DiCoDiLe on text images

This example illustrates pattern recovery on a noisy text image using DiCoDiLe algorithm.

import matplotlib.pyplot as plt
import numpy as np

from dicodile import dicodile
from dicodile.data.images import fetch_letters_pami
from dicodile.update_d.update_d import tukey_window
from dicodile.utils.csc import reconstruct
from dicodile.utils.dictionary import init_dictionary
from dicodile.utils.viz import display_dictionaries

We will first load PAMI image generated from a text of 5000 characters drawn uniformly from the 4 letters P A M I and 2 whitespaces and assign it to X.

We will also load the images of the four characters used to generate X and assign it to variable D.

X_original, D = fetch_letters_pami()

Downloading data from https://s3-eu-west-1.amazonaws.com/pfigshare-u-files/26750168/text_4_5000_PAMI.npz?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAIYCQYOYV5JSSROOA/20230519/eu-west-1/s3/aws4_request&X-Amz-Date=20230519T144532Z&X-Amz-Expires=10&X-Amz-SignedHeaders=host&X-Amz-Signature=191d18eef2ccf9605e9769e1b71ce3ec5dd9989129c830669fe5a66e3fbfd0f5 (17.8 MB)


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Successfully downloaded file to /github/home/data/dicodile/images/text/text_4_5000_PAMI.npz

We will work on the copy X of the original image and we need to reshape image data X to fit to the expected signal shape of dicodile:

(n_channels, *sig_support)

X = X_original.copy()
X = X.reshape(1, *X.shape)
print(X.shape)

(1, 2321, 2004)

Reshape D to fit to dictionary format:

(n_atoms, n_channels, *atom_support)

D = D.reshape(4, 1, *D.shape[-2:])

# We pad `D` in order to have atoms with 0 on the border,
# compatible with windowed dictionary.
D = np.pad(D, [(0, 0), (0, 0), (4, 4), (4, 4)])
print(D.shape)

(4, 1, 37, 33)

Let’s display an extract of the original text image X_original and all the images of characters from D.

zoom_x = X_original[190:490, 250:750]
plt.axis('off')
plt.imshow(zoom_x, cmap='gray')

display_dictionaries(D)

• 
• 

<Figure size 640x480 with 4 Axes>

We add some Gaussian white noise with standard deviation std 3 times larger than X.std to X.

std = 3
rng = np.random.default_rng(None)

X += std * X.std() * rng.standard_normal(X.shape)

We will create a random dictionary of K = 10 patches from the noisy image.

# set number of patches
n_atoms = 10
# set individual atom (patch) size
atom_support = np.array(D.shape[-2:])

D_init = init_dictionary(X, n_atoms=n_atoms, atom_support=atom_support,
                         random_state=60)

# window the dictionary, this helps make sure that the border values are 0
atom_support = D_init.shape[-2:]
tw = tukey_window(atom_support)[None, None]
D_init *= tw

print(D_init.shape)

(10, 1, 37, 33)

Let’s display an extract of noisy X and random dictionary D_init generated from X.

zoom_x = X[0][190:490, 250:750]
plt.axis('off')
plt.imshow(zoom_x, cmap='gray')

display_dictionaries(D_init)

• 
• 

<Figure size 640x480 with 12 Axes>

Set model parameters.

# regularization parameter
reg = .2
# maximum number of iterations
n_iter = 100
# when True, makes sure that the borders of the atoms are 0
window = True
# when True, requires all activations Z to be positive
z_positive = True
# number of workers to be used for computations
n_workers = 10
# number of jobs per row
w_world = 'auto'
# tolerance for minimal update size
tol = 1e-3

Fit the dictionary with dicodile.

D_hat, z_hat, pobj, times = dicodile(X, D_init, reg=reg, n_iter=n_iter,
                                     window=window, z_positive=z_positive,
                                     n_workers=n_workers,
                                     dicod_kwargs={"max_iter": 10000},
                                     w_world=w_world, tol=tol, verbose=6)

print("[DICOD] final cost : {}".format(pobj))

[DEBUG:DICODILE] Lambda_max = 23.816136424875115
Started 10 workers in 4.22s
[INFO:DICODILE] - CD iterations 0 / 100 (0s)
[DEBUG:DICODILE] lambda = 4.763e+00

[INFO:DICOD-10] converged in 9.692s (7.917s) with 105507 iterations (13831 updates).
[DEBUG:DICODILE] Objective (z) : 3.322e+06 (12s)

[PROGRESS:Update D] 2s -   1.00% iterations (1.321e+00)
[PROGRESS:Update D] 4s -   2.00% iterations (2.180e-03)
[PROGRESS:Update D] 4s -   3.00% iterations (1.089e-03)
[PROGRESS:Update D] 5s -   4.00% iterations (1.089e-03)
[PROGRESS:Update D] 5s -   5.00% iterations (1.089e-03)
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[PROGRESS:Update D] 10s -  18.00% iterations (1.089e-03)
[PROGRESS:Update D] 11s -  19.00% iterations (1.089e-03)
[PROGRESS:Update D] 11s -  20.00% iterations (2.301e-04)
[INFO:Update D]: 21 iterations
[DEBUG:DICODILE] Objective (d) : 3.289e+06  (21s)
[INFO:DICODILE] - CD iterations 1 / 100 (39s)
[DEBUG:DICODILE] lambda = 4.763e+00

[INFO:DICOD-10] converged in 105.136s (86.757s) with 768381 iterations (162517 updates).
[DEBUG:DICODILE] Objective (z) : 3.226e+06 (110s)

