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## Automatically adapted for scipy Oct 21, 2005 by convertcode.py
import numpy as sb
import scipy as s
def daub(p):
"""The coefficients for the FIR low-pass filter producing Daubechies wavelets.
p>=1 gives the order of the zero at f=1/2. There are 2p filter coefficients.
"""
sqrt = sb.sqrt
assert(p>=1)
if p==1:
c = 1/sqrt(2)
return sb.array([c,c])
elif p==2:
f = sqrt(2)/8
c = sqrt(3)
return f*sb.array([1+c,3+c,3-c,1-c])
elif p==3:
tmp = 12*sqrt(10)
z1 = 1.5 + sqrt(15+tmp)/6 - 1j*(sqrt(15)+sqrt(tmp-15))/6
z1c = sb.conj(z1)
f = sqrt(2)/8
d0 = sb.real((1-z1)*(1-z1c))
a0 = sb.real(z1*z1c)
a1 = 2*sb.real(z1)
return f/d0*sb.array([a0, 3*a0-a1, 3*a0-3*a1+1, a0-3*a1+3, 3-a1, 1])
elif p<35:
# construct polynomial and factor it
if p<35:
P = [s.comb(p-1+k,k,exact=1) for k in range(p)][::-1]
yj = sb.roots(P)
else: # try different polynomial --- needs work
P = [s.comb(p-1+k,k,exact=1)/4.0**k for k in range(p)][::-1]
yj = sb.roots(P) / 4
# for each root, compute two z roots, select the one with |z|>1
# Build up final polynomial
c = sb.poly1d([1,1])**p
q = sb.poly1d([1])
for k in range(p-1):
yval = yj[k]
part = 2*sqrt(yval*(yval-1))
const = 1-2*yval
z1 = const + part
if (abs(z1)) < 1:
z1 = const - part
q = q * [1,-z1]
q = sb.real(q) * c
# Normalize result
q = q / sb.sum(q) * sqrt(2)
return q.c[::-1]
else:
raise ValueError, "Polynomial factorization does not work "\
"well for p too large."
def qmf(hk):
"""Return high-pass qmf filter from low-pass
"""
N = len(hk)-1
asgn = [{0:1,1:-1}[k%2] for k in range(N+1)]
return hk[::-1]*sb.array(asgn)
def wavedec(amn,hk):
gk = qmf(hk)
return NotImplemented
def cascade(hk,J=7):
"""(x,phi,psi) at dyadic points K/2**J from filter coefficients.
Inputs:
hk -- coefficients of low-pass filter
J -- values will be computed at grid points $K/2^J$
Outputs:
x -- the dyadic points $K/2^J$ for $K=0...N*2^J-1$
where len(hk)=len(gk)=N+1
phi -- the scaling function phi(x) at x
$\phi(x) = \sum_{k=0}^{N} h_k \phi(2x-k)$
psi -- the wavelet function psi(x) at x
$\psi(x) = \sum_{k=0}^N g_k \phi(2x-k)$
Only returned if gk is not None
Algorithm:
Uses the vector cascade algorithm described by Strang and Nguyen in
"Wavelets and Filter Banks"
Builds a dictionary of values and slices for quick reuse.
Then inserts vectors into final vector at then end
"""
N = len(hk)-1
if (J > 30 - sb.log2(N+1)):
raise ValueError, "Too many levels."
if (J < 1):
raise ValueError, "Too few levels."
# construct matrices needed
nn,kk = sb.ogrid[:N,:N]
s2 = sb.sqrt(2)
# append a zero so that take works
thk = sb.r_[hk,0]
gk = qmf(hk)
tgk = sb.r_[gk,0]
indx1 = sb.clip(2*nn-kk,-1,N+1)
indx2 = sb.clip(2*nn-kk+1,-1,N+1)
m = sb.zeros((2,2,N,N),'d')
m[0,0] = sb.take(thk,indx1,0)
m[0,1] = sb.take(thk,indx2,0)
m[1,0] = sb.take(tgk,indx1,0)
m[1,1] = sb.take(tgk,indx2,0)
m *= s2
# construct the grid of points
x = sb.arange(0,N*(1<<J),dtype=sb.Float) / (1<<J)
phi = 0*x
psi = 0*x
# find phi0, and phi1
lam, v = s.linalg.eig(m[0,0])
ind = sb.argmin(sb.absolute(lam-1))
# a dictionary with a binary representation of the
# evaluation points x < 1 -- i.e. position is 0.xxxx
v = sb.real(v[:,ind])
# need scaling function to integrate to 1 so find
# eigenvector normalized to sum(v,axis=0)=1
sm = sb.sum(v)
if sm < 0: # need scaling function to integrate to 1
v = -v
sm = -sm
bitdic = {}
bitdic['0'] = v / sm
bitdic['1'] = sb.dot(m[0,1],bitdic['0'])
step = 1<<J
phi[::step] = bitdic['0']
phi[(1<<(J-1))::step] = bitdic['1']
psi[::step] = sb.dot(m[1,0],bitdic['0'])
psi[(1<<(J-1))::step] = sb.dot(m[1,1],bitdic['0'])
# descend down the levels inserting more and more values
# into bitdic -- store the values in the correct location once we
# have computed them -- stored in the dictionary
# for quicker use later.
prevkeys = ['1']
for level in range(2,J+1):
newkeys = ['%d%s' % (xx,yy) for xx in [0,1] for yy in prevkeys]
fac = 1<<(J-level)
for key in newkeys:
# convert key to number
num = 0
for pos in range(level):
if key[pos] == '1':
num += (1<<(level-1-pos))
pastphi = bitdic[key[1:]]
ii = int(key[0])
temp = sb.dot(m[0,ii],pastphi)
bitdic[key] = temp
phi[num*fac::step] = temp
psi[num*fac::step] = sb.dot(m[1,ii],pastphi)
prevkeys = newkeys
return x, phi, psi
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