International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B4, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
adopted to store and compute DTMs. Section 2 discusses our 
new approach and its storage requirements. In Sect. 3, the 
performances of the different models are analyzed, by 
comparing their application to a case study. The conclusions 
and the future developments follow. 
1.1 Data based Models 
Different data models can be adopted to store and transmit a 
DTM. Contour lines, grids (or elevation matrix) and Triangular 
Irregular Networks (TIN) are the standard. 
Contour lines are obtained by connecting with a line all the 
points with the same height. The lines are drawn at given, 
equally spaced in height, intervals. Contour lines are useful to 
visualize heights on maps in 2D applications, but seldom are 
used for analysis purposes, and are stored and transmitted 
following the general rules of vector objects. 
Gridded DTMs (in the following DTMGrip) are georeferenced 
as regular grids of nodes, whose heights are stored. The storage 
of a grid requires a set of metadata that allow its georeferencing 
(see Sect. 3) and are listed in the so called header. The heights 
are stored in an ordered sequence. DTMorm is a very simple 
conceptual model and can be casily accessed, visualized and 
spatially analyzed by map algebra. However, the choice of the 
grid resolution is a crucial point, because the storage size is 
inversely proportional to the square of the gridding interval. If 
rough terrain (for example mountains) alternates to flat terrain 
(plains), the high resolution needed to accurately describe the 
first causes a useless redundancy in the second. To 
continuously describe the heights between the nodes, either a 
bilinear or a bicubic interpolation is typically applied. 
In a TIN, the DTM (in the following DTMqq) is described by a 
set of planar triangular faces that are obtained by connecting 
sparse points, whose horizontal coordinates and heights are 
given. Usually, the Delaunay criterion is applied to triangulate 
the points. By a TIN model, more points can be stored where 
the terrain is rough and less points are used in flat areas. Each 
point of a TIN is represented by its three (X, Y, height) 
coordinates. Moreover, to reconstruct the topology of the 
triangles, the labels of the three vertices of each triangle are 
needed. This simple data model requires long computation 
times for the processing and the analysis of the 3D surface: 
therefore, in the practice, more complex topological models are 
applied, like for example the node based, the triangle based or 
the edge based data structures. These models reduce 
computation times but require an overhead of information that 
is stored and transmitted to the clients. When a TIN model is 
used, the height within each triangle is linearly interpolated 
from its three vertices. 
1.2 Interpolation techniques 
To produce a DTM, several interpolation techniques exist: a 
first classification can be into exact and approximate 
interpolations. An exact interpolator passes for all the 
observations and allows the complete reconstruction of all the 
discontinuities existing in the dataset. However, the observation 
errors are not filtered and propagate into the model. A classical 
example of exact interpolator is given by the /nverse Distance 
Weighting (IDW). Approximate interpolators apply statistical 
methods to estimate a smoother function from the observations: 
in this way, the errors can be filtered and both the observations 
accuracy and the function correctness can be assessed. 
However, actual details and discontinuities can be lost in the 
smoothing. Local Polynomial (POL) is an approximate 
interpolator when the coefficients are fewer than the 
observations and are estimated by least squares. 
In the deterministic interpolation, either exact or approximate, 
the analytical model of the surface is a priori chosen and the 
observations are used to estimate it: IDW and POL are 
examples of deterministic interpolators. In the stochastic 
interpolation  (Christakos, 1992), the observations are 
considered as a sample of a random field (the surface) that is 
completely described by spatial stochastic properties like, for 
example, the covariance function. The stochastic properties are 
estimated analyzing the observations and then applied to 
interpolate the surface. Collocation and kriging are the classical 
examples of stochastic interpolators. 
Note that the most popular interpolation techniques, as reported 
in scientific and technical literature, cannot be easily and 
efficiently used to implement an analytical model because the 
interpolating functions cannot be described by a small number 
of parameters or coefficients. In IDW and POL, the 
interpolation coefficients and domain are a function of the 
positions of both the interpolation point and the observations: 
to reply the model, all the observations must be stored and 
distributed. Radial Basis technique uses a linear combination of 
radial functions that interpolate exactly the observations and 
are characterized by the minimum curvature. These methods 
(Regularized Spline, Spline with Tension, Thin Plate Spline) 
differ in the function choice, all of them could be analytically 
described by a finite set of coefficients but the needed 
coefficients are at least as many as the raw observations. 
Let consider a stochastic interpolator, for example the 
collocation. The height in a point is provided by the 
h(P) =c'&, where & is the vector of the observations 
multiplied by its inverse covariance matrix, c is the cross- 
covariance vector between the point and the observations. c 
can be built by knowing the covariance function of the surface 
and the positions of the observations, while & needs to be 
stored: also in this case, an analytical model would require as 
many data as the original observations. 
The classical bilinear splines estimated by least squares provide 
a twofold interpretation, because they can be thought as both 
data based and analytical models. Given the required spatial 
resolution, the observations are interpolated to estimate the 
coefficients of the splines, that are used to predict the heights 
on a regular grid, that represents the data based model. If the 
splines and the grid have the same spatial resolution, the 
coefficients of the splines and the heights of the relevant grid 
nodes are equal. Moreover, the coefficients of the bilinear 
splines used to interpolate from the four neighboring nodes of a 
regular grid are exactly the relevant four heights: indeed each 
local bilinear spline assumes the maximum in its node, while 
annihilates on all the other nodes. In this case, the analytical 
model has exactly the same complexity of the data based 
model. 
The adoption of a new multi-resolution splines interpolation 
has been studied, that represents a true analytical model and 
provides actual storage and distribution saving with respect to a 
data based model. 
2. THE MULTI-RESOLUTION SPLINES APPROACH 
The approach has been previously discussed in a preliminary 
way by (Brovelli and Zamboni, 2009): it is an approximate 
deterministic method whose estimation principle is based on 
Least Squares (LS, Kock, 1987). 
The founding idea is to combine splines with different width in 
order to guarantee the resolution adequate to the data density in 
every region of the field, exploiting all available information 
implicitly stored in the sample. Different levels of splines, 
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