Multiresolution Analysis of Spatial Environmental ... - ima.ro file1. Discrete Data. Grid Points. 2....

1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension

3. Multiresolution Analysis4. Numerical Results

Multiresolution Analysis of Spatial EnvironmentalData

Stelian ION∗, Dorin MARINESCU∗, Virgil IORDACHE∗∗,Stefan-Gicu CRUCEANU∗

∗”Gh. Mihoc - C. Iacob” Institute of Mathematical Statistics andApplied Mathematics of ROMANIAN ACADEMY

∗∗Research Center for Ecological Services, University of Bucharest

Conference on Applied and Industrial MathematicsBacau, September 18-21, 2014

Partially Supported by ANCS, CNDI - UEFISCDI PNII programme, 50/2012

S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data



Table of Contents

1 1. Discrete Data. Grid Points.

2 2. 2D Bi-cubic Extension

3 3. Multiresolution Analysis

4 4. Numerical Results




GeneralitiesRough classification of Grid Points

1D case: Pi : a ≤ x0 < x1 < x2 < · · · xn ≤ bregular grid: xi+1 − xi = h, ∀i = 1, n;irregular grid.

2D case: Pi ,j = (xi , yj), i = 1, n, j = 1,m.Cartesian grid: there are two 1D grids {xi}i=1,n, {yi}i=1,mwhose Cartezian product can be identified with N = {Pi ,j};regular Cartesian grid: the two 1D grids are both regular;quasi-regular grid: the points from N can be gruped in afinite number of rows (columns) y = yj , j = 1,m(x = xi , i = 1, n).




GeneralitiesRough classification of Grid Points

regular Cartesian

y

x

Cartesian

quasi−regular




The problem of bi-cubic extension

D a rectangular domain in R2;N a finite set of points (xi , yj) in D;ωa the elements of a cell partition of D.

The problemGiven a reticulated function

g : N → R, N =⋃

i ,j {(xi , yj)} , g(xi , yj) = gi ,j ,

define the extension function

g : D → R, D =⋃

a ωa, g(x , y)∣∣∣ωa∈ π3,3

that approximates the model function G : D → R (g = G∣∣N ).






The problem

Given a reticulated function

g : N → R, N =⋃

i ,j {(xi , yj)} , g(xi , yj) = gi ,j ,


g : D → R, D =⋃

a ωa, g(x , y)∣∣∣ωa∈ π3,3







The problemGiven a reticulated function

g : N → R, N =⋃

i ,j {(xi , yj)} , g(xi , yj) = gi ,j ,


g : D → R, D =⋃

a ωa, g(x , y)∣∣∣ωa∈ π3,3





1D InterpolationGiven {(ξi , fi )}i=1,N a reticulated function.

Newton’s form of the Lagrange interpolation polynomial

P f ;(i3,i4)(i1,i2) (ξ) := fi1 + (ξ − ξi1)[ξi1 ; ξi2 ]f + (ξ − ξi1)(ξ − ξi2)[ξi1 ; ξi2 ; ξi3 ]f

+(ξ − ξi1)(ξ − ξi2)(ξ − ξi3)[ξi1 ; ξi2 ; ξi3 ; ξi4 ]f ,P f ;(i3,i4)

(i1,i2) (ξm) = fm, ∀m ∈ {i1, i2, i3, i4}.

The divided difference operator:

[ξi1 ; ξi2 ] f = fi2 − fi1ξi2 − ξi1

,

[ξi1 ; . . . ; ξin+1

]f =

[ξi2 ; . . . ; ξin+1

]f − [ξi1 ; . . . ; ξin ] f

ξin+1 − ξi1.




1D Essentially Non-Oscillating Extension (ENO) Algorithm

f

f

f

ξ ξ ξ ξ

f f

k k+2

k

k+1

k+2

k+3k−1

fk−2

k+3

k+1

ξξk−1

k−2

Q





f

f

f

ξ ξ ξ ξ

P0

Q

f f

k k+2

k

k+1

k+2

k+3k−1

fk−2

k+3

k+1

ξξk−1

k−2





f

f

f

ξ ξ ξ ξ

Q

P−1

f f

k k+2

k

k+1

k+2

k+3k−1

fk−2

k+3

k+1

ξξk−1

k−2





f

f

f

ξ ξ ξ ξ

P1

f f

k k+2

k

k+1

k+2

k+3k−1

fk−2

k+3

k+1

ξξk−1

k−2

Q





f

f

f

ξ ξ ξ ξ

P0

Q

P1

P−1

f f

k k+2

k

k+1

k+2

k+3k−1

fk−2

k+3

k+1

ξξk−1

k−2

{(ξi , fi )}i=1,N . For each Ik = [ξk , ξk+1], find Pa that solvesmin

b∈{−1,0,1}‖Pb −Q‖L2(Ik). Set f ENO := Pa on Ik .




2D Extension SchemeData Input: D,N , g : N → R; (x , y) ∈ DStep 1.

x

y




2D Extension SchemeData Input: D,N , g : N → R; (x , y) ∈ DStep 2.

x

y




2D Extension SchemeData Input: D,N , g : N → R; (x , y) ∈ DStep 3. Data Output: g(x , y)

x

y

g(x,y)




Properties of the 2D extension operator

D := {(x , y) ∈ R2 | a ≤ x ≤ b, c ≤ y ≤ d};N := {Pi ,j = (x j

i , yj) ∈ D | i = 1,Nj , j = 1,M};L∞(D) - the space of bounded functions on D;R := {g : N → R} - the space of reticulated functions.