[PROGRESS:Update D] 3s -   1.00% iterations (2.064e-02)
[PROGRESS:Update D] 4s -   2.00% iterations (2.181e-03)
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[PROGRESS:Update D] 19s -  37.00% iterations (1.438e-05)
[INFO:Update D]: 38 iterations
[DEBUG:DICODILE] Objective (d) : 3.218e+06  (29s)
[INFO:DICODILE] - CD iterations 2 / 100 (185s)
[DEBUG:DICODILE] lambda = 4.763e+00

[INFO:DICOD-10] converged in 98.236s (80.487s) with 729887 iterations (154125 updates).
[DEBUG:DICODILE] Objective (z) : 3.209e+06 (105s)

[PROGRESS:Update D] 3s -   1.00% iterations (1.032e-02)
[PROGRESS:Update D] 4s -   2.00% iterations (5.159e-03)
[PROGRESS:Update D] 5s -   3.00% iterations (1.363e-04)
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[PROGRESS:Update D] 6s -   5.00% iterations (6.812e-05)
[PROGRESS:Update D] 6s -   6.00% iterations (6.812e-05)
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[PROGRESS:Update D] 7s -   8.00% iterations (3.406e-05)
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[PROGRESS:Update D] 8s -  10.00% iterations (3.406e-05)
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[PROGRESS:Update D] 9s -  12.00% iterations (1.703e-05)
[PROGRESS:Update D] 10s -  13.00% iterations (1.703e-05)
[INFO:Update D]: 14 iterations
[DEBUG:DICODILE] Objective (d) : 3.206e+06  (20s)
[INFO:DICODILE] - CD iterations 3 / 100 (316s)
[DEBUG:DICODILE] lambda = 4.763e+00

[INFO:DICOD-10] converged in 84.464s (69.503s) with 492200 iterations (137070 updates).
[DEBUG:DICODILE] Objective (z) : 3.200e+06 (91s)

[PROGRESS:Update D] 3s -   1.00% iterations (2.064e-02)
[PROGRESS:Update D] 3s -   2.00% iterations (1.032e-02)
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[PROGRESS:Update D] 27s -  58.00% iterations (1.702e-05)
[PROGRESS:Update D] 27s -  59.00% iterations (1.702e-05)
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[PROGRESS:Update D] 29s -  63.00% iterations (1.702e-05)
[PROGRESS:Update D] 29s -  64.00% iterations (1.702e-05)
[INFO:Update D]: 65 iterations
[DEBUG:DICODILE] Objective (d) : 3.195e+06  (40s)
[INFO:DICODILE] - CD iterations 4 / 100 (453s)
[DEBUG:DICODILE] lambda = 4.763e+00

[INFO:DICOD-10] converged in 73.848s (62.045s) with 591172 iterations (117856 updates).
[DEBUG:DICODILE] Objective (z) : 3.187e+06 (80s)

[PROGRESS:Update D] 3s -   1.00% iterations (4.127e-02)
[PROGRESS:Update D] 4s -   2.00% iterations (2.181e-03)
[PROGRESS:Update D] 5s -   3.00% iterations (5.756e-05)
[PROGRESS:Update D] 6s -   4.00% iterations (5.756e-05)
[PROGRESS:Update D] 6s -   5.00% iterations (2.877e-05)
[PROGRESS:Update D] 6s -   6.00% iterations (2.877e-05)
[PROGRESS:Update D] 7s -   7.00% iterations (2.877e-05)
[PROGRESS:Update D] 7s -   8.00% iterations (2.877e-05)
[PROGRESS:Update D] 8s -   9.00% iterations (1.438e-05)
[PROGRESS:Update D] 8s -  10.00% iterations (1.438e-05)
[INFO:Update D]: 11 iterations
[DEBUG:DICODILE] Objective (d) : 3.178e+06  (19s)
[INFO:DICODILE] - CD iterations 5 / 100 (558s)
[DEBUG:DICODILE] lambda = 4.763e+00

[INFO:DICOD-10] converged in 69.356s (57.637s) with 419915 iterations (110032 updates).
[DEBUG:DICODILE] Objective (z) : 3.167e+06 (76s)

[PROGRESS:Update D] 4s -   1.00% iterations (1.611e-04)
[PROGRESS:Update D] 5s -   2.00% iterations (8.041e-05)
[PROGRESS:Update D] 5s -   3.00% iterations (4.011e-05)
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[PROGRESS:Update D] 7s -   7.00% iterations (2.005e-05)
[PROGRESS:Update D] 8s -   8.00% iterations (2.005e-05)
[INFO:Update D]: 9 iterations
[DEBUG:DICODILE] Objective (d) : 3.157e+06  (18s)
[INFO:DICODILE] - CD iterations 6 / 100 (658s)
[DEBUG:DICODILE] lambda = 4.763e+00

[INFO:DICOD-10] converged in 60.628s (52.248s) with 396445 iterations (94710 updates).
[DEBUG:DICODILE] Objective (z) : 3.151e+06 (67s)

[PROGRESS:Update D] 4s -   1.00% iterations (8.051e-05)
[PROGRESS:Update D] 5s -   2.00% iterations (3.997e-05)
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[INFO:Update D]: 13 iterations
[DEBUG:DICODILE] Objective (d) : 3.146e+06  (19s)
[INFO:DICODILE] - CD iterations 7 / 100 (750s)
[DEBUG:DICODILE] lambda = 4.763e+00

[INFO:DICOD-10] converged in 53.584s (45.535s) with 492653 iterations (81762 updates).
[DEBUG:DICODILE] Objective (z) : 3.144e+06 (60s)

[PROGRESS:Update D] 4s -   1.00% iterations (1.612e-04)
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[PROGRESS:Update D] 12s -  15.00% iterations (5.884e-09)
[INFO:Update D]: 16 iterations
[DEBUG:DICODILE] Objective (d) : 3.142e+06  (20s)
[INFO:DICODILE] - CD iterations 8 / 100 (835s)
[DEBUG:DICODILE] lambda = 4.763e+00

