getting_started_0.ipynb 18.2 KB
 Philipp Arras committed Feb 01, 2018 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 committed Feb 04, 2018 27  "Typically, $s$ is a continuous field, $d$ a discrete data vector. Particularly, $R$ is not invertible.\n",  Philipp Arras committed Feb 01, 2018 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 committed Feb 04, 2018 34  "NIFTy (Numerical Information Field Theory) is a Python framework in which IFT problems can be tackled easily.\n",  Philipp Arras committed Feb 01, 2018 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 committed Feb 04, 2018 40  "- **Operators**: Acting on fields."  Philipp Arras committed Feb 01, 2018 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60  ] }, { "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",  Philipp Arras committed Jul 10, 2018 61  "$$\\mathcal P (s|d) \\propto P(s,d) = \\mathcal G(d-Rs,N) \\,\\mathcal G(s,S) \\propto \\mathcal G (s-m,D)$$\n",  Philipp Arras committed Feb 01, 2018 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82  "\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 committed Feb 06, 2018 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 committed Feb 04, 2018 85  "- $N = 0.2 \\cdot \\mathbb{1}$.\n",  Martin Reinecke committed Feb 04, 2018 86 87  "- Number of data points $N_{pix} = 512$.\n", "- reconstruction in harmonic space.\n",  Philipp Arras committed Feb 01, 2018 88  "- Response operator:\n",  Martin Reinecke committed Feb 04, 2018 89  "$$R = FFT_{\\text{harmonic} \\rightarrow \\text{position}}$$\n"  Philipp Arras committed Feb 01, 2018 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 committed Feb 04, 2018 105 106  " P0, k0, gamma = [.2, 5, 4]\n", " return P0 / ((1. + (k/k0)**2)**(gamma / 2))"  Philipp Arras committed Feb 01, 2018 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  ] }, { "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": {  Martin Reinecke committed Jun 21, 2018 135  "scrolled": true,  Philipp Arras committed Feb 01, 2018 136 137 138 139 140 141 142  "slideshow": { "slide_type": "-" } }, "outputs": [], "source": [ "import numpy as np\n",  Philipp Arras committed Jun 11, 2021 143  "import nifty8 as ift\n",  Martin Reinecke committed Feb 04, 2018 144 145  "import matplotlib.pyplot as plt\n", "%matplotlib inline"  Philipp Arras committed Feb 01, 2018 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 committed Feb 05, 2018 169  "def Curvature(R, N, Sh):\n",  Martin Reinecke committed Feb 04, 2018 170  " IC = ift.GradientNormController(iteration_limit=50000,\n",  Martin Reinecke committed Feb 04, 2018 171  " tol_abs_gradnorm=0.1)\n",  Martin Reinecke committed Feb 05, 2018 172 173  " # WienerFilterCurvature is (R.adjoint*N.inverse*R + Sh.inverse) plus some handy\n", " # helper methods.\n",  Philipp Arras committed May 13, 2020 174  " return ift.WienerFilterCurvature(R,N,Sh,iteration_controller=IC,iteration_controller_sampling=IC)"  Philipp Arras committed Feb 01, 2018 175 176 177 178 179 180 181 182 183 184 185 186 187  ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "skip" } }, "source": [ "### Conjugate Gradient Preconditioning\n", "\n", "- $D$ is defined via:\n",  Martin Reinecke committed Feb 04, 2018 188  "$$D^{-1} = \\mathcal S_h^{-1} + R^\\dagger N^{-1} R.$$\n",  Philipp Arras committed Feb 01, 2018 189 190  "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 committed Feb 04, 2018 191  ""  Philipp Arras committed Feb 01, 2018 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224  ] }, { "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 committed Jun 21, 2018 225 226 227  "metadata": { "scrolled": true },  Philipp Arras committed Feb 01, 2018 228 229  "outputs": [], "source": [  Martin Reinecke committed Feb 04, 2018 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",  Philipp Arras committed Feb 01, 2018 233 234  "\n", "# Operators\n",  Martin Reinecke committed Feb 04, 2018 235  "Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)\n",  Philipp Arras committed Sep 14, 2020 236  "R = HT # @ ift.create_harmonic_smoothing_operator((h_space,), 0, 0.02)\n",  Philipp Arras committed Feb 01, 2018 237 238  "\n", "# Fields and data\n",  Philipp Arras committed May 12, 2020 239  "sh = Sh.draw_sample_with_dtype(dtype=np.float64)\n",  Martin Reinecke committed Feb 04, 2018 240  "noiseless_data=R(sh)\n",  Martin Reinecke committed Feb 04, 2018 241  "noise_amplitude = np.sqrt(0.2)\n",  Gordian Edenhofer committed Dec 05, 2019 242  "N = ift.ScalingOperator(s_space, noise_amplitude**2)\n",  Martin Reinecke committed Feb 04, 2018 243 244  "\n", "n = ift.Field.from_random(domain=s_space, random_type='normal',\n",  Martin Reinecke committed Feb 04, 2018 245  " std=noise_amplitude, mean=0)\n",  Martin Reinecke committed Feb 04, 2018 246 247  "d = noiseless_data + n\n", "j = R.adjoint_times(N.inverse_times(d))\n",  Martin Reinecke committed Feb 05, 2018 248 249  "curv = Curvature(R=R, N=N, Sh=Sh)\n", "D = curv.inverse"  Philipp Arras committed Feb 01, 2018 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  ] }, { "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": [  Martin Reinecke committed Feb 18, 2018 284  "### Signal Reconstruction"  Philipp Arras committed Feb 01, 2018 285 286 287 288 289 290 291 292 293 294 295 296 297  ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [], "source": [ "# Get signal data and reconstruction data\n",  Martin Reinecke committed Dec 05, 2019 298 299 300  "s_data = HT(sh).val\n", "m_data = HT(m).val\n", "d_data = d.val\n",  Philipp Arras committed Feb 01, 2018 301  "\n",  Martin Reinecke committed Feb 04, 2018 302  "plt.figure(figsize=(15,10))\n",  Martin Reinecke committed Feb 06, 2018 303 304 305  "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 committed Feb 01, 2018 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320  "plt.title(\"Reconstruction\")\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [], "source": [  Martin Reinecke committed Feb 04, 2018 321  "plt.figure(figsize=(15,10))\n",  Martin Reinecke committed Feb 06, 2018 322 323 324  "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",  Martin Reinecke committed Feb 04, 2018 325  "plt.axhspan(-noise_amplitude,noise_amplitude, facecolor='0.9', alpha=.5)\n",  Philipp Arras committed Feb 01, 2018 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  "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 committed Dec 05, 2019 352 353  "s_power_data = ift.power_analyze(sh).val\n", "m_power_data = ift.power_analyze(m).val\n",  Martin Reinecke committed Feb 04, 2018 354  "plt.figure(figsize=(15,10))\n",  Philipp Arras committed Feb 01, 2018 355 356 357 358 359  "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 committed Feb 06, 2018 360 361 362  "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",  Martin Reinecke committed Feb 04, 2018 363 364  "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 committed Feb 01, 2018 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  "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",  Martin Reinecke committed Feb 04, 2018 392  "Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)\n",  Gordian Edenhofer committed Dec 05, 2019 393  "N = ift.ScalingOperator(s_space, noise_amplitude**2)\n",  Philipp Arras committed Feb 01, 2018 394 395 396  "# R is defined below\n", "\n", "# Fields\n",  Philipp Arras committed May 12, 2020 397  "sh = Sh.draw_sample_with_dtype(dtype=np.float64)\n",  Martin Reinecke committed Feb 04, 2018 398 399 400  "s = HT(sh)\n", "n = ift.Field.from_random(domain=s_space, random_type='normal',\n", " std=noise_amplitude, mean=0)"  Philipp Arras committed Feb 01, 2018 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424  ] }, { "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",  Martin Reinecke committed Feb 04, 2018 425  "h = int(N_pixels * 0.2 * 2)\n",  Philipp Arras committed Feb 01, 2018 426  "\n",  Martin Reinecke committed Feb 18, 2018 427 428  "mask = np.full(s_space.shape, 1.)