Commit 6f781aee authored by Martin Reinecke's avatar Martin Reinecke
Browse files

Merge branch 'NIFTy_5' into 11-models-multifields-top-level-documentation

parents 80667408 7585873e
......@@ -63,133 +63,6 @@ pages:
before_script:
- export MPLBACKEND="agg"
run_critical_filtering:
stage: demo_runs
script:
- ls
- python setup.py install --user -f
- python3 setup.py install --user -f
- python demos/critical_filtering.py
- python3 demos/critical_filtering.py
artifacts:
paths:
- '*.png'
run_nonlinear_critical_filter:
stage: demo_runs
script:
- python setup.py install --user -f
- python3 setup.py install --user -f
- python demos/nonlinear_critical_filter.py
- python3 demos/nonlinear_critical_filter.py
artifacts:
paths:
- '*.png'
run_nonlinear_wiener_filter:
stage: demo_runs
script:
- python setup.py install --user -f
- python3 setup.py install --user -f
- python demos/nonlinear_wiener_filter.py
- python3 demos/nonlinear_wiener_filter.py
only:
- run_demos
artifacts:
paths:
- '*.png'
# FIXME: disable for now. Fixing it is part of issue #244.
#run_poisson_demo:
# stage: demo_runs
# script:
# - python setup.py install --user -f
# - python3 setup.py install --user -f
# - python demos/poisson_demo.py
# - python3 demos/poisson_demo.py
# artifacts:
# paths:
# - '*.png'
run_probing:
stage: demo_runs
script:
- python setup.py install --user -f
- python3 setup.py install --user -f
- python demos/probing.py
- python3 demos/probing.py
artifacts:
paths:
- '*.png'
run_sampling:
stage: demo_runs
script:
- python setup.py install --user -f
- python3 setup.py install --user -f
- python demos/sampling.py
- python3 demos/sampling.py
artifacts:
paths:
- '*.png'
run_tomography:
stage: demo_runs
script:
- python setup.py install --user -f
- python3 setup.py install --user -f
- python demos/tomography.py
- python3 demos/tomography.py
artifacts:
paths:
- '*.png'
run_wiener_filter_data_space_noiseless:
stage: demo_runs
script:
- python setup.py install --user -f
- python3 setup.py install --user -f
- python demos/wiener_filter_data_space_noiseless.py
- python3 demos/wiener_filter_data_space_noiseless.py
artifacts:
paths:
- '*.png'
run_wiener_filter_easy.py:
stage: demo_runs
script:
- python setup.py install --user -f
- python3 setup.py install --user -f
- python demos/wiener_filter_easy.py
- python3 demos/wiener_filter_easy.py
artifacts:
paths:
- '*.png'
run_wiener_filter_via_curvature.py:
stage: demo_runs
script:
- pip install --user numericalunits
- pip3 install --user numericalunits
- python setup.py install --user -f
- python3 setup.py install --user -f
- python demos/wiener_filter_via_curvature.py
- python3 demos/wiener_filter_via_curvature.py
artifacts:
paths:
- '*.png'
run_wiener_filter_via_hamiltonian.py:
stage: demo_runs
script:
- python setup.py install --user -f
- python3 setup.py install --user -f
- python demos/wiener_filter_via_hamiltonian.py
- python3 demos/wiener_filter_via_hamiltonian.py
artifacts:
paths:
- '*.png'
run_ipynb:
stage: demo_runs
script:
......
%% 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
- 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.library.WienerFilterCurvature(R,N,Sh,iteration_controller=IC,iteration_controller_sampling=IC)
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*).
<!--
- 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
$$T = \mathcal F^\dagger S_h^{-1} \mathcal F.$$
-->
%% 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()
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()
# 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)
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!**
......
import nifty5 as ift
import numpy as np
from nifty5 import Exponential, Linear, Tanh
np.random.seed(42)
def adjust_zero_mode(m0, t0):
mtmp = m0.to_global_data().copy()
zero_position = len(m0.shape)*(0,)
zero_mode = mtmp[zero_position]
mtmp[zero_position] = zero_mode / abs(zero_mode)
ttmp = t0.to_global_data().copy()
ttmp[0] += 2 * np.log(abs(zero_mode))
return (ift.Field.from_global_data(m0.domain, mtmp),
ift.Field.from_global_data(t0.domain, ttmp))
if __name__ == "__main__":
noise_level = 1.
p_spec = (lambda k: (.5 / (k + 1) ** 3))
nonlinearity = Linear()
# Set up position space
s_space = ift.RGSpace((128, 128))
h_space = s_space.get_default_codomain()
# Define harmonic transformation and associated harmonic space
HT = ift.HarmonicTransformOperator(h_space, target=s_space)
# Setting up power space
p_space = ift.PowerSpace(h_space,
binbounds=ift.PowerSpace.useful_binbounds(
h_space, logarithmic=True))
# Choosing the prior correlation structure and defining
# correlation operator
p = ift.PS_field(p_space, p_spec)
log_p = ift.log(p)
S = ift.create_power_operator(h_space, power_spectrum=lambda k: 1e-5)
# Drawing a sample sh from the prior distribution in harmonic space
sh = S.draw_sample()
# Choosing the measurement instrument
# Instrument = SmoothingOperator(s_space, sigma=0.01)
mask = np.ones(s_space.shape)
# mask[6000:8000] = 0.
mask[30:70, 30:70] = 0.
