Wiener Filter.ipynb 20.8 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 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 { "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", "Typically, $s$ continuous field, $d$ discrete data vector. Particularily, $R$ is not invertible.\n", "\n", "IFT aims at **inverting** the above uninvertible problem in the **best possible way** using Bayesian statistics.\n", "\n", "\n", "## NIFTy\n", "\n", "NIFTy (Numerical Information Field Theory, en. raffiniert) is a Python framework in which IFT problems can be tackeled easily.\n", "\n", "Main Interfaces:\n", "\n", "- **Spaces**: Cartesian, 2-Spheres (Healpix, Gauss-Legendre) and their respective harmonic spaces.\n", "- **Fields**: Defined on spaces.\n", "- **Operators**: Acting on spaces." ] }, { "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", "- One-dimensional signal with powerspectrum: $$P(k) = P_0\\,\\left(1+\\left(\\frac{k}{k_0}\\right)^2\\right)^{-\\gamma /2},$$\n", "with $P_0 = 0.2, k_0 = 5, \\gamma = 4$. Recall: $P(k)$ defines an isotropic and homogeneous $S$.\n", "- $N = 0.5 \\cdot \\text{id}$.\n", "- Number data points $N_{pix} = 512$.\n", "- Response operator:\n", "$$R_x=\\begin{pmatrix} \\delta(x-0)\\\\\\delta(x-1)\\\\\\ldots\\\\ \\delta(x-511) \\end{pmatrix}.$$\n", "However, the signal space is also discrete on the computer and $R = \\text{id}$." ] }, { "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", " P0, k0, gamma = [.2, 5, 6]\n", " return P0 * (1. + (k/k0)**2)**(- gamma / 2)" ] }, { "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 committed Feb 04, 2018 142 143 144 145  "np.random.seed(42)\n", "import nifty4 as ift\n", "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 04, 2018 169 170 171 172 173 174 175 176  "def PropagatorOperator(R, N, Sh):\n", " IC = ift.GradientNormController(name=\"inverter\", iteration_limit=50000,\n", " 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", " return ift.InversionEnabler(D, inverter)\n", " #return ift.library.wiener_filter_curvature.WienerFilterCurvature(R,N,Sh,inverter).inverse\n"  Philipp Arras committed Feb 01, 2018 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224  ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "skip" } }, "source": [ "### Conjugate Gradient Preconditioning\n", "\n", "- $D$ is defined via:\n", "$$D^{-1} = \\mathcal F^\\dagger S_h^{-1}\\mathcal F + R^\\dagger N^{-1} R.$$\n", "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", "- 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", "- By default, conjugate gradient solves: $D^{-1} m = j$ for $m$, where $D^{-1}$ can be bad conditioned. If one knows a non-singular matrix $T$ for which $TD^{-1}$ is better conditioned, one can solve the equivalent problem:\n", "$$\\tilde A m = \\tilde j,$$\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", "$$T = \\mathcal F^\\dagger S_h^{-1} \\mathcal F.$$" ] }, { "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 Feb 04, 2018 225  "metadata": {},  Philipp Arras committed Feb 01, 2018 226 227  "outputs": [], "source": [  Martin Reinecke committed Feb 04, 2018 228 229 230 231  "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 committed Feb 01, 2018 232 233  "\n", "# Operators\n",  Martin Reinecke committed Feb 04, 2018 234 235  "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 236 237  "\n", "# Fields and data\n",  Martin Reinecke committed Feb 04, 2018 238 239 240 241 242 243 244 245 246 247 248 249 250 251  "sh = ift.power_synthesize(ift.PS_field(p_space, pow_spec),real_signal=True)\n", "noiseless_data=R(sh)\n", "signal_to_noise = 5\n", "noise_amplitude = noiseless_data.std()/signal_to_noise\n", "N = ift.ScalingOperator(noise_amplitude**2, s_space)\n", "\n", "n = ift.Field.from_random(domain=s_space, random_type='normal',\n", " std=noise_amplitude, mean=0)\n", "ift.plot(n)\n", "d = noiseless_data + n\n", "ift.plot(d)\n", "j = R.adjoint_times(N.inverse_times(d))\n", "ift.plot(HT(j))\n", "D = PropagatorOperator(R=R, N=N, Sh=Sh)"  Philipp Arras committed Feb 01, 2018 