-
Notifications
You must be signed in to change notification settings - Fork 81
/
Copy pathplot_digits_denoising.html
447 lines (406 loc) · 37.1 KB
/
plot_digits_denoising.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
<!DOCTYPE html>
<!--[if IE 8]><html class="no-js lt-ie9" lang="en" > <![endif]-->
<!--[if gt IE 8]><!--> <html class="no-js" lang="en" > <!--<![endif]-->
<head>
<meta charset="utf-8">
<meta name="generator" content="Docutils 0.19: https://docutils.sourceforge.io/" />
<meta property="og:title" content="Image denoising using kernel PCA" />
<meta property="og:type" content="website" />
<meta property="og:url" content="https://scikit-learn/stable/auto_examples/applications/plot_digits_denoising.html" />
<meta property="og:site_name" content="scikit-learn" />
<meta property="og:description" content="This example shows how to use KernelPCA to denoise images. In short, we take advantage of the approximation function learned during fit to reconstruct the original image. We will compare the result..." />
<meta property="og:image" content="https://scikit-learn.org/stable/_static/scikit-learn-logo-small.png" />
<meta property="og:image:alt" content="scikit-learn" />
<meta name="description" content="This example shows how to use KernelPCA to denoise images. In short, we take advantage of the approximation function learned during fit to reconstruct the original image. We will compare the result..." />
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Image denoising using kernel PCA — scikit-learn 1.2.2 documentation</title>
<link rel="canonical" href="http://scikit-learn.org/stable/auto_examples/applications/plot_digits_denoising.html" />
<link rel="shortcut icon" href="../../_static/favicon.ico"/>
<link rel="stylesheet" href="../../_static/css/vendor/bootstrap.min.css" type="text/css" />
<link rel="stylesheet" href="../../_static/pygments.css" type="text/css" />
<link rel="stylesheet" href="../../_static/css/theme.css" type="text/css" />
<link rel="stylesheet" href="../../_static/plot_directive.css" type="text/css" />
<link rel="stylesheet" href="../../_static/sg_gallery.css" type="text/css" />
<link rel="stylesheet" href="../../_static/sg_gallery-binder.css" type="text/css" />
<link rel="stylesheet" href="../../_static/sg_gallery-dataframe.css" type="text/css" />
<link rel="stylesheet" href="../../_static/sg_gallery-rendered-html.css" type="text/css" />
<link rel="stylesheet" href="../../_static/css/theme.css" type="text/css" />
<script id="documentation_options" data-url_root="../../" src="../../_static/documentation_options.js"></script>
<script src="../../_static/js/vendor/jquery-3.6.3.slim.min.js"></script>
</head>
<body>
<nav id="navbar" class="sk-docs-navbar navbar navbar-expand-md navbar-light bg-light py-0">
<div class="container-fluid sk-docs-container px-0">
<a class="navbar-brand py-0" href="../../index.html">
<img
class="sk-brand-img"
src="../../_static/scikit-learn-logo-small.png"
alt="logo"/>
</a>
<button
id="sk-navbar-toggler"
class="navbar-toggler"
type="button"
data-toggle="collapse"
data-target="#navbarSupportedContent"
aria-controls="navbarSupportedContent"
aria-expanded="false"
aria-label="Toggle navigation"
>
<span class="navbar-toggler-icon"></span>
</button>
<div class="sk-navbar-collapse collapse navbar-collapse" id="navbarSupportedContent">
<ul class="navbar-nav mr-auto">
<li class="nav-item">
<a class="sk-nav-link nav-link" href="../../install.html">Install</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" href="../../user_guide.html">User Guide</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" href="../../modules/classes.html">API</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" href="../index.html">Examples</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" target="_blank" rel="noopener noreferrer" href="https://blog.scikit-learn.org/">Community</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../getting_started.html" >Getting Started</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../tutorial/index.html" >Tutorial</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../whats_new/v1.2.html" >What's new</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../glossary.html" >Glossary</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://scikit-learn.org/dev/developers/index.html" target="_blank" rel="noopener noreferrer">Development</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../faq.html" >FAQ</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../support.html" >Support</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../related_projects.html" >Related packages</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../roadmap.html" >Roadmap</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../governance.html" >Governance</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../about.html" >About us</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://github.com/scikit-learn/scikit-learn" >GitHub</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://scikit-learn.org/dev/versions.html" >Other Versions and Download</a>
</li>
<li class="nav-item dropdown nav-more-item-dropdown">
<a class="sk-nav-link nav-link dropdown-toggle" href="#" id="navbarDropdown" role="button" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false">More</a>