[INFO:DICOD-10] converged in 39.560s (33.106s) with 266047 iterations (60648 updates).
[DEBUG:DICODILE] Objective (z) : 3.141e+06 (46s)

[PROGRESS:Update D] 1s -   3.00% iterations (1.554e-08)
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[PROGRESS:Update D] 7s -  18.00% iterations (1.554e-08)
[PROGRESS:Update D] 7s -  19.00% iterations (1.554e-08)
[PROGRESS:Update D] 8s -  20.00% iterations (1.554e-08)
[PROGRESS:Update D] 8s -  21.00% iterations (1.554e-08)
[PROGRESS:Update D] 9s -  22.00% iterations (1.554e-08)
[PROGRESS:Update D] 9s -  23.00% iterations (1.554e-08)
[PROGRESS:Update D] 9s -  24.00% iterations (1.554e-08)
[PROGRESS:Update D] 10s -  25.00% iterations (1.554e-08)
[PROGRESS:Update D] 10s -  26.00% iterations (1.554e-08)
[PROGRESS:Update D] 11s -  27.00% iterations (1.554e-08)
[PROGRESS:Update D] 11s -  28.00% iterations (1.554e-08)
[PROGRESS:Update D] 11s -  29.00% iterations (1.554e-08)
[PROGRESS:Update D] 12s -  30.00% iterations (1.554e-08)
[PROGRESS:Update D] 12s -  31.00% iterations (1.554e-08)
[PROGRESS:Update D] 13s -  32.00% iterations (1.554e-08)
[PROGRESS:Update D] 13s -  33.00% iterations (1.554e-08)
[PROGRESS:Update D] 13s -  34.00% iterations (1.554e-08)
[PROGRESS:Update D] 14s -  35.00% iterations (1.554e-08)
[PROGRESS:Update D] 14s -  36.00% iterations (1.554e-08)
[PROGRESS:Update D] 15s -  37.00% iterations (1.554e-08)
[PROGRESS:Update D] 15s -  38.00% iterations (1.554e-08)
[PROGRESS:Update D] 15s -  39.00% iterations (1.554e-08)
[PROGRESS:Update D] 16s -  40.00% iterations (1.554e-08)
[PROGRESS:Update D] 16s -  41.00% iterations (1.554e-08)
[PROGRESS:Update D] 17s -  42.00% iterations (1.554e-08)
[PROGRESS:Update D] 17s -  43.00% iterations (1.554e-08)
[PROGRESS:Update D] 17s -  44.00% iterations (1.554e-08)
[PROGRESS:Update D] 18s -  45.00% iterations (1.554e-08)
[PROGRESS:Update D] 18s -  46.00% iterations (1.554e-08)
[PROGRESS:Update D] 19s -  47.00% iterations (1.554e-08)
[PROGRESS:Update D] 19s -  48.00% iterations (1.554e-08)
[PROGRESS:Update D] 19s -  49.00% iterations (1.554e-08)
[PROGRESS:Update D] 20s -  50.00% iterations (1.554e-08)
[PROGRESS:Update D] 20s -  51.00% iterations (1.554e-08)
[PROGRESS:Update D] 21s -  52.00% iterations (1.554e-08)
[PROGRESS:Update D] 21s -  53.00% iterations (1.554e-08)
[PROGRESS:Update D] 21s -  54.00% iterations (1.554e-08)
[PROGRESS:Update D] 22s -  55.00% iterations (1.554e-08)
[PROGRESS:Update D] 22s -  56.00% iterations (1.554e-08)
[PROGRESS:Update D] 23s -  57.00% iterations (1.554e-08)
[PROGRESS:Update D] 23s -  58.00% iterations (1.554e-08)
[PROGRESS:Update D] 23s -  59.00% iterations (1.554e-08)
[PROGRESS:Update D] 24s -  60.00% iterations (1.554e-08)
[PROGRESS:Update D] 24s -  61.00% iterations (1.554e-08)
[PROGRESS:Update D] 24s -  62.00% iterations (1.554e-08)
[PROGRESS:Update D] 25s -  63.00% iterations (1.554e-08)
[PROGRESS:Update D] 25s -  64.00% iterations (1.554e-08)
[PROGRESS:Update D] 26s -  65.00% iterations (1.554e-08)
[PROGRESS:Update D] 26s -  66.00% iterations (1.554e-08)
[PROGRESS:Update D] 26s -  67.00% iterations (1.554e-08)
[PROGRESS:Update D] 27s -  68.00% iterations (1.554e-08)
[PROGRESS:Update D] 27s -  69.00% iterations (1.554e-08)
[PROGRESS:Update D] 28s -  70.00% iterations (1.554e-08)
[PROGRESS:Update D] 28s -  71.00% iterations (1.554e-08)
[PROGRESS:Update D] 28s -  72.00% iterations (1.554e-08)
[PROGRESS:Update D] 29s -  73.00% iterations (1.554e-08)
[PROGRESS:Update D] 29s -  74.00% iterations (1.554e-08)
[PROGRESS:Update D] 30s -  75.00% iterations (1.554e-08)
[PROGRESS:Update D] 30s -  76.00% iterations (1.554e-08)
[PROGRESS:Update D] 30s -  77.00% iterations (1.554e-08)
[PROGRESS:Update D] 31s -  78.00% iterations (1.554e-08)
[PROGRESS:Update D] 31s -  79.00% iterations (1.554e-08)
[PROGRESS:Update D] 32s -  80.00% iterations (1.554e-08)
[PROGRESS:Update D] 32s -  81.00% iterations (1.554e-08)
[PROGRESS:Update D] 32s -  82.00% iterations (1.554e-08)
[PROGRESS:Update D] 33s -  83.00% iterations (1.554e-08)
[PROGRESS:Update D] 33s -  84.00% iterations (1.554e-08)
[PROGRESS:Update D] 34s -  85.00% iterations (1.554e-08)
[PROGRESS:Update D] 34s -  86.00% iterations (1.554e-08)
[PROGRESS:Update D] 34s -  87.00% iterations (1.554e-08)
[PROGRESS:Update D] 35s -  88.00% iterations (1.554e-08)
[PROGRESS:Update D] 35s -  89.00% iterations (1.554e-08)
[PROGRESS:Update D] 36s -  90.00% iterations (1.554e-08)
[PROGRESS:Update D] 36s -  91.00% iterations (1.554e-08)
[PROGRESS:Update D] 36s -  92.00% iterations (1.554e-08)
[PROGRESS:Update D] 37s -  93.00% iterations (1.554e-08)
[PROGRESS:Update D] 37s -  94.00% iterations (1.554e-08)
[PROGRESS:Update D] 37s -  95.00% iterations (1.554e-08)
[PROGRESS:Update D] 38s -  96.00% iterations (1.554e-08)
[PROGRESS:Update D] 38s -  97.00% iterations (1.554e-08)
[PROGRESS:Update D] 39s -  98.00% iterations (1.554e-08)
[PROGRESS:Update D] 39s -  99.00% iterations (1.554e-08)
[INFO:Update D] update did not converge

