Commit 73b5bbab authored by Martin Reinecke's avatar Martin Reinecke
Browse files

re-style the notebook a bit

parent 5e9551b1
...@@ -80,8 +80,8 @@ ...@@ -80,8 +80,8 @@
"source": [ "source": [
"## Wiener Filter: Example\n", "## Wiener Filter: Example\n",
"\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", "- 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 $P_0 = 0.2, k_0 = 5, \\gamma = 4$. Recall: $P(k)$ defines an isotropic and homogeneous $S$.\n", "with $P_0 = 0.2, k_0 = 5, \\gamma = 4$.\n",
"- $N = 0.2 \\cdot \\mathbb{1}$.\n", "- $N = 0.2 \\cdot \\mathbb{1}$.\n",
"- Number of data points $N_{pix} = 512$.\n", "- Number of data points $N_{pix} = 512$.\n",
"- reconstruction in harmonic space.\n", "- reconstruction in harmonic space.\n",
...@@ -328,9 +328,9 @@ ...@@ -328,9 +328,9 @@
"outputs": [], "outputs": [],
"source": [ "source": [
"plt.figure(figsize=(15,10))\n", "plt.figure(figsize=(15,10))\n",
"plt.plot(s_data, 'g', label=\"Signal\")\n", "plt.plot(s_data, 'r', label=\"Signal\", linewidth=3)\n",
"plt.plot(d_data, 'k+', label=\"Data\")\n", "plt.plot(d_data, 'k.', label=\"Data\")\n",
"plt.plot(m_data, 'r', label=\"Reconstruction\")\n", "plt.plot(m_data, 'k', label=\"Reconstruction\",linewidth=3)\n",
"plt.title(\"Reconstruction\")\n", "plt.title(\"Reconstruction\")\n",
"plt.legend()\n", "plt.legend()\n",
"plt.show()" "plt.show()"
...@@ -347,9 +347,9 @@ ...@@ -347,9 +347,9 @@
"outputs": [], "outputs": [],
"source": [ "source": [
"plt.figure(figsize=(15,10))\n", "plt.figure(figsize=(15,10))\n",
"plt.plot(s_data - s_data, 'g', label=\"Signal\")\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(d_data - s_data, 'k.', label=\"Data\")\n",
"plt.plot(m_data - s_data, 'r', label=\"Reconstruction\")\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.axhspan(-noise_amplitude,noise_amplitude, facecolor='0.9', alpha=.5)\n",
"plt.title(\"Residuals\")\n", "plt.title(\"Residuals\")\n",
"plt.legend()\n", "plt.legend()\n",
...@@ -383,9 +383,9 @@ ...@@ -383,9 +383,9 @@
"ymin = min(m_power_data)\n", "ymin = min(m_power_data)\n",
"plt.ylim(ymin, 1)\n", "plt.ylim(ymin, 1)\n",
"xs = np.arange(1,int(N_pixels/2),.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(xs, pow_spec(xs), label=\"True Power Spectrum\", color='k',alpha=0.5)\n",
"plt.plot(s_power_data, 'g', label=\"Signal\")\n", "plt.plot(s_power_data, 'r', label=\"Signal\")\n",
"plt.plot(m_power_data, 'r', label=\"Reconstruction\")\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.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.axhspan(noise_amplitude**2 / N_pixels, ymin, facecolor='0.9', alpha=.5)\n",
"plt.title(\"Power Spectrum\")\n", "plt.title(\"Power Spectrum\")\n",
...@@ -545,8 +545,8 @@ ...@@ -545,8 +545,8 @@
"outputs": [], "outputs": [],
"source": [ "source": [
"plt.figure(figsize=(15,10))\n", "plt.figure(figsize=(15,10))\n",
"plt.plot(s_data, 'g', label=\"Signal\", linewidth=1)\n", "plt.plot(s_data, 'r', label=\"Signal\", linewidth=3)\n",
"plt.plot(d_data, 'k+', label=\"Data\", alpha=1)\n", "plt.plot(d_data, 'k.', label=\"Data\")\n",
"plt.axvspan(l, h, facecolor='0.8', alpha=.5)\n", "plt.axvspan(l, h, facecolor='0.8', alpha=.5)\n",
"plt.title(\"Incomplete Data\")\n", "plt.title(\"Incomplete Data\")\n",
"plt.legend()" "plt.legend()"
...@@ -563,11 +563,11 @@ ...@@ -563,11 +563,11 @@