<div class="dropdown-menu" aria-labelledby="navbarDropdown">
<a class="sk-nav-dropdown-item dropdown-item" href="../../getting_started.html" >Getting Started</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../../tutorial/index.html" >Tutorial</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../../whats_new/v1.2.html" >What's new</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../../glossary.html" >Glossary</a>
<a class="sk-nav-dropdown-item dropdown-item" href="https://scikit-learn.org/dev/developers/index.html" target="_blank" rel="noopener noreferrer">Development</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../../faq.html" >FAQ</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../../support.html" >Support</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../../related_projects.html" >Related packages</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../../roadmap.html" >Roadmap</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../../governance.html" >Governance</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../../about.html" >About us</a>
<a class="sk-nav-dropdown-item dropdown-item" href="https://github.com/scikit-learn/scikit-learn" >GitHub</a>
<a class="sk-nav-dropdown-item dropdown-item" href="https://scikit-learn.org/dev/versions.html" >Other Versions and Download</a>
</div>
</li>
</ul>
<div id="searchbox" role="search">
<div class="searchformwrapper">
<form class="search" action="../../search.html" method="get">
<input class="sk-search-text-input" type="text" name="q" aria-labelledby="searchlabel" />
<input class="sk-search-text-btn" type="submit" value="Go" />
</form>
</div>
</div>
</div>
</div>
</nav>
<div class="d-flex" id="sk-doc-wrapper">
<input type="checkbox" name="sk-toggle-checkbox" id="sk-toggle-checkbox">
<label id="sk-sidemenu-toggle" class="sk-btn-toggle-toc btn sk-btn-primary" for="sk-toggle-checkbox">Toggle Menu</label>
<div id="sk-sidebar-wrapper" class="border-right">
<div class="sk-sidebar-toc-wrapper">
<div class="btn-group w-100 mb-2" role="group" aria-label="rellinks">
<a href="plot_face_recognition.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="Faces recognition example using eigenfaces and SVMs">Prev</a><a href="index.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="Examples based on real world datasets">Up</a>
<a href="svm_gui.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="Libsvm GUI">Next</a>
</div>
<div class="alert alert-danger p-1 mb-2" role="alert">
<p class="text-center mb-0">
<strong>scikit-learn 1.2.2</strong><br/>
<a href="http://scikit-learn.org/dev/versions.html">Other versions</a>
</p>
</div>
<div class="alert alert-warning p-1 mb-2" role="alert">
<p class="text-center mb-0">
Please <a class="font-weight-bold" href="../../about.html#citing-scikit-learn"><string>cite us</string></a> if you use the software.
</p>
</div>
<div class="sk-sidebar-toc">
<ul>
<li><a class="reference internal" href="#">Image denoising using kernel PCA</a><ul>
<li><a class="reference internal" href="#load-the-dataset-via-openml">Load the dataset via OpenML</a></li>
<li><a class="reference internal" href="#learn-the-pca-basis">Learn the <code class="docutils literal notranslate"><span class="pre">PCA</span></code> basis</a></li>
<li><a class="reference internal" href="#reconstruct-and-denoise-test-images">Reconstruct and denoise test images</a></li>
</ul>
</li>
</ul>
</div>
</div>
</div>
<div id="sk-page-content-wrapper">
<div class="sk-page-content container-fluid body px-md-3" role="main">
<div class="sphx-glr-download-link-note admonition note">
<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-applications-plot-digits-denoising-py"><span class="std std-ref">here</span></a>
to download the full example code or to run this example in your browser via Binder</p>
</div>
<section class="sphx-glr-example-title" id="image-denoising-using-kernel-pca">
<span id="sphx-glr-auto-examples-applications-plot-digits-denoising-py"></span><h1>Image denoising using kernel PCA<a class="headerlink" href="#image-denoising-using-kernel-pca" title="Permalink to this heading">¶</a></h1>
<p>This example shows how to use <a class="reference internal" href="../../modules/generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA" title="sklearn.decomposition.KernelPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">KernelPCA</span></code></a> to
denoise images. In short, we take advantage of the approximation function
learned during <code class="docutils literal notranslate"><span class="pre">fit</span></code> to reconstruct the original image.</p>
<p>We will compare the results with an exact reconstruction using
<a class="reference internal" href="../../modules/generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a>.</p>
<p>We will use USPS digits dataset to reproduce presented in Sect. 4 of <a class="footnote-reference brackets" href="#id2" id="id1" role="doc-noteref"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></a>.</p>
<aside class="topic">
<p class="topic-title">References</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id2" role="note">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">1</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://papers.nips.cc/paper/2003/file/ac1ad983e08ad3304a97e147f522747e-Paper.pdf">Bakır, Gökhan H., Jason Weston, and Bernhard Schölkopf.
