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