"outputs": [], "outputs": [],
"source": [ "source": [
"fig = plt.figure(figsize=(15,10))\n", "fig = plt.figure(figsize=(15,10))\n",
"plt.plot(s_data, 'g', label=\"Signal\", alpha=1, linewidth=4)\n", "plt.axvspan(l, h, facecolor='0.8',alpha=0.5)\n",
"plt.plot(d_data, 'k+', label=\"Data\", alpha=.5)\n", "plt.fill_between(range(N_pixels), m_data - uncertainty, m_data + uncertainty, facecolor='0.5', alpha=0.5)\n",
"plt.plot(m_data, 'r', label=\"Reconstruction\")\n", "plt.plot(s_data, 'r', label=\"Signal\", alpha=1, linewidth=3)\n",
"plt.axvspan(l, h, facecolor='0.8', alpha=.5)\n", "plt.plot(d_data, 'k.', label=\"Data\")\n",
"plt.fill_between(range(N_pixels), m_data - uncertainty, m_data + uncertainty, facecolor='0')\n", "plt.plot(m_data, 'k', label=\"Reconstruction\", linewidth=3)\n",
"plt.title(\"Reconstruction of incomplete data\")\n", "plt.title(\"Reconstruction of incomplete data\")\n",
"plt.legend()" "plt.legend()"
] ]
...@@ -702,10 +702,12 @@ ...@@ -702,10 +702,12 @@
"mi = np.min(s_data)\n", "mi = np.min(s_data)\n",
"ma = np.max(s_data)\n", "ma = np.max(s_data)\n",
"\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", "\n",
"data = [s_data, m_data, s_data - m_data, uncertainty]\n", "data = [s_data, m_data, samp1, samp2, s_data - m_data, uncertainty]\n",
"caption = [\"Signal\", \"Reconstruction\", \"Residuals\", \"Uncertainty Map\"]\n", "caption = [\"Signal\", \"Reconstruction\", \"Sample 1\", \"Sample 2\", \"Residuals\", \"Uncertainty Map\"]\n",
"\n", "\n",
"for ax in axes.flat:\n", "for ax in axes.flat:\n",
" im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cm, vmin=mi, vmax=ma)\n", " im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cm, vmin=mi, vmax=ma)\n",
......
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
# A NIFTy demonstration # A NIFTy demonstration
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
## IFT: Big Picture ## IFT: Big Picture
IFT starting point: IFT starting point:
$$d = Rs+n$$ $$d = Rs+n$$
Typically, $s$ is a continuous field, $d$ a discrete data vector. Particularly, $R$ is not invertible. 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. IFT aims at **inverting** the above uninvertible problem in the **best possible way** using Bayesian statistics.
## NIFTy ## NIFTy
NIFTy (Numerical Information Field Theory) is a Python framework in which IFT problems can be tackled easily. NIFTy (Numerical Information Field Theory) is a Python framework in which IFT problems can be tackled easily.
Main Interfaces: Main Interfaces:
- **Spaces**: Cartesian, 2-Spheres (Healpix, Gauss-Legendre) and their respective harmonic spaces. - **Spaces**: Cartesian, 2-Spheres (Healpix, Gauss-Legendre) and their respective harmonic spaces.
- **Fields**: Defined on spaces. - **Fields**: Defined on spaces.
- **Operators**: Acting on fields. - **Operators**: Acting on fields.
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
## Wiener Filter: Formulae ## Wiener Filter: Formulae
### Assumptions ### Assumptions
- $d=Rs+n$, $R$ linear operator. - $d=Rs+n$, $R$ linear operator.
- $\mathcal P (s) = \mathcal G (s,S)$, $\mathcal P (n) = \mathcal G (n,N)$ where $S, N$ are positive definite matrices. - $\mathcal P (s) = \mathcal G (s,S)$, $\mathcal P (n) = \mathcal G (n,N)$ where $S, N$ are positive definite matrices.
