Wiener_Filter.ipynb 19.9 KB
Newer Older
Philipp Arras's avatar
Philipp Arras committed
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
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# A NIFTy demonstration"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## IFT: Big Picture\n",
    "IFT starting point:\n",
    "\n",
    "$$d = Rs+n$$\n",
    "\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
27
    "Typically, $s$ is a continuous field, $d$ a discrete data vector. Particularly, $R$ is not invertible.\n",
Philipp Arras's avatar
Philipp Arras committed
28
29
30
31
32
33
    "\n",
    "IFT aims at **inverting** the above uninvertible problem in the **best possible way** using Bayesian statistics.\n",
    "\n",
    "\n",
    "## NIFTy\n",
    "\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
34
    "NIFTy (Numerical Information Field Theory) is a Python framework in which IFT problems can be tackled easily.\n",
Philipp Arras's avatar
Philipp Arras committed
35
36
37
38
39
    "\n",
    "Main Interfaces:\n",
    "\n",
    "- **Spaces**: Cartesian, 2-Spheres (Healpix, Gauss-Legendre) and their respective harmonic spaces.\n",
    "- **Fields**: Defined on spaces.\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
40
    "- **Operators**: Acting on fields."
Philipp Arras's avatar
Philipp Arras committed
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
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Wiener Filter: Formulae\n",
    "\n",
    "### Assumptions\n",
    "\n",
    "- $d=Rs+n$, $R$ linear operator.\n",
    "- $\\mathcal P (s) = \\mathcal G (s,S)$, $\\mathcal P (n) = \\mathcal G (n,N)$ where $S, N$ are positive definite matrices.\n",
    "\n",
    "### Posterior\n",
    "The Posterior is given by:\n",
    "\n",
    "$$\\mathcal P (s|d) \\propto P(s,d) = \\mathcal G(d-Rs,N) \\,\\mathcal G(s,S) \\propto \\mathcal G (m,D) $$\n",
    "\n",
    "where\n",
    "$$\\begin{align}\n",
    "m &= Dj \\\\\n",
    "D^{-1}&= (S^{-1} +R^\\dagger N^{-1} R )\\\\\n",
    "j &= R^\\dagger N^{-1} d\n",
    "\\end{align}$$\n",
    "\n",
    "Let us implement this in NIFTy!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Wiener Filter: Example\n",
    "\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
83
    "- One-dimensional signal with power spectrum: $$P(k) = P_0\\,\\left(1+\\left(\\frac{k}{k_0}\\right)^2\\right)^{-\\gamma /2},$$\n",
Philipp Arras's avatar
Philipp Arras committed
84
    "with $P_0 = 0.2, k_0 = 5, \\gamma = 4$. Recall: $P(k)$ defines an isotropic and homogeneous $S$.\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
85
    "- $N = 0.2 \\cdot \\mathbb{1}$.\n",
Martin Reinecke's avatar
Martin Reinecke committed
86
87
    "- Number of data points $N_{pix} = 512$.\n",
    "- reconstruction in harmonic space.\n",
Philipp Arras's avatar
Philipp Arras committed
88
    "- Response operator:\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
89
    "$$R = FFT_{\\text{harmonic} \\rightarrow \\text{position}}$$\n"
Philipp Arras's avatar
Philipp Arras committed
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "N_pixels = 512     # Number of pixels\n",
    "\n",
    "def pow_spec(k):\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
105
106
    "    P0, k0, gamma = [.2, 5, 4]\n",
    "    return P0 / ((1. + (k/k0)**2)**(gamma / 2))"
Philipp Arras's avatar
Philipp Arras committed
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
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Wiener Filter: Implementation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "### Import Modules"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
142
    "np.random.seed(40)\n",
143
144
145
    "import nifty4 as ift\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
Philipp Arras's avatar