[INFO:Update D]: 100 iterations
[DEBUG:DICODILE] Objective (d) : 3.141e+06  (45s)
[INFO:DICODILE] - CD iterations 9 / 100 (932s)
[DEBUG:DICODILE] lambda = 4.763e+00

[INFO:DICOD-10] converged in 6.856s (5.725s) with 62873 iterations (9580 updates).
[DEBUG:DICODILE] Objective (z) : 3.141e+06 (13s)

[PROGRESS:Update D] 1s -   3.00% iterations (1.554e-06)
[PROGRESS:Update D] 2s -   4.00% iterations (1.554e-06)
[PROGRESS:Update D] 2s -   5.00% iterations (1.554e-06)
[PROGRESS:Update D] 2s -   6.00% iterations (1.554e-06)
[PROGRESS:Update D] 3s -   7.00% iterations (1.554e-06)
[PROGRESS:Update D] 3s -   8.00% iterations (1.554e-06)
[PROGRESS:Update D] 4s -   9.00% iterations (1.554e-06)
[PROGRESS:Update D] 4s -  10.00% iterations (1.554e-06)
[PROGRESS:Update D] 4s -  11.00% iterations (1.554e-06)
[PROGRESS:Update D] 5s -  12.00% iterations (1.554e-06)
[PROGRESS:Update D] 5s -  13.00% iterations (1.554e-06)
[PROGRESS:Update D] 5s -  14.00% iterations (1.554e-06)
[PROGRESS:Update D] 6s -  15.00% iterations (1.554e-06)
[PROGRESS:Update D] 6s -  16.00% iterations (1.554e-06)
[PROGRESS:Update D] 7s -  17.00% iterations (1.554e-06)
[PROGRESS:Update D] 7s -  18.00% iterations (1.554e-06)
[PROGRESS:Update D] 7s -  19.00% iterations (1.554e-06)
[PROGRESS:Update D] 8s -  20.00% iterations (1.554e-06)
[PROGRESS:Update D] 8s -  21.00% iterations (1.554e-06)
[PROGRESS:Update D] 9s -  22.00% iterations (1.554e-06)
[PROGRESS:Update D] 9s -  23.00% iterations (1.554e-06)
[PROGRESS:Update D] 9s -  24.00% iterations (1.554e-06)
[PROGRESS:Update D] 10s -  25.00% iterations (1.554e-06)
[PROGRESS:Update D] 10s -  26.00% iterations (1.554e-06)
[PROGRESS:Update D] 11s -  27.00% iterations (1.554e-06)
[PROGRESS:Update D] 11s -  28.00% iterations (1.554e-06)
[PROGRESS:Update D] 11s -  29.00% iterations (1.554e-06)
[PROGRESS:Update D] 12s -  30.00% iterations (1.554e-06)
[PROGRESS:Update D] 12s -  31.00% iterations (1.554e-06)
[PROGRESS:Update D] 13s -  32.00% iterations (1.554e-06)
[PROGRESS:Update D] 13s -  33.00% iterations (1.554e-06)
[PROGRESS:Update D] 13s -  34.00% iterations (1.554e-06)
[PROGRESS:Update D] 14s -  35.00% iterations (1.554e-06)
[PROGRESS:Update D] 14s -  36.00% iterations (1.554e-06)
[PROGRESS:Update D] 15s -  37.00% iterations (1.554e-06)
[PROGRESS:Update D] 15s -  38.00% iterations (1.554e-06)
[PROGRESS:Update D] 15s -  39.00% iterations (1.554e-06)
[PROGRESS:Update D] 16s -  40.00% iterations (1.554e-06)
[PROGRESS:Update D] 16s -  41.00% iterations (1.554e-06)
[PROGRESS:Update D] 17s -  42.00% iterations (1.554e-06)
[PROGRESS:Update D] 17s -  43.00% iterations (1.554e-06)
[PROGRESS:Update D] 17s -  44.00% iterations (1.554e-06)
[PROGRESS:Update D] 18s -  45.00% iterations (1.554e-06)
[PROGRESS:Update D] 18s -  46.00% iterations (1.554e-06)
[PROGRESS:Update D] 19s -  47.00% iterations (1.554e-06)
[PROGRESS:Update D] 19s -  48.00% iterations (1.554e-06)
[PROGRESS:Update D] 19s -  49.00% iterations (1.554e-06)
[PROGRESS:Update D] 20s -  50.00% iterations (1.554e-06)
[PROGRESS:Update D] 20s -  51.00% iterations (1.554e-06)
[PROGRESS:Update D] 20s -  52.00% iterations (1.554e-06)
[PROGRESS:Update D] 21s -  53.00% iterations (1.554e-06)
[PROGRESS:Update D] 21s -  54.00% iterations (1.554e-06)
[PROGRESS:Update D] 22s -  55.00% iterations (1.554e-06)
[PROGRESS:Update D] 22s -  56.00% iterations (1.554e-06)
[PROGRESS:Update D] 22s -  57.00% iterations (1.554e-06)
[PROGRESS:Update D] 23s -  58.00% iterations (1.554e-06)
[PROGRESS:Update D] 23s -  59.00% iterations (1.554e-06)
[PROGRESS:Update D] 24s -  60.00% iterations (1.554e-06)
[PROGRESS:Update D] 24s -  61.00% iterations (1.554e-06)
[PROGRESS:Update D] 24s -  62.00% iterations (1.554e-06)
[PROGRESS:Update D] 25s -  63.00% iterations (1.554e-06)
[PROGRESS:Update D] 25s -  64.00% iterations (1.554e-06)
