Commit 73b5bbab by Martin Reinecke

### re-style the notebook a bit

parent 5e9551b1
 ... ... @@ -80,8 +80,8 @@ "source": [ "## Wiener Filter: Example\n", "\n", "- One-dimensional signal with power spectrum: $$P(k) = P_0\\,\\left(1+\\left(\\frac{k}{k_0}\\right)^2\\right)^{-\\gamma /2},$$\n", "with $P_0 = 0.2, k_0 = 5, \\gamma = 4$. Recall: $P(k)$ defines an isotropic and homogeneous $S$.\n", "- 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", "- $N = 0.2 \\cdot \\mathbb{1}$.\n", "- Number of data points $N_{pix} = 512$.\n", "- reconstruction in harmonic space.\n", ... ... @@ -328,9 +328,9 @@ "outputs": [], "source": [ "plt.figure(figsize=(15,10))\n", "plt.plot(s_data, 'g', label=\"Signal\")\n", "plt.plot(d_data, 'k+', label=\"Data\")\n", "plt.plot(m_data, 'r', label=\"Reconstruction\")\n", "plt.plot(s_data, 'r', label=\"Signal\", linewidth=3)\n", "plt.plot(d_data, 'k.', label=\"Data\")\n", "plt.plot(m_data, 'k', label=\"Reconstruction\",linewidth=3)\n", "plt.title(\"Reconstruction\")\n", "plt.legend()\n", "plt.show()" ... ... @@ -347,9 +347,9 @@ "outputs": [], "source": [ "plt.figure(figsize=(15,10))\n", "plt.plot(s_data - s_data, 'g', label=\"Signal\")\n", "plt.plot(d_data - s_data, 'k+', label=\"Data\")\n", "plt.plot(m_data - s_data, 'r', label=\"Reconstruction\")\n", "plt.plot(s_data - s_data, 'r', label=\"Signal\", linewidth=3)\n", "plt.plot(d_data - s_data, 'k.', label=\"Data\")\n", "plt.plot(m_data - s_data, 'k', label=\"Reconstruction\",linewidth=3)\n", "plt.axhspan(-noise_amplitude,noise_amplitude, facecolor='0.9', alpha=.5)\n", "plt.title(\"Residuals\")\n", "plt.legend()\n", ... ... @@ -383,9 +383,9 @@ "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, 'g', label=\"Signal\")\n", "plt.plot(m_power_data, 'r', label=\"Reconstruction\")\n", "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", "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", "plt.title(\"Power Spectrum\")\n", ... ... @@ -545,8 +545,8 @@ "outputs": [], "source": [ "plt.figure(figsize=(15,10))\n", "plt.plot(s_data, 'g', label=\"Signal\", linewidth=1)\n", "plt.plot(d_data, 'k+', label=\"Data\", alpha=1)\n", "plt.plot(s_data, 'r', label=\"Signal\", linewidth=3)\n", "plt.plot(d_data, 'k.', label=\"Data\")\n", "plt.axvspan(l, h, facecolor='0.8', alpha=.5)\n", "plt.title(\"Incomplete Data\")\n", "plt.legend()" ... ... @@ -563,11 +563,11 @@ "outputs": [], "source": [ "fig = plt.figure(figsize=(15,10))\n", "plt.plot(s_data, 'g', label=\"Signal\", alpha=1, linewidth=4)\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.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", "plt.plot(s_data, 'r', label=\"Signal\", alpha=1, linewidth=3)\n", "plt.plot(d_data, 'k.', label=\"Data\")\n", "plt.plot(m_data, 'k', label=\"Reconstruction\", linewidth=3)\n", "plt.title(\"Reconstruction of incomplete data\")\n", "plt.legend()" ] ... ... @@ -702,10 +702,12 @@ "mi = np.min(s_data)\n", "ma = np.max(s_data)\n", "\n", "fig, axes = plt.subplots(2, 2, figsize=(15, 15))\n", "fig, axes = plt.subplots(3, 2, figsize=(15, 22.5))\n", "samp1 = HT(curv.draw_sample()+m).val\n", "samp2 = HT(curv.draw_sample()+m).val\n", "\n", "data = [s_data, m_data, s_data - m_data, uncertainty]\n", "caption = [\"Signal\", \"Reconstruction\", \"Residuals\", \"Uncertainty Map\"]\n", "data = [s_data, m_data, samp1, samp2, s_data - m_data, uncertainty]\n", "caption = [\"Signal\", \"Reconstruction\", \"Sample 1\", \"Sample 2\", \"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", ... ...
