Rapid evaluation of Daubechies Scaling function
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Let the Daubechies 4-tap scaling function $phiin C_{0}([0,3])$ be defined by
begin{align}
phi(x) &= frac{1+sqrt{3}}{4} phi(2x) + frac{3+sqrt{3}}{4}phi(2x-1) + frac{3-sqrt{3}}{4} phi(2x-2) + frac{1-sqrt{3}}{4}phi(2x -3) \
phi(1) &= frac{1+sqrt{3}}{2} qquad phi(2) = frac{1-sqrt{3}}{2}
end{align}
This definition allows us to compute $phi$ on the dyadic numbers $mathbb{D}_{j}$ (numbers of the form $k/2^{j}$), but to evaluate any $x in mathbb{D}_{j}$, first the values from all previous levels must be evaluated and stored, making this algorithm require $mathcal{O}(2^{j})$ storage. If we are to recover even float precision, storage requirements become truly obscene. If instead of storing all possible floats in the interval $(0,3)$, we store $j_{max}$ levels, and then interpolate, we trade some storage for computation, but since the 4-tap scaling function is only continuous (not differentiable), smooth interpolators do the wrong thing, and (from numerical experiments I've performed) linear interpolation is really not that much help.
Another option is proposed here, equation 120:
begin{align*}
phi(n/2^{m}) = sum_{ell_1, ell_2,cdots, ell_m} 2^{m/2}h_{ell_1}h_{ell_2}cdots h_{ell_m}phi(n- 2^{m-1}ell_1 - 2^{m-2} ell_2 -cdots 2ell_{m-1} - ell_{m})
end{align*}
This requires no storage, but requires $4^{m}$ computations. For a 32 bit float, the worst case is $4^{23}$ operations.
Are there more efficient algorithms to evaluate $phi$?
reference-request computational-mathematics wavelets
asked Nov 21 at 19:36
user14717
3,7731020
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Let the Daubechies 4-tap scaling function $phiin C_{0}([0,3])$ be defined by
begin{align}
phi(x) &= frac{1+sqrt{3}}{4} phi(2x) + frac{3+sqrt{3}}{4}phi(2x-1) + frac{3-sqrt{3}}{4} phi(2x-2) + frac{1-sqrt{3}}{4}phi(2x -3) \
phi(1) &= frac{1+sqrt{3}}{2} qquad phi(2) = frac{1-sqrt{3}}{2}
end{align}
This definition allows us to compute $phi$ on the dyadic numbers $mathbb{D}_{j}$ (numbers of the form $k/2^{j}$), but to evaluate any $x in mathbb{D}_{j}$, first the values from all previous levels must be evaluated and stored, making this algorithm require $mathcal{O}(2^{j})$ storage. If we are to recover even float precision, storage requirements become truly obscene. If instead of storing all possible floats in the interval $(0,3)$, we store $j_{max}$ levels, and then interpolate, we trade some storage for computation, but since the 4-tap scaling function is only continuous (not differentiable), smooth interpolators do the wrong thing, and (from numerical experiments I've performed) linear interpolation is really not that much help.
Another option is proposed here, equation 120:
begin{align*}
phi(n/2^{m}) = sum_{ell_1, ell_2,cdots, ell_m} 2^{m/2}h_{ell_1}h_{ell_2}cdots h_{ell_m}phi(n- 2^{m-1}ell_1 - 2^{m-2} ell_2 -cdots 2ell_{m-1} - ell_{m})
end{align*}
This requires no storage, but requires $4^{m}$ computations. For a 32 bit float, the worst case is $4^{23}$ operations.
