Wiener_Filter.ipynb 19.1 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
Martin Reinecke committed
83
84
    "- We assume statistical homogeneity and isotropy. Therefore the signal covariance $S$ is diagonal in harmonic space, and is described by a one-dimensional power spectrum, assumed here as $$P(k) = P_0\\,\\left(1+\\left(\\frac{k}{k_0}\\right)^2\\right)^{-\\gamma /2},$$\n",
    "with $P_0 = 0.2, k_0 = 5, \\gamma = 4$.\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": [
Martin Reinecke's avatar
cleanup    
Martin Reinecke committed
169
    "def Curvature(R, N, Sh):\n",
Martin Reinecke's avatar
Martin Reinecke committed
170
    "    IC = ift.GradientNormController(iteration_limit=50000,\n",
171
172
    "                                    tol_abs_gradnorm=0.1)\n",
    "    inverter = ift.ConjugateGradient(controller=IC)\n",
Martin Reinecke's avatar
cleanup    
Martin Reinecke committed
173
174
    "    # WienerFilterCurvature is (R.adjoint*N.inverse*R + Sh.inverse) plus some handy\n",
    "    # helper methods.\n",
Martin Reinecke's avatar
Martin Reinecke committed
175
    "    return ift.library.WienerFilterCurvature(R,N,Sh,inverter)"
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,
Martin Reinecke's avatar
Martin Reinecke committed
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
    "d = noiseless_data + n\n",
    "j = R.adjoint_times(N.inverse_times(d))\n",
Martin Reinecke's avatar
cleanup    
Martin Reinecke committed
248
249
    "curv = Curvature(R=R, N=N, Sh=Sh)\n",
    "D = curv.inverse"
Philipp Arras's avatar
Philipp Arras committed
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
   ]
  },
  {
   "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": [
297
298
    "s_power = ift.power_analyze(sh)\n",
    "m_power = ift.power_analyze(m)\n",
Martin Reinecke's avatar
Martin Reinecke committed
299
300
    "s_power_data = s_power.to_global_data()\n",
    "m_power_data = m_power.to_global_data()\n",
Philipp Arras's avatar
Philipp Arras committed
301
302
    "\n",
    "# Get signal data and reconstruction data\n",
Martin Reinecke's avatar
Martin Reinecke committed
303
304
    "s_data = HT(sh).to_global_data()\n",
    "m_data = HT(m).to_global_data()\n",
Philipp Arras's avatar
Philipp Arras committed
305
    "\n",
Martin Reinecke's avatar
Martin Reinecke committed
306
    "d_data = d.to_global_data()"
Philipp Arras's avatar
Philipp Arras committed
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
   ]
  },
  {
   "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
330
    "plt.figure(figsize=(15,10))\n",
Martin Reinecke's avatar
Martin Reinecke committed
331
332
333
    "plt.plot(s_data, 'r', label=\"Signal\", linewidth=3)\n",
    "plt.plot(d_data, 'k.', label=\"Data\")\n",
    "plt.plot(m_data, 'k', label=\"Reconstruction\",linewidth=3)\n",
Philipp Arras's avatar
Philipp Arras committed
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
    "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
349
    "plt.figure(figsize=(15,10))\n",
Martin Reinecke's avatar
Martin Reinecke committed
350
351
352
    "plt.plot(s_data - s_data, 'r', label=\"Signal\", linewidth=3)\n",
    "plt.plot(d_data - s_data, 'k.', label=\"Data\")\n",
    "plt.plot(m_data - s_data, 'k', label=\"Reconstruction\",linewidth=3)\n",
353
    "plt.axhspan(-noise_amplitude,noise_amplitude, facecolor='0.9', alpha=.5)\n",
Philipp Arras's avatar
Philipp Arras committed
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
    "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
380
    "plt.figure(figsize=(15,10))\n",
Philipp Arras's avatar
Philipp Arras committed
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",
Martin Reinecke's avatar
Martin Reinecke committed
386
387
388
    "plt.plot(xs, pow_spec(xs), label=\"True Power Spectrum\", color='k',alpha=0.5)\n",
    "plt.plot(s_power_data, 'r', label=\"Signal\")\n",
    "plt.plot(m_power_data, 'k', label=\"Reconstruction\")\n",
389
390
    "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
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
    "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",
418
419
    "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
420
421
422
    "# R is defined below\n",
    "\n",
    "# Fields\n",
423
424
425
426
    "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
