Wiener_Filter.ipynb 18.3 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 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 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",  Martin Reinecke committed Feb 04, 2018 143  "np.random.seed(40)\n",  Philipp Arras committed Jun 18, 2018 144  "import nifty5 as ift\n",  Martin Reinecke committed Feb 04, 2018 145 146  "import matplotlib.pyplot as plt\n", "%matplotlib inline"  Philipp Arras committed Feb 01, 2018 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169  ] }, { "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 170  "def Curvature(R, N, Sh):\n",  Martin Reinecke committed Feb 04, 2018 171  " IC = ift.GradientNormController(iteration_limit=50000,\n",  Martin Reinecke committed Feb 04, 2018 172  " tol_abs_gradnorm=0.1)\n",  Martin Reinecke committed Feb 05, 2018 173 174  " # WienerFilterCurvature is (R.adjoint*N.inverse*R + Sh.inverse) plus some handy\n", " # helper methods.\n",  Martin Reinecke committed Jun 21, 2018 175  " return ift.library.WienerFilterCurvature(R,N,Sh,iteration_controller=IC,iteration_controller_sampling=IC)"  Philipp Arras committed Feb 01, 2018 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 committed Feb 04, 2018 189  "$$D^{-1} = \\mathcal S_h^{-1} + R^\\dagger N^{-1} R.$$\n",  Philipp Arras committed Feb 01, 2018 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 committed Feb 04, 2018 192  ""  Philipp Arras committed Feb 01, 2018 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 committed Jun 21, 2018 226 227 228  "metadata": { "scrolled": true },  Philipp Arras committed Feb 01, 2018 229 230  "outputs": [], "source": [  Martin Reinecke committed Feb 04, 2018 231 232 233  "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 234 235  "\n", "# Operators\n",  Martin Reinecke committed Feb 04, 2018 236 237  "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 committed Feb 01, 2018 238 239  "\n", "# Fields and data\n",  Martin Reinecke committed Feb 21, 2018 240  "sh = Sh.draw_sample()\n",  Martin Reinecke committed Feb 04, 2018 241  "noiseless_data=R(sh)\n",  Martin Reinecke committed Feb 04, 2018 242  "noise_amplitude = np.sqrt(0.2)\n",  Martin Reinecke committed Feb 04, 2018 243 244 245  "N = ift.ScalingOperator(noise_amplitude**2, s_space)\n", "\n", "n = ift.Field.from_random(domain=s_space, random_type='normal',\n",  Martin Reinecke committed Feb 04, 2018 246  " std=noise_amplitude, mean=0)\n",  Martin Reinecke committed Feb 04, 2018 247 248  "d = noiseless_data + n\n", "j = R.adjoint_times(N.inverse_times(d))\n",  Martin Reinecke committed Feb 05, 2018 249 250  "curv = Curvature(R=R, N=N, Sh=Sh)\n", "D = curv.inverse"  Philipp Arras committed Feb 01, 2018 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  ] }, { "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 285  "### Signal Reconstruction"  Philipp Arras committed Feb 01, 2018 286 287 288 289 290 291 292 293 294 295 296 297 298  ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [], "source": [ "# Get signal data and reconstruction data\n",  Martin Reinecke committed Feb 18, 2018 299 300  "s_data = HT(sh).to_global_data()\n", "m_data = HT(m).to_global_data()\n",  Martin Reinecke committed Feb 18, 2018 301  "d_data = d.to_global_data()\n",  Philipp Arras committed Feb 01, 2018 302  "\n",  Martin Reinecke committed Feb 04, 2018 303  "plt.figure(figsize=(15,10))\n",  Martin Reinecke committed Feb 06, 2018 304 305 306  "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 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321  "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 322  "plt.figure(figsize=(15,10))\n",  Martin Reinecke committed Feb 06, 2018 323 324 325  "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 326  "plt.axhspan(-noise_amplitude,noise_amplitude, facecolor='0.9', alpha=.5)\n",  Philipp Arras committed Feb 01, 2018 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  "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 Feb 18, 2018 353 354  "s_power_data = ift.power_analyze(sh).to_global_data()\n", "m_power_data = ift.power_analyze(m).to_global_data()\n",  Martin Reinecke committed Feb 04, 2018 355  "plt.figure(figsize=(15,10))\n",  Philipp Arras committed Feb 01, 2018 356 357 358 359 360  "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 361 362 363  "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 364 365  "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 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  "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 393 394  "Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)\n", "N = ift.ScalingOperator(noise_amplitude**2,s_space)\n",  Philipp Arras committed Feb 01, 2018 395 396 397  "# R is defined below\n", "\n", "# Fields\n",  Martin Reinecke committed Feb 21, 2018 398  "sh = Sh.draw_sample()\n",  Martin Reinecke committed Feb 04, 2018 399 400 401  "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 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425  ] }, { "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 426  "h = int(N_pixels * 0.2 * 2)\n",  Philipp Arras committed Feb 01, 2018 427  "\n",  Martin Reinecke committed Feb 18, 2018 428 429 430  "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 committed Feb 01, 2018 431  "\n",  Martin Reinecke committed Feb 04, 2018 432  "R = ift.DiagonalOperator(mask)*HT\n",  Martin Reinecke committed Feb 18, 2018 433 434 435  "n = n.to_global_data()\n", "n[l:h] = 0\n", "n = ift.Field.from_global_data(s_space, n)\n",  Philipp Arras committed Feb 01, 2018 436  "\n",  Martin Reinecke committed Feb 04, 2018 437  "d = R(sh) + n"  Philipp Arras committed Feb 01, 2018 438 439 440 441 442 443 444 445 446 447 448 449  ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [  Martin Reinecke committed Feb 05, 2018 450 451  "curv = Curvature(R=R, N=N, Sh=Sh)\n", "D = curv.inverse\n",  Philipp Arras committed Feb 01, 2018 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470  "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 471  "scrolled": true  Philipp Arras committed Feb 01, 2018 472 473 474  }, "outputs": [], "source": [  Martin Reinecke committed Feb 18, 2018 475  "m_mean, m_var = ift.probe_with_posterior_samples(curv, HT, 200)"  Philipp Arras committed Feb 01, 2018 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499  ] }, { "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 Feb 18, 2018 500 501 502  "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 committed Feb 06, 2018 503  "uncertainty = np.sqrt(m_var_data)\n",  Martin Reinecke committed Feb 18, 2018 504  "d_data = d.to_global_data()\n",  Philipp Arras committed Feb 01, 2018 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520  "\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 521 522 523 524 525  "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 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543  "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 544  "metadata": {},  Philipp Arras committed Feb 01, 2018 545 546 547  "outputs": [], "source": [ "N_pixels = 256 # Number of pixels\n",  Martin Reinecke committed Feb 06, 2018 548  "sigma2 = 2. # Noise variance\n",  Philipp Arras committed Feb 01, 2018 549 550  "\n", "def pow_spec(k):\n",  Martin Reinecke committed Feb 04, 2018 551  " P0, k0, gamma = [.2, 2, 4]\n",  Martin Reinecke committed Feb 06, 2018 552  " return P0 * (1. + (k/k0)**2)**(-gamma/2)\n",  Philipp Arras committed Feb 01, 2018 553  "\n",  Martin Reinecke committed Feb 04, 2018 554  "s_space = ift.RGSpace([N_pixels, N_pixels])"  Philipp Arras committed Feb 01, 2018 555 556 557 558 559 560 561 562 563 564 565 566  ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [  Martin Reinecke committed Feb 04, 2018 567  "h_space = s_space.get_default_codomain()\n",  Martin Reinecke committed Feb 04, 2018 568  "HT = ift.HarmonicTransformOperator(h_space,s_space)\n",  