International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
1) In the method of the Divided difference of Newton, by
adding new points before the first point or after the last
point of the table, a few extra operations are needed to
correct and adjust the previous interpolation polynomial
with the new situation. Whereas, in the Lagrange method,
all of the operations must be recommenced.
In our case, this feature is not important. Because, the
number of points determined in the evaluation phase and
after that time will be constant.
2) Although the fault of both methods is equal, the number
of the division operations in the latter method is more than
the former.
In the Lagrange method, for n points we need n division
operations, but in the Newton method we have n(n-1)/2 of
such operations.
As we see, for more than three points (that it will be so) the
number of the divisions in Newton case is more than that
of the other one. Division causes floating point error as in
digital computers, so the faults in the Newton method will
be more the faults in the Lagrange method.
3) In Newton method, sometimes useless operations are
observed.
The following table can be considered as an example:
UP) to [RSI ANA UE
0
2 2
3 0
2 5 2 0
5 0
3 10 2
7
4 135
Table 3. Useless operations in Newton method
There are five points in this table and it is expected to have
a fourth power polynomial as the Newton interpolation
function. But really, there is a second power polynomial.
So some of the operations will be useless, while in the
Lagrange method, the number of operations is fixed and
determined.
4) In the Lagrange method, it is possible to have parallel
calculations, because the calculation phases are individual.
726
But in the Newton method, each phase needs the result of
the previous phase to complete its calculation. Therefore,
although the number of operations in the Lagrange
interpolation may be, because of parallel processing, more
than the Newton one, the total computation time will be
less than the second one's.
Reviewing the above reasons, it can be concluded that the
Lagrange interpolation method is better than the Newton
method in our case.
S. CONCLUSION
The introduced method has some advantages such as
simplicity, accuracy, needing no auxiliary devices and no
dependency on the camera parameters, compared with the
previous methods.
The limitation of this method is the dependency on
primitive height and horizontal angle. But the effect of
changing these items isn't considerable, and there are some
ways to decrease the effect of these faults.
It has also been proved that the Lagrange interpolation
method's efficiency is better than the Newton one's in this
method.
This method can be used for applications in which more
accuracy in a limited domain for depth perception is
needed.
6. . REFERENCES
[1] H. Guo and Y. Lu , "Depth Detection of Targets in a
Monocular Image Sequence,” 18th Digital Avionic
Systems Conference , 1999.
[2] JM. Mathews, “Numerical Methods for
Mathematics, Science and Engineering", printice-Hall,
1992.
[3] M. Mirzabaki and A. Aghagolzadeh, * Introducing a
New Method for Depth Detection by Camera using
Lagrange Interpolation, " The Second Iranian Conference
On Machine Vision, Image Processing & Applications,
2003.
[4] M. Subbarao and N. Gurumorthy , “Depth Recovery
from Blurred Edges,” In Proceeding of IEEE
International Conference on Computer Vision and
Pattern Recognition , 1988.
[5] M. Takatsuka and et al. , “Low-Cost Interactive
Active Monocular Range Finder, ” IEEE Computer Society
Conference on Computer Vision and Pattern Recognition ,
1999.
[6]. N. Yamaguti , Sh. Oe and K. Terada, *A Method of
Distance Measurement by Using Monocular Camera", The
36th SICE Annual Conference , 1997.
[7] Y. L. Murphey and et al. , “Depth Finder : A
Real-time Depth Detection System for Aided
Driving,"IEEE ntelligent Vehicles Symposium ,
2000.
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