getting_started_0.ipynb 17.6 KB
 Philipp Arras committed Feb 01, 2018 1 2 3 4 5 6 7 8 9 10 { "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [  Philipp Arras committed Jul 20, 2021 11  "# Code example: Wiener filter"  Philipp Arras committed Feb 01, 2018 12 13 14 15 16 17 18 19 20 21  ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [  Philipp Arras committed Jul 20, 2021 22  "## Introduction\n",  Philipp Arras committed Feb 01, 2018 23 24 25 26  "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  "\n", "IFT aims at **inverting** the above uninvertible problem in the **best possible way** using Bayesian statistics.\n", "\n",  Martin Reinecke committed Feb 04, 2018 31  "NIFTy (Numerical Information Field Theory) is a Python framework in which IFT problems can be tackled easily.\n",  Philipp Arras committed Feb 01, 2018 32 33 34 35 36  "\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 37  "- **Operators**: Acting on fields."  Philipp Arras committed Feb 01, 2018 38 39 40 41 42 43 44 45 46 47  ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [  Philipp Arras committed Jul 20, 2021 48  "## Wiener filter on one-dimensional fields\n",  Philipp Arras committed Feb 01, 2018 49 50 51 52 53 54 55 56 57  "\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 58  "$$\\mathcal P (s|d) \\propto P(s,d) = \\mathcal G(d-Rs,N) \\,\\mathcal G(s,S) \\propto \\mathcal G (s-m,D)$$\n",  Philipp Arras committed Feb 01, 2018 59 60  "\n", "where\n",  Philipp Arras committed Jul 20, 2021 61 62 63  "$$m = Dj$$\n", "with\n", "$$D = (S^{-1} +R^\\dagger N^{-1} R)^{-1} , \\quad j = R^\\dagger N^{-1} d.$$\n",  Philipp Arras committed Feb 01, 2018 64 65 66 67 68 69 70 71 72 73 74 75  "\n", "Let us implement this in NIFTy!" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [  Philipp Arras committed Jul 20, 2021 76  "### In NIFTy\n",  Philipp Arras committed Feb 01, 2018 77  "\n",  Martin Reinecke committed Feb 06, 2018 78 79  "- 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 80  "- $N = 0.2 \\cdot \\mathbb{1}$.\n",  Martin Reinecke committed Feb 04, 2018 81 82  "- Number of data points $N_{pix} = 512$.\n", "- reconstruction in harmonic space.\n",  Philipp Arras committed Feb 01, 2018 83  "- Response operator:\n",  Martin Reinecke committed Feb 04, 2018 84  "$$R = FFT_{\\text{harmonic} \\rightarrow \\text{position}}$$\n"  Philipp Arras committed Feb 01, 2018 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99  ] }, { "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 100 101  " P0, k0, gamma = [.2, 5, 4]\n", " return P0 / ((1. + (k/k0)**2)**(gamma / 2))"  Philipp Arras committed Feb 01, 2018 102 103 104 105 106 107 108 109 110 111  ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [  Philipp Arras committed Jul 20, 2021 112  "### Implementation"  Philipp Arras committed Feb 01, 2018 113 114 115 116 117 118 119 120 121 122  ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [  Philipp Arras committed Jul 20, 2021 123  "#### Import Modules"  Philipp Arras committed Feb 01, 2018 124 125 126 127 128 129  ] }, { "cell_type": "code", "execution_count": null, "metadata": {  Martin Reinecke committed Jun 21, 2018 130  "scrolled": true,  Philipp Arras committed Feb 01, 2018 131 132 133 134 135 136 137  "slideshow": { "slide_type": "-" } }, "outputs": [], "source": [ "import numpy as np\n",  Philipp Arras committed Jun 11, 2021 138  "import nifty8 as ift\n",  Martin Reinecke committed Feb 04, 2018 139  "import matplotlib.pyplot as plt\n",  Philipp Arras committed Jul 20, 2021 140  "plt.style.use(\"seaborn-notebook\")"  Philipp Arras committed Feb 01, 2018 141 142 143 144 145 146 147 148 149 150  ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [  Philipp Arras committed Jul 20, 2021 151  "#### Implement Propagator"  Philipp Arras committed Feb 01, 2018 152 153 154 155 156 157 158 159 160 161 162 163  ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [], "source": [  Martin Reinecke committed Feb 05, 2018 164  "def Curvature(R, N, Sh):\n",  Martin Reinecke committed Feb 04, 2018 165  " IC = ift.GradientNormController(iteration_limit=50000,\n",  Martin Reinecke committed Feb 04, 2018 166  " tol_abs_gradnorm=0.1)\n",  Martin Reinecke committed Feb 05, 2018 167 168  " # WienerFilterCurvature is (R.adjoint*N.inverse*R + Sh.inverse) plus some handy\n", " # helper methods.\n",  Philipp Arras committed May 13, 2020 169  " return ift.WienerFilterCurvature(R,N,Sh,iteration_controller=IC,iteration_controller_sampling=IC)"  Philipp Arras committed Feb 01, 2018 170 171 172 173 174 175 176 177 178 179  ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "skip" } }, "source": [  Philipp Arras committed Jul 20, 2021 180  "#### Conjugate Gradient Preconditioning\n",  Philipp Arras committed Feb 01, 2018 181 182  "\n", "- $D$ is defined via:\n",  Martin Reinecke committed Feb 04, 2018 183  "$$D^{-1} = \\mathcal S_h^{-1} + R^\\dagger N^{-1} R.$$\n",  Philipp Arras committed Feb 01, 2018 184 185  "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 186  ""  Philipp Arras committed Feb 01, 2018 201 202 203 204 205 206 207 208 209 210  ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [  Philipp Arras committed Jul 20, 2021 211  "#### Generate Mock data\n",  Philipp Arras committed Feb 01, 2018 212 213 214 215 216 217 218 219  "\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 220 221 222  "metadata": { "scrolled": true },  Philipp Arras committed Feb 01, 2018 223 224  "outputs": [], "source": [  Martin Reinecke committed Feb 04, 2018 225 226 227  "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 228 229  "\n", "# Operators\n",  Martin Reinecke committed Feb 04, 2018 230  "Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)\n",  Philipp Arras committed Sep 14, 2020 231  "R = HT # @ ift.create_harmonic_smoothing_operator((h_space,), 0, 0.02)\n",  Philipp Arras committed Feb 01, 2018 232 233  "\n", "# Fields and data\n",  Philipp Arras committed May 12, 2020 234  "sh = Sh.draw_sample_with_dtype(dtype=np.float64)\n",  Martin Reinecke committed Feb 04, 2018 235  "noiseless_data=R(sh)\n",  Martin Reinecke committed Feb 04, 2018 236  "noise_amplitude = np.sqrt(0.2)\n",  Gordian Edenhofer committed Dec 05, 2019 237  "N = ift.ScalingOperator(s_space, noise_amplitude**2)\n",  Martin Reinecke committed Feb 04, 2018 238 239  "\n", "n = ift.Field.from_random(domain=s_space, random_type='normal',\n",  Martin Reinecke committed Feb 04, 2018 240  " std=noise_amplitude, mean=0)\n",  Martin Reinecke committed Feb 04, 2018 241 242  "d = noiseless_data + n\n", "j = R.adjoint_times(N.inverse_times(d))\n",  Martin Reinecke committed Feb 05, 2018 243 244  "curv = Curvature(R=R, N=N, Sh=Sh)\n", "D = curv.inverse"  Philipp Arras committed Feb 01, 2018 245 246 247 248 249 250 251 252 253 254  ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [  Philipp Arras committed Jul 20, 2021 255  "#### Run Wiener Filter"  Philipp Arras committed Feb 01, 2018 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278  ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [], "source": [ "m = D(j)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [  Philipp Arras committed Jul 20, 2021 279  "#### Results"  Philipp Arras committed Feb 01, 2018 280 281 282 283 284 285 286 287 288 289 290 291 292  ] }, { "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 