\n", "mask[l:h] = 0\n",  Martin Reinecke committed Dec 05, 2019 429  "mask = ift.Field.from_raw(s_space, mask)\n",  Philipp Arras committed Feb 01, 2018 430  "\n",  Martin Reinecke committed Aug 05, 2018 431  "R = ift.DiagonalOperator(mask)(HT)\n",  Martin Reinecke committed Dec 05, 2019 432  "n = n.val_rw()\n",  Martin Reinecke committed Feb 18, 2018 433  "n[l:h] = 0\n",  Martin Reinecke committed Dec 05, 2019 434  "n = ift.Field.from_raw(s_space, n)\n",  Philipp Arras committed Feb 01, 2018 435  "\n",  Martin Reinecke committed Feb 04, 2018 436  "d = R(sh) + n"  Philipp Arras committed Feb 01, 2018 437 438 439 440 441 442 443 444 445 446 447 448  ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [  Martin Reinecke committed Feb 05, 2018 449 450  "curv = Curvature(R=R, N=N, Sh=Sh)\n", "D = curv.inverse\n",  Philipp Arras committed Feb 01, 2018 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469  "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": {  Martin Reinecke committed Feb 04, 2018 470  "scrolled": true  Philipp Arras committed Feb 01, 2018 471 472 473  }, "outputs": [], "source": [  Martin Reinecke committed Apr 26, 2020 474  "m_mean, m_var = ift.probe_with_posterior_samples(curv, HT, 200, np.float64)"  Philipp Arras committed Feb 01, 2018 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498  ] }, { "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 committed Dec 05, 2019 499 500 501  "s_data = s.val\n", "m_data = HT(m).val\n", "m_var_data = m_var.val\n",  Martin Reinecke committed Feb 06, 2018 502  "uncertainty = np.sqrt(m_var_data)\n",  Martin Reinecke committed Dec 05, 2019 503  "d_data = d.val_rw()\n",  Philipp Arras committed Feb 01, 2018 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519  "\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 committed Feb 06, 2018 520 521 522 523 524  "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 committed Feb 01, 2018 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542  "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 committed Feb 05, 2018 543  "metadata": {},  Philipp Arras committed Feb 01, 2018 544 545 546  "outputs": [], "source": [ "N_pixels = 256 # Number of pixels\n",  Martin Reinecke committed Feb 06, 2018 547  "sigma2 = 2. # Noise variance\n",  Philipp Arras committed Feb 01, 2018 548 549  "\n", "def pow_spec(k):\n",  Martin Reinecke committed Feb 04, 2018 550  " P0, k0, gamma = [.2, 2, 4]\n",  Martin Reinecke committed Feb 06, 2018 551  " return P0 * (1. + (k/k0)**2)**(-gamma/2)\n",  Philipp Arras committed Feb 01, 2018 552  "\n",  Martin Reinecke committed Feb 04, 2018 553  "s_space = ift.RGSpace([N_pixels, N_pixels])"  Philipp Arras committed Feb 01, 2018 554 555 556 557 558 559 560 561 562 563 564 565  ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [  Martin Reinecke committed Feb 04, 2018 566  "h_space = s_space.get_default_codomain()\n",  Martin Reinecke committed Feb 04, 2018 567  "HT = ift.HarmonicTransformOperator(h_space,s_space)\n",  Philipp Arras committed Feb 01, 2018 568 569  "\n", "# Operators\n",  Martin Reinecke committed Feb 04, 2018 570  "Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)\n",  Gordian Edenhofer committed Dec 05, 2019 571  "N = ift.ScalingOperator(s_space, sigma2)\n",  Philipp Arras committed Feb 01, 2018 572 573  "\n", "# Fields and data\n",  Philipp Arras committed May 12, 2020 574  "sh = Sh.draw_sample_with_dtype(dtype=np.float64)\n",  Martin Reinecke committed Feb 04, 2018 575  "n = ift.Field.from_random(domain=s_space, random_type='normal',\n",  Philipp Arras committed Feb 01, 2018 576 577 578 579  " std=np.sqrt(sigma2), mean=0)\n", "\n", "# Lose some data\n", "\n",  Martin Reinecke committed Feb 04, 2018 580 581  "l = int(N_pixels * 0.33)\n", "h = int(N_pixels * 0.33 * 2)\n",  Philipp Arras committed Feb 01, 2018 582  "\n",  Martin Reinecke committed Feb 18, 2018 583 584  "mask = np.full(s_space.shape, 1.)