Commit 4e7ef6d8 by Philipp Arras

### Merge remote-tracking branch 'nifty-dev/NIFTy_5' into presentation

parents b95de987 8c34584a
 ... ... @@ -96,3 +96,12 @@ run_getting_started_3: script: - python demos/getting_started_3.py - python3 demos/getting_started_3.py run_bernoulli: stage: demo_runs script: - python demos/bernoulli_demo.py - python3 demos/bernoulli_demo.py artifacts: paths: - '*.png'
 %% Cell type:markdown id: tags: # A NIFTy demonstration %% Cell type:markdown id: tags: ## IFT: Big Picture IFT starting point: \$\$d = Rs+n\$\$ Typically, \$s\$ is a continuous field, \$d\$ a discrete data vector. Particularly, \$R\$ is not invertible. IFT aims at **inverting** the above uninvertible problem in the **best possible way** using Bayesian statistics. ## NIFTy NIFTy (Numerical Information Field Theory) is a Python framework in which IFT problems can be tackled easily. Main Interfaces: - **Spaces**: Cartesian, 2-Spheres (Healpix, Gauss-Legendre) and their respective harmonic spaces. - **Fields**: Defined on spaces. - **Operators**: Acting on fields. %% Cell type:markdown id: tags: ## Wiener Filter: Formulae ### Assumptions - \$d=Rs+n\$, \$R\$ linear operator. - \$\mathcal P (s) = \mathcal G (s,S)\$, \$\mathcal P (n) = \mathcal G (n,N)\$ where \$S, N\$ are positive definite matrices. ### Posterior The Posterior is given by: \$\$\mathcal P (s|d) \propto P(s,d) = \mathcal G(d-Rs,N) \,\mathcal G(s,S) \propto \mathcal G (m,D) \$\$ \$\$\mathcal P (s|d) \propto P(s,d) = \mathcal G(d-Rs,N) \,\mathcal G(s,S) \propto \mathcal G (s-m,D) \$\$ where \$\$\begin{align} m &= Dj \\ D^{-1}&= (S^{-1} +R^\dagger N^{-1} R )\\ j &= R^\dagger N^{-1} d \end{align}\$\$ Let us implement this in NIFTy! %% Cell type:markdown id: tags: ## Wiener Filter: Example - 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},\$\$ with \$P_0 = 0.2, k_0 = 5, \gamma = 4\$. - \$N = 0.2 \cdot \mathbb{1}\$. - Number of data points \$N_{pix} = 512\$. - reconstruction in harmonic space. - Response operator: \$\$R = FFT_{\text{harmonic} \rightarrow \text{position}}\$\$ %% Cell type:code id: tags: ``` python N_pixels = 512 # Number of pixels def pow_spec(k): P0, k0, gamma = [.2, 5, 4] return P0 / ((1. + (k/k0)**2)**(gamma / 2)) ``` %% Cell type:markdown id: tags: ## Wiener Filter: Implementation %% Cell type:markdown id: tags: ### Import Modules %% Cell type:code id: tags: ``` python import numpy as np np.random.seed(40) import nifty5 as ift import matplotlib.pyplot as plt %matplotlib inline ``` %% Cell type:markdown id: tags: ### Implement Propagator %% Cell type:code id: tags: ``` python def Curvature(R, N, Sh): IC = ift.GradientNormController(iteration_limit=50000, tol_abs_gradnorm=0.1) # WienerFilterCurvature is (R.adjoint*N.inverse*R + Sh.inverse) plus some handy # helper methods. return ift.WienerFilterCurvature(R,N,Sh,iteration_controller=IC,iteration_controller_sampling=IC) ``` %% Cell type:markdown id: tags: ### Conjugate Gradient Preconditioning - \$D\$ is defined via: \$\$D^{-1} = \mathcal S_h^{-1} + R^\dagger N^{-1} R.