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
   ]
  },
  {
   "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",
451
    "h = int(N_pixels * 0.2 * 2)\n",
Philipp Arras's avatar
Philipp Arras committed
452
    "\n",
Martin Reinecke's avatar
Martin Reinecke committed
453
454
455
    "mask = np.full(s_space.shape, 1.)\n",
    "mask[l:h] = 0\n",
    "mask = ift.Field.from_global_data(s_space, mask)\n",
Philipp Arras's avatar
Philipp Arras committed
456
    "\n",
457
    "R = ift.DiagonalOperator(mask)*HT\n",
Martin Reinecke's avatar
Martin Reinecke committed
458
459
460
    "n = n.to_global_data()\n",
    "n[l:h] = 0\n",
    "n = ift.Field.from_global_data(s_space, n)\n",
Philipp Arras's avatar
Philipp Arras committed
461
    "\n",
462
    "d = R(sh) + n"
Philipp Arras's avatar
Philipp Arras committed
463
464
465
466
467
468
469
470
471
472
473
474
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
Martin Reinecke's avatar
cleanup    
Martin Reinecke committed
475
476
    "curv = Curvature(R=R, N=N, Sh=Sh)\n",
    "D = curv.inverse\n",
Philipp Arras's avatar
Philipp Arras committed
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
    "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": {
496
    "scrolled": true
Philipp Arras's avatar
Philipp Arras committed
497
498
499
   },
   "outputs": [],
   "source": [
Martin Reinecke's avatar
Martin Reinecke committed
500
    "m_mean, m_var = ift.probe_with_posterior_samples(curv, HT, 200)"
Philipp Arras's avatar
Philipp Arras committed
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "### Get data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "# Get signal data and reconstruction data\n",
Martin Reinecke's avatar
Martin Reinecke committed
525
526
527
    "s_data = s.to_global_data()\n",
    "m_data = HT(m).to_global_data()\n",
    "m_var_data = m_var.to_global_data()\n",
Martin Reinecke's avatar
Martin Reinecke committed
528
    "uncertainty = np.sqrt(m_var_data)\n",
Martin Reinecke's avatar
Martin Reinecke committed
529
    "d_data = d.to_global_data()\n",
Philipp Arras's avatar
Philipp Arras committed
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
    "\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": [
    "fig = plt.figure(figsize=(15,10))\n",
Martin Reinecke's avatar
Martin Reinecke committed
546
547
548
549
550
    "plt.axvspan(l, h, facecolor='0.8',alpha=0.5)\n",
    "plt.fill_between(range(N_pixels), m_data - uncertainty, m_data + uncertainty, facecolor='0.5', alpha=0.5)\n",
    "plt.plot(s_data, 'r', label=\"Signal\", alpha=1, linewidth=3)\n",
    "plt.plot(d_data, 'k.', label=\"Data\")\n",
    "plt.plot(m_data, 'k', label=\"Reconstruction\", linewidth=3)\n",
Philipp Arras's avatar
Philipp Arras committed
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
    "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,
Martin Reinecke's avatar
Martin Reinecke committed
569
   "metadata": {},
Philipp Arras's avatar
Philipp Arras committed
570
571
572
   "outputs": [],
   "source": [
    "N_pixels = 256      # Number of pixels\n",
Martin Reinecke's avatar
Martin Reinecke committed
573
    "sigma2 = 2.         # Noise variance\n",
Philipp Arras's avatar
Philipp Arras committed
574
575
    "\n",
    "def pow_spec(k):\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
576
    "    P0, k0, gamma = [.2, 2, 4]\n",
Martin Reinecke's avatar
Martin Reinecke committed
577
    "    return P0 * (1. + (k/k0)**2)**(-gamma/2)\n",
Philipp Arras's avatar
Philipp Arras committed
578
    "\n",
579
    "s_space = ift.RGSpace([N_pixels, N_pixels])"
Philipp Arras's avatar
Philipp Arras committed
580
581
582
583
584
585
586
587
588
589
590
591
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
592
    "h_space = s_space.get_default_codomain()\n",
Martin Reinecke's avatar
Martin Reinecke committed
593
    "HT = ift.HarmonicTransformOperator(h_space,s_space)\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
594
    "p_space = ift.PowerSpace(h_space)\n",
Philipp Arras's avatar
Philipp Arras committed
595
596
    "\n",
    "# Operators\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
597
598
    "Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)\n",
    "N = ift.ScalingOperator(sigma2,s_space)\n",
Philipp Arras's avatar
Philipp Arras committed
599
600
    "\n",
