er row) and without
ects the step lengths
1orm would increase,
in), and converges in
ion (upper right col-
investigated by ap-
essively more distant
their success or fail-
111 = bii = 97.95,
; = 934.33. r; — 0.1.
tch position a 11,511
starting approxima-
rs were aio = b91 =
values for the radio-
d from the picture at
convergence for the
) and 7, respectively.
oward the correct so-
le the damped algo-
tions. The undamped
inima at 5 occasions.
imped algorithm was
Iness of a line-search
st Squares Matching
1e damped algorithm
extra residual calcu-
t which is more than
ons the undamped al-
or the same problem,
orithm is higher than
er, the repetitive pat-
ased the convergence
is likely to be less of
es.
Figure 6: Pull-in range results for the damped (left) and
undamped (right) algorithms. The bright pixels indicate
for which displacements of the patch center the algorithms
converged toward the correct solution.
Finally, this paper illustrates but one example of the large
potential of combining the theories and experiences of the
two research fields photogrammetry and non-linear least
squares optimization. This potential will be further ex-
plored in the future.
REFERENCES
Borlin, N., 2002. Adaptive least squares matching and the
Gauss-Newton method. Technical Report UMINF-02.02,
Department of Computing Science, Umed University.
Dennis Jr., D. E. and Schnabel, R. B., 1983. Numerical
Methods for Unconstrained Optimization and Nonlinear
Equations. Prentice-Hall.
Eriksson, J., 1996. Optimization and Regularization of
Nonlinear Least Squares Problems. PhD thesis, Depart-
ment of Computing Science, Umeà University, Sweden.
Eriksson, J., Gulliksson, M., Lindstróm, P. and Wedin,
P-À., 1998. Regularization tools for training large feed-
forward neural networks using automatic differentiation.
Optimization Methods and Software 10, pp. 49-69.
Gruen, A., 1996. Least squares matching: a fundamen-
tal measurement algorithm. In: K. B. Atkinson (ed.),
Close Range Photogrammetry and Machine Vision, Whit-
tles, Caithness, Scotland, chapter 8, pp. 217—255.
Figure 7: Convergence toward side minima for the damped
(left) and undamped (right) algorithm. Bright pixels indi-
cate for which displacements of the patch center the al-
gorithms converged toward a side minima with the same
residual as the correct solution.
Sóderkvist, L, 1993. Computing Parameters in Nonlinear
Least Squares Models. PhD thesis, Department of Com-
puting Science, University of Umeä, Sweden.
Gruen, A. W., 1985. Adaptive least squares correlation:
A powerful image matching technique. S Afr J of Pho-
togrammetry 14(3), pp. 175-187.
Gulliksson, M., 1993. Algorithms for Overdetermined
Systems of Equations. PhD thesis, Department of Com-
puting Science, Umeä University, Sweden.
Gulliksson, M. and Sóderkvist, L, 1995. Algorithms for
constrained and weighted nonlinear least squares. Opti-
mization Methods and Software 5, pp. 247—269.
Gulliksson, M. and Wedin, P.-A., 1992. Modifying the QR-
decomposition to constrained and weighted linear least
squares. SIAM Journal on Matrix Analysis and Applica-
tions 13(4), pp. 1298-1313.
Gulliksson, M., Sóderkvist, I. and Wedin, P.-À., 1997.
Algorithms for constrained and weighted nonlinear least
squares. SIAM J. OPTIM. 7(1), pp. 208—224.
Nash, S. G. and Sofer, A., 1996. Linear and Nonlinear
Programming. McGraw-Hill.
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