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Simple Function. Complicated Function. Signal Image Solution to PDE. Polynomials Splines Rational Func. Metric:. Approximation Theory. 1. Linear space of dimension n. 2. Nonlinear manifold of dimension n. 3. Highly nonlinear: Highly
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Simple Function Complicated Function Signal Image Solution to PDE Polynomials Splines Rational Func Metric: Approximation Theory
1. Linear space of dimension n. 2. Nonlinear manifold of dimension n. 3. Highly nonlinear: Highly redundant dictionary. Functions g chosen from:
Linear: , , , . . . . . . , n 2 1 Examples: (i) . -- Alg. poly. of degree (ii) -- Trig. poly. of degree . Splines -- piecewise poly. of degree r, pieces. (iii) 0 1 (iv) , span CONS
0 1 pieces I N Nonlinear : n dimensional manifold (i) . : Rational function (ii) free knots. Splines with (iii) - term approximation CONS
arbitrary, , Bases B1, B2, . . . Bm, . . . best n-term Bj Highly Nonlinear choose best basis choose n-term approximation
Characterize Main Question We shall restrict ourselves to approximation by piecewise constants in what follows.
Linear Piecewise Constants 0 1/n 1 Theorem (DeVore-Richards) Fix . close to
, , Theorem (DeVore-Richards) . for
Nonlinear Linear . Noninear Theorem (Kahane) . Know (Petrushev)
1 0 I n - term 1 Haar Basis 1 0 -1 Dyadic Interval
Theorem (DeVore-Jawerth-Popov) known. Simple strategy: Choose n terms where largest.
Nonlinear Linear
Image Application Image Compression Piecewise constant function (Haar) Threshold Problem:Need to encode positions. Dominate Bits
Cohen-Dahmen-Daubechies-DeVore: are almost the same requirements. Tree Approximation
Generate tree as follows: 1) Threshold: 2) Complete to Tree: 1 1 3) Encode the subtree: 1 1 0 0 (Each bit tells whether the child is in the tree.) 1 1 0 0 0 0 0 0
Progressive • Universal • Optimal • Burn In Features of Tree Encoder
Position Bits of = P k B B P B B B P P B . . . 00 10 0 11 20 21 22 1 2 Encoder } B { bit b of = , jk j
Cohen-Dahmen-DeVore Elliptic Equation Wavelet transform gives - positive definite. - has decay properties. CDDgives an adaptive algorithm
, then the Theorem If adaptive algorithm produces : Theorem If , then using n computations the adaptive algorithm produces :
residual Refinement:Let be the smallest Error: “Error Indicators”: set of indices such that . Define new set