increment is dominant in x direction in the middle and the 
right part of the area (represented by B and C in figure 3 and 
4). Contour lines in area B do not closely match the texture 
located diagonally in the image, because they are calculated 
from just 20x20 grid points using interpolation. 
Two passes of the shape from shading process were needed to 
reconstruct a digital surface model with 172x172 points. In the 
first pass the original grid with 20x20 points was densified to 
an intermediate grid with 58x58 points. From this intermediate 
grid we reached the final grid with 172x172 points in the 
second pass, which matches the resolution of the sonar image. 
The regularization parameter À in (4) was set to 25. The first 
criterion to stop the iterative procedure of relaxation 
represented in (4) is the average difference between estimated 
and known depths of all known points; the second criterion is 
the average difference between calculated and original image 
gray values of all pixels. Figure 5 depicts the contour lines of 
the reconstructed sea floor surface derived from shape from 
shading. 
By comparing figure 4 and 5 we can see that there is no 
significant change in global surface relief. But the sea floor 
surface variation described by image texture in diagonal 
direction in the area B in figure 3 was reconstructed in figure 
5 by shape from shading. Less changes occurred in areas A 
and C where only homogeneous image texture exists. 
Therefore contour lines in figure 5 look more harmonious with 
image features than contours in figure 4. Figure 6 shows a 
shaded relief image of the bathymetric data improved by shape 
from shading technique. Again the surface relief matches 
image features, especially in the area B. 
4. DISCUSSION 
The shape from shading method provides a way to improve 
bathymetric data bases if sonar images with higher resolution 
of the same surveyed area are available. But the quality of the 
results using this technique is influenced by many factors. 
First of all, accurate navigational data are needed to determine 
the transformation from object space to image space so that 
the calculated gray values of points on the sea floor can be 
compared to the appropriate pixels in the original image. 
Secondly, the backscattering model plays a very important 
role because it converts image intensity values to surface slope 
values. A more complex reflectance map should be applied to 
approach sonar image backscattering modeling. Classification 
results of the seafloor may be used to identify where gray 
value changes are caused not by surface slope variations but 
by the high heterogeneity of reflectance of seafloor materials. 
Finally, unlike the reconstruction of a land terrain model, an 
accurate surface model is often not available as a reference to 
check results obtained by shape from shading. Therefore, in 
cases where the original grid is relatively dense and the depths 
of these points are reliable, a low regularization parameter 
should be selected, and strict depth constraints should be 
employed. Thus, the depths of additional points will be 
calculated by using sonar images and the reconstructed sea 
floor surface model will fit the original gridded data. 
Certainly further theoretical and practical work is needed to 
apply the method to map production and for applications in 
ocean related fields. Future efforts may include: a) application 
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of more complex backscattering models, b) the use of multiple 
images if an area is covered by more than one image, and c) 
improving software in order to handle large bathymetric data 
files for mapping purposes. 
ACKNOWLEDGEMENT 
The support of USGS, NOAA and PICHTR is gratefully 
acknowledged. Data applied in this paper are supplied by 
USGS and NOAA. Dr. E. Wingert is acknowledged for his 
photographic work and Ms. S. Pai for her editing assistance. 
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