Wiener Filter.ipynb 20 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 04, 2018 83  "- One-dimensional signal with power spectrum: $$P(k) = P_0\\,\\left(1+\\left(\\frac{k}{k_0}\\right)^2\\right)^{-\\gamma /2},$$\n",  Philipp Arras committed Feb 01, 2018 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104  "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",  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 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141  ] }, { "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 committed Feb 04, 2018 142  "np.random.seed(40)\n",  Martin Reinecke committed Feb 04, 2018 143 144 145  "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  "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",  Martin Reinecke committed Feb 04, 2018 175  " return ift.InversionEnabler(D, inverter)\n"  Philipp Arras committed Feb 01, 2018 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197  ] }, { "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",  Martin Reinecke committed Feb 04, 2018 198  "- 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 committed Feb 01, 2018 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  "$$\\tilde A m = \\tilde j,$$\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 224  "metadata": {},  Philipp Arras committed Feb 01, 2018 225 226  "outputs": [], "source": [  Martin Reinecke committed Feb 04, 2018 227 228 229 230  "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 231 232  "\n", "# Operators\n",  Martin Reinecke committed Feb 04, 2018 233 234  "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 235 236  "\n", "# Fields and data\n",  Martin Reinecke committed Feb 04, 2018 237 238  "sh = ift.power_synthesize(ift.PS_field(p_space, pow_spec),real_signal=True)\n", "noiseless_data=R(sh)\n",  Martin Reinecke committed Feb 04, 2018 239  "noise_amplitude = np.sqrt(0.05)\n",  Martin Reinecke committed Feb 04, 2018 240 241 242  "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 243  " std=noise_amplitude, mean=0)\n",  Martin Reinecke committed Feb 04, 2018 244 245 246  "d = noiseless_data + n\n", "j = R.adjoint_times(N.inverse_times(d))\n", "D = PropagatorOperator(R=R, N=N, Sh=Sh)"  Philipp Arras committed Feb 01, 2018 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293  ] }, { "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 294 295 296 297  "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 298 299  "\n", "# Get signal data and reconstruction data\n",  Martin Reinecke committed Feb 04, 2018 300 301  "s_data = HT(sh).val.real\n", "m_data = HT(m).val.real\n",  Philipp Arras committed Feb 01, 2018 302  "\n",  Martin Reinecke committed Feb 04, 2018 303  "d_data = d.val.real"  Philipp Arras committed Feb 01, 2018 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326  ] }, { "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 committed Feb 04, 2018 327  "plt.plot(s_data, 'g', label=\"Signal\")\n",  Philipp Arras committed Feb 01, 2018 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345  "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",  Martin Reinecke committed Feb 04, 2018 346  "plt.plot(s_data - s_data, 'g', label=\"Signal\")\n",  Philipp Arras committed Feb 01, 2018 347 348  "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 349  "plt.axhspan(-noise_amplitude,noise_amplitude, facecolor='0.9', alpha=.5)\n",  Philipp Arras committed Feb 01, 2018 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383  "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 384 385  "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 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412  "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 413 414  "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 415 416 417  "# R is defined below\n", "\n", "# Fields\n",  Martin Reinecke committed Feb 04, 2018 418 419 420 421  "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 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445  ] }, { "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 446  "h = int(N_pixels * 0.2 * 