getting_started_0.ipynb 17.8 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  "slideshow": { "slide_type": "-" } }, "outputs": [], "source": [  Philipp Arras committed Jul 23, 2021 137  "%matplotlib inline\n",  Philipp Arras committed Feb 01, 2018 138  "import numpy as np\n",  Philipp Arras committed Jun 11, 2021 139  "import nifty8 as ift\n",  Martin Reinecke committed Feb 04, 2018 140  "import matplotlib.pyplot as plt\n",  Martin Reinecke committed Jul 20, 2021 141  "plt.rcParams['figure.dpi'] = 100\n",  Philipp Arras committed Jul 20, 2021 142  "plt.style.use(\"seaborn-notebook\")"  Philipp Arras committed Feb 01, 2018 143 144 145 146 147 148 149 150 151 152  ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [  Philipp Arras committed Jul 20, 2021 153  "#### Implement Propagator"  Philipp Arras committed Feb 01, 2018 154 155 156 157 158 159 160 161 162 163 164 165  ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [], "source": [  Martin Reinecke committed Feb 05, 2018 166  "def Curvature(R, N, Sh):\n",  Martin Reinecke committed Feb 04, 2018 167  " IC = ift.GradientNormController(iteration_limit=50000,\n",  Martin Reinecke committed Feb 04, 2018 168  " tol_abs_gradnorm=0.1)\n",  Philipp Arras committed Aug 16, 2021 169 170  " N = ift.SamplingDtypeSetter(N, np.float64)\n", " Sh = ift.SamplingDtypeSetter(Sh, np.float64)\n",  Martin Reinecke committed Feb 05, 2018 171 172  " # WienerFilterCurvature is (R.adjoint*N.inverse*R + Sh.inverse) plus some handy\n", " # helper methods.\n",  Philipp Arras committed May 13, 2020 173  " return ift.WienerFilterCurvature(R,N,Sh,iteration_controller=IC,iteration_controller_sampling=IC)"  Philipp Arras committed Feb 01, 2018 174 175 176 177 178 179 180 181 182 183  ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "skip" } }, "source": [  Philipp Arras committed Jul 20, 2021 184  "#### Conjugate Gradient Preconditioning\n",  Philipp Arras committed Feb 01, 2018 185 186  "\n", "- $D$ is defined via:\n",  Martin Reinecke committed Feb 04, 2018 187  "$$D^{-1} = \\mathcal S_h^{-1} + R^\\dagger N^{-1} R.$$\n",  Philipp Arras committed Feb 01, 2018 188 189  "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 190  ""  Philipp Arras committed Feb 01, 2018 205 206 207 208 209 210 211 212 213 214  ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [  Philipp Arras committed Jul 20, 2021 215  "#### Generate Mock data\n",  Philipp Arras committed Feb 01, 2018 216 217 218 219 220 221 222 223  "\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 224 225 226  "metadata": { "scrolled": true },  Philipp Arras committed Feb 01, 2018 227 228  "outputs": [], "source": [  Martin Reinecke committed Feb 04, 2018 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",  Philipp Arras committed Feb 01, 2018 232 233  "\n", "# Operators\n",  Martin Reinecke committed Feb 04, 2018 234  "Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)\n",  Philipp Arras committed Sep 14, 2020 235  "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",  Philipp Arras committed May 12, 2020 238  "sh = Sh.draw_sample_with_dtype(dtype=np.float64)\n",  Martin Reinecke committed Feb 04, 2018 239  "noiseless_data=R(sh)\n",  Martin Reinecke committed Feb 04, 2018 240  "noise_amplitude = np.sqrt(0.2)\n",  Gordian Edenhofer committed Dec 05, 2019 241  "N = ift.ScalingOperator(s_space, noise_amplitude**2)\n",  Martin Reinecke committed Feb 04, 2018 242 243  "\n", "n = ift.Field.from_random(domain=s_space, random_type='normal',\n",  Martin Reinecke committed Feb 04, 2018 244  " std=noise_amplitude, mean=0)\n",  Martin Reinecke committed Feb 04, 2018 245 246  "d = noiseless_data + n\n", "j = R.adjoint_times(N.inverse_times(d))\n",  Martin Reinecke committed Feb 05, 2018 247 248  "curv = Curvature(R=R, N=N, Sh=Sh)\n", "D = curv.inverse"  Philipp Arras committed Feb 01, 2018 249 250 251 252 253 254 255 256 257 258  ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [  Philipp Arras committed Jul 20, 2021 259  "#### Run Wiener Filter"  Philipp Arras committed Feb 01, 2018 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282  ] }, { "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 283  "#### Results"  Philipp Arras committed Feb 01, 2018 284 285 286 287 288 289 290 291 292 293 294 295 296  ] }, { "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 297 298 299  "s_data = HT(sh).val\n", "m_data = HT(m).val\n", "d_data = d.val\n",  Philipp Arras committed Feb 01, 2018 300  "\n",  Martin Reinecke committed Jul 20, 2021 301  "plt.plot(s_data, 'r', label=\"Signal\", linewidth=2)\n",  Martin Reinecke committed Feb 06, 2018 302  "plt.plot(d_data, 'k.', label=\"Data\")\n",  Martin Reinecke committed Jul 20, 2021 303  "plt.plot(m_data, 'k', label=\"Reconstruction\",linewidth=2)\n",  Philipp Arras committed Feb 01, 2018 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318  "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 319  "plt.plot(s_data - s_data, 'r', label=\"Signal\", linewidth=2)\n",  Martin Reinecke committed Feb 06, 2018 320  "plt.plot(d_data - s_data, 'k.', label=\"Data\")\n",  Martin Reinecke committed Jul 20, 2021 321  "plt.plot(m_data - s_data, 'k', label=\"Reconstruction\",linewidth=2)\n",  Martin Reinecke committed Feb 04, 2018 322  "plt.axhspan(-noise_amplitude,noise_amplitude, facecolor='0.9', alpha=.5)\n",  Philipp Arras committed Feb 01, 2018 323 324 325 326 327 328 329 330 331 332 333 334 335  "plt.title(\"Residuals\")\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [  Philipp Arras committed Jul 20, 2021 336  "#### Power Spectrum"  Philipp Arras committed Feb 01, 2018 337 338 339 340 341 342 343 344 345 346 347 348  ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [], "source": [  Martin Reinecke committed Dec 05, 2019 349 350  "s_power_data = ift.power_analyze(sh).val\n", "m_power_data = ift.power_analyze(m).val\n",  Philipp Arras committed Feb 01, 2018 351 352 353 354 355  "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 356 357 358  "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 359 360  "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 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  "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 388  "Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)\n",  Gordian Edenhofer committed Dec 05, 2019 389  "N = ift.ScalingOperator(s_space, noise_amplitude**2)\n",  Philipp Arras committed Feb 01, 2018 390 391 392  "# R is defined below\n", "\n", "# Fields\n",  Philipp Arras committed May 12, 2020 393  "sh = Sh.draw_sample_with_dtype(dtype=np.float64)\n",  Martin Reinecke committed Feb 04, 2018 394 395 396  "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 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420  ] }, { "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 421  "h = int(N_pixels * 0.2 * 2)\n",  Philipp Arras committed Feb 01, 2018 422  "\n",  Martin Reinecke committed Feb 18, 2018 423 424  "mask = np.full(s_space.shape, 1.)