PATTERNS OF AERIAL TRIANGULATION BY INDEPENDENT MODELS
R. S. ROSTOM
Professor, University of Nairobi, Kenya, Commission III
ABSTRACT
The position of the non-zero elements in the coefficient matrix of the reduced normal equationsare directly
defined based on the existence of tie points between the models.
The shape and the size of the matrix and
the fill-in elements during its factorization depend on the geometric configuration of the models in the
network, as well as the order of sequencing the models in the block.
homogeneous networks is investigated for variable geometrical parameters.
The ordering scheme of the models of
With the aid of computer
graphics, the pattern and size of the reduced matrix could be established from the topology of the network
before any measurements take place.
Manipulation of the ordering of models, beside other considerations
of the method and strategy of the matrix decomposition, and the computation facilities would lead to the
most efficient use of the computer storage and economy in computation time.
KEY WORDS: Aerotriangulation, Analytical, Computer Graphics.
1. INTRODUCTION
In aerial triangulation projects the computation
and the decomposition of the matrix of normal
equations play a decisive role in the costing and
execution time of the project. The size of this
matrix for large networks is usually of large
order. However, this size can eventually be
reduced to involve either the models' transform-
ation parameters only or the ground coordinates
of the tie points only. This procedure is well
established and explained in many literature,e.g.
Wong (1980). The number of models' transformation
parameters is generally less than the number of
the ground coordinates of tie points as unknowns
in a block. Therefore, the reduced normal
equations of the models' transformation parameters
M is the one which is oftenly formed for economic
computations and the one which is considered in
this paper. Further-more M is sparse, symmetric
and positive definite. Therefore, advantage
Should be taken of these properties in order to
reduce both storage requirements and computation
time.
The shape and the size of the reduced matrix M,
particularly the non-zero entries, depend on a
number of parameters. The configuration of the
block, the number of the strips s and their
direction w.r.t. the block, the number of photo-
graphs in each strip g, the percentages of fore-
laps p and side laps q and the ordering scheme of
the models are the main parameters. With the
development and spread of microcomputers, the
computation of large triangulation networks using
such device became an objective (Julia, 1984;Klein,
1988). This necessitates the intorduction of
economical storage schemes to make most efficient
use of the limited core and memory capacity. The
use of perephiral storage might be employed to
overcome the problem (Lucas, 84). An attempt was
made (Lucas, 84) to identify the structure and the
pattern of the banded matrix M for p-q-67$, while
the system was proposed to be solved by recursive
partitioning. To avoid arithmetic operations with
zero and storage of zeros within the regular band
structure of M a nested dissection ordering
technique is recommended (Stark and Steidler,79).
The same ordering technique is proposed by Shan
(1988) for combined networks. Another proposal to
arrange the sequence of unknowns to reduce the fill
-in of new elements during factorization of M was
to use the graph theory (Kruck, 84). A search
routine had been deviced (Julia', 86) to identify
which point connect which models, and how many
models share a particular point.
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This study is an attempt to investigate the
possibility of automatic definition of the shape
and size of M from the input data,and to establish
the number and location of the non-zero elements m
of M for variable parameters. The defined pattern
of M gives in addition an insight to the required
computing device and facilities, which form an
important aspect in project planning. The
conditions for particular ordering to achieve the
minimum band width of M, or the least fill-in
during its factorization to be established.
2. BASIC CONCEPTS
By virtue of its symmetry the matrix M would be
presented by its main diagonal and the upper
triangle only. A model which contributes to the
structure of M should be joined at least to another
model. This contribution is summarised in the
following: -
+ Any model I(i) of order i contributes to M a
matrix m(i,i) on its main diagonal
as can be seen in figure 1. mG,i) is
conventionally called basic variance matrix. It is
47x7 symmetric positive definite matrix, which is
sparse on its own.
* Any model I(i) which is joined with another model
I(j) by one tie point, or more, contributes to M an
off-diagonal matrix m(i,j) (figure 1), which is
conventionally called basic covariance matrix.
is a 7x7 non-symmetric sparse matrix,
It
However, a number of models which are joined in a
sequential order would form a line of models L(K).
(i)
Figure |. i
Basic elements 3
of M. m(i,i m(i, j)
b(k,k) b(k,kel)
L(k) BUD.
=. | |
Non-zero basic bik, k+2)
element m. Symm.