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298  ] }, { "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": [  Martin Reinecke committed Feb 04, 2018 299 300 301 302  "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 committed Feb 01, 2018 303 304  "\n", "# Get signal data and reconstruction data\n",  Martin Reinecke committed Feb 04, 2018 305 306  "s_data = HT(sh).val.real\n", "m_data = HT(m).val.real\n",  Philipp Arras committed Feb 01, 2018 307  "\n",  Martin Reinecke committed Feb 04, 2018 308  "d_data = d.val.real"  Philipp Arras committed Feb 01, 2018 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353  ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Signal Reconstruction" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [], "source": [ "plt.plot(s_data, 'k', label=\"Signal\", alpha=.5, linewidth=.5)\n", "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": [ "plt.figure()\n", "plt.plot(s_data - s_data, 'k', label=\"Signal\", alpha=.5, linewidth=.5)\n", "plt.plot(d_data - s_data, 'k+', label=\"Data\")\n", "plt.plot(m_data - s_data, 'r', label=\"Reconstruction\")\n",  Martin Reinecke committed Feb 04, 2018 354  "plt.axhspan(-noise_amplitude,noise_amplitude, facecolor='0.9', alpha=.5)\n",  Philipp Arras committed Feb 01, 2018 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388  "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": [ "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", "plt.plot(s_power_data, 'k', label=\"Signal\", alpha=.5, linewidth=.5)\n", "plt.plot(m_power_data, 'r', label=\"Reconstruction\")\n",  Martin Reinecke committed Feb 04, 2018 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 committed Feb 01, 2018 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",  Martin Reinecke committed Feb 04, 2018 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 committed Feb 01, 2018 420 421 422  "# R is defined below\n", "\n", "# Fields\n",  Martin Reinecke committed Feb 04, 2018 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 committed Feb 01, 2018 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",  Martin Reinecke committed Feb 04, 2018 451  "h = int(N_pixels * 0.2 * 2)\n",  Philipp Arras committed Feb 01, 2018 452  "\n",  Martin Reinecke committed Feb 04, 2018 453  "mask = ift.Field(s_space, val=1)\n",  Philipp Arras committed Feb 01, 2018 454 455  "mask.val[ l : h] = 0\n", "\n",  Martin Reinecke committed Feb 04, 2018 456  "R = ift.DiagonalOperator(mask)*HT\n",  Philipp Arras committed Feb 01, 2018 457 458  "n.val[l:h] = 0\n", "\n",  Martin Reinecke committed Feb 04, 2018 459  "d = R(sh) + n"  Philipp Arras committed Feb 01, 2018 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491  ] }, { "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": {  Martin Reinecke committed Feb 04, 2018 492  "scrolled": true  Philipp Arras committed Feb 01, 2018 493 494 495  }, "outputs": [], "source": [  Martin Reinecke committed Feb 04, 2018 496  "sc = ift.probing.utils.StatCalculator()\n",  Philipp Arras committed Feb 01, 2018 497  "\n",  Martin Reinecke committed Feb 04, 2018 498 499 500 501 502 503 504 505 506  "IC = ift.GradientNormController(name=\"inverter\", iteration_limit=50000,\n", " 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", " sc.add(HT(curv.generate_posterior_sample()))\n", "\n", "m_var = sc.var"  Philipp Arras committed Feb 01, 2018 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": [  Martin Reinecke committed Feb 04, 2018 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 committed Feb 01, 2018 534 535  "\n", "# Get signal data and reconstruction data\n",  Martin Reinecke committed Feb 04, 2018 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 committed Feb 01, 2018 539  "uncertainty = np.sqrt(np.abs(m_var_data))\n",  Martin Reinecke committed Feb 04, 2018 540 541  "ift.plot(ift.sqrt(m_var))\n", "d_data = d.val.real\n",  Philipp Arras committed Feb 01, 2018 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624  "\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", "plt.plot(s_data, 