residuals, belonging to the inner core of data, is minimized
in order to obtain the expected results:
h
Ö = min X vi.
k=]
In other words, only a part of the observations are pro-
cessed in each step of linear adjustment.
The use of a sequential updating of a preliminary com-
puted solution is a possible alternative to repeating many
times the whole adjustment (Lawson, Hanson 74). The
formulae for updating, in terms of parameters as well as in
term of observations, of the Cholesky factor and the inverse
matrix are shown in Tables 1 and 2.
Sequential updating of the Cholesky factor!
observation:
ÜS moi (uly
wy = 0
tig moti suu yy (gio)
ue. = CU -wPuy (1)
wt Zu is VI
parameter:
Az Gehiej«h ish
“in” only
1—1
hm n > tnt nto “ty
kd
(à « h)
h—1 :
Inn = | Chh 7 > n = tu
kl
h—1 à;
thi = (ej- rts = ty;
L1
(j » h)
ty os (wy (t>h)
we n
1
"In the following formulae, the symbol a; indicates a
generic element of the row of the design matrix A, to be
added to or dropped by the system, the symbols cii, ci;
indicate, respectively, a generic main-diagonal element and
a generic off-diagonal element of the normal matrix C, the
symbols t;;, t;; indicate the same elements of the Cholesky's
factor T.
146
ho mous pl NE
odd)
wii) a (ug : mé) pe,
(7>1)
wi! = th; (> h)
Table 1
Sequential updating of inverse matrix?
observation:
(Ck dlpayite: = OF
gh SB | er Baia WAGE dk
parameter:
Sy
quits Cri pa ear irt Ci.
+. CAs CT Ir) ipie
DZ —s yr
qim s. (b o)
“out”
Ce CC
+ Clr(s + i ME RCE fie
Table 2
The weighted least trimmed squares could be minimized,
avoiding a rough partition between inliers and outliers,
where the weighted average of the squares of the residuals
takes into account the inner core of the data with weights
equal to one, an intermediate doubt region with weights
ranging from one to zero, whilst the data in the tails get
weights equal to zero:
h
d > WAV.
k=l
Least median of squares and least trimmed squares (or
weighted least trimmed squares) have the same breakdown
point near to 0.5, when the number À is around m/2, ie.
only the best half part of the observations are processed in
each step of linear adjustment.
’In the following formulae, the symbol a indicates a
generic row of the design matrix A, to be added to or
dropped by the system, the symbol p indicates the weight
of the corresponding observation; furthermore the normal
matrix is split in four parts, being their sub-blocks C, r, r'
and s, and their inverse matrix is split again in four parts,
being their sub-blocks y, p, p‘ and c.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B1. Vienna 1996
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