A
REVIEW OF STRIP AND BLOCK ADJUSTMENT DURING 1964-1967
2. TREATMENT OF CONTROL POINTS
Both Brown [11] and Schmid [5] to [7] have
now introduced the use of a covariance matrix
for the coordinates of the control points in a
block of aerial photographs. Unknown coordi
nates are given a large variance and for the
sake of simplicity, although unrealistically,
one assumes that their estimated (observed)
values are uncorrelated. If a rectangular
coordinate system is used other than the one
formed from planimetric coordinates and ter
rain heights, the proper covariance matrix in
that system can be easily computed from the
relation between the two coordinate systems.
With this procedure, the vector 5 contains
also the corrections to the coordinates, known
as well as unknown, of all control points. To
retain the advantage that only one correction
to an observation occurs in each condition
equation, Equation 1 is supplemented with
the equation
v a = 0 (3)
in which v g is the vector of corrections to
observed coordinates of the control points and
is the vector of corrections to their esti
mated or given coordinates. The equation ex
presses that one identifies the observed values
with the estimated or given values.
This procedure increases the size of the
normal equations. However, it makes their
formation simpler because now each terrain
point receives three coordinate corrections,
independent of whether it is completely
known, partially known, or unknown.
3. PARTITIONING OF THE NORMAL EQUATIONS
The components of v, 5, and k may be par
titioned into groups. If the condition equa
tions contain a set of constraint equations in
which no observations occur, the correspond
ing ^-components will be equal to zero. In
addition, the sequence of the groups in the
normal equations may be rearranged. This
can lead to a variety of formulas.
It is customary to partition the vector 5 of
Equation 2c into a vector 5i of corrections
for improvement of the camera orientations
and a vector 5 2 of corrections to approximate
coordinates of terrain points. The attitude
corrections included in 5i can be either the
parameters of a correction matrix or correc
tions to approximate values of attitude
parameters.
The corresponding partitioning of Equation
2c gives:
MiSx + Nl2§2 = £i
Nnhi + N 2i ft2 = £2
and elimination of d 2 gives
(Nn — 2V12^2V22 — Wi 2 )5i = £i — Ni 2 t No2~ 1 £2- (2e)
The components of 5i and of 5 2 can be par
titioned into groups each of which contains
only the corrections for one photograph or
for one terrain point, respectively. The cor
respondingly partitioned matrices Affi and
N22 contain non-zero submatrices along their
main diagonal only, while the submatrix of
ÌV12 which corresponds to point i and photo
graph j has non-zero elements only if point i
has been measured in photograph j.
Accordingly, the inverse of N2 2 can be
computed by inverting its submatrices sep
arately and Equation 2e can be computed
directly from the Equation 1 by treating all
the condition equations for one terrain point
as a group and computing the contribution of
this group to the normal equations separately.
In this way, no space need be reserved for the
matrices N22 and N i2 . If two photographs
have no measured point in common, the cor
responding off-diagonal submatrix in Equa
tion 2e is equal to zero.
Each of the block adjustment programs
that have been coded along these lines em
ploys a housekeeping routine, collapsing al
gorithm, or indexing technique to avoid com
puting and storing zero-submatrices and to
keep track of the locations of the non-zero
ones.
4. DIRECT SOLUTION OF THE NORMAL EQUA
TIONS
Even for large blocks, the direct solution
of the normal equations has proved to be
entirely practical provided that a block-
elimination procedure is used. This means
that instead of the matrix elements the above-
mentioned submatrices of Equations 2d or 2e
are used as units in the computations.
Both a Gauss-Cholesky type of elimination
([91] and [21]) and a Gauss-Jordan type [16]
are used. If S is an on-diagonal submatrix
which is to be used as pivot in the elimination,
the Gauss-Cholesky elimination involves the
simple computation of an upper triangular
matrix T such that T l T = S and of its in
verse. The Gauss-Jordan elimination requires
the computation of the inverse of S. The two
elimination procedures perform equally well.
5. ITERATIVE SOLUTION OF THE NORMAL
EQUATIONS
The iterative solution of a system of equa
tions has the basic disadvantage that it is
difficult to know when a sufficiently good
approximation of the exact solution has been