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 
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following S(a) is the storage space required by the model type 
a. To compute it, the following formulas are used. 
S(DTMa)= S(metadata DTM a) + S(data DTM a) 
To georeference a grid, the minimum needed metadata are four 
geographic coordinates and two integers: typically, the X 
(LL X) e Y (LL Y) coordinates of the lower left node of the 
grid, the spacing in X and Y between the nodes (DX and DY) 
and the number of the nodes in X and Y directions (N X and 
N Y) are provided. Other ways can be adopted but the number 
of minimally needed metadata does not change. Moreover, a 
field should be devoted to the conventional identifier of no-data 
(ND). So, the following holds 
S(metadata GR1D ) = 
S(LLJi)+S(LL_Y)+S(DX)+S(DY)+S(N_X)+S(N_Y)+S(ND) 
= 7><64 bits = 56 bytes 
S(data GR1D ) = N X x N_Y x S(height) = N x 8 bytes 
As TINs are concerned, the minimal model, without additional 
topological information, is discussed. 
S(metadata xlN ) = S(N V)+S(N F) 
S(data-n N )=S(data N o DE s)+S(data FAC Es) = 
N_Vx3x64bits+N_Fx3x Ceil[log 2 (N_V)]bits 
N_V and N_F are the number of vertices and faces. The 
vertices are stored as 3D points (X, Y and height). The 
triangular faces are stored simply by the list of the three 
relevant vertices. Consider that k bits can address 2 k points: if 
the number of vertices is N V, the size in bits needed for each 
label is Ceil(log 2 (N_V)), where Ceil is the rounding to the 
greater integer. 
DTM mr requires the storing of metadata relevant to the 
coefficients, that are needed to define the position and the 
resolution of each activated spline. Once defined the global 
interpolation domain (lower left and upper right comers), the 
record corresponding to a particular level is stored in the 
following way: 
Nh ’ h x ’ hy ’ ^hjlx.ily ’ h x ’ fv,,X ’ *N h r ’ ^h,iN x ,iNy 
where N h is the number of activated splines in the level; i h 
and ij are the row and column indexes of the node occupied 
by the J-th spline; u is the coefficient of the J-th spline. 
At level h, the maximum number of active splines is 
A^max ~(2 A+I +1) 2 • It is easy to show that 
Ci?//|log 2 (2 ; ' 1 + l)j = (/z + 2). The storage requirements for 
level h ,(h = 0,...M ) are: 
• 2x(h + 2) bits to store the number of active splines, 
. {h + 2) x 2 x N h bits needed to store the row and column 
indexes, 
• 64 x N h bits needed to store the coefficients. 
3. A CASE STUDY 
In order to evaluate the proposed approach and compare it with 
the data based models, we have analyzed one case study. The 
data stem from a LiDAR survey of a promontory overlooking 
the lake of Como in Northern Italy. The horizontal spacing of 
the pre-processed grid is 2 m x 2 m and its vertical accuracy 
(Rood, 2004) is of about 20 cm. 
The first step is to extract three different samples in order to 
simulate three dataset of sparse observations with different 
accuracies from which extract the relevant DTMs. 
For this reason, four TINs have been extracted from the grid, 
with different sampling tolerances. By fixing the tolerance 
equal to 5m, 2m and lm, we have created respectively the 
training datasets TR5, TR2 and TR1 containing scattered data 
(i. e. the nodes of the TIN's) . By fixing the tolerance equal to 
20 cm (and removing TR5, TR2 and TR1), we have finally 
created the test dataset TE to use for cross-validate the results. 
The original dataset is shown in Figure 2. In Table 1 the 
statistics of the datasets are reported. 
Figure 2. The original DTM 
Using the three training sets as raw observations, TIN and grid 
models corresponding to the height accuracies of 1, 2 and 5 m 
have been built. By construction, the training sets directly 
provide the DTM T1N at the different accuracy levels: in Table 2 
the storage requirements of the DTM tin are reported. 
To produce DTM GR1D , five deterministic interpolation 
techniques have been tested: the Inverse Distance Weighting 
(IDW), the 1° Order Local Polynomial (POL), the Completely 
Regularized Spline (CRS), the Spline with Tension (SWT) and 
the Thin Plate Spline (TPS). The interpolations have been 
computed in ArcGIS, applying the parameters automatically 
optimized by the software itself. 
DTM 
TR5 
TR2 
TRI 
TE 
Count 
422610 
3274 
9256 
21656 
81869 
Min 
197.44 
197.44 
197.44 
197.44 
197.47 
Max 
332.27 
332.27 
332.27 
332.27 
332.23 
Mean 
225.27 
214.33 
225.75 
230.81 
235.59 
RMS 
27.80 
28.58 
30.85 
30.36 
27.83 
Table 1. Statistics of the sampled datasets. Values in m. 
TR 
N_V 
S_V 
(bytes) 
N_F 
S_F 
(bytes) 
S 
(KB) 
lm 
21656 
519744 
41343 
294374 
795 
2m 
9256 
222144 
16580 
110430 
325 
5m 
3274 
78576 
4644 
28320 
104 
Table 2. Characteristics of the three sampled TIN. N_V: 
number of vertices; S_V: storage space for vertices; N_F: 
number of faces; S_F: storage space for faces; S: total storage 
size. 
If both the accuracy and the storage size of a grid have to be 
considered, the optimal compromise is given by the coarser 
grid that guarantees the desired accuracy. Therefore, different