Philipp Arras committed Feb 01, 2018 569 570  "\n", "# Operators\n",  Martin Reinecke committed Feb 04, 2018 571 572  "Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)\n", "N = ift.ScalingOperator(sigma2,s_space)\n",  Philipp Arras committed Feb 01, 2018 573 574  "\n", "# Fields and data\n",  Martin Reinecke committed Feb 21, 2018 575  "sh = Sh.draw_sample()\n",  Martin Reinecke committed Feb 04, 2018 576  "n = ift.Field.from_random(domain=s_space, random_type='normal',\n",  Philipp Arras committed Feb 01, 2018 577 578 579 580  " std=np.sqrt(sigma2), mean=0)\n", "\n", "# Lose some data\n", "\n",  Martin Reinecke committed Feb 04, 2018 581 582  "l = int(N_pixels * 0.33)\n", "h = int(N_pixels * 0.33 * 2)\n",  Philipp Arras committed Feb 01, 2018 583  "\n",  Martin Reinecke committed Feb 18, 2018 584 585 586  "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 committed Feb 01, 2018 587  "\n",  Martin Reinecke committed Feb 04, 2018 588  "R = ift.DiagonalOperator(mask)*HT\n",  Martin Reinecke committed Feb 18, 2018 589 590 591  "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 committed Feb 05, 2018 592 593  "curv = Curvature(R=R, N=N, Sh=Sh)\n", "D = curv.inverse\n",  Philipp Arras committed Feb 01, 2018 594  "\n",  Martin Reinecke committed Feb 04, 2018 595  "d = R(sh) + n\n",  Philipp Arras committed Feb 01, 2018 596 597 598 599 600 601  "j = R.adjoint_times(N.inverse_times(d))\n", "\n", "# Run Wiener filter\n", "m = D(j)\n", "\n", "# Uncertainty\n",  Martin Reinecke committed Feb 18, 2018 602  "m_mean, m_var = ift.probe_with_posterior_samples(curv, HT, 20)\n",  Philipp Arras committed Feb 01, 2018 603 604  "\n", "# Get data\n",  Martin Reinecke committed Feb 18, 2018 605 606 607 608  "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 committed Feb 01, 2018 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654  "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 655  "fig, axes = plt.subplots(3, 2, figsize=(15, 22.5))\n",  Martin Reinecke committed Apr 02, 2018 656  "sample = HT(curv.draw_sample(from_inverse=True)+m).to_global_data()\n",  Martin Reinecke committed Feb 18, 2018 657  "post_mean = (m_mean + HT(m)).to_global_data()\n",  Philipp Arras committed Feb 01, 2018 658  "\n",  Martin Reinecke committed Feb 18, 2018 659 660  "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 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691  "\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 692  "precise = (np.abs(s_data-m_data) < uncertainty)\n",  Philipp Arras committed Feb 01, 2018 693 694  "print(\"Error within uncertainty map bounds: \" + str(np.sum(precise) * 100 / N_pixels**2) + \"%\")\n", "\n",  Martin Reinecke committed Feb 04, 2018 695  "plt.figure(figsize=(15,10))\n",  Philipp Arras committed Feb 01, 2018 696  "plt.imshow(precise.astype(float), cmap=\"brg\")\n",  Martin Reinecke committed Feb 04, 2018 697  "plt.colorbar()"  Philipp Arras committed Feb 01, 2018 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712  ] }, { "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 committed Jun 19, 2018 713  "NIFTy v5 **more or less stable!**"  Philipp Arras committed Feb 01, 2018 714 715 716 717 718 719  ] } ], "metadata": { "celltoolbar": "Slideshow", "kernelspec": {  Martin Reinecke committed Jun 21, 2018 720  "display_name": "Python 2",  Philipp Arras committed Feb 01, 2018 721  "language": "python",  Martin Reinecke committed Jun 21, 2018 722  "name": "python2"  Philipp Arras committed Feb 01, 2018 723 724 725 726  }, "language_info": { "codemirror_mode": { "name": "ipython",  Martin Reinecke committed Jun 21, 2018 727  "version": 2  Philipp Arras committed Feb 01, 2018 728 729 730 731 732  }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python",  Martin Reinecke committed Jun 21, 2018 733 734  "pygments_lexer": "ipython2", "version": "2.7.15"  Philipp Arras committed Feb 01, 2018 735 736 737 738 739  } }, "nbformat": 4, "nbformat_minor": 2 }