\n", "mask[l:h] = 0\n",  Martin Reinecke committed Dec 05, 2019 425  "mask = ift.Field.from_raw(s_space, mask)\n",  Philipp Arras committed Feb 01, 2018 426  "\n",  Martin Reinecke committed Aug 05, 2018 427  "R = ift.DiagonalOperator(mask)(HT)\n",  Martin Reinecke committed Dec 05, 2019 428  "n = n.val_rw()\n",  Martin Reinecke committed Feb 18, 2018 429  "n[l:h] = 0\n",  Martin Reinecke committed Dec 05, 2019 430  "n = ift.Field.from_raw(s_space, n)\n",  Philipp Arras committed Feb 01, 2018 431  "\n",  Martin Reinecke committed Feb 04, 2018 432  "d = R(sh) + n"  Philipp Arras committed Feb 01, 2018 433 434 435 436 437 438 439 440 441 442 443 444  ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [  Martin Reinecke committed Feb 05, 2018 445 446  "curv = Curvature(R=R, N=N, Sh=Sh)\n", "D = curv.inverse\n",  Philipp Arras committed Feb 01, 2018 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465  "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 466  "scrolled": true  Philipp Arras committed Feb 01, 2018 467 468 469  }, "outputs": [], "source": [  Martin Reinecke committed Apr 26, 2020 470  "m_mean, m_var = ift.probe_with_posterior_samples(curv, HT, 200, np.float64)"  Philipp Arras committed Feb 01, 2018 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494  ] }, { "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 495 496 497  "s_data = s.val\n", "m_data = HT(m).val\n", "m_var_data = m_var.val\n",  Martin Reinecke committed Feb 06, 2018 498  "uncertainty = np.sqrt(m_var_data)\n",  Martin Reinecke committed Dec 05, 2019 499  "d_data = d.val_rw()\n",  Philipp Arras committed Feb 01, 2018 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514  "\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 515 516  "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 517  "plt.plot(s_data, 'r', label=\"Signal\", alpha=1, linewidth=2)\n",  Martin Reinecke committed Feb 06, 2018 518  "plt.plot(d_data, 'k.', label=\"Data\")\n",  Martin Reinecke committed Jul 20, 2021 519  "plt.plot(m_data, 'k', label=\"Reconstruction\", linewidth=2)\n",  Philipp Arras committed Feb 01, 2018 520 521 522 523 524 525 526 527 528 529 530 531  "plt.title(\"Reconstruction of incomplete data\")\n", "plt.legend()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [  Philipp Arras committed Jul 20, 2021 532  "## Wiener filter on two-dimensional field"  Philipp Arras committed Feb 01, 2018 533 534 535 536 537  ] }, { "cell_type": "code", "execution_count": null,  Martin Reinecke committed Feb 05, 2018 538  "metadata": {},  Philipp Arras committed Feb 01, 2018 539 540 541  "outputs": [], "source": [ "N_pixels = 256 # Number of pixels\n",  Martin Reinecke committed Feb 06, 2018 542  "sigma2 = 2. # Noise variance\n",  Philipp Arras committed Feb 01, 2018 543 544  "\n", "def pow_spec(k):\n",  Martin Reinecke committed Feb 04, 2018 545  " P0, k0, gamma = [.2, 2, 4]\n",  Martin Reinecke committed Feb 06, 2018 546  " return P0 * (1. + (k/k0)**2)**(-gamma/2)\n",  Philipp Arras committed Feb 01, 2018 547  "\n",  Martin Reinecke committed Feb 04, 2018 548  "s_space = ift.RGSpace([N_pixels, N_pixels])"  Philipp Arras committed Feb 01, 2018 549 550 551 552 553 554 555 556 557 558 559 560  ] }, { "cell_type": "code", "execution_count": null, "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [  Martin Reinecke committed Feb 04, 2018 561  "h_space = s_space.get_default_codomain()\n",  Martin Reinecke committed Feb 04, 2018 562  "HT = ift.HarmonicTransformOperator(h_space,s_space)\n",  Philipp Arras committed Feb 01, 2018 563 564  "\n", "# Operators\n",  Martin Reinecke committed Feb 04, 2018 565  "Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)\n",  Gordian Edenhofer committed Dec 05, 2019 566  "N = ift.ScalingOperator(s_space, sigma2)\n",  Philipp Arras committed Feb 01, 2018 567 568  "\n", "# Fields and data\n",  Philipp Arras committed May 12, 2020 569  "sh = Sh.draw_sample_with_dtype(dtype=np.float64)\n",  Martin Reinecke committed Feb 04, 2018 570  "n = ift.Field.from_random(domain=s_space, random_type='normal',\n",  Philipp Arras committed Feb 01, 2018 571 572 573 574  " std=np.sqrt(sigma2), mean=0)\n", "\n", "# Lose some data\n", "\n",  Martin Reinecke committed Feb 04, 2018 575 576  "l = int(N_pixels * 0.33)\n", "h = int(N_pixels * 0.33 * 2)\n",  Philipp Arras committed Feb 01, 2018 577  "\n",  Martin Reinecke committed Feb 18, 2018 578 579  "mask = np.full(s_space.shape, 1.)