“Learning to find pre-images.”
Advances in neural information processing systems 16 (2004): 449-456.</a></p>
</aside>
</aside>
</aside>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># Authors: Guillaume Lemaitre <[email protected]></span>
<span class="c1"># Licence: BSD 3 clause</span>
</pre></div>
</div>
<section id="load-the-dataset-via-openml">
<h2>Load the dataset via OpenML<a class="headerlink" href="#load-the-dataset-via-openml" title="Permalink to this heading">¶</a></h2>
<p>The USPS digits datasets is available in OpenML. We use
<a class="reference internal" href="../../modules/generated/sklearn.datasets.fetch_openml.html#sklearn.datasets.fetch_openml" title="sklearn.datasets.fetch_openml"><code class="xref py py-func docutils literal notranslate"><span class="pre">fetch_openml</span></code></a> to get this dataset. In addition, we
normalize the dataset such that all pixel values are in the range (0, 1).</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.datasets.fetch_openml.html#sklearn.datasets.fetch_openml" title="sklearn.datasets.fetch_openml" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">fetch_openml</span></a>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.preprocessing.MinMaxScaler.html#sklearn.preprocessing.MinMaxScaler" title="sklearn.preprocessing.MinMaxScaler" class="sphx-glr-backref-module-sklearn-preprocessing sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">MinMaxScaler</span></a>
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.model_selection.train_test_split.html#sklearn.model_selection.train_test_split" title="sklearn.model_selection.train_test_split" class="sphx-glr-backref-module-sklearn-model_selection sphx-glr-backref-type-py-function"><span class="n">train_test_split</span></a>
<span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.datasets.fetch_openml.html#sklearn.datasets.fetch_openml" title="sklearn.datasets.fetch_openml" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">fetch_openml</span></a><span class="p">(</span><span class="n">data_id</span><span class="o">=</span><span class="mi">41082</span><span class="p">,</span> <span class="n">as_frame</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">return_X_y</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">parser</span><span class="o">=</span><span class="s2">"pandas"</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.preprocessing.MinMaxScaler.html#sklearn.preprocessing.MinMaxScaler" title="sklearn.preprocessing.MinMaxScaler" class="sphx-glr-backref-module-sklearn-preprocessing sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">MinMaxScaler</span></a><span class="p">()</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
</pre></div>
</div>
<p>The idea will be to learn a PCA basis (with and without a kernel) on
noisy images and then use these models to reconstruct and denoise these
images.</p>
<p>Thus, we split our dataset into a training and testing set composed of 1,000
samples for the training and 100 samples for testing. These images are
noise-free and we will use them to evaluate the efficiency of the denoising
approaches. In addition, we create a copy of the original dataset and add a
Gaussian noise.</p>
<p>The idea of this application, is to show that we can denoise corrupted images
by learning a PCA basis on some uncorrupted images. We will use both a PCA
and a kernel-based PCA to solve this problem.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.model_selection.train_test_split.html#sklearn.model_selection.train_test_split" title="sklearn.model_selection.train_test_split" class="sphx-glr-backref-module-sklearn-model_selection sphx-glr-backref-type-py-function"><span class="n">train_test_split</span></a><span class="p">(</span>
<span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">stratify</span><span class="o">=</span><span class="n">y</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">train_size</span><span class="o">=</span><span class="mi">1_000</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mi">100</span>
<span class="p">)</span>
<span class="n">rng</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/random/legacy.html#numpy.random.RandomState" title="numpy.random.RandomState" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">RandomState</span></a><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">noise</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">scale</span><span class="o">=</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">X_test</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">X_test_noisy</span> <span class="o">=</span> <span class="n">X_test</span> <span class="o">+</span> <span class="n">noise</span>