Philipp Arras committed
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Implement Propagator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
169
    "def PropagatorOperator(R, N, Sh):\n",
Martin Reinecke's avatar
Martin Reinecke committed
170
    "    IC = ift.GradientNormController(iteration_limit=50000,\n",
171
172
173
174
    "                                    tol_abs_gradnorm=0.1)\n",
    "    inverter = ift.ConjugateGradient(controller=IC)\n",
    "    D = (R.adjoint*N.inverse*R + Sh.inverse).inverse\n",
    "    # MR FIXME: we can/should provide a preconditioner here as well!\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
175
    "    return ift.InversionEnabler(D, inverter)\n"
Philipp Arras's avatar
Philipp Arras committed
176
177
178
179
180
181
182
183
184
185
186
187
188
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "### Conjugate Gradient Preconditioning\n",
    "\n",
    "- $D$ is defined via:\n",
Martin Reinecke's avatar
Martin Reinecke committed
189
    "$$D^{-1} = \\mathcal S_h^{-1} + R^\\dagger N^{-1} R.$$\n",
Philipp Arras's avatar
Philipp Arras committed
190
191
    "In the end, we want to apply $D$ to $j$, i.e. we need the inverse action of $D^{-1}$. This is done numerically (algorithm: *Conjugate Gradient*). \n",
    "\n",
Martin Reinecke's avatar
Martin Reinecke committed
192
    "<!--\n",
Philipp Arras's avatar
Philipp Arras committed
193
194
195
196
197
198
    "- One can define the *condition number* of a non-singular and normal matrix $A$:\n",
    "$$\\kappa (A) := \\frac{|\\lambda_{\\text{max}}|}{|\\lambda_{\\text{min}}|},$$\n",
    "where $\\lambda_{\\text{max}}$ and $\\lambda_{\\text{min}}$ are the largest and smallest eigenvalue of $A$, respectively.\n",
    "\n",
    "- The larger $\\kappa$ the slower Conjugate Gradient.\n",
    "\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
199
    "- By default, conjugate gradient solves: $D^{-1} m = j$ for $m$, where $D^{-1}$ can be badly conditioned. If one knows a non-singular matrix $T$ for which $TD^{-1}$ is better conditioned, one can solve the equivalent problem:\n",
Philipp Arras's avatar
Philipp Arras committed
200
201
202
203
204
    "$$\\tilde A m = \\tilde j,$$\n",
    "where $\\tilde A = T D^{-1}$ and $\\tilde j = Tj$.\n",
    "\n",
    "- In our case $S^{-1}$ is responsible for the bad conditioning of $D$ depending on the chosen power spectrum. Thus, we choose\n",
    "\n",
Martin Reinecke's avatar
Martin Reinecke committed
205
206
    "$$T = \\mathcal F^\\dagger S_h^{-1} \\mathcal F.$$\n",
    "-->"
Philipp Arras's avatar
Philipp Arras committed
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Generate Mock data\n",
    "\n",
    "- Generate a field $s$ and $n$ with given covariances.\n",
    "- Calculate $d$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
226
   "metadata": {},
Philipp Arras's avatar
Philipp Arras committed
227
228
   "outputs": [],
   "source": [
229
230
231
232
    "s_space = ift.RGSpace(N_pixels)\n",
    "h_space = s_space.get_default_codomain()\n",
    "HT = ift.HarmonicTransformOperator(h_space, target=s_space)\n",
    "p_space = ift.PowerSpace(h_space)\n",
Philipp Arras's avatar
Philipp Arras committed
233
234
    "\n",
    "# Operators\n",
235
236
    "Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)\n",
    "R = HT #*ift.create_harmonic_smoothing_operator((h_space,), 0, 0.02)\n",
Philipp Arras's avatar
Philipp Arras committed
237
238
    "\n",
    "# Fields and data\n",
239
240
    "sh = ift.power_synthesize(ift.PS_field(p_space, pow_spec),real_signal=True)\n",
    "noiseless_data=R(sh)\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
241
    "noise_amplitude = np.sqrt(0.2)\n",
242
243
244
    "N = ift.ScalingOperator(noise_amplitude**2, s_space)\n",
    "\n",