293 294 295  "s_data = HT(sh).val\n", "m_data = HT(m).val\n", "d_data = d.val\n",  Philipp Arras committed Feb 01, 2018 296  "\n",  Martin Reinecke committed Jul 20, 2021 297  "plt.plot(s_data, 'r', label=\"Signal\", linewidth=2)\n",  Martin Reinecke committed Feb 06, 2018 298  "plt.plot(d_data, 'k.', label=\"Data\")\n",  Martin Reinecke committed Jul 20, 2021 299  "plt.plot(m_data, 'k', label=\"Reconstruction\",linewidth=2)\n",  Philipp Arras committed Feb 01, 2018 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314  "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 Jul 20, 2021 315  "plt.plot(s_data - s_data, 'r', label=\"Signal\", linewidth=2)\n",  Martin Reinecke committed Feb 06, 2018 316  "plt.plot(d_data - s_data, 'k.', label=\"Data\")\n",  Martin Reinecke committed Jul 20, 2021 317  "plt.plot(m_data - s_data, 'k', label=\"Reconstruction\",linewidth=2)\n",  Martin Reinecke committed Feb 04, 2018 318  "plt.axhspan(-noise_amplitude,noise_amplitude, facecolor='0.9', alpha=.5)\n",  Philipp Arras committed Feb 01, 2018 319 320 321 322 323 324 325 326 327 328 329 330 331  "plt.title(\"Residuals\")\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [  Philipp Arras committed Jul 20, 2021 332  "#### Power Spectrum"  Philipp Arras committed Feb 01, 2018 333 334 335 336 337 338 339 340 341 342 343 344  ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [], "source": [  Martin Reinecke committed Dec 05, 2019 345 346  "s_power_data = ift.power_analyze(sh).val\n", "m_power_data = ift.power_analyze(m).val\n",  Philipp Arras committed Feb 01, 2018 347 348 349 350 351  "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 352 353 354  "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 355 356  "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 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(\"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 384  "Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)\n",  Gordian Edenhofer committed Dec 05, 2019 385  "N = ift.ScalingOperator(s_space, noise_amplitude**2)\n",  Philipp Arras committed Feb 01, 2018 386 387 388  "# R is defined below\n", "\n", "# Fields\n",  Philipp Arras committed May 12, 2020 389  "sh = Sh.draw_sample_with_dtype(dtype=np.float64)\n",  Martin Reinecke committed Feb 04, 2018 390 391 392  "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 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416  ] }, { "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 417  "h = int(N_pixels * 0.2 * 2)\n",  Philipp Arras committed Feb 01, 2018 418  "\n",  Martin Reinecke committed Feb 18, 2018 419 420  "mask = np.full(s_space.shape, 1.)\n", "mask[l:h] = 0\n",  Martin Reinecke committed Dec 05, 2019 421  "mask = ift.Field.from_raw(s_space, mask)\n",  Philipp Arras committed Feb 01, 2018 422  "\n",  Martin Reinecke committed Aug 05, 2018 423  "R = ift.DiagonalOperator(mask)(HT)\n",  Martin Reinecke committed Dec 05, 2019 424  "n = n.val_rw()\n",  Martin Reinecke committed Feb 18, 2018 425  "n[l:h] = 0\n",  Martin Reinecke committed Dec 05, 2019 426  "n = ift.Field.from_raw(s_space, n)\n",  Philipp Arras committed Feb 01, 2018 427  "\n",  Martin Reinecke committed Feb 04, 2018 428  "d = R(sh) + n"  Philipp Arras committed Feb 01, 2018 429 430 431 432 433 434 435 436 437 438 439 440  ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [  Martin Reinecke committed Feb 05, 2018 441 442  "curv = Curvature(R=R, N=N, Sh=Sh)\n", "D = curv.inverse\n",  Philipp Arras committed Feb 01, 2018 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461  "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 462  "scrolled": true  Philipp Arras committed Feb 