<span class="n">noise</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">scale</span><span class="o">=</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">X_train_noisy</span> <span class="o">=</span> <span class="n">X_train</span> <span class="o">+</span> <span class="n">noise</span>
</pre></div>
</div>
<p>In addition, we will create a helper function to qualitatively assess the
image reconstruction by plotting the test images.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="k">def</span> <span class="nf">plot_digits</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">title</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Small helper function to plot 100 digits."""</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="n">nrows</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
<span class="k">for</span> <span class="n">img</span><span class="p">,</span> <span class="n">ax</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">axs</span><span class="o">.</span><span class="n">ravel</span><span class="p">()):</span>
<span class="n">ax</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">img</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="mi">16</span><span class="p">,</span> <span class="mi">16</span><span class="p">)),</span> <span class="n">cmap</span><span class="o">=</span><span class="s2">"Greys"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s2">"off"</span><span class="p">)</span>
<span class="n">fig</span><span class="o">.</span><span class="n">suptitle</span><span class="p">(</span><span class="n">title</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">24</span><span class="p">)</span>
</pre></div>
</div>
<p>In addition, we will use the mean squared error (MSE) to quantitatively
assess the image reconstruction.</p>
<p>Let’s first have a look to see the difference between noise-free and noisy
images. We will check the test set in this regard.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">plot_digits</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="s2">"Uncorrupted test images"</span><span class="p">)</span>
<span class="n">plot_digits</span><span class="p">(</span>
<span class="n">X_test_noisy</span><span class="p">,</span> <span class="sa">f</span><span class="s2">"Noisy test images</span><span class="se">\n</span><span class="s2">MSE: </span><span class="si">{</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.mean.html#numpy.mean" title="numpy.mean" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">mean</span></a><span class="p">((</span><span class="n">X_test</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">X_test_noisy</span><span class="p">)</span><span class="w"> </span><span class="o">**</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">"</span>
<span class="p">)</span>
</pre></div>
</div>
<ul class="sphx-glr-horizontal">
<li><img src="../../_images/sphx_glr_plot_digits_denoising_001.png" srcset="../../_images/sphx_glr_plot_digits_denoising_001.png" alt="Uncorrupted test images" class = "sphx-glr-multi-img"/></li>
<li><img src="../../_images/sphx_glr_plot_digits_denoising_002.png" srcset="../../_images/sphx_glr_plot_digits_denoising_002.png" alt="Noisy test images MSE: 0.06" class = "sphx-glr-multi-img"/></li>
</ul>
</section>
<section id="learn-the-pca-basis">
<h2>Learn the <code class="docutils literal notranslate"><span class="pre">PCA</span></code> basis<a class="headerlink" href="#learn-the-pca-basis" title="Permalink to this heading">¶</a></h2>
<p>We can now learn our PCA basis using both a linear PCA and a kernel PCA that
uses a radial basis function (RBF) kernel.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.decomposition</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA" class="sphx-glr-backref-module-sklearn-decomposition sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">PCA</span></a><span class="p">,</span> <a href="../../modules/generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA" title="sklearn.decomposition.KernelPCA" class="sphx-glr-backref-module-sklearn-decomposition sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">KernelPCA</span></a>
<span class="n">pca</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA" class="sphx-glr-backref-module-sklearn-decomposition sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">PCA</span></a><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">32</span><span class="p">)</span>
<span class="n">kernel_pca</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA" title="sklearn.decomposition.KernelPCA" class="sphx-glr-backref-module-sklearn-decomposition sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">KernelPCA</span></a><span class="p">(</span>