\$\$ 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*). %% Cell type:markdown id: tags: ### Generate Mock data - Generate a field \$s\$ and \$n\$ with given covariances. - Calculate \$d\$. %% Cell type:code id: tags: ``` python s_space = ift.RGSpace(N_pixels) h_space = s_space.get_default_codomain() HT = ift.HarmonicTransformOperator(h_space, target=s_space) # Operators Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec) R = HT #*ift.create_harmonic_smoothing_operator((h_space,), 0, 0.02) # Fields and data sh = Sh.draw_sample() noiseless_data=R(sh) noise_amplitude = np.sqrt(0.2) N = ift.ScalingOperator(noise_amplitude**2, s_space) n = ift.Field.from_random(domain=s_space, random_type='normal', std=noise_amplitude, mean=0) d = noiseless_data + n j = R.adjoint_times(N.inverse_times(d)) curv = Curvature(R=R, N=N, Sh=Sh) D = curv.inverse ``` %% Cell type:markdown id: tags: ### Run Wiener Filter %% Cell type:code id: tags: ``` python m = D(j) ``` %% Cell type:markdown id: tags: ### Signal Reconstruction %% Cell type:code id: tags: ``` python # Get signal data and reconstruction data s_data = HT(sh).to_global_data() m_data = HT(m).to_global_data() d_data = d.to_global_data() plt.figure(figsize=(15,10)) plt.plot(s_data, 'r', label="Signal", linewidth=3) plt.plot(d_data, 'k.', label="Data") plt.plot(m_data, 'k', label="Reconstruction",linewidth=3) plt.title("Reconstruction") plt.legend() plt.show() ``` %% Cell type:code id: tags: ``` python plt.figure(figsize=(15,10)) plt.plot(s_data - s_data, 'r', label="Signal", linewidth=3) plt.plot(d_data - s_data, 'k.', label="Data") plt.plot(m_data - s_data, 'k', label="Reconstruction",linewidth=3) plt.axhspan(-noise_amplitude,noise_amplitude, facecolor='0.9', alpha=.5) plt.title("Residuals") plt.legend() plt.show() ``` %% Cell type:markdown id: tags: ### Power Spectrum %% Cell type:code id: tags: ``` python s_power_data = ift.power_analyze(sh).to_global_data() m_power_data = ift.power_analyze(m).to_global_data() plt.figure(figsize=(15,10)) plt.loglog() plt.xlim(1, int(N_pixels/2)) ymin = min(m_power_data) plt.ylim(ymin, 1) xs = np.arange(1,int(N_pixels/2),.1) plt.plot(xs, pow_spec(xs), label="True Power Spectrum", color='k',alpha=0.5) plt.plot(s_power_data, 'r', label="Signal") plt.plot(m_power_data, 'k', label="Reconstruction") plt.axhline(noise_amplitude**2 / N_pixels, color="k", linestyle='--', label="Noise level", alpha=.5) plt.axhspan(noise_amplitude**2 / N_pixels, ymin, facecolor='0.9', alpha=.5) plt.title("Power Spectrum") plt.legend() plt.show() ``` %% Cell type:markdown id: tags: ## Wiener Filter on Incomplete Data %% Cell type:code id: tags: ``` python # Operators Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec) N = ift.ScalingOperator(noise_amplitude**2,s_space) # R is defined below # Fields sh = Sh.draw_sample() s = HT(sh) n = ift.Field.from_random(domain=s_space, random_type='normal', std=noise_amplitude, mean=0) ``` %% Cell type:markdown id: tags: ### Partially Lose Data %% Cell type:code id: tags: ``` python l = int(N_pixels * 0.2) h = int(N_pixels * 0.2 * 2) mask = np.full(s_space.shape, 1.) mask[l:h] = 0 mask = ift.Field.from_global_data(s_space, mask) R = ift.DiagonalOperator(mask)*HT n = n.to_global_data().copy() n[l:h] = 0 n = ift.Field.from_global_data(s_space, n) d = R(sh) + n ``` %% Cell type:code id: tags: ``` python curv = Curvature(R=R, N=N, Sh=Sh) D = curv.inverse j = R.adjoint_times(N.inverse_times(d)) m = D(j) ``` %% Cell type:markdown id: tags: ### Compute Uncertainty %% Cell type:code id: tags: ``` python m_mean, m_var = ift.probe_with_posterior_samples(curv, HT, 200) ``` %% Cell type:markdown id: tags: ### Get data %% Cell type:code id: tags: ``` python # Get signal data and reconstruction data s_data = s.to_global_data() m_data = HT(m).to_global_data() m_var_data = m_var.to_global_data() uncertainty = np.sqrt(m_var_data) d_data = d.to_global_data().copy() # Set lost data to