 %% 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: $$\mathcal P (s|d) \propto P(s,d) = \mathcal G(d-Rs,N) \,\mathcal G(s,S) \propto \mathcal G (m,D)$$ 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 - One-dimensional signal with power spectrum: $$P(k) = P_0\,\left(1+\left(\frac{k}{k_0}\right)^2\right)^{-\gamma /2},$$ with $P_0 = 0.2, k_0 = 5, \gamma = 4$. Recall: $P(k)$ defines an isotropic and homogeneous $S$. - 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 nifty4 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) inverter = ift.ConjugateGradient(controller=IC) # WienerFilterCurvature is (R.adjoint*N.inverse*R + Sh.inverse) plus some handy # helper methods. return ift.library.WienerFilterCurvature(R,N,Sh,inverter)  %% 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) p_space = ift.PowerSpace(h_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 = ift.power_synthesize(ift.PS_field(p_space, pow_spec),real_signal=True) 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: ### Create Power Spectra of Signal and Reconstruction %% Cell type:code id: tags:  python s_power = ift.power_analyze(sh) m_power = ift.power_analyze(m) s_power_data = s_power.val.real m_power_data = m_power.val.real # Get signal data and reconstruction data s_data = HT(sh).val.real m_data = HT(m).val.real d_data = d.val.real  %% Cell type:markdown id: tags: ### Signal Reconstruction %% Cell type:code id: tags:  python plt.figure(figsize=(15,10)) plt.plot(s_data, 'g', label="Signal") plt.plot(d_data, 'k+', label="Data") plt.plot(m_data, 'r', label="Reconstruction") 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, 'g', label="Signal") plt.plot(d_data - s_data, 'k+', label="Data") plt.plot(m_data - s_data, 'r', label="Reconstruction") 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 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", linewidth=.7, color='k') plt.plot(s_power_data, 'g', label="Signal") plt.plot(m_power_data, 'r', label="Reconstruction") 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 = ift.power_synthesize(ift.PS_field(p_space,pow_spec),real_signal=True) 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 = ift.Field(s_space, val=1) mask.val[ l : h] = 0 R = ift.DiagonalOperator(mask)*HT n.val[l:h] = 0 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, m, HT, 200)  %% Cell type:markdown id: tags: ### Get data %% Cell type:code id: tags:  python s_power = ift.power_analyze(sh) m_power = ift.power_analyze(m) s_power_data = s_power.val.real m_power_data = m_power.val.real # Get signal data and reconstruction data s_data = s.val.real m_data = HT(m).val.real m_var_data = m_var.val.real uncertainty = np.sqrt(np.abs(m_var_data)) d_data = d.val.real # Set lost data to NaN for proper plotting d_data[d_data == 0] = np.nan  %% Cell type:code id: tags:  python plt.figure(figsize=(15,10)) plt.plot(s_data, 'g', label="Signal", linewidth=1) plt.plot(d_data, 'k+', label="Data", alpha=1) plt.plot(s_data, 'r', label="Signal", linewidth=3) plt.plot(d_data, 'k.', label="Data") plt.axvspan(l, h, facecolor='0.8', alpha=.5) plt.title("Incomplete Data") plt.legend()  %% Cell type:code id: tags:  python fig = plt.figure(figsize=(15,10)) plt.plot(s_data, 'g', label="Signal", alpha=1, linewidth=4) plt.plot(d_data, 'k+', label="Data", alpha=.5) plt.plot(m_data, 'r', label="Reconstruction") plt.axvspan(l, h, facecolor='0.8', alpha=.5) plt.fill_between(range(N_pixels), m_data - uncertainty, m_data + uncertainty, facecolor='0') 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) p_space = ift.PowerSpace(h_space) # Operators Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec) N = ift.ScalingOperator(sigma2,s_space) # Fields and data sh = ift.power_synthesize(ift.PS_field(p_space,pow_spec),real_signal=True) 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 = ift.Field(s_space, val=1) mask.val[l:h,l:h] = 0 R = ift.DiagonalOperator(mask)*HT n.val[l:h, l:h] = 0 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, m, HT, 20) # Get data s_power = ift.power_analyze(sh) m_power = ift.power_analyze(m) s_power_data = s_power.val.real m_power_data = m_power.val.real s_data = HT(sh).val.real m_data = HT(m).val.real m_var_data = m_var.val.real d_data = d.val.real 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(2, 2, figsize=(15, 15)) fig, axes = plt.subplots(3, 2, figsize=(15, 22.5)) samp1 = HT(curv.draw_sample()+m).val samp2 = HT(curv.draw_sample()+m).val data = [s_data, m_data, s_data - m_data, uncertainty] caption = ["Signal", "Reconstruction", "Residuals", "Uncertainty Map"] data = [s_data, m_data, samp1, samp2, s_data - m_data, uncertainty] caption = ["Signal", "Reconstruction", "Sample 1", "Sample 2", "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 v4 **more or less stable!** ... ...
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