[PROGRESS:Update D] 26s -  65.00% iterations (1.554e-06)
[PROGRESS:Update D] 26s -  66.00% iterations (1.554e-06)
[PROGRESS:Update D] 26s -  67.00% iterations (1.554e-06)
[PROGRESS:Update D] 27s -  68.00% iterations (1.554e-06)
[PROGRESS:Update D] 27s -  69.00% iterations (1.554e-06)
[PROGRESS:Update D] 28s -  70.00% iterations (1.554e-06)
[PROGRESS:Update D] 28s -  71.00% iterations (1.554e-06)
[PROGRESS:Update D] 28s -  72.00% iterations (1.554e-06)
[PROGRESS:Update D] 29s -  73.00% iterations (1.554e-06)
[PROGRESS:Update D] 29s -  74.00% iterations (1.554e-06)
[PROGRESS:Update D] 30s -  75.00% iterations (1.554e-06)
[PROGRESS:Update D] 30s -  76.00% iterations (1.554e-06)
[PROGRESS:Update D] 30s -  77.00% iterations (1.554e-06)
[PROGRESS:Update D] 31s -  78.00% iterations (1.554e-06)
[PROGRESS:Update D] 31s -  79.00% iterations (1.554e-06)
[PROGRESS:Update D] 32s -  80.00% iterations (1.554e-06)
[PROGRESS:Update D] 32s -  81.00% iterations (1.554e-06)
[PROGRESS:Update D] 32s -  82.00% iterations (1.554e-06)
[PROGRESS:Update D] 33s -  83.00% iterations (1.554e-06)
[PROGRESS:Update D] 33s -  84.00% iterations (1.554e-06)
[PROGRESS:Update D] 34s -  85.00% iterations (1.554e-06)
[PROGRESS:Update D] 34s -  86.00% iterations (1.554e-06)
[PROGRESS:Update D] 34s -  87.00% iterations (1.554e-06)
[PROGRESS:Update D] 35s -  88.00% iterations (1.554e-06)
[PROGRESS:Update D] 35s -  89.00% iterations (1.554e-06)
[PROGRESS:Update D] 36s -  90.00% iterations (1.554e-06)
[PROGRESS:Update D] 36s -  91.00% iterations (1.554e-06)
[PROGRESS:Update D] 36s -  92.00% iterations (1.554e-06)
[PROGRESS:Update D] 37s -  93.00% iterations (1.554e-06)
[PROGRESS:Update D] 37s -  94.00% iterations (1.554e-06)
[PROGRESS:Update D] 38s -  95.00% iterations (1.554e-06)
[PROGRESS:Update D] 38s -  96.00% iterations (1.554e-06)
[PROGRESS:Update D] 38s -  97.00% iterations (1.554e-06)
[PROGRESS:Update D] 39s -  98.00% iterations (1.554e-06)
[PROGRESS:Update D] 39s -  99.00% iterations (1.554e-06)
[INFO:Update D] update did not converge

[INFO:Update D]: 100 iterations
[DEBUG:DICODILE] Objective (d) : 3.141e+06  (45s)
[INFO:DICODILE] - CD iterations 10 / 100 (996s)
[DEBUG:DICODILE] lambda = 4.763e+00

[INFO:DICOD-10] converged in 29.556s (24.876s) with 206477 iterations (44748 updates).
[DEBUG:DICODILE] Objective (z) : 3.140e+06 (36s)

[PROGRESS:Update D] 1s -   2.00% iterations (3.284e-05)
[PROGRESS:Update D] 2s -   3.00% iterations (1.642e-05)
[PROGRESS:Update D] 2s -   4.00% iterations (1.642e-05)
[PROGRESS:Update D] 3s -   5.00% iterations (1.642e-05)
[PROGRESS:Update D] 3s -   6.00% iterations (1.642e-05)
[PROGRESS:Update D] 3s -   7.00% iterations (1.642e-05)
[PROGRESS:Update D] 4s -   8.00% iterations (1.642e-05)
[PROGRESS:Update D] 4s -   9.00% iterations (1.642e-05)
[PROGRESS:Update D] 5s -  10.00% iterations (1.642e-05)
[PROGRESS:Update D] 6s -  11.00% iterations (8.674e-07)
[PROGRESS:Update D] 6s -  12.00% iterations (8.674e-07)
[PROGRESS:Update D] 7s -  13.00% iterations (8.674e-07)
[PROGRESS:Update D] 7s -  14.00% iterations (4.337e-07)
[PROGRESS:Update D] 8s -  15.00% iterations (2.169e-07)
[PROGRESS:Update D] 8s -  16.00% iterations (2.169e-07)
[PROGRESS:Update D] 9s -  17.00% iterations (1.084e-07)
[PROGRESS:Update D] 9s -  18.00% iterations (5.422e-08)
[PROGRESS:Update D] 10s -  19.00% iterations (2.711e-08)
[PROGRESS:Update D] 10s -  20.00% iterations (1.355e-08)
[PROGRESS:Update D] 11s -  21.00% iterations (1.355e-08)
[PROGRESS:Update D] 11s -  22.00% iterations (6.777e-09)
[PROGRESS:Update D] 12s -  23.00% iterations (1.432e-09)
[PROGRESS:Update D] 13s -  24.00% iterations (1.432e-09)
[PROGRESS:Update D] 13s -  25.00% iterations (1.432e-09)
[INFO:Update D]: 26 iterations
[DEBUG:DICODILE] Objective (d) : 3.140e+06  (19s)
[INFO:DICODILE] - CD iterations 11 / 100 (1057s)
[DEBUG:DICODILE] lambda = 4.763e+00

