138
DYNAMIC MODELLING OF SCENE DEFORMATION
FOR CRUSTAL MOVEMENT MONITORING
Bruno Crippa, Eva Savina Malinverni, Luigi Mussio
DIIAR. Politecnico di Milano, P.zza Leonardo da Vinci 32, 20133 Milano (Italy).
Phone: ++39.2.23996501, Fax: ++39 2.23996530, e-mail: Bruno@ipmtf2.topo.polimi.it
KEY WORDS: Deformation Monitoring, Dynamic Modelling, Scene Analysis.
ABSTRACT.
A statistical and numerical procedure on dynamic modelling of scene deformation is presented and applied to crustal movement
monitoring. The same procedure may be applied, with minor changes, to different fields of application, like image sequence
analysis, navigation and fusion of various information.
The procedure consists of the adjustment of observables of photogrammetry or other sensors and systeins, taking into account
displacements of a certain number of points and some shape descriptors to give a linkage among one point and the neighbouring
ones. Therefore this procedure performs a kinematic adjustment and a spatial analysis of the displacements, that can be executed
together by using a modern integrated approach.
1. INTRODUCTION ( Cunietti, et al., 1985 ).
Those interested both in control measurements and in photogrammetry can often meet the following problem: is it possible to
control by photogrammetry ? The question is pertinent and senseful for two basic reasons:
» because it is possible to measure many points (even very close to each others) at the same time by photogrammetry;
» because metric images, base of photogrammetry, are a document on the state of the object to be controlled, that can be filed and
that (under certain conditions) can always be used.
A positive answer can be given; therefore it is necessary to define its terms, to analyze it in a general way and then to draw
conclusions on condition that some processes are improved.
It must be clear that the authors don't want certainly to exhaust the subject. The main task is only to report a research, that was
carried out for a long time in order to insert it rightly in the general field of photogrammetry.
A modern trend in the analysis of samples of phenomena is to model them by statistical methods: this approach can be applied also
to the control problems. Two opposite situations seems to happen more frequently:
» the number of points to be monitored is small and the rate of repetitions high;
» the number of points to be monitored is high, but the rate of repetitions is low.
The first situation occurs often when the velocity of deformation is high. Since the control network is supposed to be small, each
repetition can be executed in a very short time, so that the hypothesis, that the vertices are reasonably stable in this span of time, is
acceptable and the adjustment can be performed separately. After the adjustment the displacements of each vertex can be analysed
supposing that these are sampled from realisations of stochastic processes. Thus an interpolation removes the non-stationary trend
and a filtering splits then the stochastic signal from the random noise. In this phase the eventual linkage among the vertices can be
described by the definition of suitable crosscovariance functions. Note that the processing after the adjustment is the classical
object of the time series analysis.
The second situation occurs often in the monitoring of a slow motion. The execution of the measurements requires generally a long
time, because the control network is wide. Therefore the measurement of a repetition cannot any more be considered as executed at
the same time and the time span of a repetition is comparable to the period between two repetitions. Thus an unique adjustment of
all repetitions has to be done by applying the proper kinematic model. In this case, there is typically a small number of repetitions
that doesn't allow for a modelling in terms of time series; however a deterministic functional model per each vertex can be suitably
applied. After the adjustment the additional parameters of the functional model (i.e. velocities, accelerations, ....) can be analysed
as samples from realisations of generally stationary stochastic processes, depending on the coordinates of each vertex.
2. KINEMATIC ADJUSTMENT AND SPATIAL ANALYSIS ( Forlani, et al., 1986; De Haan, et al., 1987 ).
The movements in the slow moving control cases, where number of vertices of a control network 1s high, but the rate of repetitions
is small, often present one of the following characteristics:
» the magnitude of the shifts is always increasing (or decreasing);
» the amplitude of the shifts changes almost regularly from positive to negative and viceversa;
» acombination of the two preceding cases occurs.
The first type represents a dilatation (or a contraction) of the controlled object, following a typical law of approach to the steady
state:
s(t) = ae to) (1)
the second one represents a periodical movement expressed in the most regular case by a formula of the type:
s(f) 2acos (c(t —t9) - d) (2)
the third type combines the two preceding movements:
s(£) 2 aet? 070) cos (c(t — ty) +d) (3)
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop "From Pixels to Sequences”, Zurich, March 22-24 1995
