Abstract:

In this work, we present an algorithm for phaseless reconstruction from magnitude-only wavelet coefficients. The method relies on an explicit relation between the log-magnitude and phase gradients of analytic wavelet transforms and an extension of the Phase-Gradient Heap Integration (PGHI) algorithm recently introduced for Gabor phaseless reconstruction. This relation is exact for a certain family of mother wavelets including Cauchy wavelets of arbitrary order, but only holds approximately otherwise. The presented experiments show that, in practice, the proposed wavelet PGHI method provides competitive quality for various mother wavelets. Furthermore, wavelet PGHI is a non-iterative scheme and thus computational performance is significantly better than established alternate projection methods.

The preprint is available here.

Sound examples - Experiment 1 (alpha = 30): The playback can be started by selecting one of the table cells (the cells turn yellow when the cursor hovers over them). Your browser must support HTML5 audio player. Alternativelly, the file path is shown below the player and it can be downloaded by Save Link As ...

Table 1: Reconstruction comparison

Loaded file: None
Cauchy Morlet Morse, gam=2 Morse, gam=3
Original
WPGHI
0-FGLIM
W-FGLIM
WPGHI
0-FGLIM
W-FGLIM
WPGHI
0-FGLIM
W-FGLIM
WPGHI
0-FGLIM
W-FGLIM
Alignment signals:
01 Sine wave, 1 kHz X X X X X X X X X X X X
02 Pink noise X X X X X X X X X X X X
Artificial signals:
04 Electronic gong X X X X X X X X X X X X
Wind instruments:
14 Oboe X X X X X X X X X X X X
15 Cor anglais X X X X X X X X X X X X
16 Clarinet X X X X X X X X X X X X
Percussion instruments:
27 Castanets X X X X X X X X X X X X
Keyboard instruments:
39 Grand piano X X X X X X X X X X X X
Speech:
49 Female eng. X X X X X X X X X X X X
50 Male eng. X X X X X X X X X X X X
51 Female fra. X X X X X X X X X X X X
52 male fra. X X X X X X X X X X X X
53 Female ger. X X X X X X X X X X X X
54 Male ger. X X X X X X X X X X X X
Pop music:
70 Eddie Rabbitt X X X X X X X X X X X X
Sound examples - Experiment 1 (alpha = 300): The playback can be started by selecting one of the table cells (the cells turn yellow when the cursor hovers over them). Your browser must support HTML5 audio player. Alternativelly, the file path is shown below the player and it can be downloaded by Save Link As ...

Table 2: Reconstruction comparison

Loaded file: None
Cauchy Morlet Morse, gam=2 Morse, gam=3 FBSP, xi=3 FBSP, xi=5
Original
WPGHI
0-FGLIM
W-FGLIM
WPGHI
0-FGLIM
W-FGLIM
WPGHI
0-FGLIM
W-FGLIM
WPGHI
0-FGLIM
W-FGLIM
WPGHI
0-FGLIM
W-FGLIM
WPGHI
0-FGLIM
W-FGLIM
Alignment signals:
01 Sine wave, 1 kHz X X X X X X X X X X X X X X X X X X
02 Pink noise X X X X X X X X X X X X X X X X X X
Artificial signals:
04 Electronic gong X X X X X X X X X X X X X X X X X X
Wind instruments:
14 Oboe X X X X X X X X X X X X X X X X X X
15 Cor anglais X X X X X X X X X X X X X X X X X X
16 Clarinet X X X X X X X X X X X X X X X X X X
Percussion instruments:
27 Castanets X X X X X X X X X X X X X X X X X X
Keyboard instruments:
39 Grand piano X X X X X X X X X X X X X X X X X X
Speech:
49 Female eng. X X X X X X X X X X X X X X X X X X
50 Male eng. X X X X X X X X X X X X X X X X X X
51 Female fra. X X X X X X X X X X X X X X X X X X
52 male fra. X X X X X X X X X X X X X X X X X X
53 Female ger. X X X X X X X X X X X X X X X X X X
54 Male ger. X X X X X X X X X X X X X X X X X X
Pop music:
70 Eddie Rabbitt X X X X X X X X X X X X X X X X X X
Sound examples - Experiment 1 (alpha = 3000): The playback can be started by selecting one of the table cells (the cells turn yellow when the cursor hovers over them). Your browser must support HTML5 audio player. Alternativelly, the file path is shown below the player and it can be downloaded by Save Link As ...