[INFO:DICOD-10] converged in 24.468s (20.313s) with 207850 iterations (37604 updates).
[DEBUG:DICODILE] Objective (z) : 3.140e+06 (30s)

[PROGRESS:Update D] 1s -   3.00% iterations (7.161e-08)
[PROGRESS:Update D] 2s -   4.00% iterations (7.161e-08)
[PROGRESS:Update D] 2s -   5.00% iterations (7.161e-08)
[PROGRESS:Update D] 2s -   6.00% iterations (7.161e-08)
[PROGRESS:Update D] 3s -   7.00% iterations (7.161e-08)
[PROGRESS:Update D] 3s -   8.00% iterations (7.161e-08)
[PROGRESS:Update D] 4s -   9.00% iterations (7.161e-08)
[PROGRESS:Update D] 4s -  10.00% iterations (7.161e-08)
[PROGRESS:Update D] 4s -  11.00% iterations (7.161e-08)
[PROGRESS:Update D] 5s -  12.00% iterations (7.161e-08)
[PROGRESS:Update D] 5s -  13.00% iterations (7.161e-08)
[PROGRESS:Update D] 6s -  14.00% iterations (7.161e-08)
[PROGRESS:Update D] 6s -  15.00% iterations (7.161e-08)
[PROGRESS:Update D] 6s -  16.00% iterations (7.161e-08)
[PROGRESS:Update D] 7s -  17.00% iterations (7.161e-08)
[PROGRESS:Update D] 7s -  18.00% iterations (7.161e-08)
[PROGRESS:Update D] 8s -  19.00% iterations (7.161e-08)
[PROGRESS:Update D] 8s -  20.00% iterations (7.161e-08)
[PROGRESS:Update D] 8s -  21.00% iterations (7.161e-08)
[PROGRESS:Update D] 9s -  22.00% iterations (7.161e-08)
[PROGRESS:Update D] 9s -  23.00% iterations (7.161e-08)
[PROGRESS:Update D] 10s -  24.00% iterations (7.161e-08)
[PROGRESS:Update D] 10s -  25.00% iterations (7.161e-08)
[PROGRESS:Update D] 10s -  26.00% iterations (7.161e-08)
[PROGRESS:Update D] 11s -  27.00% iterations (7.161e-08)
[PROGRESS:Update D] 11s -  28.00% iterations (7.161e-08)
[PROGRESS:Update D] 12s -  29.00% iterations (7.161e-08)
[PROGRESS:Update D] 12s -  30.00% iterations (7.161e-08)
[PROGRESS:Update D] 12s -  31.00% iterations (7.161e-08)
[PROGRESS:Update D] 13s -  32.00% iterations (7.161e-08)
[PROGRESS:Update D] 13s -  33.00% iterations (7.161e-08)
[PROGRESS:Update D] 14s -  34.00% iterations (7.161e-08)
[PROGRESS:Update D] 14s -  35.00% iterations (7.161e-08)
[PROGRESS:Update D] 14s -  36.00% iterations (7.161e-08)
[PROGRESS:Update D] 15s -  37.00% iterations (7.161e-08)
[PROGRESS:Update D] 15s -  38.00% iterations (7.161e-08)
[PROGRESS:Update D] 15s -  39.00% iterations (7.161e-08)
[PROGRESS:Update D] 16s -  40.00% iterations (7.161e-08)
[PROGRESS:Update D] 16s -  41.00% iterations (7.161e-08)
[PROGRESS:Update D] 17s -  42.00% iterations (7.161e-08)
[PROGRESS:Update D] 17s -  43.00% iterations (7.161e-08)
[PROGRESS:Update D] 17s -  44.00% iterations (7.161e-08)
[PROGRESS:Update D] 18s -  45.00% iterations (7.161e-08)
[PROGRESS:Update D] 18s -  46.00% iterations (7.161e-08)
[PROGRESS:Update D] 19s -  47.00% iterations (7.161e-08)
[PROGRESS:Update D] 19s -  48.00% iterations (7.161e-08)
[PROGRESS:Update D] 19s -  49.00% iterations (7.161e-08)
[PROGRESS:Update D] 20s -  50.00% iterations (7.161e-08)