01, 2018 463 464 465  }, "outputs": [], "source": [  Martin Reinecke committed Apr 26, 2020 466  "m_mean, m_var = ift.probe_with_posterior_samples(curv, HT, 200, np.float64)"  Philipp Arras committed Feb 01, 2018 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490  ] }, { "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 491 492 493  "s_data = s.val\n", "m_data = HT(m).val\n", "m_var_data = m_var.val\n",  Martin Reinecke committed Feb 06, 2018 494  "uncertainty = np.sqrt(m_var_data)\n",  Martin Reinecke committed Dec 05, 2019 495  "d_data = d.val_rw()\n",  Philipp Arras committed Feb 01, 2018 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510  "\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": [  Martin Reinecke committed Feb 06, 2018 511 512  "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",  Martin Reinecke committed Jul 20, 2021 513  "plt.plot(s_data, 'r', label=\"Signal\", alpha=1, linewidth=2)\n",  Martin Reinecke committed Feb 06, 2018 514  "plt.plot(d_data, 'k.', label=\"Data\")\n",  Martin Reinecke committed Jul 20, 2021 515  "plt.plot(m_data, 'k', label=\"Reconstruction\", linewidth=2)\n",  Philipp Arras committed Feb 01, 2018 516 517 518 519 520 521 522 523 524 525 526 527  "plt.title(\"Reconstruction of incomplete data\")\n", "plt.legend()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [  Philipp Arras committed Jul 20, 2021 528  "## Wiener filter on two-dimensional field"  Philipp Arras committed Feb 01, 2018 529 530 531 532 533  ] }, { "cell_type": "code", "execution_count": null,  Martin Reinecke committed Feb 05, 2018 534  "metadata": {},  Philipp Arras committed Feb 01, 2018 535 536 537  "outputs": [], "source": [ "N_pixels = 256 # Number of pixels\n",  Martin Reinecke committed Feb 06, 2018 538  "sigma2 = 2. # Noise variance\n",  Philipp Arras committed Feb 01, 2018 539 540  "\n", "def pow_spec(k):\n",  Martin Reinecke committed Feb 04, 2018 541  " P0, k0, gamma = [.2, 2, 4]\n",  Martin Reinecke committed Feb 06, 2018 542  " return P0 * (1. + (k/k0)**2)**(-gamma/2)\n",  Philipp Arras committed Feb 01, 2018 543  "\n",  Martin Reinecke committed Feb 04, 2018 544  "s_space = ift.RGSpace([N_pixels, N_pixels])"  Philipp Arras committed Feb 01, 2018 545 546 547 548 549 550 551 552 553 554 555 556  ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [  Martin Reinecke committed Feb 04, 2018 557  "h_space = s_space.get_default_codomain()\n",  Martin Reinecke committed Feb 04, 2018 558  "HT = ift.HarmonicTransformOperator(h_space,s_space)\n",  Philipp Arras committed Feb 01, 2018 559 560  "\n", "# Operators\n",  Martin Reinecke committed Feb 04, 2018 561  "Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)\n",  Gordian Edenhofer committed Dec 05, 2019 562  "N = ift.ScalingOperator(s_space, sigma2)\n",  Philipp Arras committed Feb 01, 2018 563 564  "\n", "# Fields and data\n",  Philipp Arras committed May 12, 2020 565  "sh = Sh.draw_sample_with_dtype(dtype=np.float64)\n",  Martin Reinecke committed Feb 04, 2018 566  "n = ift.Field.from_random(domain=s_space, random_type='normal',\n",  Philipp Arras committed Feb 01, 2018 567 568 569 570  " std=np.sqrt(sigma2), mean=0)\n", "\n", "# Lose some data\n", "\n",  Martin Reinecke committed Feb 04, 2018 571 572  "l = int(N_pixels * 0.33)\n", "h = int(N_pixels * 0.33 * 2)\n",  Philipp Arras committed Feb 01, 2018 573  "\n",  Martin Reinecke committed Feb 18, 2018 574 575  "mask = np.full(s_space.shape, 1.)\n", "mask[l:h,l:h] = 0.