OPTIMIZATION OF LEAST-SQUARES COMPUTATIONS
IN ON-LINE PHOTOGRAMMETRY
Dr. J.A.R. Blais
Division of Surveying Engineering
The University of Calgary
Calgary, Alberta, Canada T2N 1N4
Commission III
Abstract:
Least-squares computations can be carried out in several different ways for
adjustment, prediction, filtering and smoothing purposes. The orthogonal
and square-root algorithms are especially advantageous for on-line computa-
tions where computational efficiency, storage requirements and numerical
accuracy are critical. Different algorithms for such least-squares compu-
tations are briefly discussed with their principal characteristics in inter-
active environments. Some aspects of the corresponding firmware implementa-
tions are also mentioned for photogrammetric applications.
1. INTRODUCTION
Over the past decade, there has been much research and development in the
field of least-squares computations for adjustment, prediction, filtering
and smoothing purposes. A great deal of motivation for these efforts has
been related to complex navigation problems in the space programme. The
critical situations often encountered in real-time navigation require the
best numerical techniques available.
The estimation problems in the context of on-line photogrammetry are ob-
viously much simpler than in space navigation. However in order to opti-
mize the benefits of recursive least-squares computations with limited
computer facilities and quality control requirements, it is imperative that
proper attention be given to the computational algorithms and related numeri-
cal matters. Otherwise the advantages of on-line computations can easily be
eroded away to the point that their implementation becomes debatable in
practical situations.
The principal objective of the following discussion is to emphasize that
least-squares computations can indeed be optimized in terms of computational
efficiency, storage requirements and numerical accuracy without sacrificing
any aspect of the mathematical rigour of the least-squares formulations.
2. LEAST-SQUARES ESTIMATION PROBLEMS
Photogrammetric estimation problems are quite varied in different applica-
tion contexts. For the purposes of the following discussion, they are going
to be classified as adjustment, prediction, filtering and smoothing problems
of least-squares estimation. Appropriate linearizations of the mathematical
models are also assumed for the sequel.
Adjustment problems for orientation and triangulation purposes are definitely
the most common least-squares problems in photogrammetry. The corresponding
mathematical model consists in forming an overdetermined system of linear
equations
T
Ax =f +e E[e] = 0 E[ee ] = Co