### Posterior ### Posterior
The Posterior is given by: The Posterior is given by:
$$\mathcal P (s|d) \propto P(s,d) = \mathcal G(d-Rs,N) \,\mathcal G(s,S) \propto \mathcal G (m,D) $$ $$\mathcal P (s|d) \propto P(s,d) = \mathcal G(d-Rs,N) \,\mathcal G(s,S) \propto \mathcal G (m,D) $$
where where
$$\begin{align} $$\begin{align}
m &= Dj \\ m &= Dj \\
D^{-1}&= (S^{-1} +R^\dagger N^{-1} R )\\ D^{-1}&= (S^{-1} +R^\dagger N^{-1} R )\\
j &= R^\dagger N^{-1} d j &= R^\dagger N^{-1} d
\end{align}$$ \end{align}$$
Let us implement this in NIFTy! Let us implement this in NIFTy!
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
## Wiener Filter: Example ## Wiener Filter: Example
- One-dimensional signal with power spectrum: $$P(k) = P_0\,\left(1+\left(\frac{k}{k_0}\right)^2\right)^{-\gamma /2},$$ - 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 $P_0 = 0.2, k_0 = 5, \gamma = 4$. Recall: $P(k)$ defines an isotropic and homogeneous $S$. with $P_0 = 0.2, k_0 = 5, \gamma = 4$.
- $N = 0.2 \cdot \mathbb{1}$. - $N = 0.2 \cdot \mathbb{1}$.
- Number of data points $N_{pix} = 512$. - Number of data points $N_{pix} = 512$.
- reconstruction in harmonic space. - reconstruction in harmonic space.
- Response operator: - Response operator:
$$R = FFT_{\text{harmonic} \rightarrow \text{position}}$$ $$R = FFT_{\text{harmonic} \rightarrow \text{position}}$$
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
N_pixels = 512 # Number of pixels N_pixels = 512 # Number of pixels
def pow_spec(k): def pow_spec(k):
P0, k0, gamma = [.2, 5, 4] P0, k0, gamma = [.2, 5, 4]
return P0 / ((1. + (k/k0)**2)**(gamma / 2)) return P0 / ((1. + (k/k0)**2)**(gamma / 2))
``` ```
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
## Wiener Filter: Implementation ## Wiener Filter: Implementation
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
### Import Modules ### Import Modules
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
import numpy as np import numpy as np
np.random.seed(40) np.random.seed(40)
import nifty4 as ift import nifty4 as ift
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
%matplotlib inline %matplotlib inline
``` ```
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
### Implement Propagator ### Implement Propagator
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
def Curvature(R, N, Sh): def Curvature(R, N, Sh):
IC = ift.GradientNormController(iteration_limit=50000, IC = ift.GradientNormController(iteration_limit=50000,
tol_abs_gradnorm=0.1) tol_abs_gradnorm=0.1)
inverter = ift.ConjugateGradient(controller=IC) inverter = ift.ConjugateGradient(controller=IC)
# WienerFilterCurvature is (R.adjoint*N.inverse*R + Sh.inverse) plus some handy # WienerFilterCurvature is (R.adjoint*N.inverse*R + Sh.inverse) plus some handy
# helper methods. # helper methods.
return ift.library.WienerFilterCurvature(R,N,Sh,inverter) return ift.library.WienerFilterCurvature(R,N,Sh,inverter)
``` ```
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
### Conjugate Gradient Preconditioning ### Conjugate Gradient Preconditioning
- $D$ is defined via: - $D$ is defined via:
$$D^{-1} = \mathcal S_h^{-1} + R^\dagger N^{-1} R.$$ $$D^{-1} = \mathcal S_h^{-1} + R^\dagger N^{-1} R.$$
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*). 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 $A$: - One can define the *condition number* of a non-singular and normal matrix $A$:
$$\kappa (A) := \frac{|\lambda_{\text{max}}|}{|\lambda_{\text{min}}|},$$ $$\kappa (A) := \frac{|\lambda_{\text{max}}|}{|\lambda_{\text{min}}|},$$
where $\lambda_{\text{max}}$ and $\lambda_{\text{min}}$ are the largest and smallest eigenvalue of $A$, respectively. where $\lambda_{\text{max}}$ and $\lambda_{\text{min}}$ are the largest and smallest eigenvalue of $A$, respectively.
