def orthonormalize_wavelets(g0, g1, k):
'''
USE LONGFLOATS
Orthonormalize in reverse order
'''
for i in range(k - 1, -1, -1):
for attempt in range(1): # was 2
# Orthog to previous functions
for m in range(k - 1, i, -1):
dotim = dot(g0[i] + g1[i], g0[m] + g1[m])
for j in range(k):
g0[i][j] = g0[i][j] - dotim * g0[m][j]
g1[i][j] = g1[i][j] - dotim * g1[m][j]
norm = dot(g0[i] + g1[i], g0[i] + g1[i])
rnorm = norm.rsqrt()
for j in range(k):
g0[i][j] = g0[i][j] * rnorm
g1[i][j] = g1[i][j] * rnorm
def orthonormalize_wavelets(g0,g1,k):
'''
USE LONGFLOATS
Orthonormalize in reverse order
'''
for i in range(k-1,-1,-1):
for attempt in range(1): # was 2
# Orthog to previous functions
for m in range(k-1,i,-1):
dotim = dot(g0[i]+g1[i],g0[m]+g1[m])
for j in range(k):
g0[i][j] = g0[i][j] - dotim*g0[m][j]
g1[i][j] = g1[i][j] - dotim*g1[m][j]
norm = dot(g0[i]+g1[i],g0[i]+g1[i])
rnorm = norm.rsqrt()
for j in range(k):
g0[i][j] = g0[i][j]*rnorm
g1[i][j] = g1[i][j]*rnorm
def g0g1(k):
'''
USE LONGFLOATS
Compute the g0 & g1 two-scale coefficients which define the
multi-wavelet in terms of the scaling functions at the finer
level.
psi_i(x) = sqrt(2)*sum(j) g0_ij*phi_j(2x) + g1_ij*phi_j(2x-1)
Projection with sqrt(2)*phi_j(2x) and sqrt(2)*phi_j(2x-1), which
are the normalized scaling functions on the finer level, yields
g0_ij = sqrt(2)*int(psi_i(x)*phi_j(2x),x=0..1/2)
g1_ij = sqrt(2)*int(psi_i(x)*phi_j(2x-1),x=1/2..1)
With the original Alpert conditions we have
g1_ij = (-1)^(i+j+k) g0_ij from symmetry properties of the psi
but you only get this if you impose the condition of additional
vanishing moments.
The projection of the additional moments is very numerically
unstable and forces the use of higher precision for k >= 6.
'''
g0 = zeromatrix(k, k)
g1 = zeromatrix(k, k)
# Initialize the basis
rroot2 = longfloat(2).rsqrt()
for i in range(0, k):
g0[i][i] = rroot2
g1[i][i] = -rroot2
(h0, h1) = h0h1(k)
# Project out the moments == orthog to the scaling functions.
# Need to use the two-scale relation for the scaling functions.
for i in range(k):
for attempt in range(1): # was 2
for m in range(k):
dotim = dot(g0[i], h0[m]) + dot(g1[i], h1[m])
for j in range(k):
g0[i][j] = g0[i][j] - dotim * h0[m][j]
g1[i][j] = g1[i][j] - dotim * h1[m][j]
# Orthog to x^k, x^k+1, ... , x^2k-2
first = 0
root2 = longfloat(2).sqrt()
(hmom1, hmom2) = half_moments(2 * k - 2, k)
test = longfloat(1).eps().sqrt()
for power in range(k, 2 * k - 1):
# Transform to the wavelet basis
overlap = zerovector(k)
for i in range(k):
for j in range(k):
overlap[i] = overlap[i] + root2*\
(g0[i][j]*hmom1[power][j] + g1[i][j]*hmom2[power][j])
# Put first function with non-zero overlap in front of the list
for i in range(first, k):
if abs(overlap[i]) > test: # Is there a better test?
(g0[first], g0[i]) = (g0[i], g0[first])
(g1[first], g1[i]) = (g1[i], g1[first])
(overlap[first], overlap[i]) = (overlap[i], overlap[first])
break
# Project out the first component as necessary.
if abs(overlap[first]) > test: # Is there a better test?
for i in range(first + 1, k):
scale = overlap[i] / overlap[first]
for j in range(k):
g0[i][j] = g0[i][j] - scale * g0[first][j]
g1[i][j] = g1[i][j] - scale * g1[first][j]
first = first + 1
orthonormalize_wavelets(g0, g1, k)
return (g0, g1)
def g0g1(k):
'''
USE LONGFLOATS
Compute the g0 & g1 two-scale coefficients which define the
multi-wavelet in terms of the scaling functions at the finer
level.
psi_i(x) = sqrt(2)*sum(j) g0_ij*phi_j(2x) + g1_ij*phi_j(2x-1)
Projection with sqrt(2)*phi_j(2x) and sqrt(2)*phi_j(2x-1), which
are the normalized scaling functions on the finer level, yields
g0_ij = sqrt(2)*int(psi_i(x)*phi_j(2x),x=0..1/2)
g1_ij = sqrt(2)*int(psi_i(x)*phi_j(2x-1),x=1/2..1)
With the original Alpert conditions we have
g1_ij = (-1)^(i+j+k) g0_ij from symmetry properties of the psi
but you only get this if you impose the condition of additional
vanishing moments.
The projection of the additional moments is very numerically
unstable and forces the use of higher precision for k >= 6.
'''
g0 = zeromatrix(k,k)
g1 = zeromatrix(k,k)
# Initialize the basis
rroot2 = longfloat(2).rsqrt()
for i in range(0,k):
g0[i][i] = rroot2
g1[i][i] = -rroot2
(h0, h1) = h0h1(k)
# Project out the moments == orthog to the scaling functions.
# Need to use the two-scale relation for the scaling functions.
for i in range(k):
for attempt in range(1): # was 2
for m in range(k):
dotim = dot(g0[i],h0[m])+dot(g1[i],h1[m])
for j in range(k):
g0[i][j] = g0[i][j] - dotim*h0[m][j]
g1[i][j] = g1[i][j] - dotim*h1[m][j]
# Orthog to x^k, x^k+1, ... , x^2k-2
first = 0
root2 = longfloat(2).sqrt()
(hmom1,hmom2) = half_moments(2*k-2,k)
test = longfloat(1).eps().sqrt()
for power in range(k,2*k-1):
# Transform to the wavelet basis
overlap = zerovector(k)
for i in range(k):
for j in range(k):
overlap[i] = overlap[i] + root2*\
(g0[i][j]*hmom1[power][j] + g1[i][j]*hmom2[power][j])
# Put first function with non-zero overlap in front of the list
for i in range(first,k):
if abs(overlap[i]) > test: # Is there a better test?
(g0[first],g0[i]) = (g0[i],g0[first])
(g1[first],g1[i]) = (g1[i],g1[first])
(overlap[first],overlap[i]) = (overlap[i],overlap[first])
break
# Project out the first component as necessary.
if abs(overlap[first]) > test: # Is there a better test?
for i in range(first+1,k):
scale = overlap[i]/overlap[first]
for j in range(k):
g0[i][j] = g0[i][j] - scale*g0[first][j]
g1[i][j] = g1[i][j] - scale*g1[first][j]
first = first + 1
orthonormalize_wavelets(g0,g1,k)
return (g0, g1)
