139 
These models are non-linear and the search for the preliminary values presents often same difficulties. However the group of the 
first terms of the Taylor expansion of these models eliminates the problem of the search for the preliminary values and furnishes an 
other model equivalent to represent the same movements. 
If the number of the repetitions is very low or the movements are very quiet, a polynomial of third order can be fitted on the shifts 
of each vertex: 
s(t) 2a * b(t - ty) - e(t- ty - d(t - tj 2a bT -cT? «dT? (4) 
The first coefficient is constant, like the position at the reference time ( t, ); the second and the third ones represent the speed and 
the acceleration of the position variation; the last coefficient is the first variation of the acceleration: its sign is equal to the sign of 
ment the speed in the movements of first type, otherwise in opposition. 
lence If the number of repetitions is not so small or the movement is not very slow, a polynomial of higher order can be applied. Note 
that the use of spline functions of the third order must be preferred for the local regularity of the interpolating line. 
count . 
uring The procedure consists of three steps, which imply the solution of three systems of equations: 
cuted .  D)kinematic adjustment, which furnishes the kinematic parameters; the related system is formed by observation equations of 
differences of position of two points ( i, j ) at a fixed time T: 
à; -bT «GT? cd? -à; -bjT - 61? -d,T? - AS (T) - 5j (5) 
le to 
2)interpolation of the kinematic parameters by bicubic spline functions and computation of the residuals; the related system is 
formed by pseudo-observation equations: 
1 and 
draw X mannm.) =D (6) 
| was where: 
| also Bm n(k) is the value of the bicubic spline function with origin in ( m, n ) in the vertex k; 
Sm, n is the coefficient of the B, , spline function; 
each Vi is the velocity, or the acceleration, or ... of the vertex k; 
ne, is 
lysed 3)filtering of the residuals by collocation, after having estimated the covariance function of each set of residuals; the system related 
re to the filtering is formed by the following equations: 
in be 
sical A Ri M 
Sy tn. = Vi (7) 
long 
ed at While in this approach the kinematic adjustment and the spatial analysis are completely separated, an integrated approach can be 
nt of conceived, which provides for the contemporary execution of the kinematic adjustment and the spatial analysis. 
tions 
tably 3. INTEGRATED APPROACH TO KINEMATIC ADJUSTMENT AND SPATIAL ANALYSIS 
lysed ( Barzaghi, et al., 1988; De Haan, et al.,1994 ). 
The integrated approach requires all the systems of equations to be solved simultaneously. To this aim, equations (6) and (7) can 
be compacted in the following pseudo-observation equation: 
tions 4 ne 
mn mann) Ye 7 Sk = Nu, (8) 
so that the unique system to be solved is: 
eady B5-n2a =0 (9) 
(1) where B is the design matrix, o? the observed quantities, 7? the vector of the residuals (noise) and 5 the vector of the parameters, 
which are both non-stochastic (positions, position variations, shape descriptors) and stochastic (signal). 
Therefore the covariance matrix of the signal, assumed to be known a priori, consists of four blocks: two diagonal blocks 
containing the covariance matrices of the stochastic and non-stochastic parameters, and two off-diagonal blocks containing the 
(2) crossvariance matrices between the two types of parameters, which are identically equal to zero. 
The block of the stochastic parameters is determined by one or more auto and crosscovariance functions, which can be estimated 
empirically by the results of preceding separate adjustments. The block of the non-stochastic parameters is a diagonal matrix, 
whose elements have to be chosen in balance with the variances of the stochastic parameters, in such a way that the solution is not 
too weak or constrained too much to other type of parameters. 
(3) The general variance of the noise also has to be known a priori; it is assumed equal to the estimated square sigma naught of the 
5 IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop "From Pixels to Sequences", Zurich, March 22-24 1995