- The larger $\kappa$ the slower Conjugate Gradient. - 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: - 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:
$$\tilde A m = \tilde j,$$ $$\tilde A m = \tilde j,$$
where $\tilde A = T D^{-1}$ and $\tilde j = Tj$. 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 - In our case $S^{-1}$ is responsible for the bad conditioning of $D$ depending on the chosen power spectrum. Thus, we choose
$$T = \mathcal F^\dagger S_h^{-1} \mathcal F.$$ $$T = \mathcal F^\dagger S_h^{-1} \mathcal F.$$
--> -->
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
### Generate Mock data ### Generate Mock data
- Generate a field $s$ and $n$ with given covariances. - Generate a field $s$ and $n$ with given covariances.
- Calculate $d$. - Calculate $d$.
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
s_space = ift.RGSpace(N_pixels) s_space = ift.RGSpace(N_pixels)
h_space = s_space.get_default_codomain() h_space = s_space.get_default_codomain()
HT = ift.HarmonicTransformOperator(h_space, target=s_space) HT = ift.HarmonicTransformOperator(h_space, target=s_space)
p_space = ift.PowerSpace(h_space) p_space = ift.PowerSpace(h_space)
# Operators # Operators
Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec) Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)
R = HT #*ift.create_harmonic_smoothing_operator((h_space,), 0, 0.02) R = HT #*ift.create_harmonic_smoothing_operator((h_space,), 0, 0.02)
# Fields and data # Fields and data
sh = ift.power_synthesize(ift.PS_field(p_space, pow_spec),real_signal=True) sh = ift.power_synthesize(ift.PS_field(p_space, pow_spec),real_signal=True)
noiseless_data=R(sh) noiseless_data=R(sh)
noise_amplitude = np.sqrt(0.2) noise_amplitude = np.sqrt(0.2)
N = ift.ScalingOperator(noise_amplitude**2, s_space) N = ift.ScalingOperator(noise_amplitude**2, s_space)
n = ift.Field.from_random(domain=s_space, random_type='normal', n = ift.Field.from_random(domain=s_space, random_type='normal',
std=noise_amplitude, mean=0) std=noise_amplitude, mean=0)
d = noiseless_data + n d = noiseless_data + n
j = R.adjoint_times(N.inverse_times(d)) j = R.adjoint_times(N.inverse_times(d))
curv = Curvature(R=R, N=N, Sh=Sh) curv = Curvature(R=R, N=N, Sh=Sh)
D = curv.inverse D = curv.inverse
``` ```
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
### Run Wiener Filter ### Run Wiener Filter
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
m = D(j) m = D(j)
``` ```
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
### Create Power Spectra of Signal and Reconstruction ### Create Power Spectra of Signal and Reconstruction
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
s_power = ift.power_analyze(sh) s_power = ift.power_analyze(sh)
m_power = ift.power_analyze(m) m_power = ift.power_analyze(m)
s_power_data = s_power.val.real s_power_data = s_power.val.real
m_power_data = m_power.val.real m_power_data = m_power.val.real
# Get signal data and reconstruction data # Get signal data and reconstruction data
s_data = HT(sh).val.real s_data = HT(sh).val.real
m_data = HT(m).val.real m_data = HT(m).val.real
d_data = d.val.real d_data = d.val.real
``` ```
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
### Signal Reconstruction ### Signal Reconstruction
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
plt.figure(figsize=(15,10)) plt.figure(figsize=(15,10))
plt.plot(s_data, 'g', label="Signal") plt.plot(s_data, 'r', label="Signal", linewidth=3)
plt.plot(d_data, 'k+', label="Data") plt.plot(d_data, 'k.', label="Data")
plt.plot(m_data, 'r', label="Reconstruction") plt.plot(m_data, 'k', label="Reconstruction",linewidth=3)
plt.title("Reconstruction") plt.title("Reconstruction")
plt.legend() plt.legend()
plt.show() plt.show()
``` ```