    "n = ift.Field.from_random(domain=s_space, random_type='normal',\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
245
    "                          std=noise_amplitude, mean=0)\n",
246
247
248
    "d = noiseless_data + n\n",
    "j = R.adjoint_times(N.inverse_times(d))\n",
    "D = PropagatorOperator(R=R, N=N, Sh=Sh)"
Philipp Arras's avatar
Philipp Arras committed
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
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Run Wiener Filter"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "m = D(j)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Create Power Spectra of Signal and Reconstruction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
296
297
298
299
    "s_power = ift.power_analyze(sh)\n",
    "m_power = ift.power_analyze(m)\n",
    "s_power_data = s_power.val.real\n",
    "m_power_data = m_power.val.real\n",
Philipp Arras's avatar
Philipp Arras committed
300
301
    "\n",
    "# Get signal data and reconstruction data\n",
302
303
    "s_data = HT(sh).val.real\n",
    "m_data = HT(m).val.real\n",
Philipp Arras's avatar
Philipp Arras committed
304
    "\n",
305
    "d_data = d.val.real"
Philipp Arras's avatar
Philipp Arras committed
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Signal Reconstruction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
329
    "plt.figure(figsize=(15,10))\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
330
    "plt.plot(s_data, 'g', label=\"Signal\")\n",
Philipp Arras's avatar
Philipp Arras committed
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
    "plt.plot(d_data, 'k+', label=\"Data\")\n",
    "plt.plot(m_data, 'r', label=\"Reconstruction\")\n",
    "plt.title(\"Reconstruction\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
348
    "plt.figure(figsize=(15,10))\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
349
    "plt.plot(s_data - s_data, 'g', label=\"Signal\")\n",
Philipp Arras's avatar
Philipp Arras committed
350
351
    "plt.plot(d_data - s_data, 'k+', label=\"Data\")\n",
    "plt.plot(m_data - s_data, 'r', label=\"Reconstruction\")\n",
352
    "plt.axhspan(-noise_amplitude,noise_amplitude, facecolor='0.9', alpha=.5)\n",
Philipp Arras's avatar
Philipp Arras committed
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
    "plt.title(\"Residuals\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Power Spectrum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
379
    "plt.figure(figsize=(15,10))\n",
Philipp Arras's avatar
Philipp Arras committed
380
381
382
383
384
385
    "plt.loglog()\n",
    "plt.xlim(1, int(N_pixels/2))\n",
    "ymin = min(m_power_data)\n",
    "plt.ylim(ymin, 1)\n",
    "xs = np.arange(1,int(N_pixels/2),.1)\n",
    "plt.plot(xs, pow_spec(xs), label=\"True Power Spectrum\", linewidth=.7, color='k')\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
386
    "plt.plot(s_power_data, 'g', label=\"Signal\")\n",
Philipp Arras's avatar
Philipp Arras committed
387
    "plt.plot(m_power_data, 'r', label=\"Reconstruction\")\n",
388
389
    "plt.axhline(noise_amplitude**2 / N_pixels, color=\"k\", linestyle='--', label=\"Noise level\", alpha=.5)\n",
    "plt.axhspan(noise_amplitude**2 / N_pixels, ymin, facecolor='0.9', alpha=.5)\n",
Philipp Arras's avatar
Philipp Arras committed
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
    "plt.title(\"Power Spectrum\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Wiener Filter on Incomplete Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "# Operators\n",
417
418
    "Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)\n",
    "N = ift.ScalingOperator(noise_amplitude**2,s_space)\n",
Philipp Arras's avatar
Philipp Arras committed
419
420
421
    "# R is defined below\n",
    "\n",
    "# Fields\n",
422
423
424
425