2)\n",  Philipp Arras committed Feb 01, 2018 447  "\n",  Martin Reinecke committed Feb 04, 2018 448  "mask = ift.Field(s_space, val=1)\n",  Philipp Arras committed Feb 01, 2018 449 450  "mask.val[ l : h] = 0\n", "\n",  Martin Reinecke committed Feb 04, 2018 451  "R = ift.DiagonalOperator(mask)*HT\n",  Philipp Arras committed Feb 01, 2018 452 453  "n.val[l:h] = 0\n", "\n",  Martin Reinecke committed Feb 04, 2018 454  "d = R(sh) + n"  Philipp Arras committed Feb 01, 2018 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486  ] }, { "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 487  "scrolled": true  Philipp Arras committed Feb 01, 2018 488 489 490  }, "outputs": [], "source": [  Martin Reinecke committed Feb 04, 2018 491  "sc = ift.probing.utils.StatCalculator()\n",  Philipp Arras committed Feb 01, 2018 492  "\n",  Martin Reinecke committed Feb 04, 2018 493  "IC = ift.GradientNormController(iteration_limit=50000,\n",  Martin Reinecke committed Feb 04, 2018 494 495 496 497 498  " tol_abs_gradnorm=0.1)\n", "inverter = ift.ConjugateGradient(controller=IC)\n", "curv = ift.library.wiener_filter_curvature.WienerFilterCurvature(R,N,Sh,inverter)\n", "\n", "for i in range(200):\n",  Martin Reinecke committed Feb 04, 2018 499  " print i\n",  Martin Reinecke committed Feb 04, 2018 500 501 502  " sc.add(HT(curv.generate_posterior_sample()))\n", "\n", "m_var = sc.var"  Philipp Arras committed Feb 01, 2018 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525  ] }, { "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 526 527 528 529  "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 530 531  "\n", "# Get signal data and reconstruction data\n",  Martin Reinecke committed Feb 04, 2018 532 533 534  "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 535  "uncertainty = np.sqrt(np.abs(m_var_data))\n",  Martin Reinecke committed Feb 04, 2018 536  "d_data = d.val.real\n",  Philipp Arras committed Feb 01, 2018 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593  "\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": "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": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# 2d Example" ] }, { "cell_type": "code", "execution_count": null,  Martin Reinecke committed Feb 04, 2018 594  "metadata": {},  Philipp Arras committed Feb 01, 2018 595 596 597  "outputs": [], "source": [ "N_pixels = 256 # Number of pixels\n",  Martin Reinecke committed Feb 04, 2018 598  "sigma2 = 10. # Noise variance\n",  Philipp Arras committed Feb 01, 2018 599 600 601  "\n", "\n", "def pow_spec(k):\n",  Martin Reinecke committed Feb 04, 2018 602  " P0, k0, gamma = [.2, 5, 4]\n",  Philipp Arras committed Feb 01, 2018 603 604 605  " return P0 * (1. + (k/k0)**2)**(- gamma / 2)\n", "\n", "\n",  Martin Reinecke committed Feb 04, 2018 606  "s_space = ift.RGSpace([N_pixels, N_pixels])"  Philipp Arras committed Feb 01, 2018 607 608 609 610 611 612 613 614 615 616 617 618  ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [  Martin Reinecke committed Feb 04, 2018 619  "h_space = s_space.get_default_codomain()\n",  Martin Reinecke committed Feb 04, 2018 620  "HT = ift.HarmonicTransformOperator(h_space,s_space)\n",  Martin Reinecke committed Feb 04, 2018 621  "p_space = ift.PowerSpace(h_space)\n",  Philipp Arras committed Feb 01, 2018 622 623  "\n", "# Operators\n",  Martin Reinecke committed Feb 04, 2018 624 625 626 627  "Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)\n", "N = ift.ScalingOperator(sigma2,s_space)\n", "R = ift.FFTSmoothingOperator(s_space, sigma=.01)\n", "#D = PropagatorOperator(R=R, N=N, Sh=Sh)\n",  Philipp Arras committed Feb 01, 2018 628 629  "\n", "# Fields and data\n",  Martin Reinecke committed Feb 04, 2018 630 631  "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 committed Feb 01, 2018 632 633 634 635 636 637 638  " 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",  Martin Reinecke committed Feb 04, 2018 639  "mask = ift.Field(s_space, val=1)\n",  Philipp Arras committed Feb 01, 2018 640 641  "mask.val[l:h,l:h] = 0\n", "\n",  Martin Reinecke committed Feb 04, 2018 642  "R = ift.DiagonalOperator(mask)*HT\n",  Philipp Arras committed Feb 01, 2018 643  "n.val[l:h, l:h] = 0\n",  Martin Reinecke committed Feb 04, 2018 644  "D = PropagatorOperator(R=R, N=N, Sh=Sh)\n",  Philipp Arras committed Feb 01, 2018 645  "\n",  Martin Reinecke committed Feb 04, 2018 646  "d = R(sh) + n\n",  Philipp Arras committed Feb 01, 2018 647 648 649 650 651 652  "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 04, 2018 653 654  "sc = ift.probing.utils.StatCalculator()\n", "\n",  Martin Reinecke committed Feb 04, 2018 655 656  "IC = ift.GradientNormController(iteration_limit=50000,\n", " tol_abs_gradnorm=0.1)\n",  Martin Reinecke committed Feb 04, 2018 657  "inverter = ift.ConjugateGradient(controller=IC)\n",  Martin Reinecke committed Feb 04, 2018 658  "curv = ift.library.wiener_filter_curvature.WienerFilterCurvature(R,N,Sh,inverter)\n",  Martin Reinecke committed Feb 04, 2018 659 660  "\n", "for i in range(20):\n",  Martin Reinecke committed Feb 04, 2018 661  " print i\n",  Martin Reinecke committed Feb 04, 2018 662 663 664  " sc.add(HT(curv.generate_posterior_sample()))\n", "\n", "m_var = sc.var\n",  Philipp Arras committed Feb 01, 2018 665 666  "\n", "# Get data\n",  Martin Reinecke committed Feb 04, 2018 667 668 669 670 671 672 673 674  "s_power = ift.power_analyze(sh)\n", "m_power = ift.power_analyze(m)\n", "s_power_data = s_power.val.real\n", "m_power_data = m_power.val.real\n", "s_data = HT(sh).val.real\n", "m_data = HT(m).val.real\n", "m_var_data = m_var.val.real\n", "d_data = d.val.real\n",  Philipp Arras committed Feb 01, 2018 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  "\n", "uncertainty = np.sqrt(np.abs(m_var_data))" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [ "cm = ['magma', 'inferno', 'plasma', 'viridis'][1]\n", "\n", "mi = np.min(s_data)\n", "ma = np.max(s_data)\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(15, 7))\n", "\n", "data = [s_data, d_data]\n", "caption = [\"Signal\", \"Data\"]\n", "\n", "for ax in axes.flat:\n", " im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cm, vmin=mi,\n", " vmax=ma)\n", " ax.set_title(caption.pop(0))\n", "\n", "fig.subplots_adjust(right=0.8)\n", "cbar_ax = fig.add_axes([0.85, 0.15, 0.05, 0.7])\n", "fig.colorbar(im, cax=cbar_ax)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [ "mi = np.min(s_data)\n", "ma = np.max(s_data)\n", "\n", "fig, axes = plt.subplots(2, 2, figsize=(15, 15))\n", "\n", "data = [s_data, m_data, s_data - m_data, uncertainty]\n", "caption = [\"Signal\", \"Reconstruction\", \"Residuals\", \"Uncertainty Map\"]\n", "\n", "for ax in axes.flat:\n", " im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cm, vmin=mi, vmax=ma)\n", " ax.set_title(caption.pop(0))\n", "\n", "fig.subplots_adjust(right=0.8)\n", "cbar_ax = fig.add_axes([.85, 0.15, 0.05, 0.7])\n", "fig.colorbar(im, cax=cbar_ax)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Is the uncertainty map reliable?" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [], "source": [ "precise = (np.abs(s_data-m_data) < uncertainty )\n", "print(\"Error within uncertainty map bounds: \" + str(np.sum(precise) * 100 / N_pixels**2) + \"%\")\n", "\n", "fig = plt.figure()\n", "plt.imshow(precise.astype(float), cmap=\"brg\")\n",  Martin Reinecke committed Feb 04, 2018 762  "plt.colorbar()"  Philipp Arras committed Feb 01, 2018 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786  ] }, { "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 787  "display_name": "Python 2",  Philipp Arras committed Feb 01, 2018 788  "language": "python",  Martin Reinecke committed Feb 04, 2018 789  "name": "python2"  Philipp Arras committed Feb 01, 2018 790 791 792 793  }, "language_info": { "codemirror_mode": { "name": "ipython",  Martin Reinecke committed Feb 04, 2018 794  "version": 2  Philipp Arras committed Feb 01, 2018 795 796 797 798 799  }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python",  Martin Reinecke committed Feb 04, 2018 800 801  "pygments_lexer": "ipython2", "version": "2.7.12"  Philipp Arras committed Feb 01, 2018 802 803 804 805 806  } }, "nbformat": 4, "nbformat_minor": 2 }