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
plt.figure(figsize=(15,10)) plt.figure(figsize=(15,10))
plt.plot(s_data - s_data, 'g', label="Signal") plt.plot(s_data - s_data, 'r', label="Signal", linewidth=3)
plt.plot(d_data - s_data, 'k+', label="Data") plt.plot(d_data - s_data, 'k.', label="Data")
plt.plot(m_data - s_data, 'r', label="Reconstruction") plt.plot(m_data - s_data, 'k', label="Reconstruction",linewidth=3)
plt.axhspan(-noise_amplitude,noise_amplitude, facecolor='0.9', alpha=.5) plt.axhspan(-noise_amplitude,noise_amplitude, facecolor='0.9', alpha=.5)
plt.title("Residuals") plt.title("Residuals")
plt.legend() plt.legend()
plt.show() plt.show()
``` ```
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
### Power Spectrum ### Power Spectrum
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
plt.figure(figsize=(15,10)) plt.figure(figsize=(15,10))
plt.loglog() plt.loglog()
plt.xlim(1, int(N_pixels/2)) plt.xlim(1, int(N_pixels/2))
ymin = min(m_power_data) ymin = min(m_power_data)
plt.ylim(ymin, 1) plt.ylim(ymin, 1)
xs = np.arange(1,int(N_pixels/2),.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(xs, pow_spec(xs), label="True Power Spectrum", color='k',alpha=0.5)
plt.plot(s_power_data, 'g', label="Signal") plt.plot(s_power_data, 'r', label="Signal")
plt.plot(m_power_data, 'r', label="Reconstruction") plt.plot(m_power_data, 'k', label="Reconstruction")
plt.axhline(noise_amplitude**2 / N_pixels, color="k", linestyle='--', label="Noise level", alpha=.5) 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.axhspan(noise_amplitude**2 / N_pixels, ymin, facecolor='0.9', alpha=.5)
plt.title("Power Spectrum") plt.title("Power Spectrum")
plt.legend() plt.legend()
plt.show() plt.show()
``` ```
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
## Wiener Filter on Incomplete Data ## Wiener Filter on Incomplete Data
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
# Operators # Operators
Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec) Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)
N = ift.ScalingOperator(noise_amplitude**2,s_space) N = ift.ScalingOperator(noise_amplitude**2,s_space)
# R is defined below # R is defined below
# Fields # Fields
sh = ift.power_synthesize(ift.PS_field(p_space,pow_spec),real_signal=True) sh = ift.power_synthesize(ift.PS_field(p_space,pow_spec),real_signal=True)
s = HT(sh) s = HT(sh)
n = ift.Field.from_random(domain=s_space, random_type='normal', n = ift.Field.from_random(domain=s_space, random_type='normal',
std=noise_amplitude, mean=0) std=noise_amplitude, mean=0)
``` ```
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
### Partially Lose Data ### Partially Lose Data
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
l = int(N_pixels * 0.2) l = int(N_pixels * 0.2)
h = int(N_pixels * 0.2 * 2) h = int(N_pixels * 0.2 * 2)
mask = ift.Field(s_space, val=1) mask = ift.Field(s_space, val=1)
mask.val[ l : h] = 0 mask.val[ l : h] = 0
R = ift.DiagonalOperator(mask)*HT R = ift.DiagonalOperator(mask)*HT
n.val[l:h] = 0 n.val[l:h] = 0
d = R(sh) + n d = R(sh) + n
``` ```
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
curv = Curvature(R=R, N=N, Sh=Sh) curv = Curvature(R=R, N=N, Sh=Sh)
D = curv.inverse D = curv.inverse
j = R.adjoint_times(N.inverse_times(d)) j = R.adjoint_times(N.inverse_times(d))
m = D(j) m = D(j)
``` ```
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
### Compute Uncertainty ### Compute Uncertainty
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
m_mean, m_var = ift.probe_with_posterior_samples(curv, m, HT, 200) m_mean, m_var = ift.probe_with_posterior_samples(curv, m, HT, 200)
``` ```
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
### Get data ### Get data
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