    "sh = ift.power_synthesize(ift.PS_field(p_space,pow_spec),real_signal=True)\n",
    "s = HT(sh)\n",
    "n = ift.Field.from_random(domain=s_space, random_type='normal',\n",
    "                      std=noise_amplitude, mean=0)"
Philipp Arras's avatar
Philipp Arras committed
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "### Partially Lose Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "l = int(N_pixels * 0.2)\n",
450
    "h = int(N_pixels * 0.2 * 2)\n",
Philipp Arras's avatar
Philipp Arras committed
451
    "\n",
452
    "mask = ift.Field(s_space, val=1)\n",
Philipp Arras's avatar
Philipp Arras committed
453
454
    "mask.val[ l : h] = 0\n",
    "\n",
455
    "R = ift.DiagonalOperator(mask)*HT\n",
Philipp Arras's avatar
Philipp Arras committed
456
457
    "n.val[l:h] = 0\n",
    "\n",
458
    "d = R(sh) + n"
Philipp Arras's avatar
Philipp Arras committed
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "D = PropagatorOperator(R=R, N=N, Sh=Sh)\n",
    "j = R.adjoint_times(N.inverse_times(d))\n",
    "m = D(j)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Compute Uncertainty\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
491
    "scrolled": true
Philipp Arras's avatar
Philipp Arras committed
492
493
494
   },
   "outputs": [],
   "source": [
495
    "sc = ift.probing.utils.StatCalculator()\n",
Philipp Arras's avatar
Philipp Arras committed
496
    "\n",
Martin Reinecke's avatar
Martin Reinecke committed
497
    "IC = ift.GradientNormController(iteration_limit=50000,\n",
498
499
500
501
502
    "                                    tol_abs_gradnorm=0.1)\n",
    "inverter = ift.ConjugateGradient(controller=IC)\n",
    "curv = ift.library.wiener_filter_curvature.WienerFilterCurvature(R,N,Sh,inverter)\n",
    "\n",
    "for i in range(200):\n",
Martin Reinecke's avatar
Martin Reinecke committed
503
    "    print i\n",
504
505
506
    "    sc.add(HT(curv.generate_posterior_sample()))\n",
    "\n",
    "m_var = sc.var"
Philipp Arras's avatar
Philipp Arras committed
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "### Get data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
530
531
532
533
    "s_power = ift.power_analyze(sh)\n",
    "m_power = ift.power_analyze(m)\n",
    "s_power_data = s_power.val.real\n",
    "m_power_data = m_power.val.real\n",
Philipp Arras's avatar
Philipp Arras committed
534
535
    "\n",
    "# Get signal data and reconstruction data\n",
536
537
538
    "s_data = s.val.real\n",
    "m_data = HT(m).val.real\n",
    "m_var_data = m_var.val.real\n",
Philipp Arras's avatar
Philipp Arras committed
539
    "uncertainty = np.sqrt(np.abs(m_var_data))\n",
540
    "d_data = d.val.real\n",
Philipp Arras's avatar
Philipp Arras committed
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
    "\n",
    "# Set lost data to NaN for proper plotting\n",
    "d_data[d_data == 0] = np.nan"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
556
557
    "plt.figure(figsize=(15,10))\n",
    "plt.plot(s_data, 'g', label=\"Signal\", linewidth=1)\n",
Philipp Arras's avatar
Philipp Arras committed
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
    "plt.plot(d_data, 'k+', label=\"Data\", alpha=1)\n",
    "plt.axvspan(l, h, facecolor='0.8', alpha=.5)\n",
    "plt.title(\"Incomplete Data\")\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "fig = plt.figure(figsize=(15,10))\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
575
    "plt.plot(s_data, 'g', label=\"Signal\", alpha=1, linewidth=1)\n",
Philipp Arras's avatar
Philipp Arras committed
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
    "plt.plot(d_data, 'k+', label=\"Data\", alpha=.5)\n",
    "plt.plot(m_data, 'r', label=\"Reconstruction\")\n",
    "plt.axvspan(l, h, facecolor='0.8', alpha=.5)\n",
    "plt.fill_between(range(N_pixels), m_data - uncertainty, m_data + uncertainty, facecolor='0')\n",
