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• a surface referred to a plane can be analyzed by surface

reconstruction.

• the contour of a curve referred to a suitable approximation

of the contour itself can be studied by using two one-

dimensional form descriptors of the two components of the

contour.

• the surface of an object referred to a suitable

approximation of the surface itself can be studied by using

three two-dimensional form descriptors for the three

components of the surface.

Notice that figures or objects delimited respectively by a

contour or a surface are in this context simply geometric

quantities, independent of their physical nature. Therefore

contours and surfaces have a geometric meaning only and they

differ from the first two cases.

There are many mathematical approaches useful to solve

problem: in particular, the tessellation method and finite

elements (e.g. splines) represent the most interesting methods.

Stochastic and/or mixed approaches (i.e. covariance estimation

and optimal filtering and/or their suitable combination with

deterministic methods) could be applied offering more

sophisticated strategies to model the behavior of phenomena.

However whilst these methods present some advantages from

the point of view of data understanding, they suffer for some

disadvantages from the point of view of data compression.

Let remember that 3D models overcome intrinsic constraints of

2D models, for which height data are processed as attributions,

although metrical. Indeed, this approach, typical of classical

cartography, is quite limited when complex objects are to be

shown, such as hollow, non-stellar, possibly pluri-connected

bodies. Voronoi’s diagrams, triangulation/tetrahedration by

Delaunay, Thicsscns’ polygons and finite elements (splines on

triangular patches) by Bezier prove as well suited tools for the

said needs. All this is apt to modelization of objects, previously

detected by acquisition operations and recorded by restitution

techniques.

Discrete network description

A set of points may be connected by a network of straightline

segments which assumes the feature of the Delaunay’s

triangulation in 2D (tetrahedron network in 3D), if following

conditions arc satisfied:

• there exists, at least, a circle by two points which doesn’t

contain any point inside;

• three points are vertices of a triangle, if and only if there

exists a circle by them which doesn’t contain any point

inside;

• an additional condition suggest to perform equilateral

triangles, as much as possible, avoiding to have ill-

conditioned configuration.

The passage from 2D to 3D substitutes the sphere to the circle,

the tetrahedron to the triangle which connects four points,

instead of three ones, and where the regularity of the solid

elements is always highly appreciated. A set of triangles around

a points in 2D (of the tetrahedrons around a point in 3D) is

called Thiessen’ polygon (polyhedron). The complementary

region around a point passing by the mid point of each

straightline segments is called Voronoi’s or Dirichlet’s

diagrams.

Continuous field description

A set of points may be approximated by a continuous

interpolation, in forma of suitable lines in 2D and suitable lines

and surfaces in 3D. As the points arc generally irregular

distributed, particular kinds of interpolation methods should be

used.

'file Catmull-Rom’s curves and the Bezier’s splines on

triangular patches respond positively, to this aim.

Indeed the former gives a local interpolation (in 2D or in 3D),

hiking into account three points only; the curve passes throw

these points and has the tangent vector in the mid point parallel

to tlie cord vector between the beginning point and the end one.

The latter is defined on triangular patches and has the analytical

form given by Bernstein’s polynoms. The surface is obtained by

multiplying three quadratic families of curves parallel to each

edge of a triangular and it has a degree of sestic order.

In such a way, special types of piecewise polynoms are apt to

model the phenomena, according to the finite element method.

6. Robust procedures (Bcllone, ct al., 1996)

The most promising robust estimators are, among the

downweighting methods, the redescending estimators,

especially when their breakdown point is very high. In fact

outliers of bigger size (e.g. leverages) and/or in a large number

may be considered; moreover different explanations ca be set

up, when the anomalous data, after rejection, show a

homogeneous behavior.

The basic idea follows some suggestions of Hampel to

introduce a rejection point in the loss function, so that the data

outside the interval get, automatically, weight equal to zero. On

the contrary, the data inside get weight equal to one, it they

belong to the inner core of data, or ranging from one to zero, it

they stay in intermediate region of doubt.

There are many ways to concretize these suggestions. 'ITie

easiest one is represented by the Generalized M-estimators,

where some suitable weight functions correct the behavior ol

tlie M-estimators, as defined by Ilubcr. Unfortunately this

strategy (called by some authoritative authors: minimax),

although it increases the breakdown point, is unable to raise it

substantially.

'Hie most refined and conservative modality is represented by

the least median of squares, where the median of tlie squares of

the residuals in minimized in order to obtain the expected

results.

Unfortunately this strategy is, at present time, computationally

to expensive, because no efficient algorithm is known to solve

an enormous number of systems, selecting a subset ot

observations among the whole set of observations, forming a

sample whose dimension is exactly equal to the number of tlie

unknown parameters.

An advantageous alternative is represented by the least

trimmed squares, where the average of squares of tlie residuals,

belonging to the inner core of data, is minimized in order to

obtain tlie expected results.

hi other words, only a part of the observations are processed in

each step or linear adjustment. The use of a sequential updating

of a preliminary computed solution is possible alternative to

repeating many times tlie whole adjustment.

The weighted least trimmed squares could be minimized,

avoiding a rough partition between inliers and outliers. 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.

Least median of squares and least trimmed squares (or weighted

least trimmed squares) have the same breakdown point near to

05, when the number of the observations actually processed is