    "# Fields and data\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
601
602
    "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
603
604
605
606
    "                      std=np.sqrt(sigma2), mean=0)\n",
    "\n",
    "# Lose some data\n",
    "\n",
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
607
608
    "l = int(N_pixels * 0.33)\n",
    "h = int(N_pixels * 0.33 * 2)\n",
Philipp Arras's avatar
Philipp Arras committed
609
    "\n",
Martin Reinecke's avatar
Martin Reinecke committed
610
611
612
    "mask = np.full(s_space.shape, 1.)\n",
    "mask[l:h,l:h] = 0.\n",
    "mask = ift.Field.from_global_data(s_space, mask)\n",
Philipp Arras's avatar
Philipp Arras committed
613
    "\n",
Martin Reinecke's avatar
Martin Reinecke committed
614
    "R = ift.DiagonalOperator(mask)*HT\n",
Martin Reinecke's avatar
Martin Reinecke committed
615
616
617
    "n = n.to_global_data()\n",
    "n[l:h, l:h] = 0\n",
    "n = ift.Field.from_global_data(s_space, n)\n",
Martin Reinecke's avatar
cleanup    
Martin Reinecke committed
618
619
    "curv = Curvature(R=R, N=N, Sh=Sh)\n",
    "D = curv.inverse\n",
Philipp Arras's avatar
Philipp Arras committed
620
    "\n",
Martin Reinecke's avatar
Martin Reinecke committed
621
    "d = R(sh) + n\n",
Philipp Arras's avatar
Philipp Arras committed
622
623
624
625
626
627
    "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
Martin Reinecke committed
628
    "m_mean, m_var = ift.probe_with_posterior_samples(curv, HT, 20)\n",
Philipp Arras's avatar
Philipp Arras committed
629
630
    "\n",
    "# Get data\n",
Martin Reinecke's avatar
Martin Reinecke committed
631
632
    "s_power = ift.power_analyze(sh)\n",
    "m_power = ift.power_analyze(m)\n",
Martin Reinecke's avatar
Martin Reinecke committed
633
634
635
636
637
638
    "s_power_data = s_power.to_global_data()\n",
    "m_power_data = m_power.to_global_data()\n",
    "s_data = HT(sh).to_global_data()\n",
    "m_data = HT(m).to_global_data()\n",
    "m_var_data = m_var.to_global_data()\n",
    "d_data = d.to_global_data()\n",
Philipp Arras's avatar
Philipp Arras committed
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
    "\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",
Martin Reinecke's avatar
Martin Reinecke committed
686
    "fig, axes = plt.subplots(3, 2, figsize=(15, 22.5))\n",
Martin Reinecke's avatar
Martin Reinecke committed
687
688
    "samp1 = HT(curv.draw_sample()+m).to_global_data()\n",
    "samp2 = HT(curv.draw_sample()+m).to_global_data()\n",
Philipp Arras's avatar
Philipp Arras committed
689
    "\n",
Martin Reinecke's avatar
Martin Reinecke committed
690
691
    "data = [s_data, m_data, samp1, samp2, s_data - m_data, uncertainty]\n",
    "caption = [\"Signal\", \"Reconstruction\", \"Sample 1\", \"Sample 2\", \"Residuals\", \"Uncertainty Map\"]\n",
Philipp Arras's avatar
Philipp Arras committed
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
    "\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
726
    "plt.figure(figsize=(15,10))\n",
Philipp Arras's avatar
Philipp Arras committed
727
    "plt.imshow(precise.astype(float), cmap=\"brg\")\n",
Martin Reinecke's avatar
Martin Reinecke committed
728
    "plt.colorbar()"
Philipp Arras's avatar
Philipp Arras committed
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
   ]
  },
  {
   "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
744
    "NIFTy v4 **more or less stable!**"
Philipp Arras's avatar
Philipp Arras committed
745
746
747
748
749
750
   ]
  }
 ],
 "metadata": {
  "celltoolbar": "Slideshow",
  "kernelspec": {
Martin Reinecke's avatar
Martin Reinecke committed
751
   "display_name": "Python 2",
Philipp Arras's avatar
Philipp Arras committed
752
   "language": "python",
Martin Reinecke's avatar
Martin Reinecke committed
753
   "name": "python2"
Philipp Arras's avatar
Philipp Arras committed
754
755
756
757
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
Martin Reinecke's avatar
Martin Reinecke committed
758
    "version": 2
Philipp Arras's avatar
Philipp Arras committed
759
760
761
762
763
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
Martin Reinecke's avatar
Martin Reinecke committed
764
765
   "pygments_lexer": "ipython2",
   "version": "2.7.12"
Philipp Arras's avatar
Philipp Arras committed
766
767
768
769
770
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}