[PROGRESS:Update D] 20s -  51.00% iterations (7.161e-08)
[PROGRESS:Update D] 21s -  52.00% iterations (7.161e-08)
[PROGRESS:Update D] 21s -  53.00% iterations (7.161e-08)
[PROGRESS:Update D] 21s -  54.00% iterations (7.161e-08)
[PROGRESS:Update D] 22s -  55.00% iterations (7.161e-08)
[PROGRESS:Update D] 22s -  56.00% iterations (7.161e-08)
[PROGRESS:Update D] 23s -  57.00% iterations (7.161e-08)
[PROGRESS:Update D] 23s -  58.00% iterations (7.161e-08)
[PROGRESS:Update D] 23s -  59.00% iterations (7.161e-08)
[PROGRESS:Update D] 24s -  60.00% iterations (7.161e-08)
[PROGRESS:Update D] 24s -  61.00% iterations (7.161e-08)
[PROGRESS:Update D] 25s -  62.00% iterations (7.161e-08)
[PROGRESS:Update D] 25s -  63.00% iterations (7.161e-08)
[PROGRESS:Update D] 25s -  64.00% iterations (7.161e-08)
[PROGRESS:Update D] 26s -  65.00% iterations (7.161e-08)
[PROGRESS:Update D] 26s -  66.00% iterations (7.161e-08)
[PROGRESS:Update D] 27s -  67.00% iterations (7.161e-08)
[PROGRESS:Update D] 27s -  68.00% iterations (7.161e-08)
[PROGRESS:Update D] 27s -  69.00% iterations (7.161e-08)
[PROGRESS:Update D] 28s -  70.00% iterations (7.161e-08)
[PROGRESS:Update D] 28s -  71.00% iterations (7.161e-08)
[PROGRESS:Update D] 29s -  72.00% iterations (7.161e-08)
[PROGRESS:Update D] 29s -  73.00% iterations (7.161e-08)
[PROGRESS:Update D] 29s -  74.00% iterations (7.161e-08)
[PROGRESS:Update D] 30s -  75.00% iterations (7.161e-08)
[PROGRESS:Update D] 30s -  76.00% iterations (7.161e-08)
[PROGRESS:Update D] 31s -  77.00% iterations (7.161e-08)
[PROGRESS:Update D] 31s -  78.00% iterations (7.161e-08)
[PROGRESS:Update D] 31s -  79.00% iterations (7.161e-08)
[PROGRESS:Update D] 32s -  80.00% iterations (7.161e-08)
[PROGRESS:Update D] 32s -  81.00% iterations (7.161e-08)
[PROGRESS:Update D] 33s -  82.00% iterations (7.161e-08)
[PROGRESS:Update D] 33s -  83.00% iterations (7.161e-08)
[PROGRESS:Update D] 33s -  84.00% iterations (7.161e-08)
[PROGRESS:Update D] 34s -  85.00% iterations (7.161e-08)
[PROGRESS:Update D] 34s -  86.00% iterations (7.161e-08)
[PROGRESS:Update D] 35s -  87.00% iterations (7.161e-08)
[PROGRESS:Update D] 35s -  88.00% iterations (7.161e-08)
[PROGRESS:Update D] 35s -  89.00% iterations (7.161e-08)
[PROGRESS:Update D] 36s -  90.00% iterations (7.161e-08)
[PROGRESS:Update D] 36s -  91.00% iterations (7.161e-08)
[PROGRESS:Update D] 37s -  92.00% iterations (7.161e-08)
[PROGRESS:Update D] 37s -  93.00% iterations (7.161e-08)
[PROGRESS:Update D] 37s -  94.00% iterations (7.161e-08)
[PROGRESS:Update D] 38s -  95.00% iterations (7.161e-08)
[PROGRESS:Update D] 38s -  96.00% iterations (7.161e-08)
[PROGRESS:Update D] 39s -  97.00% iterations (7.161e-08)
[PROGRESS:Update D] 39s -  98.00% iterations (7.161e-08)
[PROGRESS:Update D] 39s -  99.00% iterations (7.161e-08)
[INFO:Update D] update did not converge