    "plt.title(\"Reconstruction of incomplete data\")\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# 2d Example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
598
   "metadata": {},
Philipp Arras's avatar
Philipp Arras committed
599
600
601
   "outputs": [],
   "source": [
    "N_pixels = 256      # Number of pixels\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
602
    "sigma2 = 2.        # Noise variance\n",
Philipp Arras's avatar
Philipp Arras committed
603
604
605
    "\n",
    "\n",
    "def pow_spec(k):\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
606
    "    P0, k0, gamma = [.2, 2, 4]\n",
Philipp Arras's avatar
Philipp Arras committed
607
608
609
    "    return P0 * (1. + (k/k0)**2)**(- gamma / 2)\n",
    "\n",
    "\n",
610
    "s_space = ift.RGSpace([N_pixels, N_pixels])"
Philipp Arras's avatar
Philipp Arras committed
611
612
613
614
615
616
617
618
619
620
621
622
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
623
    "h_space = s_space.get_default_codomain()\n",
Martin Reinecke's avatar
Martin Reinecke committed
624
    "HT = ift.HarmonicTransformOperator(h_space,s_space)\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
625
    "p_space = ift.PowerSpace(h_space)\n",
Philipp Arras's avatar
Philipp Arras committed
626
627
    "\n",
    "# Operators\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
628
629
630
631
    "Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)\n",
    "N = ift.ScalingOperator(sigma2,s_space)\n",
    "R = ift.FFTSmoothingOperator(s_space, sigma=.01)\n",
    "#D = PropagatorOperator(R=R, N=N, Sh=Sh)\n",
Philipp Arras's avatar
Philipp Arras committed
632
633
    "\n",
    "# Fields and data\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
634
635
    "sh = ift.power_synthesize(ift.PS_field(p_space,pow_spec),real_signal=True)\n",
    "n = ift.Field.from_random(domain=s_space, random_type='normal',\n",
Philipp Arras's avatar
Philipp Arras committed
636
637
638
639
    "                      std=np.sqrt(sigma2), mean=0)\n",
    "\n",
    "# Lose some data\n",
    "\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
640
641
    "l = int(N_pixels * 0.33)\n",
    "h = int(N_pixels * 0.33 * 2)\n",
Philipp Arras's avatar
Philipp Arras committed
642
    "\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
643
    "mask = ift.Field(s_space, val=1)\n",
Philipp Arras's avatar
Philipp Arras committed
644
645
    "mask.val[l:h,l:h] = 0\n",
    "\n",
Martin Reinecke's avatar
Martin Reinecke committed
646
    "R = ift.DiagonalOperator(mask)*HT\n",
Philipp Arras's avatar
Philipp Arras committed
647
    "n.val[l:h, l:h] = 0\n",
Martin Reinecke's avatar
Martin Reinecke committed
648
    "D = PropagatorOperator(R=R, N=N, Sh=Sh)\n",
Philipp Arras's avatar
Philipp Arras committed
649
    "\n",
Martin Reinecke's avatar
Martin Reinecke committed
650
    "d = R(sh) + n\n",
Philipp Arras's avatar
Philipp Arras committed
651
652
653
654
655
656
    "j = R.adjoint_times(N.inverse_times(d))\n",
    "\n",
    "# Run Wiener filter\n",
    "m = D(j)\n",
    "\n",
    "# Uncertainty\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
657
658
    "sc = ift.probing.utils.StatCalculator()\n",
    "\n",
Martin Reinecke's avatar
Martin Reinecke committed
659
660
    "IC = ift.GradientNormController(iteration_limit=50000,\n",
    "                                tol_abs_gradnorm=0.1)\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
661
    "inverter = ift.ConjugateGradient(controller=IC)\n",
Martin Reinecke's avatar
Martin Reinecke committed
662
    "curv = ift.library.wiener_filter_curvature.WienerFilterCurvature(R,N,Sh,inverter)\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
663
664
    "\n",
    "for i in range(20):\n",