NaN for proper plotting d_data[d_data == 0] = np.nan ``` %% Cell type:code id: tags: ``` python fig = plt.figure(figsize=(15,10)) plt.axvspan(l, h, facecolor='0.8',alpha=0.5) plt.fill_between(range(N_pixels), m_data - uncertainty, m_data + uncertainty, facecolor='0.5', alpha=0.5) plt.plot(s_data, 'r', label="Signal", alpha=1, linewidth=3) plt.plot(d_data, 'k.', label="Data") plt.plot(m_data, 'k', label="Reconstruction", linewidth=3) plt.title("Reconstruction of incomplete data") plt.legend() ``` %% Cell type:markdown id: tags: # 2d Example %% Cell type:code id: tags: ``` python N_pixels = 256 # Number of pixels sigma2 = 2. # Noise variance def pow_spec(k): P0, k0, gamma = [.2, 2, 4] return P0 * (1. + (k/k0)**2)**(-gamma/2) s_space = ift.RGSpace([N_pixels, N_pixels]) ``` %% Cell type:code id: tags: ``` python h_space = s_space.get_default_codomain() HT = ift.HarmonicTransformOperator(h_space,s_space) # Operators Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec) N = ift.ScalingOperator(sigma2,s_space) # Fields and data sh = Sh.draw_sample() n = ift.Field.from_random(domain=s_space, random_type='normal', std=np.sqrt(sigma2), mean=0) # Lose some data l = int(N_pixels * 0.33) h = int(N_pixels * 0.33 * 2) mask = np.full(s_space.shape, 1.) mask[l:h,l:h] = 0. mask = ift.Field.from_global_data(s_space, mask) R = ift.DiagonalOperator(mask)*HT n = n.to_global_data().copy() n[l:h, l:h] = 0 n = ift.Field.from_global_data(s_space, n) curv = Curvature(R=R, N=N, Sh=Sh) D = curv.inverse d = R(sh) + n j = R.adjoint_times(N.inverse_times(d)) # Run Wiener filter m = D(j) # Uncertainty m_mean, m_var = ift.probe_with_posterior_samples(curv, HT, 20) # Get data s_data = HT(sh).to_global_data() m_data = HT(m).to_global_data() m_var_data = m_var.to_global_data() d_data = d.to_global_data() uncertainty = np.sqrt(np.abs(m_var_data)) ``` %% Cell type:code id: tags: ``` python cm = ['magma', 'inferno', 'plasma', 'viridis'][1] mi = np.min(s_data) ma = np.max(s_data) fig, axes = plt.subplots(1, 2, figsize=(15, 7)) data = [s_data, d_data] caption = ["Signal", "Data"] for ax in axes.flat: im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cm, vmin=mi, vmax=ma) ax.set_title(caption.pop(0)) fig.subplots_adjust(right=0.8) cbar_ax = fig.add_axes([0.85, 0.15, 0.05, 0.7]) fig.colorbar(im, cax=cbar_ax) ``` %% Cell type:code id: tags: ``` python mi = np.min(s_data) ma = np.max(s_data) fig, axes = plt.subplots(3, 2, figsize=(15, 22.5)) sample = HT(curv.draw_sample(from_inverse=True)+m).to_global_data() post_mean = (m_mean + HT(m)).to_global_data() data = [s_data, m_data, post_mean, sample, s_data - m_data, uncertainty] caption = ["Signal", "Reconstruction", "Posterior mean", "Sample", "Residuals", "Uncertainty Map"] for ax in axes.flat: im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cm, vmin=mi, vmax=ma) ax.set_title(caption.pop(0)) fig.subplots_adjust(right=0.8) cbar_ax = fig.add_axes([.85, 0.15, 0.05, 0.7]) fig.colorbar(im, cax=cbar_ax) ``` %% Cell type:markdown id: tags: ### Is the uncertainty map reliable? %% Cell type:code id: tags: ``` python precise = (np.abs(s_data-m_data) < uncertainty) print("Error within uncertainty map bounds: " + str(np.sum(precise) * 100 / N_pixels**2) + "%") plt.figure(figsize=(15,10)) plt.imshow(precise.astype(float), cmap="brg") plt.colorbar() ``` %% Cell type:markdown id: tags: # Start Coding ## NIFTy Repository + Installation guide https://gitlab.mpcdf.mpg.de/ift/NIFTy NIFTy v5 **more or less stable!** ... ...
 # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 3 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # You should have received a copy of the GNU General Public License # along with this program. If not, see . # # Copyright(C) 2013-2018 Max-Planck-Society # # NIFTy is being developed at the Max-Planck-Institut fuer Astrophysik # and financially supported by the Studienstiftung des deutschen Volkes. import nifty5 as ift import numpy as np if __name__ == '__main__': # ABOUT THIS CODE # FIXME ABOUT THIS CODE np.random.seed(41) # Set up the position space of the signal ... ... @@ -70,6 +88,6 @@ if __name__ == '__main__': reconstruction = sky.at(H.position).value ift.plot(reconstruction, title='reconstruction', name='reconstruction.pdf') ift.plot(GR.adjoint_times(data), title='data', name='data.pdf') ift.plot(sky.at(mock_position).value, title='truth', name='truth.pdf') ift.plot(reconstruction, title='reconstruction', name='reconstruction.png') ift.plot(GR.adjoint_times(data), title='data', name='data.png') ift.plot(sky.at(mock_position).value, title='truth', name='truth.png')
 # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 3 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # You should have received a copy of the GNU General Public License # along with this program. If not, see . # # Copyright(C) 2013-2018 Max-Planck-Society # # NIFTy is being developed at the Max-Planck-Institut fuer Astrophysik # and financially supported by the Studienstiftung des deutschen Volkes. import nifty5 as ift import numpy as np ... ...
 # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 3 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # You should have received a copy of the GNU General Public License # along with this program. If not, see . # # Copyright(C) 2013-2018 Max-Planck-Society # # NIFTy is being developed at the Max-Planck-Institut fuer Astrophysik # and financially supported by the Studienstiftung des deutschen Volkes. import nifty5 as ift import numpy as np ... ... @@ -18,7 +36,7 @@ def get_2D_exposure(): if __name__ == '__main__': # ABOUT THIS CODE # FIXME description of the tutorial np.random.seed(41) # Set up the position space of the signal ... ...
 # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 3 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # You should have received a copy of the GNU General Public License # along with this program. If not, see . # # Copyright(C) 2013-2018 Max-Planck-Society # # NIFTy is being developed at the Max-Planck-Institut fuer Astrophysik # and financially supported by the Studienstiftung des deutschen Volkes. import nifty5 as ift import numpy as np ... ... @@ -9,7 +27,7 @@ def get_random_LOS(n_los): if __name__ == '__main__': # ## ABOUT THIS TUTORIAL # FIXME description of the tutorial np.random.seed(42) position_space = ift.RGSpace([128, 128]) ... ...
 ... ... @@ -17,11 +17,14 @@ # and financially supported by the Studienstiftung des deutschen Volkes. from __future__ import absolute_import, division, print_function from ..compat import * import sys import numpy as np from .random import Random from mpi4py import MPI import sys from ..compat import * from .random import Random _comm = MPI.COMM_WORLD ntask = _comm.Get_size() ... ...
 ... ... @@ -19,9 +19,10 @@ # Data object module for NIFTy that uses simple numpy ndarrays. import numpy as np from numpy import empty, empty_like, exp, full, log from numpy import ndarray as data_object from numpy import full, empty, empty_like, sqrt, ones, zeros, vdot, \ exp, log, tanh from numpy import ones, sqrt, tanh, vdot, zeros from .random import Random ntask = 1 ... ...
 ... ... @@ -17,9 +17,11 @@ # and financially supported by the Studienstiftung des deutschen Volkes.