\n",  Martin Reinecke committed Dec 05, 2019 576  "mask = ift.Field.from_raw(s_space, mask)\n",  Philipp Arras committed Feb 01, 2018 577  "\n",  Martin Reinecke committed Aug 05, 2018 578  "R = ift.DiagonalOperator(mask)(HT)\n",  Martin Reinecke committed Dec 05, 2019 579  "n = n.val_rw()\n",  Martin Reinecke committed Feb 18, 2018 580  "n[l:h, l:h] = 0\n",  Martin Reinecke committed Dec 05, 2019 581  "n = ift.Field.from_raw(s_space, n)\n",  Martin Reinecke committed Feb 05, 2018 582 583  "curv = Curvature(R=R, N=N, Sh=Sh)\n", "D = curv.inverse\n",  Philipp Arras committed Feb 01, 2018 584  "\n",  Martin Reinecke committed Feb 04, 2018 585  "d = R(sh) + n\n",  Philipp Arras committed Feb 01, 2018 586 587 588 589 590 591  "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 592  "m_mean, m_var = ift.probe_with_posterior_samples(curv, HT, 20, np.float64)\n",  Philipp Arras committed Feb 01, 2018 593 594  "\n", "# Get data\n",  Martin Reinecke committed Dec 05, 2019 595 596 597 598  "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 599 600 601 602 603 604 605 606 607 608 609 610 611  "uncertainty = np.sqrt(np.abs(m_var_data))" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [  Martin Reinecke committed Jul 20, 2021 612  "cmap = ['magma', 'inferno', 'plasma', 'viridis'][1]\n",  Philipp Arras committed Feb 01, 2018 613 614 615 616  "\n", "mi = np.min(s_data)\n", "ma = np.max(s_data)\n", "\n",  Philipp Arras committed Jul 20, 2021 617  "fig, axes = plt.subplots(1, 2)\n",  Philipp Arras committed Feb 01, 2018 618 619 620 621 622  "\n", "data = [s_data, d_data]\n", "caption = [\"Signal\", \"Data\"]\n", "\n", "for ax in axes.flat:\n",  Martin Reinecke committed Jul 20, 2021 623  " im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cmap, vmin=mi,\n",  Philipp Arras committed Feb 01, 2018 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644  " 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",  Philipp Arras committed Jul 20, 2021 645  "fig, axes = plt.subplots(3, 2, figsize=(10, 15))\n",  Philipp Arras committed May 12, 2020 646  "sample = HT(curv.draw_sample(from_inverse=True)+m).val\n",  Martin Reinecke committed Dec 05, 2019 647  "post_mean = (m_mean + HT(m)).val\n",  Philipp Arras committed Feb 01, 2018 648  "\n",  Martin Reinecke committed Feb 18, 2018 649 650  "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 651 652  "\n", "for ax in axes.flat:\n",  Martin Reinecke committed Jul 20, 2021 653  " im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cmap, vmin=mi, vmax=ma)\n",  Philipp Arras committed Feb 01, 2018 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  " 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 682  "precise = (np.abs(s_data-m_data) < uncertainty)\n",  Philipp Arras committed Feb 01, 2018 683 684 685  "print(\"Error within uncertainty map bounds: \" + str(np.sum(precise) * 100 / N_pixels**2) + \"%\")\n", "\n", "plt.imshow(precise.astype(float), cmap=\"brg\")\n",  Martin Reinecke committed Feb 04, 2018 686  "plt.colorbar()"  Philipp Arras committed Feb 01, 2018 687 688 689 690 691 692  ] } ], "metadata": { "celltoolbar": "Slideshow", "kernelspec": {  Philipp Arras committed Jun 28, 2018 693  "display_name": "Python 3",  Philipp Arras committed Feb 01, 2018 694  "language": "python",  Philipp Arras committed Jun 28, 2018 695  "name": "python3"  Philipp Arras committed Feb 01, 2018 696 697 698 699  }, "language_info": { "codemirror_mode": { "name": "ipython",  Philipp Arras committed Jun 28, 2018 700  "version": 3  Philipp Arras committed Feb 01, 2018 701 702 703 704 705  }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python",  Philipp Arras committed Jun 28, 2018 706  "pygments_lexer": "ipython3",  Martin Reinecke committed Jul 20, 2021 707  "version": "3.9.2"  Philipp Arras committed Feb 01, 2018 708 709 710 711 712  } }, "nbformat": 4, "nbformat_minor": 2 }