Martin Reinecke's avatar
Martin Reinecke committed
665
    "    print i\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
666
667
668
    "    sc.add(HT(curv.generate_posterior_sample()))\n",
    "\n",
    "m_var = sc.var\n",
Philipp Arras's avatar
Philipp Arras committed
669
670
    "\n",
    "# Get data\n",
Martin Reinecke's avatar
Martin Reinecke committed
671
672
673
674
675
676
677
678
    "s_power = ift.power_analyze(sh)\n",
    "m_power = ift.power_analyze(m)\n",
    "s_power_data = s_power.val.real\n",
    "m_power_data = m_power.val.real\n",
    "s_data = HT(sh).val.real\n",
    "m_data = HT(m).val.real\n",
    "m_var_data = m_var.val.real\n",
    "d_data = d.val.real\n",
Philipp Arras's avatar
Philipp Arras committed
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
    "\n",
    "uncertainty = np.sqrt(np.abs(m_var_data))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "cm = ['magma', 'inferno', 'plasma', 'viridis'][1]\n",
    "\n",
    "mi = np.min(s_data)\n",
    "ma = np.max(s_data)\n",
    "\n",
    "fig, axes = plt.subplots(1, 2, figsize=(15, 7))\n",
    "\n",
    "data = [s_data, d_data]\n",
    "caption = [\"Signal\", \"Data\"]\n",
    "\n",
    "for ax in axes.flat:\n",
    "    im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cm, vmin=mi,\n",
    "                   vmax=ma)\n",
    "    ax.set_title(caption.pop(0))\n",
    "\n",
    "fig.subplots_adjust(right=0.8)\n",
    "cbar_ax = fig.add_axes([0.85, 0.15, 0.05, 0.7])\n",
    "fig.colorbar(im, cax=cbar_ax)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "mi = np.min(s_data)\n",
    "ma = np.max(s_data)\n",
    "\n",
    "fig, axes = plt.subplots(2, 2, figsize=(15, 15))\n",
    "\n",
    "data = [s_data, m_data, s_data - m_data, uncertainty]\n",
    "caption = [\"Signal\", \"Reconstruction\", \"Residuals\", \"Uncertainty Map\"]\n",
    "\n",
    "for ax in axes.flat:\n",
    "    im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cm, vmin=mi, vmax=ma)\n",
    "    ax.set_title(caption.pop(0))\n",
    "\n",
    "fig.subplots_adjust(right=0.8)\n",
    "cbar_ax = fig.add_axes([.85, 0.15, 0.05, 0.7])\n",
    "fig.colorbar(im, cax=cbar_ax)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Is the uncertainty map reliable?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "precise = (np.abs(s_data-m_data) < uncertainty )\n",
    "print(\"Error within uncertainty map bounds: \" + str(np.sum(precise) * 100 / N_pixels**2) + \"%\")\n",
    "\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
764
    "plt.figure(figsize=(15,10))\n",
Philipp Arras's avatar
Philipp Arras committed
765
    "plt.imshow(precise.astype(float), cmap=\"brg\")\n",
Martin Reinecke's avatar
Martin Reinecke committed
766
    "plt.colorbar()"
Philipp Arras's avatar
Philipp Arras committed
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Start Coding\n",
    "## NIFTy Repository + Installation guide\n",
    "\n",
    "https://gitlab.mpcdf.mpg.de/ift/NIFTy\n",
    "\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
782
    "NIFTy v4 **more or less stable!**"
Philipp Arras's avatar
Philipp Arras committed
783
784
785
786
787
788
   ]
  }
 ],
 "metadata": {
  "celltoolbar": "Slideshow",
  "kernelspec": {
789
   "display_name": "Python 2",
Philipp Arras's avatar
Philipp Arras committed
790
   "language": "python",
791
   "name": "python2"
Philipp Arras's avatar
Philipp Arras committed
792
793
794
795
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
796
    "version": 2
Philipp Arras's avatar
Philipp Arras committed
797
798
799
800
801
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
802
803
   "pygments_lexer": "ipython2",
   "version": "2.7.12"
Philipp Arras's avatar
Philipp Arras committed
804
805
806
807
808
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}