s_power = ift.power_analyze(sh) s_power = ift.power_analyze(sh)
m_power = ift.power_analyze(m) m_power = ift.power_analyze(m)
s_power_data = s_power.val.real s_power_data = s_power.val.real
m_power_data = m_power.val.real m_power_data = m_power.val.real
# Get signal data and reconstruction data # Get signal data and reconstruction data
s_data = s.val.real s_data = s.val.real
m_data = HT(m).val.real m_data = HT(m).val.real
m_var_data = m_var.val.real m_var_data = m_var.val.real
uncertainty = np.sqrt(np.abs(m_var_data)) uncertainty = np.sqrt(np.abs(m_var_data))
d_data = d.val.real d_data = d.val.real
# Set lost data to NaN for proper plotting # Set lost data to NaN for proper plotting
d_data[d_data == 0] = np.nan d_data[d_data == 0] = np.nan
``` ```
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
plt.figure(figsize=(15,10)) plt.figure(figsize=(15,10))
plt.plot(s_data, 'g', label="Signal", linewidth=1) plt.plot(s_data, 'r', label="Signal", linewidth=3)
plt.plot(d_data, 'k+', label="Data", alpha=1) plt.plot(d_data, 'k.', label="Data")
plt.axvspan(l, h, facecolor='0.8', alpha=.5) plt.axvspan(l, h, facecolor='0.8', alpha=.5)
plt.title("Incomplete Data") plt.title("Incomplete Data")
plt.legend() plt.legend()
``` ```
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
fig = plt.figure(figsize=(15,10)) fig = plt.figure(figsize=(15,10))
plt.plot(s_data, 'g', label="Signal", alpha=1, linewidth=4) plt.axvspan(l, h, facecolor='0.8',alpha=0.5)
plt.plot(d_data, 'k+', label="Data", alpha=.5) plt.fill_between(range(N_pixels), m_data - uncertainty, m_data + uncertainty, facecolor='0.5', alpha=0.5)
plt.plot(m_data, 'r', label="Reconstruction") plt.plot(s_data, 'r', label="Signal", alpha=1, linewidth=3)
plt.axvspan(l, h, facecolor='0.8', alpha=.5) plt.plot(d_data, 'k.', label="Data")
plt.fill_between(range(N_pixels), m_data - uncertainty, m_data + uncertainty, facecolor='0') plt.plot(m_data, 'k', label="Reconstruction", linewidth=3)
plt.title("Reconstruction of incomplete data") plt.title("Reconstruction of incomplete data")
plt.legend() plt.legend()
``` ```
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
# 2d Example # 2d Example
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
N_pixels = 256 # Number of pixels N_pixels = 256 # Number of pixels
sigma2 = 2. # Noise variance sigma2 = 2. # Noise variance
def pow_spec(k): def pow_spec(k):
P0, k0, gamma = [.2, 2, 4] P0, k0, gamma = [.2, 2, 4]
return P0 * (1. + (k/k0)**2)**(- gamma / 2) return P0 * (1. + (k/k0)**2)**(- gamma / 2)
s_space = ift.RGSpace([N_pixels, N_pixels]) s_space = ift.RGSpace([N_pixels, N_pixels])
``` ```
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
h_space = s_space.get_default_codomain() h_space = s_space.get_default_codomain()
HT = ift.HarmonicTransformOperator(h_space,s_space) HT = ift.HarmonicTransformOperator(h_space,s_space)
p_space = ift.PowerSpace(h_space) p_space = ift.PowerSpace(h_space)
# Operators # Operators
Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec) Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)
N = ift.ScalingOperator(sigma2,s_space) N = ift.ScalingOperator(sigma2,s_space)
# Fields and data # Fields and data
sh = ift.power_synthesize(ift.PS_field(p_space,pow_spec),real_signal=True) 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', n = ift.Field.from_random(domain=s_space, random_type='normal',
std=np.sqrt(sigma2), mean=0) std=np.sqrt(sigma2), mean=0)
# Lose some data # Lose some data
l = int(N_pixels * 0.33) l = int(N_pixels * 0.33)
h = int(N_pixels * 0.33 * 2) h = int(N_pixels * 0.33 * 2)
mask = ift.Field(s_space, val=1) mask = ift.Field(s_space, val=1)
mask.val[l:h,l:h] = 0 mask.val[l:h,l:h] = 0
R = ift.DiagonalOperator(mask)*HT R = ift.DiagonalOperator(mask)*HT
