedure (4). So
boundary constraints can be described as:
Z(j-Z64,)-Z]j) i1. N-Lj-..N,
Zi, j) = Z(i, j+1)-Z,j), _i=1,.,N, j=1,.,N,-1.
65)
with
N, - 4 and N, - 4.
Generally, if the subarea to be reconstructed has a dimension
of N, by N, and 4 known depth points are available, there will
be 2N,N,-N,-N, equations in (5) and N,N,-4 unknown depths.
Here the number of equations is greater than that of
unknowns. Therefore the equation system (5) is
overdetermined and can be solved by the least squares method.
Depths and slopes thus obtained ensure that the depth of a
point calculated from any point and through any path will
have the same value, and the sum of slope corrections is the
minimum.
Figure 2. Depths of unknown points are calculated
by an adjustment of slopes.
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