<span class="n">n_components</span><span class="o">=</span><span class="mi">400</span><span class="p">,</span> <span class="n">kernel</span><span class="o">=</span><span class="s2">"rbf"</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">1e-3</span><span class="p">,</span> <span class="n">fit_inverse_transform</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">5e-3</span>
<span class="p">)</span>
<span class="n">pca</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_noisy</span><span class="p">)</span>
<span class="n">_</span> <span class="o">=</span> <span class="n">kernel_pca</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_noisy</span><span class="p">)</span>
</pre></div>
</div>
</section>
<section id="reconstruct-and-denoise-test-images">
<h2>Reconstruct and denoise test images<a class="headerlink" href="#reconstruct-and-denoise-test-images" title="Permalink to this heading">¶</a></h2>
<p>Now, we can transform and reconstruct the noisy test set. Since we used less
components than the number of original features, we will get an approximation
of the original set. Indeed, by dropping the components explaining variance
in PCA the least, we hope to remove noise. Similar thinking happens in kernel
PCA; however, we expect a better reconstruction because we use a non-linear
kernel to learn the PCA basis and a kernel ridge to learn the mapping
function.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">X_reconstructed_kernel_pca</span> <span class="o">=</span> <span class="n">kernel_pca</span><span class="o">.</span><span class="n">inverse_transform</span><span class="p">(</span>
<span class="n">kernel_pca</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test_noisy</span><span class="p">)</span>
<span class="p">)</span>
<span class="n">X_reconstructed_pca</span> <span class="o">=</span> <span class="n">pca</span><span class="o">.</span><span class="n">inverse_transform</span><span class="p">(</span><span class="n">pca</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test_noisy</span><span class="p">))</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">plot_digits</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="s2">"Uncorrupted test images"</span><span class="p">)</span>
<span class="n">plot_digits</span><span class="p">(</span>
<span class="n">X_reconstructed_pca</span><span class="p">,</span>
<span class="sa">f</span><span class="s2">"PCA reconstruction</span><span class="se">\n</span><span class="s2">MSE: </span><span class="si">{</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.mean.html#numpy.mean" title="numpy.mean" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">mean</span></a><span class="p">((</span><span class="n">X_test</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">X_reconstructed_pca</span><span class="p">)</span><span class="w"> </span><span class="o">**</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">"</span><span class="p">,</span>
<span class="p">)</span>
<span class="n">plot_digits</span><span class="p">(</span>
<span class="n">X_reconstructed_kernel_pca</span><span class="p">,</span>
<span class="s2">"Kernel PCA reconstruction</span><span class="se">\n</span><span class="s2">"</span>
<span class="sa">f</span><span class="s2">"MSE: </span><span class="si">{</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.mean.html#numpy.mean" title="numpy.mean" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">mean</span></a><span class="p">((</span><span class="n">X_test</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">X_reconstructed_kernel_pca</span><span class="p">)</span><span class="w"> </span><span class="o">**</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="si">:</span><span class="s2">.2f</span><span class="si">}</span><span class="s2">"</span><span class="p">,</span>
<span class="p">)</span>
</pre></div>
</div>
<ul class="sphx-glr-horizontal">
<li><img src="../../_images/sphx_glr_plot_digits_denoising_003.png" srcset="../../_images/sphx_glr_plot_digits_denoising_003.png" alt="Uncorrupted test images" class = "sphx-glr-multi-img"/></li>
<li><img src="../../_images/sphx_glr_plot_digits_denoising_004.png" srcset="../../_images/sphx_glr_plot_digits_denoising_004.png" alt="PCA reconstruction MSE: 0.01" class = "sphx-glr-multi-img"/></li>
<li><img src="../../_images/sphx_glr_plot_digits_denoising_005.png" srcset="../../_images/sphx_glr_plot_digits_denoising_005.png" alt="Kernel PCA reconstruction MSE: 0.03" class = "sphx-glr-multi-img"/></li>
</ul>
<p>PCA has a lower MSE than kernel PCA. However, the qualitative analysis might