\n", "mask[l:h,l:h] = 0.\n",  Martin Reinecke committed Dec 05, 2019 585  "mask = ift.Field.from_raw(s_space, mask)\n",  Philipp Arras committed Feb 01, 2018 586  "\n",  Martin Reinecke committed Aug 05, 2018 587  "R = ift.DiagonalOperator(mask)(HT)\n",  Martin Reinecke committed Dec 05, 2019 588  "n = n.val_rw()\n",  Martin Reinecke committed Feb 18, 2018 589  "n[l:h, l:h] = 0\n",  Martin Reinecke committed Dec 05, 2019 590  "n = ift.Field.from_raw(s_space, n)\n",  Martin Reinecke committed Feb 05, 2018 591 592  "curv = Curvature(R=R, N=N, Sh=Sh)\n", "D = curv.inverse\n",  Philipp Arras committed Feb 01, 2018 593  "\n",  Martin Reinecke committed Feb 04, 2018 594  "d = R(sh) + n\n",  Philipp Arras committed Feb 01, 2018 595 596 597 598 599 600  "j = R.adjoint_times(N.inverse_times(d))\n", "\n", "# Run Wiener filter\n", "m = D(j)\n", "\n", "# Uncertainty\n",  Martin Reinecke committed Apr 26, 2020 601  "m_mean, m_var = ift.probe_with_posterior_samples(curv, HT, 20, np.float64)\n",  Philipp Arras committed Feb 01, 2018 602 603  "\n", "# Get data\n",  Martin Reinecke committed Dec 05, 2019 604 605 606 607  "s_data = HT(sh).val\n", "m_data = HT(m).val\n", "m_var_data = m_var.val\n", "d_data = d.val\n",  Philipp Arras committed Feb 01, 2018 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653  "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 committed Feb 06, 2018 654  "fig, axes = plt.subplots(3, 2, figsize=(15, 22.5))\n",  Philipp Arras committed May 12, 2020 655  "sample = HT(curv.draw_sample(from_inverse=True)+m).val\n",  Martin Reinecke committed Dec 05, 2019 656  "post_mean = (m_mean + HT(m)).val\n",  Philipp Arras committed Feb 01, 2018 657  "\n",  Martin Reinecke committed Feb 18, 2018 658 659  "data = [s_data, m_data, post_mean, sample, s_data - m_data, uncertainty]\n", "caption = [\"Signal\", \"Reconstruction\", \"Posterior mean\", \"Sample\", \"Residuals\", \"Uncertainty Map\"]\n",  Philipp Arras committed Feb 01, 2018 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690  "\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": [  Martin Reinecke committed May 30, 2018 691  "precise = (np.abs(s_data-m_data) < uncertainty)\n",  Philipp Arras committed Feb 01, 2018 692 693  "print(\"Error within uncertainty map bounds: \" + str(np.sum(precise) * 100 / N_pixels**2) + \"%\")\n", "\n",  Martin Reinecke committed Feb 04, 2018 694  "plt.figure(figsize=(15,10))\n",  Philipp Arras committed Feb 01, 2018 695  "plt.imshow(precise.astype(float), cmap=\"brg\")\n",  Martin Reinecke committed Feb 04, 2018 696  "plt.colorbar()"  Philipp Arras committed Feb 01, 2018 697 698 699 700 701 702 703 704 705 706 707 708 709  ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Start Coding\n", "## NIFTy Repository + Installation guide\n", "\n",  Philipp Arras committed May 28, 2020 710  "https://gitlab.mpcdf.mpg.de/ift/NIFTy\n"  Philipp Arras committed Feb 01, 2018 711  ]  Philipp Arras committed May 12, 2020 712 713 714 715 716 717 718  }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": []  Philipp Arras committed Feb 01, 2018 719 720 721 722 723  } ], "metadata": { "celltoolbar": "Slideshow", "kernelspec": {  Philipp Arras committed Jun 28, 2018 724  "display_name": "Python 3",  Philipp Arras committed Feb 01, 2018 725  "language": "python",  Philipp Arras committed Jun 28, 2018 726  "name": "python3"  Philipp Arras committed Feb 01, 2018 727 728 729 730  }, "language_info": { "codemirror_mode": { "name": "ipython",  Philipp Arras committed Jun 28, 2018 731  "version": 3  Philipp Arras committed Feb 01, 2018 732 733 734 735 736  }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python",  Philipp Arras committed Jun 28, 2018 737  "pygments_lexer": "ipython3",  Philipp Arras committed May 12, 2020 738  "version": "3.8.2"  Philipp Arras committed Feb 01, 2018 739 740 741 742 743  } }, "nbformat": 4, "nbformat_minor": 2 }