Are there more efficient algorithms to evaluate $phi$?
reference-request computational-mathematics wavelets
asked Nov 21 at 19:36
user14717
3,7731020
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let the Daubechies 4-tap scaling function $phiin C_{0}([0,3])$ be defined by
begin{align}
phi(x) &= frac{1+sqrt{3}}{4} phi(2x) + frac{3+sqrt{3}}{4}phi(2x-1) + frac{3-sqrt{3}}{4} phi(2x-2) + frac{1-sqrt{3}}{4}phi(2x -3) \
phi(1) &= frac{1+sqrt{3}}{2} qquad phi(2) = frac{1-sqrt{3}}{2}
end{align}
This definition allows us to compute $phi$ on the dyadic numbers $mathbb{D}_{j}$ (numbers of the form $k/2^{j}$), but to evaluate any $x in mathbb{D}_{j}$, first the values from all previous levels must be evaluated and stored, making this algorithm require $mathcal{O}(2^{j})$ storage. If we are to recover even float precision, storage requirements become truly obscene. If instead of storing all possible floats in the interval $(0,3)$, we store $j_{max}$ levels, and then interpolate, we trade some storage for computation, but since the 4-tap scaling function is only continuous (not differentiable), smooth interpolators do the wrong thing, and (from numerical experiments I've performed) linear interpolation is really not that much help.
Another option is proposed here, equation 120:
begin{align*}
phi(n/2^{m}) = sum_{ell_1, ell_2,cdots, ell_m} 2^{m/2}h_{ell_1}h_{ell_2}cdots h_{ell_m}phi(n- 2^{m-1}ell_1 - 2^{m-2} ell_2 -cdots 2ell_{m-1} - ell_{m})
end{align*}
This requires no storage, but requires $4^{m}$ computations. For a 32 bit float, the worst case is $4^{23}$ operations.
Are there more efficient algorithms to evaluate $phi$?
reference-request computational-mathematics wavelets
asked Nov 21 at 19:36
user14717
3,7731020
Let the Daubechies 4-tap scaling function $phiin C_{0}([0,3])$ be defined by
begin{align}
phi(x) &= frac{1+sqrt{3}}{4} phi(2x) + frac{3+sqrt{3}}{4}phi(2x-1) + frac{3-sqrt{3}}{4} phi(2x-2) + frac{1-sqrt{3}}{4}phi(2x -3) \
phi(1) &= frac{1+sqrt{3}}{2} qquad phi(2) = frac{1-sqrt{3}}{2}
end{align}
This definition allows us to compute $phi$ on the dyadic numbers $mathbb{D}_{j}$ (numbers of the form $k/2^{j}$), but to evaluate any $x in mathbb{D}_{j}$, first the values from all previous levels must be evaluated and stored, making this algorithm require $mathcal{O}(2^{j})$ storage. If we are to recover even float precision, storage requirements become truly obscene. If instead of storing all possible floats in the interval $(0,3)$, we store $j_{max}$ levels, and then interpolate, we trade some storage for computation, but since the 4-tap scaling function is only continuous (not differentiable), smooth interpolators do the wrong thing, and (from numerical experiments I've performed) linear interpolation is really not that much help.
Another option is proposed here, equation 120:
begin{align*}
phi(n/2^{m}) = sum_{ell_1, ell_2,cdots, ell_m} 2^{m/2}h_{ell_1}h_{ell_2}cdots h_{ell_m}phi(n- 2^{m-1}ell_1 - 2^{m-2} ell_2 -cdots 2ell_{m-1} - ell_{m})
end{align*}
This requires no storage, but requires $4^{m}$ computations. For a 32 bit float, the worst case is $4^{23}$ operations.
Are there more efficient algorithms to evaluate $phi$?
reference-request computational-mathematics wavelets
reference-request computational-mathematics wavelets
asked Nov 21 at 19:36
user14717
3,7731020
asked Nov 21 at 19:36
user14717
3,7731020
edited Nov 21 at 19:49
asked Nov 21 at 19:36
user14717
3,7731020
asked Nov 21 at 19:36
user14717
3,7731020
asked Nov 21 at 19:36
user14717
3,7731020
3,7731020
add a comment |
add a comment |
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