Properties of the 2D extension operator L : R → L∞(D), L(g) = g1 L(g)(Pij) = g(Pij), ∀g ∈ R.2 L(G) = G , ∀G ∈ π3,3.3 For ENO, L(g) is always cont. with respect to y and cont.

with respect to x except for a finite number of points.4 If the 1D ENO algorithm does not have a unique solution for

a particular x , then (x , y) is possibly a discontinuity point ofg . Otherwise, g is locally continuous at (x , y).




Chui and Quak [3] MRA on a bounded interval I

The cubic spline MRA on bounded interval I relies on a set ofclosed subspaces {V j}j≥j0 and {W j}j≥j0 of functions on I with

1 V j ⊂ V j+1;

2

∞⋃j=j0

V j = L2(I);

3 V j+1 = V j ⊕W j ;4 V j0

∞⊕j=j0

W j = L2(I).

where j0 is the lowest resolution level.

Spline scaling functions {ϕjk}−3≤k≤2j−1 −→ basis for V j .

Spline wavelet functions {ψjk}−3≤k≤2j−4 −→ basis for W j .




Chui and Quak [3] MRA on a bounded interval I

As V j+1 = V j ⊕W j , any f ∈ V j+1 can be written as

f (x) =2j−1∑k=−3

f jkϕ

jk(x) +

2j−4∑k=−3

d jkψ

jk(x).

Moreover, V j+1 = V j0 ⊕W j0 ⊕W j0+1⊕ · · ·⊕W j , and thus

f (x) =2j0−1∑k=−3

f j0k ϕ

j0k (x) +

2j0−4∑k=−3

d j0k ψ

j0k (x) + · · ·+

2j−4∑k=−3

d jkψ

jk(x),

j0 - the lowest resolution level.




2D MRA on a bounded box I1 × I2

The cubic spline MRA on I1 × I2 relies on closed subspaces {V j}jand {W j}j of functions on I1 × I2 satisfying properties 1− 4 as inthe 1D case.

V j generated by {ϕjk ⊗ ϕ

jl}k,l −→ {Φj

p}p.W j generated by {ϕj

k ⊗ ψjl , ψ

jk ⊗ ψ

jl , ψ

jk ⊗ ϕ

jl}k,l −→ {Ψj

p}p.

g(x , y) =∑

pg j0

p Φj0p (x , y) +

j∑i=j0

∑p

d ipΨi

p(x , y),

where j0 is the lowest resolution level.




2D Cubic Spline Wavelet (CSW) Extension

Data Input: N ⊂ D = [0, 1]× [0, 1], g : N → R, J - max res llData Output: gCSW : D → R.

2D CSW SchemeStep 1. Use ENO (OF) to define g : [0, 1]× [0, 1] −→ R.Step 2. Choose the level J of the CSW hierarchy and solve:

g(xm, yn) =∑p

gJp ΦJ

p(xm, yn), xm = m2J , yn = n

2J .

Step 3. Set gCSW (x , y) =∑p

gJp ΦJ

p(x , y).

Step 4. MRA decomposition:

gCSW (x , y) =∑

pg j0

p Φj0p (x , y) +

J−1∑j=j0

∑p

d jpΨj

p(x , y).




Ampoi’s hydrographic basin - ENO.




Ampoi’s hydrographic basin - CRS.




Ampoi’s hydrographic basin - Wavelets j=4.




Relief from Romanian Paul’s Valley

Figure: Left - a 13× 24 regular raster data with square cells of 100 msize. Middle - a 130× 240 regular raster data with square cells of 10 msize. Right - a hexagonal raster (of 5.7735 m cell size) obtained byapplying our ENO method to the same data input as for the first figure.




Potential water accumulation zones in Paul’s Valley

Figure: Left - aerial photo for this region. Blue transparent areas -potential water determined by iterative process (cellular automata) withTarboton’s rules (left figure) for water change among square cells, andour rule (right figure) for water change among hexagonal cells.




Thank you for your kindattention!




References

S. Ion, D. Marinescu, S.G. Cruceanu, V. Iordache, A data portingtool for coupling models with different discretization needs,Accepted for publication, Environmental Modelling & Software,Elsevier, (2014).

S. Ion, D. Marinescu, Spline wavelets analysis of reticulatedfunctions on bounded interval, Mathematical Reports 4, (2002),191-205.C.K. Chui, and E. Quak, Wavelets on a Bounded Interval,Numerical Methods of Approximation Theory, (D. Braess and L.L.Schumaker, eds.), Birkhauser-Verlag, Basel, 9, (1992), 1-24.

E. Quak and N. Weyrich, Wavelets on Interval, ApproximationTheory. In: Wavelets and Applications, (S.P. Singh ed.), KluwerAcademic Publishers (1995) 247-283.


Multiresolution Analysis of Spatial Environmental ... - ima.ro file1. Discrete Data. Grid Points. 2....

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