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Approximation Theory
Simple Function
Complicated Function
Polynomials
Splines
Rational Func
Signal
Image
Solution to PDE
Metric:
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Functions g chosen from:
1. Linear space
of
dimension n.
2. Nonlinear manifold
of
dimension n.
3. Highly nonlinear: Highly
redundant dictionary.
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Linear:
,
2,
1
. . .,
n ,. . .
Examples:
(i)
-- Alg. poly. of degree
(ii)
-- Trig. poly. of degree
(iii) Splines -- piecewise poly. of
degree r, pieces.
0
1
(iv) span
,
CONS
3
.
.
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Nonlinear
: n dimensional manifold
(i)
: Rational function
(ii) Splines with free knots.
0
(iii)
.
1
pieces
- term approximation
CONS
IN
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Highly Nonlinear
arbitrary,
,
Bases
B1, B2, . . . Bm, . . .
best n-term
Bj
choose best basis choose n-term approximation
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Main Question
Characterize
We shall restrict ourselves to
approximation by piecewise
constants in what follows.
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Linear
Piecewise Constants
0
1/n
1
Theorem (DeVore-Richards) Fix
close to
7
.
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Theorem (DeVore-Richards)
,
for
,
.
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Noninear
Theorem (Kahane)
.
Nonlinear
Linear
Know (Petrushev)
.
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n - term
1
Haar Basis
1
0
-1
Dyadic Interval
0
10
I
1
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CONS
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Theorem (DeVore-Jawerth-Popov)
known.
Simple strategy:
Choose n terms where
largest.
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Application
Image Compression
Image
Piecewise constant function
(Haar)
Threshold
Problem: Need to encode positions.
Dominate Bits
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Tree Approximation
Cohen-Dahmen-Daubechies-DeVore:
are almost the same requirements.
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Generate tree as follows:
1) Threshold:
2) Complete to Tree:
1
3) Encode the subtree:
(Each bit tells whether
the child is in the tree.)
1
1 0
0 0
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0
1
1 0
0 1
0 0
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Features of Tree Encoder
• Progressive
• Universal
• Optimal
• Burn In
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Encoder
P0 B00 P1 B10 B11 P2 B 20 B21 B 22 . . .
Pk = Position Bits of
B jk = { bit bj of
18
,
}
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Cohen-Dahmen-DeVore
Elliptic Equation
Wavelet transform gives
- positive definite.
- has decay properties.
CDD gives an adaptive algorithm
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Theorem If
, then
using n computations the adaptive
algorithm produces :
, then the
Theorem If
adaptive algorithm produces
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:
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Error:
residual
“Error Indicators”:
be the smallest
Refinement: Let
set of indices such that
.
Define new set
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