'k', label=\"Signal\", alpha=.5, linewidth=1)\n", "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": "subslide" } }, "outputs": [], "source": [ "fig" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [ "fig = plt.figure(figsize=(15,10))\n", "plt.plot(s_data, 'k', label=\"Signal\", alpha=1, linewidth=1)\n", "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": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [], "source": [ "fig" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# 2d Example" ] }, { "cell_type": "code", "execution_count": null,  Martin Reinecke committed Feb 04, 2018 625  "metadata": {},  Philipp Arras committed Feb 01, 2018 626 627 628 629 630 631 632 633 634 635 636  "outputs": [], "source": [ "N_pixels = 256 # Number of pixels\n", "sigma2 = 1000 # Noise variance\n", "\n", "\n", "def pow_spec(k):\n", " P0, k0, gamma = [.2, 20, 4]\n", " return P0 * (1. + (k/k0)**2)**(- gamma / 2)\n", "\n", "\n",  Martin Reinecke committed Feb 04, 2018 637  "s_space = ift.RGSpace([N_pixels, N_pixels])"  Philipp Arras committed Feb 01, 2018 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832  ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [ "fft = FFTOperator(s_space)\n", "h_space = fft.target[0]\n", "p_space = PowerSpace(h_space)\n", "\n", "# Operators\n", "Sh = create_power_operator(h_space, power_spectrum=pow_spec)\n", "N = DiagonalOperator(s_space, diagonal=sigma2, bare=True)\n", "R = SmoothingOperator(s_space, sigma=.01)\n", "D = PropagatorOperator(R=R, N=N, Sh=Sh)\n", "\n", "# Fields and data\n", "sh = Field(p_space, val=pow_spec).power_synthesize(real_signal=True)\n", "s = fft.adjoint_times(sh)\n", "n = Field.from_random(domain=s_space, random_type='normal',\n", " std=np.sqrt(sigma2), mean=0)\n", "\n", "# Lose some data\n", "\n", "l = int(N_pixels * 0.2)\n", "h = int(N_pixels * 0.2 * 2)\n", "\n", "mask = Field(s_space, val=1)\n", "mask.val[l:h,l:h] = 0\n", "\n", "R = DiagonalOperator(s_space, diagonal = mask)\n", "n.val[l:h, l:h] = 0\n", "D = PropagatorOperator(R=R, N=N, Sh=Sh)\n", "\n", "d = R(s) + n\n", "j = R.adjoint_times(N.inverse_times(d))\n", "\n", "# Run Wiener filter\n", "m = D(j)\n", "\n", "# Uncertainty\n", "diagProber = DiagonalProber(domain=s_space, probe_dtype=np.complex, probe_count=10)\n", "diagProber(D)\n", "m_var = Field(s_space, val=diagProber.diagonal.val).weight(-1)\n", "\n", "# Get data\n", "s_power = sh.power_analyze()\n", "m_power = fft(m).power_analyze()\n", "s_power_data = s_power.val.get_full_data().real\n", "m_power_data = m_power.val.get_full_data().real\n", "s_data = s.val.get_full_data().real\n", "m_data = m.val.get_full_data().real\n", "m_var_data = m_var.val.get_full_data().real\n", "d_data = d.val.get_full_data().real\n", "\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": {}, "outputs": [], "source": [ "fig" ] }, { "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": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [], "source": [ "fig" ] }, { "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", "fig = plt.figure()\n", "plt.imshow(precise.astype(float), cmap=\"brg\")\n", "plt.colorbar()\n", "fig" ] }, { "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", "commit 1d10be4674a42945f8548f3b68688bf0f0d753fe\n", "\n", "NIFTy v3 **not (yet) stable!**" ] } ], "metadata": { "celltoolbar": "Slideshow", "kernelspec": {  Martin Reinecke committed Feb 04, 2018 833  "display_name": "Python 2",  Philipp Arras committed Feb 01, 2018 834  "language": "python",  Martin Reinecke committed Feb 04, 2018 835  "name": "python2"  Philipp Arras committed Feb 01, 2018 836 837 838 839  }, "language_info": { "codemirror_mode": { "name": "ipython",  Martin Reinecke committed Feb 04, 2018 840  "version": 2  Philipp Arras committed Feb 01, 2018 841 842 843 844 845  }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python",  Martin Reinecke committed Feb 04, 2018 846 847  "pygments_lexer": "ipython2", "version": "2.7.12"  Philipp Arras committed Feb 01, 2018 848 849 850 851 852  } }, "nbformat": 4, "nbformat_minor": 2 }