\n", "mask[l:h,l:h] = 0.\n",  Martin Reinecke committed Dec 05, 2019 580  "mask = ift.Field.from_raw(s_space, mask)\n",  Philipp Arras committed Feb 01, 2018 581  "\n",  Martin Reinecke committed Aug 05, 2018 582  "R = ift.DiagonalOperator(mask)(HT)\n",  Martin Reinecke committed Dec 05, 2019 583  "n = n.val_rw()\n",  Martin Reinecke committed Feb 18, 2018 584  "n[l:h, l:h] = 0\n",  Martin Reinecke committed Dec 05, 2019 585  "n = ift.Field.from_raw(s_space, n)\n",  Martin Reinecke committed Feb 05, 2018 586 587  "curv = Curvature(R=R, N=N, Sh=Sh)\n", "D = curv.inverse\n",  Philipp Arras committed Feb 01, 2018 588  "\n",  Martin Reinecke committed Feb 04, 2018 589  "d = R(sh) + n\n",  Philipp Arras committed Feb 01, 2018 590 591 592 593 594 595  "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 596  "m_mean, m_var = ift.probe_with_posterior_samples(curv, HT, 20, np.float64)\n",  Philipp Arras committed Feb 01, 2018 597 598  "\n", "# Get data\n",  Martin Reinecke committed Dec 05, 2019 599 600 601 602  "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 603 604 605 606 607 608 609 610 611 612 613 614 615  "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 616  "cmap = ['magma', 'inferno', 'plasma', 'viridis'][1]\n",  Philipp Arras committed Feb 01, 2018 617 618 619 620  "\n", "mi = np.min(s_data)\n", "ma = np.max(s_data)\n", "\n",  Philipp Arras committed Jul 20, 2021 621  "fig, axes = plt.subplots(1, 2)\n",  Philipp Arras committed Feb 01, 2018 622 623 624 625 626  "\n", "data = [s_data, d_data]\n", "caption = [\"Signal\", \"Data\"]\n", "\n", "for ax in axes.flat:\n",  Martin Reinecke committed Jul 20, 2021 627  " im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cmap, vmin=mi,\n",  Philipp Arras committed Feb 01, 2018 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648  " 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 649  "fig, axes = plt.subplots(3, 2, figsize=(10, 15))\n",  Philipp Arras committed May 12, 2020 650  "sample = HT(curv.draw_sample(from_inverse=True)+m).val\n",  Martin Reinecke committed Dec 05, 2019 651  "post_mean = (m_mean + HT(m)).val\n",  Philipp Arras committed Feb 01, 2018 652  "\n",  Martin Reinecke committed Feb 18, 2018 653 654  "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 655 656  "\n", "for ax in axes.flat:\n",  Martin Reinecke committed Jul 20, 2021 657  " im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cmap, vmin=mi, vmax=ma)\n",  Philipp Arras committed Feb 01, 2018 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  " 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 686  "precise = (np.abs(s_data-m_data) < uncertainty)\n",  Philipp Arras committed Feb 01, 2018 687 688 689  "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 690  "plt.colorbar()"  Philipp Arras committed Feb 01, 2018 691 692 693 694 695 696  ] } ], "metadata": { "celltoolbar": "Slideshow", "kernelspec": {  Philipp Arras committed Jun 28, 2018 697  "display_name": "Python 3",  Philipp Arras committed Feb 01, 2018 698  "language": "python",  Philipp Arras committed Jun 28, 2018 699  "name": "python3"  Philipp Arras committed Feb 01, 2018 700 701 702 703  }, "language_info": { "codemirror_mode": { "name": "ipython",  Philipp Arras committed Jun 28, 2018 704  "version": 3  Philipp Arras committed Feb 01, 2018 705 706 707 708 709  }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python",  Philipp Arras committed Jun 28, 2018 710  "pygments_lexer": "ipython3",  Martin Reinecke committed Jul 20, 2021 711  "version": "3.9.2"  Philipp Arras committed Feb 01, 2018 712 713 714 715 716  } }, "nbformat": 4, "nbformat_minor": 2 }