[INFO:Update D]: 100 iterations
[DEBUG:DICODILE] Objective (d) : 3.140e+06  (45s)
[INFO:DICODILE] - CD iterations 12 / 100 (1138s)
[DEBUG:DICODILE] lambda = 4.763e+00

[INFO:DICOD-10] converged in 8.048s (6.570s) with 54078 iterations (11741 updates).
[DEBUG:DICODILE] Objective (z) : 3.140e+06 (14s)

[PROGRESS:Update D] 1s -   3.00% iterations (7.161e-06)
[PROGRESS:Update D] 2s -   4.00% iterations (7.161e-06)
[PROGRESS:Update D] 2s -   5.00% iterations (7.161e-06)
[PROGRESS:Update D] 2s -   6.00% iterations (7.161e-06)
[PROGRESS:Update D] 3s -   7.00% iterations (7.161e-06)
[PROGRESS:Update D] 3s -   8.00% iterations (7.161e-06)
[PROGRESS:Update D] 4s -   9.00% iterations (7.161e-06)
[PROGRESS:Update D] 4s -  10.00% iterations (7.161e-06)
[PROGRESS:Update D] 4s -  11.00% iterations (7.161e-06)
[PROGRESS:Update D] 5s -  12.00% iterations (7.161e-06)
[PROGRESS:Update D] 5s -  13.00% iterations (7.161e-06)
[PROGRESS:Update D] 6s -  14.00% iterations (7.161e-06)
[PROGRESS:Update D] 6s -  15.00% iterations (7.161e-06)
[PROGRESS:Update D] 6s -  16.00% iterations (7.161e-06)
[PROGRESS:Update D] 7s -  17.00% iterations (7.161e-06)
[PROGRESS:Update D] 7s -  18.00% iterations (7.161e-06)
[PROGRESS:Update D] 8s -  19.00% iterations (7.161e-06)
[PROGRESS:Update D] 8s -  20.00% iterations (7.161e-06)
[PROGRESS:Update D] 8s -  21.00% iterations (7.161e-06)
[PROGRESS:Update D] 9s -  22.00% iterations (7.161e-06)
[PROGRESS:Update D] 9s -  23.00% iterations (7.161e-06)
[PROGRESS:Update D] 10s -  24.00% iterations (7.161e-06)
[PROGRESS:Update D] 10s -  25.00% iterations (7.161e-06)
[PROGRESS:Update D] 10s -  26.00% iterations (7.161e-06)
[PROGRESS:Update D] 11s -  27.00% iterations (7.161e-06)
[PROGRESS:Update D] 11s -  28.00% iterations (7.161e-06)
[PROGRESS:Update D] 12s -  29.00% iterations (7.161e-06)
[PROGRESS:Update D] 12s -  30.00% iterations (7.161e-06)
[PROGRESS:Update D] 13s -  31.00% iterations (7.161e-06)
[PROGRESS:Update D] 13s -  32.00% iterations (7.161e-06)
[PROGRESS:Update D] 13s -  33.00% iterations (7.161e-06)
[PROGRESS:Update D] 14s -  34.00% iterations (7.161e-06)
[PROGRESS:Update D] 14s -  35.00% iterations (7.161e-06)
[PROGRESS:Update D] 15s -  36.00% iterations (7.161e-06)
[PROGRESS:Update D] 15s -  37.00% iterations (7.161e-06)
[PROGRESS:Update D] 15s -  38.00% iterations (7.161e-06)
[PROGRESS:Update D] 16s -  39.00% iterations (7.161e-06)
[PROGRESS:Update D] 16s -  40.00% iterations (7.161e-06)
[PROGRESS:Update D] 17s -  41.00% iterations (7.161e-06)
[PROGRESS:Update D] 17s -  42.00% iterations (1.513e-06)
[PROGRESS:Update D] 18s -  43.00% iterations (7.567e-07)
[PROGRESS:Update D] 19s -  44.00% iterations (1.599e-07)
[PROGRESS:Update D] 19s -  45.00% iterations (1.599e-07)
[PROGRESS:Update D] 20s -  46.00% iterations (3.379e-08)
[PROGRESS:Update D] 20s -  47.00% iterations (1.690e-08)
[PROGRESS:Update D] 21s -  48.00% iterations (8.449e-09)
[PROGRESS:Update D] 22s -  49.00% iterations (4.224e-09)
[INFO:Update D]: 50 iterations
[DEBUG:DICODILE] Objective (d) : 3.140e+06  (28s)
[INFO:DICODILE] - CD iterations 13 / 100 (1187s)
[DEBUG:DICODILE] lambda = 4.763e+00

[INFO:DICOD-10] converged in 19.308s (16.402s) with 128506 iterations (30084 updates).
[DEBUG:DICODILE] Objective (z) : 3.139e+06 (25s)

[PROGRESS:Update D] 1s -   3.00% iterations (8.927e-08)
[PROGRESS:Update D] 2s -   4.00% iterations (8.927e-08)
[PROGRESS:Update D] 2s -   5.00% iterations (8.927e-08)
[PROGRESS:Update D] 2s -   6.00% iterations (8.927e-08)
[PROGRESS:Update D] 3s -   7.00% iterations (8.927e-08)
[PROGRESS:Update D] 3s -   8.00% iterations (8.927e-08)
[PROGRESS:Update D] 4s -   9.00% iterations (8.927e-08)
[PROGRESS:Update D] 4s -  10.00% iterations (8.927e-08)
[INFO:Update D]: 11 iterations
[DEBUG:DICODILE] Objective (d) : 3.139e+06  (12s)
[INFO:DICODILE] - CD iterations 14 / 100 (1230s)
[DEBUG:DICODILE] lambda = 4.763e+00

[INFO:DICOD-10] converged in 1.068s (0.760s) with 12321 iterations (1106 updates).
[DEBUG:DICODILE] Objective (z) : 3.139e+06 (7s)

[PROGRESS:Update D] 5s -   1.00% iterations (2.519e-06)
[INFO:Update D]: 2 iterations
[DEBUG:DICODILE] Objective (d) : 3.139e+06  (15s)
[INFO:DICODILE] Converged after 15 iteration, (dz, du) = 8.570e-08, 3.801e-08

[INFO:DICOD-10] converged in 2.532s (2.049s) with 19792 iterations (3400 updates).
[INFO:DICODILE] Finished in 1171s
[DICOD] final cost : [3328083.483032226, 3322241.735583827, 3288733.289882077, 3226087.2368888445, 3217977.3225730993, 3209002.832323461, 3205766.75786955, 3199859.673789586, 3195043.6056792084, 3187000.9523876817, 3178041.737437436, 3167250.024605673, 3156906.526775885, 3150609.1374183507, 3145849.8418757915, 3143956.091590614, 3142205.7589950897, 3141334.725412146, 3141261.1352075525, 3141256.570167604, 3140617.57811235, 3140244.1624557, 3139956.6709456914, 3139694.6416852437, 3139681.5124197137, 3139674.974150443, 3139567.4767157435, 3139418.4867491154, 3139418.4208492306, 3139418.151812008, 3139418.032470725, 3139417.033604104]

Let’s compare the initially generated random patches in D_init with the atoms in D_hat recovered with dicodile.

display_dictionaries(D_init, D_hat)

<Figure size 640x480 with 24 Axes>

Now we will reconstruct the image from z_hat and D_hat.

X_hat = reconstruct(z_hat, D_hat)
X_hat = np.clip(X_hat, 0, 1)

Let’s plot the reconstructed image X_hat together with the original image X_original and the noisy image X that was input to dicodile.

f, (ax1, ax2, ax3) = plt.subplots(3, 1, figsize=[6.4, 8])

ax1.imshow(X_original[190:490, 250:750], cmap='gray')
ax1.set_title('Original image')
ax1.axis('off')

ax2.imshow(X[0][190:490, 250:750], cmap='gray')
ax2.set_title('Noisy image')
ax2.axis('off')

ax3.imshow(X_hat[0][190:490, 250:750], cmap='gray')
ax3.set_title('Recovered image')
ax3.axis('off')
plt.tight_layout()