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 
following S(a) is the storage space required by the model type 
a. To compute it, the following formulas are used. 
S(DTMa)- S(metadatap qa) + S(dataprm) 
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(metadataggip) = 
S(LL_X)+S(LL Y)+S(DX)+S(DY)+S(N_X)+S(N_Y)+S(ND) 
= 7x64 bits = 56 bytes 
S(dataggup) " N Xx N Y x S(height) = N x 8 bytes 
As TINs are concerned, the minimal model, without additional 
topological information, is discussed. 
S(metadatarm) = S(N_V)+S(N_F) 
S(datarmn)=S(datanopes)+S(dataraces) = 
N _Vx3x64bits+N_Fx3x Ceil[log.(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* points: if 
the number of vertices is N V, the size in bits needed for each 
label is Ceil(log,(N V)), where Ceil is the rounding to the 
greater integer. 
DTMyg 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 corners), the 
record corresponding to a particular level is stored in the 
following way: 
N,, h.e h, x A 
hy ily o I oon Iu xo 
Inv > Ain, iN, 
where N, is the number of activated splines in the level; 7; 
and i, are the row and column indexes of the node occupied 
by the J-th spline; 4, ,, is the coefficient of the J-th spline. 
At level h, the maximum number of active splines is 
N.e(QU ly. "It "is easy “to show that 
Ceil (log, [^ + » =(h+2). The storage requirements for 
level h,(4#=0,...M ) are: 
° 2x(h+2) bits to store the number of active splines, 
. (h--2)x2x N, bits needed to store the row and column 
indexes, 
e 64x NN, 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 1m, we have created respectively the 
training datasets TRS, 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 TRI), 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. 
    
Height (meters) 
Elevation 
at? 
  
  
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 DTMqq at the different accuracy levels: in Table 2 
the storage requirements of the DTM are reported. 
To produce DTMgmgp, 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 TRS TR2 TRI TE 
Count| 422610 3274 9256. | 21656 81869 
Min 197.44 | 197.44 | 197.44 | 197.44 | 197.47 
Max | 33227.| 33227 | 33227. | 33227 | 33223 
Mean | 225.27 | 214.33 | 225.75 | 23031 | 2335.39 
RMS 27.80 28.58 30.85 30.36 27.83 
Table 1. Statistics of the sampled datasets. Values in m. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
SV SF S 
TR NY (bytes) NE (bytes) (KB) 
Im |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 
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