multiresolution analysis of spatial environmental ... - ima.ro file1. discrete data. grid points. 2....
TRANSCRIPT
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
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
Table of Contents
1 1. Discrete Data. Grid Points.
2 2. 2D Bi-cubic Extension
3 3. Multiresolution Analysis
4 4. Numerical Results
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. 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).
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. 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).
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. 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).
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
GeneralitiesRough classification of Grid Points
regular Cartesian
y
x
Cartesian
quasi−regular
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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 ).
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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 problem
Given 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 ).
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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 ).
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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 ).
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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 ).
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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.
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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.
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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.
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
1D Essentially Non-Oscillating Extension (ENO) Algorithm
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
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
1D Essentially Non-Oscillating Extension (ENO) Algorithm
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
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
1D Essentially Non-Oscillating Extension (ENO) Algorithm
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
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
1D Essentially Non-Oscillating Extension (ENO) Algorithm
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 .
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
2D Extension SchemeData Input: D,N , g : N → R; (x , y) ∈ DStep 1.
x
y
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
2D Extension SchemeData Input: D,N , g : N → R; (x , y) ∈ DStep 2.
x
y
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
2D Extension SchemeData Input: D,N , g : N → R; (x , y) ∈ DStep 3. Data Output: g(x , y)
x
y
g(x,y)
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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).
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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 .
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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 .
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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.
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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.
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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.
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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.
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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.
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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).
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
Ampoi’s hydrographic basin - ENO.
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
Ampoi’s hydrographic basin - CRS.
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
Ampoi’s hydrographic basin - Wavelets j=4.
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
Ampoi’s hydrographic basin - Wavelets j=6.
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
Ampoi’s hydrographic basin - Wavelets j=8.
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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.
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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.
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
Thank you for your kindattention!
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data
1. Discrete Data. Grid Points.2. 2D Bi-cubic Extension
3. Multiresolution Analysis4. Numerical Results
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.
S. Ion, D. Marinescu, V. Iordache, S.G. Cruceanu Multiresolution Analysis of Spatial Environmental Data