mask = ift.Field.from_global_data(s_space, mask)
MaskOperator = ift.DiagonalOperator(mask)
R = ift.GeometryRemover(s_space)
R = R*MaskOperator
# R = R*HT
# R = R * ift.create_harmonic_smoothing_operator((harmonic_space,), 0,
# response_sigma)
MeasurementOperator = R
d_space = MeasurementOperator.target
Distributor = ift.PowerDistributor(target=h_space, power_space=p_space)
power = Distributor(ift.exp(0.5*log_p))
# Creating the mock data
true_sky = nonlinearity(HT(power*sh))
noiseless_data = MeasurementOperator(true_sky)
noise_amplitude = noiseless_data.val.std()*noise_level
N = ift.ScalingOperator(noise_amplitude**2, d_space)
n = N.draw_sample()
# Creating the mock data
d = noiseless_data + n
m0 = ift.full(h_space, 1e-7)
t0 = ift.full(p_space, -4.)
power0 = Distributor.times(ift.exp(0.5 * t0))
plotdict = {"colormap": "Planck-like"}
zmin = true_sky.min()
zmax = true_sky.max()
ift.plot(true_sky, title="True sky", name="true_sky.png", **plotdict)
ift.plot(MeasurementOperator.adjoint_times(d), title="Data",
name="data.png", **plotdict)
IC1 = ift.GradientNormController(name="IC1", iteration_limit=100,
tol_abs_gradnorm=1e-3)
LS = ift.LineSearchStrongWolfe(c2=0.02)
minimizer = ift.RelaxedNewton(IC1, line_searcher=LS)
IC = ift.GradientNormController(iteration_limit=500,
tol_abs_gradnorm=1e-3)
for i in range(20):
power0 = Distributor(ift.exp(0.5*t0))
map0_energy = ift.library.NonlinearWienerFilterEnergy(
m0, d, MeasurementOperator, nonlinearity, HT, power0, N, S, IC)
# Minimization with chosen minimizer
map0_energy, convergence = minimizer(map0_energy)
m0 = map0_energy.position
# Updating parameters for correlation structure reconstruction
D0 = map0_energy.curvature
# Initializing the power energy with updated parameters
power0_energy = ift.library.NonlinearPowerEnergy(
position=t0, d=d, N=N, xi=m0, D=D0, ht=HT,
Instrument=MeasurementOperator, nonlinearity=nonlinearity,
Distributor=Distributor, sigma=1., samples=2,
iteration_controller=IC)
power0_energy = minimizer(power0_energy)[0]
# Setting new power spectrum
t0 = power0_energy.position
# break degeneracy between amplitude and excitation by setting
# excitation monopole to 1
m0, t0 = adjust_zero_mode(m0, t0)
ift.plot(nonlinearity(HT(power0*m0)), title="Reconstructed sky",
name="reconstructed_sky.png", zmin=zmin, zmax=zmax, **plotdict)
ymin = np.min(p.to_global_data())
ift.plot([ift.exp(t0), p], title="Power spectra",
name="ps.png", ymin=ymin, **plotdict)
import nifty5 as ift
import numpy as np
from global_newton.models_other.apply_data import ApplyData
from global_newton.models_energy.hamiltonian import Hamiltonian
from nifty5 import GaussianEnergy
if __name__ == '__main__':
# s_space = ift.RGSpace([1024])
s_space = ift.RGSpace([128,128])
# s_space = ift.HPSpace(64)
h_space = s_space.get_default_codomain()
total_domain = ift.MultiDomain.make({'xi': h_space})
HT = ift.HarmonicTransformOperator(h_space, s_space)
def sqrtpspec(k):
return 16. / (20.+k**2)
GR = ift.GeometryRemover(s_space)
d_space = GR.target
B = ift.FFTSmoothingOperator(s_space,0.1)
mask = np.ones(s_space.shape)
mask[64:89,76:100] = 0.
mask = ift.Field(s_space,val=mask)
Mask = ift.DiagonalOperator(mask)
R = GR * Mask * B
noise = 1.
