# Difference between revisions of "Short Notes on Wavelets"

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+ | 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. | ||

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+ | Both versions could be parallelised and made into in-place transforms. Enjoy! | ||

<pre>def haar_int_fwd_2d_np(d): | <pre>def haar_int_fwd_2d_np(d): |

## Latest revision as of 13:11, 27 February 2017

## 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)