In the equation (5) known points are treated as accurate points. 
No corrections are applied to these known points. In 
computational practice this often causes oscillatory "ringing" 
in the estimated surface. Therefore a "soft" mode of depth 
constraint is used, which allows known points to vary in some 
ranges (Kobert and Leberl, 1991). To prevent estimated points 
from varying within unreasonable depth ranges, maximum and 
minimum depths are set for the whole grid area. 
3. EXAMPLE 
The test data consist of a GLORIA (Geologic Long Range 
Inclined Asdic) sonar image and SEA BEAM bathymetric 
data in the area of COOK SEAMOUNT near the Island of 
Hawaii. The sonar image of the test area is of 172x172 pixels 
in size with an 8 bit gray level, and pixel size is about 
50mx50m (figure 3). In the same area covered by the 
GLORIA sonar image data, the SEA BEAM bathymetry data 
were gridded every 15 seconds with 20 x 20 points in the 
geographic system so the resolution of the sonar image is 
about 9 times higher than that of the bathymetric data. Figure 
4 shows the contour lines derived from the bathymetric data 
which were used as boundary constraints for the shape from 
shading method. 
In figure 3 and 4 we can see that the upper left part of the sea 
floor surface, indexed by A, is relative flat; the surface slope 
  
Figure 3. GLORIA side-scan sonar image used as 
input image for shape from shading 
Figure 4. Contour lines derived from SEA BEAM 
bathymetric data in the same area covered by the 
sonar image in figure 3 
  
Figure 5. Contour lines derived from the reconstructed 
seafloor surface by shape from shading technique 
Figure 6. Shaded relief image from the reconstructed 
seafloor surface by shape from shading technique