n.val[l:h, l:h] = 0 n.val[l:h, l:h] = 0
curv = Curvature(R=R, N=N, Sh=Sh) curv = Curvature(R=R, N=N, Sh=Sh)
D = curv.inverse D = curv.inverse
d = R(sh) + n d = R(sh) + n
j = R.adjoint_times(N.inverse_times(d)) j = R.adjoint_times(N.inverse_times(d))
# Run Wiener filter # Run Wiener filter
m = D(j) m = D(j)
# Uncertainty # Uncertainty
m_mean, m_var = ift.probe_with_posterior_samples(curv, m, HT, 20) m_mean, m_var = ift.probe_with_posterior_samples(curv, m, HT, 20)
# Get data # Get data
s_power = ift.power_analyze(sh) s_power = ift.power_analyze(sh)
m_power = ift.power_analyze(m) m_power = ift.power_analyze(m)
s_power_data = s_power.val.real s_power_data = s_power.val.real
m_power_data = m_power.val.real m_power_data = m_power.val.real
s_data = HT(sh).val.real s_data = HT(sh).val.real
m_data = HT(m).val.real m_data = HT(m).val.real
m_var_data = m_var.val.real m_var_data = m_var.val.real
d_data = d.val.real d_data = d.val.real
uncertainty = np.sqrt(np.abs(m_var_data)) uncertainty = np.sqrt(np.abs(m_var_data))
``` ```
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
cm = ['magma', 'inferno', 'plasma', 'viridis'][1] cm = ['magma', 'inferno', 'plasma', 'viridis'][1]
mi = np.min(s_data) mi = np.min(s_data)
ma = np.max(s_data) ma = np.max(s_data)
fig, axes = plt.subplots(1, 2, figsize=(15, 7)) fig, axes = plt.subplots(1, 2, figsize=(15, 7))
data = [s_data, d_data] data = [s_data, d_data]
caption = ["Signal", "Data"] caption = ["Signal", "Data"]
for ax in axes.flat: for ax in axes.flat:
im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cm, vmin=mi, im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cm, vmin=mi,
vmax=ma) vmax=ma)
ax.set_title(caption.pop(0)) ax.set_title(caption.pop(0))
fig.subplots_adjust(right=0.8) fig.subplots_adjust(right=0.8)
cbar_ax = fig.add_axes([0.85, 0.15, 0.05, 0.7]) cbar_ax = fig.add_axes([0.85, 0.15, 0.05, 0.7])
fig.colorbar(im, cax=cbar_ax) fig.colorbar(im, cax=cbar_ax)
``` ```
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
mi = np.min(s_data) mi = np.min(s_data)
ma = np.max(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] data = [s_data, m_data, samp1, samp2, s_data - m_data, uncertainty]
caption = ["Signal", "Reconstruction", "Residuals", "Uncertainty Map"] caption = ["Signal", "Reconstruction", "Sample 1", "Sample 2", "Residuals", "Uncertainty Map"]
for ax in axes.flat: for ax in axes.flat:
im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cm, vmin=mi, vmax=ma) im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cm, vmin=mi, vmax=ma)
ax.set_title(caption.pop(0)) ax.set_title(caption.pop(0))
fig.subplots_adjust(right=0.8) fig.subplots_adjust(right=0.8)
cbar_ax = fig.add_axes([.85, 0.15, 0.05, 0.7]) cbar_ax = fig.add_axes([.85, 0.15, 0.05, 0.7])
fig.colorbar(im, cax=cbar_ax) fig.colorbar(im, cax=cbar_ax)
``` ```
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
### Is the uncertainty map reliable? ### Is the uncertainty map reliable?
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` 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) + "%") print("Error within uncertainty map bounds: " + str(np.sum(precise) * 100 / N_pixels**2) + "%")
plt.figure(figsize=(15,10)) plt.figure(figsize=(15,10))
plt.imshow(precise.astype(float), cmap="brg") plt.imshow(precise.astype(float), cmap="brg")
plt.colorbar() plt.colorbar()
``` ```
%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
# Start Coding # Start Coding
## NIFTy Repository + Installation guide ## NIFTy Repository + Installation guide
https://gitlab.mpcdf.mpg.de/ift/NIFTy https://gitlab.mpcdf.mpg.de/ift/NIFTy
NIFTy v4 **more or less stable!** NIFTy v4 **more or less stable!**
......
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