Short Notes on Wavelets
Contents
Integer Haar Wavelets, Python implementation
The code provide is in no way optimized for speed - it's creating too many temporaries and duplicates.
This code is for illustration only, and optimization for speed is left to the reader as an exercise.
1D Case
This is a trivial implementation of Haar integer-to-integer wavelets.
The d array (list or numpy.array) has to be power of 2.
Note that resulting values typically use 1 more bit than original ones - if source values are in [0..N) interval, then resulting values are in (-N, N) interval.
Using Lists
def haar_int_fwd_1d(d):
if len(d) == 1:
return d
even = d[::2]
odd = d[1::2]
hp = [j - i for i, j in zip(even, odd)]
lp = [i + (w >> 1) + (w % 2) for i, w in zip(even, hp)]
return haar_int_fwd_1d(lp) + hp
def haar_int_inv_1d(d):
if len(d) == 1:
return d
even = haar_int_inv_1d(d[:len(d) >> 1])
odd = d[len(d) >> 1:]
lp = [i - (j >> 1) - (j % 2) for i, j in zip(even, odd)]
hp = [i + j for i, j in zip(lp, odd)]
return [x for t in zip(lp, hp) for x in t]
Using numpy Array's
import numpy as np
def haar_int_fwd_1d_np(d):
if len(d) == 1:
return d
hp = d[1::2] - d[::2]
lp = d[::2] + (hp >> 1) + (hp % 2)
return np.concatenate((haar_int_fwd_1d_np(lp), hp))
def haar_int_inv_1d_np(d):
if len(d) == 1:
return d
lp = haar_int_inv_1d_np(d[:len(d) >> 1])
hp = d[len(d) >> 1:]
even = lp - (hp >> 1) - (hp % 2)
return np.ravel(np.column_stack((even, even + hp)))
2D Extension
The list-based version is not provided.
numpy Version
This is highly un-optimized - it creates copy for each step of transformation, and both directions of transformation use np.apply_along_axis, which I suspect builds the new array row by row.
Both versions could be parallelised and made into in-place transforms. Enjoy!
def haar_int_fwd_2d_np(d):
tmp = np.apply_along_axis(haar_int_fwd_1d_np, 0, d)
return np.apply_along_axis(haar_int_fwd_1d_np, 1, tmp)
def haar_int_inv_2d_np(d):
tmp = np.apply_along_axis(haar_int_inv_1d_np, 1, d)
return np.apply_along_axis(haar_int_inv_1d_np, 0, tmp)
