4. LEAST SQUARES MDS BASED ON 
FREE NETWORK CONCEPT 
The geodesy or surveying network without the fixed 
coordinates is called a free network. The MDS for time- 
space mapping isanalogousto the trilateration free network. 
To adjust such a free network and obtain the coordinates, 
the problem of det(A'PA)=0 needs to be resolved. For 
this purpose, so-called Moore-Penrose general inverse 
(MP inverse) is applied to (17) (Rao and Mitra, 1971). 
The representation of (17) using MP inverse is 
T -(A'PA)A'PL. (18) 
where (A ‘PA is MP inverse matrix of A‘PA. It is known 
that the MP inverse realizes both min.e'Pe and 
min. T'T. (19) 
It is obvious from (19) that the MP inverse enables the 
free network to be adjusted by adding the condition of 
minimizing the norm of unknown variables. This condition 
is equivalent to the restriction of the rotation and translation 
of the free network. Therefore, MP inverse gives the 
adjusted coordinates without any fixed coordinate. 
In addition, Mittermayer (1972) proved thatthe optimization 
of (16) and (19) is equivalent to the following optimization; 
min, trace (2) (20) 
where 27 isthe variance-covariance matrix of the adjusted 
coordinates. This means that the errors of the coordinates 
are fairly minimized. The least squares MDS based on 
MP inverse is regarded as the most appropriate one to 
MDS for time-space mapping, in which the interpolation 
procedure is requested as the next task. 
Since the calculation of MP inverse is somewhat 
complicated, the approximate calculation procedure is 
provided (Rao and Mitra, 1971). With this, the MP inverse 
of an arbitrary matrix M is calculated by 
M* z lim (M'M «I ) M*. (21) 
795 
Applying the formula (21) to (18), we get the adjusted 
coordinates. The unbiased estimate of the variance of 
unit weight is given by 
oz. .£íPe 
m - rank (A) 
zc € Pg » 
m - (2n - 3) 
(22) 
Therefore, the variance-covariance matrix of the estimated 
coordinates is obtained by the error propagation law as 
follows; 
Zp20*(AÍPA) (23) 
5. INTERPOLATION PROCEDURE BASED ON 
ADJUSTED COORDINATES 
The configured points (uj, v;) (i21,2..,2) by the MDS are 
used as the control points in the interpolation. Define the 
coordinates of the physical map by (x;,y;) The following 
interpolation functions are calibrated by the least squares 
method based on the control points. 
u =f(e,y), v =g@,y) (24) 
Only for convenience, assume the interpolation functions 
to be linear to the unknown parameters. Of course, the 
functions do not need to be linear to the (x,y) coordinates. 
Let & represent the residual vector of the interpolation 
functions as follows; 
t 
& =(6; 150 Cup) £517 1£5) (25) 
Let the unknown parameter vector included in (24) be 
denoted by W z (w1,w5,..w,4) and the coefficient matrix 
for W be denoted by B. Then the observation equations 
of the interpolation problem is given by 
& =BW -S, (26) 
where S is the estimated coordinates when the adjusted 
terms, T , converges. 
Note that & depends on 27 (=25 ): hence the least 
squares estimates of W are 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996