Table 3: Reconstruction comparison

Loaded file: None
Cauchy Morlet Morse, gam=2 Morse, gam=3 FBSP, xi=3 FBSP, xi=5
Original
WPGHI
0-FGLIM
W-FGLIM
WPGHI
0-FGLIM
W-FGLIM
WPGHI
0-FGLIM
W-FGLIM
WPGHI
0-FGLIM
W-FGLIM
WPGHI
0-FGLIM
W-FGLIM
WPGHI
0-FGLIM
W-FGLIM
Alignment signals:
01 Sine wave, 1 kHz X X X X X X X X X X X X X X X X X X
02 Pink noise X X X X X X X X X X X X X X X X X X
Artificial signals:
04 Electronic gong X X X X X X X X X X X X X X X X X X
Wind instruments:
14 Oboe X X X X X X X X X X X X X X X X X X
15 Cor anglais X X X X X X X X X X X X X X X X X X
16 Clarinet X X X X X X X X X X X X X X X X X X
Percussion instruments:
27 Castanets X X X X X X X X X X X X X X X X X X
Keyboard instruments:
39 Grand piano X X X X X X X X X X X X X X X X X X
Speech:
49 Female eng. X X X X X X X X X X X X X X X X X X
50 Male eng. X X X X X X X X X X X X X X X X X X
51 Female fra. X X X X X X X X X X X X X X X X X X
52 male fra. X X X X X X X X X X X X X X X X X X
53 Female ger. X X X X X X X X X X X X X X X X X X
54 Male ger. X X X X X X X X X X X X X X X X X X
Pop music:
70 Eddie Rabbitt X X X X X X X X X X X X X X X X X X
Sound examples - Experiment 2 (redundancy 3): The playback can be started by selecting one of the table cells (the cells turn yellow when the cursor hovers over them). Your browser must support HTML5 audio player. Alternativelly, the file path is shown below the player and it can be downloaded by Save Link As ...

Table 1: Reconstruction comparison

Loaded file: None
Cauchy Morlet Morse, gam=2 Morse, gam=3 FBSP, xi=3 FBSP, xi=5
Original
WPGHI
W-FGLIM
WPGHI
W-FGLIM
WPGHI
W-FGLIM
WPGHI
W-FGLIM
WPGHI
W-FGLIM
WPGHI
W-FGLIM
Alignment signals:
01 Sine wave, 1 kHz X X X X X X X X X X X X
02 Pink noise X X X X X X X X X X X X
Artificial signals:
04 Electronic gong X X X X X X X X X X X X
Wind instruments:
14 Oboe X X X X X X X X X X X X
15 Cor anglais X X X X X X X X X X X X
16 Clarinet X X X X X X X X X X X X
Percussion instruments:
27 Castanets X X X X X X X X X X X X
Keyboard instruments:
39 Grand piano X X X X X X X X X X X X
Speech:
49 Female eng. X X X X X X X X X X X X
50 Male eng. X X X X X X X X X X X X
51 Female fra. X X X X X X X X X X X X
52 male fra. X X X X X X X X X X X X
53 Female ger. X X X X X X X X X X X X
54 Male ger. X X X X X X X X X X X X
Pop music:
70 Eddie Rabbitt X X X X X X X X X X X X
Sound examples - Experiment 2 (redundancy 5): The playback can be started by selecting one of the table cells (the cells turn yellow when the cursor hovers over them). Your browser must support HTML5 audio player. Alternativelly, the file path is shown below the player and it can be downloaded by Save Link As ...

