cosmetics

demos/Wiener_Filter.ipynb

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 # A NIFTy demonstration
 
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 ## 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.
 
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 ## 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!
 
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 ## 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))
 ```
 
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 ## Wiener Filter: Implementation
 
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 ### Import Modules
 
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 ``` python
 import numpy as np
 np.random.seed(40)
 import nifty4 as ift
 import matplotlib.pyplot as plt
 %matplotlib inline
 ```
 
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 ### Implement Propagator
 
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 ``` 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)
 ```
 
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 ### 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*).
 
 <!--
 - One can define the *condition number* of a non-singular and normal matrix $A$:
 $$\kappa (A) := \frac{|\lambda_{\text{max}}|}{|\lambda_{\text{min}}|},$$
 where $\lambda_{\text{max}}$ and $\lambda_{\text{min}}$ are the largest and smallest eigenvalue of $A$, respectively.
 
 - The larger $\kappa$ the slower Conjugate Gradient.
 
 - 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:
 $$\tilde A m = \tilde j,$$
 where $\tilde A = T D^{-1}$ and $\tilde j = Tj$.
 
 - In our case $S^{-1}$ is responsible for the bad conditioning of $D$ depending on the chosen power spectrum. Thus, we choose
 
 $$T = \mathcal F^\dagger S_h^{-1} \mathcal F.$$
 -->
 
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 ### 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
 ```
 
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 ### Run Wiener Filter
 
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 ``` python
 m = D(j)
 ```
 
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 ### Signal Reconstruction
 
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 ``` 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()
 ```
 
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 ### 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()
 ```
 
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 ## 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)
 ```
 
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 ### 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()
 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)
 ```
 
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 ### Compute Uncertainty
 
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 ``` python
 m_mean, m_var = ift.probe_with_posterior_samples(curv, HT, 200)
 ```
 
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 ### 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()
 
 # 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()
 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 )
+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!**