N = ift.ScalingOperator(noise, d_space)
p_space = ift.PowerSpace(h_space)
pd = ift.PowerDistributor(h_space, p_space)
position = ift.from_random('normal', total_domain)
xi = ift.Variable(position)['xi']
a = ift.Constant(position, ift.PS_field(p_space, sqrtpspec))
A = pd(a)
s_h = A * xi
s = HT(s_h)
Rs = R(s)
MOCK_POSITION = ift.from_random('normal',total_domain)
data = Rs.at(MOCK_POSITION).value + N.draw_sample()
NWR = ApplyData(data, ift.Field(d_space,val=noise), Rs)
INITIAL_POSITION = ift.from_random('normal',total_domain)
likelihood = GaussianEnergy(INITIAL_POSITION, NWR)
IC = ift.GradientNormController(iteration_limit=500, tol_abs_gradnorm=1e-3)
inverter = ift.ConjugateGradient(controller=IC)
IC2 = ift.GradientNormController(name='Newton', iteration_limit=15)
minimizer = ift.RelaxedNewton(IC2)
H = Hamiltonian(likelihood, inverter)
H, convergence = minimizer(H)
result = s.at(H.position).value
import nifty5 as ift
from nifty5 import Exponential
import numpy as np
np.random.seed(42)
def adjust_zero_mode(m0, t0):
mtmp = m0.to_global_data().copy()
zero_position = len(m0.shape)*(0,)
zero_mode = mtmp[zero_position]
mtmp[zero_position] = zero_mode / abs(zero_mode)
ttmp = t0.to_global_data().copy()
ttmp[0] += 2 * np.log(abs(zero_mode))
return (ift.Field.from_global_data(m0.domain, mtmp),
ift.Field.from_global_data(t0.domain, ttmp))
if __name__ == "__main__":
noise_level = 1.
p_spec = (lambda k: (1. / (k + 1) ** 2))
# nonlinearity = Linear()
nonlinearity = Exponential()
# Set up position space
# s_space = ift.RGSpace([1024])
s_space = ift.HPSpace(32)
h_space = s_space.get_default_codomain()
# Define harmonic transformation and associated harmonic space
HT = ift.HarmonicTransformOperator(h_space, target=s_space)
# Setting up power space
p_space = ift.PowerSpace(h_space,
binbounds=ift.PowerSpace.useful_binbounds(
h_space, logarithmic=True))
# Choosing the prior correlation structure and defining
# correlation operator
p = ift.PS_field(p_space, p_spec)
log_p = ift.log(p)
S = ift.create_power_operator(h_space, lambda k: 1.)
# Drawing a sample sh from the prior distribution in harmonic space
sh = S.draw_sample()
# Choosing the measurement instrument
# Instrument = SmoothingOperator(s_space, sigma=0.01)
mask = np.ones(s_space.shape)
mask[6000:8000] = 0.
mask = ift.Field.from_global_data(s_space, mask)
MaskOperator = ift.DiagonalOperator(mask)
R = ift.GeometryRemover(s_space)
R = R*MaskOperator
# R = R*HT
# R = R * ift.create_harmonic_smoothing_operator((harmonic_space,), 0,
# response_sigma)
MeasurementOperator = R
d_space = MeasurementOperator.target
Distributor = ift.PowerDistributor(target=h_space, power_space=p_space)
power = Distributor(ift.exp(0.5*log_p))
# Creating the mock data
true_sky = nonlinearity(HT(power*sh))
noiseless_data = MeasurementOperator(true_sky)
noise_amplitude = noiseless_data.val.std()*noise_level
N = ift.ScalingOperator(noise_amplitude**2, d_space)
n = N.draw_sample()
# Creating the mock data
d = noiseless_data + n
m0 = ift.full(h_space, 1e-7)
t0 = ift.full(p_space, -4.)
power0 = Distributor.times(ift.exp(0.5 * t0))
IC1 = ift.GradientNormController(name="IC1", iteration_limit=100,
tol_abs_gradnorm=1e-3)
LS = ift.LineSearchStrongWolfe(c2=0.02)
minimizer = ift.RelaxedNewton(IC1, line_searcher=LS)
IC = ift.GradientNormController(iteration_limit=500,
tol_abs_gradnorm=1e-3)
for i in range(20):
power0 = Distributor(ift.exp(0.5*t0))
map0_energy = ift.library.NonlinearWienerFilterEnergy(
m0, d, MeasurementOperator, nonlinearity, HT, power0, N, S, IC)
# Minimization with chosen minimizer
map0_energy, convergence = minimizer(map0_energy)
m0 = map0_energy.position
# Updating parameters for correlation structure reconstruction
D0 = map0_energy.curvature
# Initializing the power energy with updated parameters
power0_energy = ift.library.NonlinearPowerEnergy(
position=t0, d=d, N=N, xi=m0, D=D0, ht=HT,
Instrument=MeasurementOperator, nonlinearity=nonlinearity,
Distributor=Distributor, sigma=1., samples=2, iteration_controller=IC)
power0_energy = minimizer(power0_energy)[0]
# Setting new power spectrum
t0 = power0_energy.position
# break degeneracy between amplitude and excitation by setting
# excitation monopole to 1
m0, t0 = adjust_zero_mode(m0, t0)
plotdict = {"colormap": "Planck-like"}
ift.plot(true_sky, name="true_sky.png", **plotdict)
ift.plot(nonlinearity(HT(power0*m0)),
name="reconstructed_sky.png", **plotdict)
ift.plot(MeasurementOperator.adjoint_times(d), name="data.png", **plotdict)
ift.plot([ift.exp(t0), p], name="ps.png")
import nifty5 as ift
from nifty5.library.nonlinearities import Linear, Exponential, Tanh
import numpy as np
np.random.seed(20)
if __name__ == "__main__":
noise_level = 0.3
correlation_length = 0.1