not favor PCA instead of kernel PCA. We observe that kernel PCA is able to
remove background noise and provide a smoother image.</p>
<p>However, it should be noted that the results of the denoising with kernel PCA
will depend of the parameters <code class="docutils literal notranslate"><span class="pre">n_components</span></code>, <code class="docutils literal notranslate"><span class="pre">gamma</span></code>, and <code class="docutils literal notranslate"><span class="pre">alpha</span></code>.</p>
<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes 14.944 seconds)</p>
<div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-applications-plot-digits-denoising-py">
<div class="binder-badge docutils container">
<a class="reference external image-reference" href="https://mybinder.org/v2/gh/scikit-learn/scikit-learn/1.2.X?urlpath=lab/tree/notebooks/auto_examples/applications/plot_digits_denoising.ipynb"><img alt="Launch binder" src="../../_images/binder_badge_logo8.svg" width="150px" /></a>
</div>
<div class="sphx-glr-download sphx-glr-download-python docutils container">
<p><a class="reference download internal" download="" href="../../_downloads/32173eb704d697c23dffbbf3fd74942a/plot_digits_denoising.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_digits_denoising.py</span></code></a></p>
</div>
<div class="sphx-glr-download sphx-glr-download-jupyter docutils container">
<p><a class="reference download internal" download="" href="../../_downloads/f499e804840a40d11222872e84726eef/plot_digits_denoising.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_digits_denoising.ipynb</span></code></a></p>
</div>
</div>
<p class="sphx-glr-signature"><a class="reference external" href="https://sphinx-gallery.github.io">Gallery generated by Sphinx-Gallery</a></p>
</section>
</section>
</div>
<div class="container">
<footer class="sk-content-footer">
© 2007 - 2023, scikit-learn developers (BSD License).
<a href="../../_sources/auto_examples/applications/plot_digits_denoising.rst.txt" rel="nofollow">Show this page source</a>
</footer>
</div>
</div>
</div>
<script src="../../_static/js/vendor/bootstrap.min.js"></script>
<script>
window.ga=window.ga||function(){(ga.q=ga.q||[]).push(arguments)};ga.l=+new Date;
ga('create', 'UA-22606712-2', 'auto');
ga('set', 'anonymizeIp', true);
ga('send', 'pageview');
</script>
<script async src='https://www.google-analytics.com/analytics.js'></script>
<script defer data-domain="scikit-learn.org" src="https://views.scientific-python.org/js/script.js">
</script>
<script>
$(document).ready(function() {
/* Add a [>>>] button on the top-right corner of code samples to hide
* the >>> and ... prompts and the output and thus make the code
* copyable. */
var div = $('.highlight-python .highlight,' +
'.highlight-python3 .highlight,' +
'.highlight-pycon .highlight,' +
'.highlight-default .highlight')
var pre = div.find('pre');
// get the styles from the current theme
pre.parent().parent().css('position', 'relative');
var hide_text = 'Hide prompts and outputs';
var show_text = 'Show prompts and outputs';
// create and add the button to all the code blocks that contain >>>
div.each(function(index) {
var jthis = $(this);
if (jthis.find('.gp').length > 0) {
var button = $('<span class="copybutton">>>></span>');
button.attr('title', hide_text);
button.data('hidden', 'false');
jthis.prepend(button);
}
// tracebacks (.gt) contain bare text elements that need to be
// wrapped in a span to work with .nextUntil() (see later)
jthis.find('pre:has(.gt)').contents().filter(function() {
return ((this.nodeType == 3) && (this.data.trim().length > 0));
}).wrap('<span>');
});
// define the behavior of the button when it's clicked
$('.copybutton').click(function(e){
e.preventDefault();
var button = $(this);
if (button.data('hidden') === 'false') {
// hide the code output
button.parent().find('.go, .gp, .gt').hide();
button.next('pre').find('.gt').nextUntil('.gp, .go').css('visibility', 'hidden');
button.css('text-decoration', 'line-through');
button.attr('title', show_text);
button.data('hidden', 'true');
} else {
// show the code output
button.parent().find('.go, .gp, .gt').show();
button.next('pre').find('.gt').nextUntil('.gp, .go').css('visibility', 'visible');
button.css('text-decoration', 'none');
button.attr('title', hide_text);
button.data('hidden', 'false');
}
});
/*** Add permalink buttons next to glossary terms ***/
$('dl.glossary > dt[id]').append(function() {
return ('<a class="headerlink" href="#' +
this.getAttribute('id') +
'" title="Permalink to this term">¶</a>');
});
});
</script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"></script>
<script src="https://scikit-learn.org/versionwarning.js"></script>
</body>
</html>