Table 1: Reconstruction comparison

Loaded file: None
Cauchy Morlet Morse, gam=2 Morse, gam=3 FBSP, xi=3 FBSP, xi=5
Original
WPGHI
W-FGLIM
WPGHI
W-FGLIM
WPGHI
W-FGLIM
WPGHI
W-FGLIM
WPGHI
W-FGLIM
WPGHI
W-FGLIM
Alignment signals:
01 Sine wave, 1 kHz X X X X X X X X X X X X
02 Pink noise X X X X X X X X X X X X
Artificial signals:
04 Electronic gong X X X X X X X X X X X X
Wind instruments:
14 Oboe X X X X X X X X X X X X
15 Cor anglais X X X X X X X X X X X X
16 Clarinet X X X X X X X X X X X X
Percussion instruments:
27 Castanets X X X X X X X X X X X X
Keyboard instruments:
39 Grand piano X X X X X X X X X X X X
Speech:
49 Female eng. X X X X X X X X X X X X
50 Male eng. X X X X X X X X X X X X
51 Female fra. X X X X X X X X X X X X
52 male fra. X X X X X X X X X X X X
53 Female ger. X X X X X X X X X X X X
54 Male ger. X X X X X X X X X X X X
Pop music:
70 Eddie Rabbitt X X X X X X X X X X X X
Sound examples - Experiment 2 (redundancy 10): The playback can be started by selecting one of the table cells (the cells turn yellow when the cursor hovers over them). Your browser must support HTML5 audio player. Alternativelly, the file path is shown below the player and it can be downloaded by Save Link As ...

Table 1: Reconstruction comparison

Loaded file: None
Cauchy Morlet Morse, gam=2 Morse, gam=3 FBSP, xi=3 FBSP, xi=5
Original
WPGHI
W-FGLIM
WPGHI
W-FGLIM
WPGHI
W-FGLIM
WPGHI
W-FGLIM
WPGHI
W-FGLIM
WPGHI
W-FGLIM
Alignment signals:
01 Sine wave, 1 kHz X X X X X X X X X X X X
02 Pink noise X X X X X X X X X X X X
Artificial signals:
04 Electronic gong X X X X X X X X X X X X
Wind instruments:
14 Oboe X X X X X X X X X X X X
15 Cor anglais X X X X X X X X X X X X
16 Clarinet X X X X X X X X X X X X
Percussion instruments:
27 Castanets X X X X X X X X X X X X
Keyboard instruments:
39 Grand piano X X X X X X X X X X X X
Speech:
49 Female eng. X X X X X X X X X X X X
50 Male eng. X X X X X X X X X X X X
51 Female fra. X X X X X X X X X X X X
52 male fra. X X X X X X X X X X X X
53 Female ger. X X X X X X X X X X X X
54 Male ger. X X X X X X X X X X X X
Pop music:
70 Eddie Rabbitt X X X X X X X X X X X X
Sound examples - Experiment 3: The playback can be started by selecting one of the table cells (the cells turn yellow when the cursor hovers over them). Your browser must support HTML5 audio player. Alternativelly, the file path is shown below the player and it can be downloaded by Save Link As ...

Table 1: Reconstruction comparison

Loaded file: None
Cauchy, xi=1 Cplx spline, xi=1 Cauchy, xi=3 Cplx spline, xi=3 Cauchy, xi=10 Cplx spline, xi=10
Original
WPGHI
0-FGLIM
W-FGLIM
WPGHI
0-FGLIM
W-FGLIM
WPGHI
0-FGLIM
W-FGLIM
WPGHI
0-FGLIM
W-FGLIM
WPGHI
0-FGLIM
W-FGLIM
WPGHI
0-FGLIM
W-FGLIM
Alignment signals:
01 Sine wave, 1 kHz X X X X X X X X X X X X X X X X X X
02 Pink noise X X X X X X X X X X X X X X X X X X
Artificial signals:
04 Electronic gong X X X X X X X X X X X X X X X X X X
Wind instruments:
14 Oboe X X X X X X X X X X X X X X X X X X
15 Cor anglais X X X X X X X X X X X X X X X X X X
16 Clarinet X X X X X X X X X X X X X X X X X X
Percussion instruments:
27 Castanets X X X X X X X X X X X X X X X X X X
Keyboard instruments:
39 Grand piano X X X X X X X X X X X X X X X X X X
Speech:
49 Female eng. X X X X X X X X X X X X X X X X X X
50 Male eng. X X X X X X X X X X X X X X X X X X
51 Female fra. X X X X X X X X X X X X X X X X X X
52 male fra. X X X X X X X X X X X X X X X X X X
53 Female ger. X X X X X X X X X X X X X X X X X X
54 Male ger. X X X X X X X X X X X X X X X X X X
Pop music:
70 Eddie Rabbitt X X X X X X X X X X X X X X X X X X