1
Interpolation and 3D Visualization of GeodataInterpolation and 3D Visualization of Geodata
University of Science and Technology, AGHDepartment of Geomatics
FIG Working Week, Eilat 2009
Marek Kulczycki, Marcin Ligas
M. Kulczycki, M. Ligas – Department of Geomatics
2
M. Kulczycki, M. Ligas – Department of Geomatics
No overburden with complicated formulas
( )( )( ) ( )[ ] ( )hChsZsZCov
sZE
=+=,µ ( ) ( )( )[ ]
( ) ( )( )[ ] ( )hsZhsZVsZhsZE
γ20
=−+=−+
( ) ( ) ( ) ( )[ ]( )∑ −+=hN
sZhsZhN
h 2
21γ̂ ( ) ( ) ( ) ( )
( )
( )
+
−+
=∑
hN
sZhsZhN
h hN
494.0457.0
1
ˆ2
4
21
γ
( ) ( ) ( )( )( ) ( ) ( )( )nn
nn
zhsZzhsZzhsZP
zsZzsZzsZP
<+<+<+==<<<
,...,,,...,,
2211
2211
( )
>≤<
=
+
−+=
ahah
h
ccah
ahcch
o
o 00
21
230
,3
θγ
( )( )[ ] ( ) ( )[ ]∑
=
=−n
iii
i
i hhhhN
1
22 min,ˆ
,21 θγγ
θγ
M. Kulczycki, M. Ligas – Department of Geomatics
No overburden with complicated formulas
( ) ( )( ) ( )( )( ) 01
ˆ
=−=−=
=−=−=−
1λmλsZλsZλ
TT
TT
mm
mEmEmmE
( )[ ] ( ) ( ) ( )( )( ) ( )λCλsZλ hVar
mmCovmVarmVarmmETT ==
=−+=− ,ˆ2ˆˆ 2
( ) ( )( )
=+−=∂
Ψ∂
=−=∂
Ψ∂
022,
022,
1λλ
1λCλλ
T
h
µµ
µµ
( ) ( )
( ) ( )
=
−
1000
0111
1 1
1
1111
µλ
λ
nnnn hchc
hchcM
L
L
MMOM
L
( ) ( )[ ] ( ) ( )µµµ
σ===
==−=−=
1λ1λ
λCλsTT
To hmmVarmmE ˆˆ 22
( ) ( )[ ] ( ) ( )[ ]( ) ( )[ ] ( )( ) ( )( )
( ) ( )( ) ( ) σλλCλsελ
sελsελ
sελssZ
To
To
To
To
T
oT
oo
hs
sVarVarsE
smmEZpE
2,cov2
,2
22
−+=−
−+=−=
=−−+=−
σεεε
ε
( ) ( )
( ) ( )
=
nnnnn
n
hchc
hchc
σ
σ
λ
λMM
L
MOM
L 11
1
111
( ) ( )
( ) ( )
=
−
10111
1 11
1
1111
nnnnn hchc
hchc
σ
σ
µλ
λMM
L
L
MMOM
L
( ) ( ) ( )[ ] ( )( )
µσσµσµσσ
+−=
=−++=−++=
=−+=−=
σλσλ1λσλσλ1σλ
σλλCλssZs
To
To
TTTo
T
To
Tooo hZpE
22
2, 22
3
M. Kulczycki, M. Ligas – Department of Geomatics
No overburden with complicated formulas
( )[ ] ( ) ( ) ( )( )( ) ( )λCλsZλ hVar
mmCovmVarmVarmmETT ==
=−+=− ,ˆ2ˆˆ 2
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )oo
Tooo εεmZ sβsfsss
sεβsFsεsmsZ+=+=
+=+=
( ) ( )[ ] ( ) ( )[ ]( ) ( )( ) ( ) ( )( )[ ]
( ) ( ) 0
,
=−=
=+−+=
=−=−
βsfβsFλ
sβsfsεβsFλ
sZsZλsZsZ
oTT
ooTT
oT
oo
εE
EpE
( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ]( )( ) ( )( ) ( ) ( )( )
( ) σλλCλsεsελsεsελ
sεsελsZsZλsZsZ
To
To
To
To
To
Too
h
VarVar
EEpE
2
,cov2
, 222
−+=
=−+=
−=−=−
σ
( ) ( ) ( )( ) ( ) ( )( )
=−=∂
Ψ∂
=+−=∂
Ψ∂
0sfλsFµµλ
0µsFσλCλµλ
oT
h
2,
222,
( )
( )
( )( )
( )
( )
( )
( )( )
( )
( )
( )
( )
( )
( )
( )
( )( )
( )
=
ok
o
o
n
k
nnk
k
nn
nk
n
n
nn
n
k
n
sf
sfsf
sf
sf
sf
sf
sf
sf
sf
sfsf
hc
hc
sf
sfsf
hc
hc
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
2
1
1
2
1
11
2
12
1
11
2
1
1
1
12
11
1
11
1
0
000
0
000
0
000
0
0001
1
11σ
σ
µ
µµλ
λ
( ) ( ) ( )[ ] ( ) σλλCλsZsZs To
Tooo hpE 2, 22 −+=−= σσ
A riddle, what can you see?
M. Kulczycki, M. Ligas – Department of Geomatics
-97% (reg) -97% (ran)
4
M. Kulczycki, M. Ligas – Department of Geomatics
Solution
M. Kulczycki, M. Ligas – Department of Geomatics
-50% (reg)
5
M. Kulczycki, M. Ligas – Department of Geomatics
-50% (ran)
M. Kulczycki, M. Ligas – Department of Geomatics
-75% (reg)
6
M. Kulczycki, M. Ligas – Department of Geomatics
-75% (ran)
M. Kulczycki, M. Ligas – Department of Geomatics
-90% (reg)
7
M. Kulczycki, M. Ligas – Department of Geomatics
-90% (ran)
M. Kulczycki, M. Ligas – Department of Geomatics
-97% (reg)
8
M. Kulczycki, M. Ligas – Department of Geomatics
-97% (ran)
M. Kulczycki, M. Ligas – Department of Geomatics
Comparison50% 25% 10% 3%
100%
9
M. Kulczycki, M. Ligas – Department of Geomatics
m&m_Cumulus
Thank you for your attention
Marek Kulczycki, Marcin Ligas
University of Science and Technology,Department of Geomatics
http://geomatyka.agh.edu.pl