Commit 0649322d authored by Martin Reinecke's avatar Martin Reinecke
Browse files

Merge branch 'op_twiddling' into 'NIFTy_4'

Replace InverseOperator and AdjointOperator with OperatorAdapter, and more

See merge request ift/NIFTy!237
parents 9740e2f3 17d86163
Pipeline #26740 passed with stages
in 10 minutes and 4 seconds
......@@ -3,32 +3,41 @@ image: debian:testing-slim
stages:
- test
- release
variables:
DOCKER_DRIVER: overlay
before_script:
- apt-get update
- sh ci/install_basics.sh
- pip install --process-dependency-links -r ci/requirements.txt
- pip3 install --process-dependency-links -r ci/requirements.txt
- pip install --user .
- pip3 install --user .
test_min:
test_python2:
stage: test
script:
- nosetests3 -q
- OMP_NUM_THREADS=1 mpiexec --allow-run-as-root -n 4 nosetests -q 2>/dev/null
- OMP_NUM_THREADS=1 mpiexec --allow-run-as-root -n 4 nosetests3 -q 2>/dev/null
- apt-get update > /dev/null
- apt-get install -y git libfftw3-dev openmpi-bin libopenmpi-dev python python-pip python-dev python-nose python-numpy python-matplotlib python-future python-mpi4py python-scipy > /dev/null
- pip install --process-dependency-links parameterized coverage git+https://gitlab.mpcdf.mpg.de/ift/pyHealpix.git > /dev/null
- pip install --user .
- OMP_NUM_THREADS=1 mpiexec --allow-run-as-root -n 2 nosetests -q 2>/dev/null
- nosetests -q --with-coverage --cover-package=nifty4 --cover-branches --cover-erase
- >
coverage report | grep TOTAL | awk '{ print "TOTAL: "$6; }'
test_python3:
stage: test
script:
- apt-get update > /dev/null
- apt-get install -y git libfftw3-dev openmpi-bin libopenmpi-dev python3 python3-pip python3-dev python3-nose python3-numpy python3-matplotlib python3-future python3-mpi4py python3-scipy > /dev/null
- pip3 install --process-dependency-links parameterized git+https://gitlab.mpcdf.mpg.de/ift/pyHealpix.git > /dev/null
- pip3 install --user .
- nosetests3 -q
- OMP_NUM_THREADS=1 mpiexec --allow-run-as-root -n 2 nosetests3 -q 2>/dev/null
pages:
stage: release
script:
- sh docs/generate.sh
- mv docs/build/ public/
- apt-get update > /dev/null
- apt-get install -y git libfftw3-dev python python-pip python-dev python-numpy python-future python-sphinx python-sphinx-rtd-theme python-numpydoc > /dev/null
- pip install --user .
- sh docs/generate.sh > /dev/null
- mv docs/build/ public/
artifacts:
paths:
- public
......
apt-get install -y git libfftw3-dev openmpi-bin libopenmpi-dev \
python python-pip python-dev python-nose python-numpy python-matplotlib python-future python-mpi4py python-scipy \
python3 python3-pip python3-dev python3-nose python3-numpy python3-matplotlib python3-future python3-mpi4py python3-scipy
parameterized
coverage
git+https://gitlab.mpcdf.mpg.de/ift/pyHealpix.git
sphinx
sphinx_rtd_theme
numpydoc
%% 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 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*).
<!--
- 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()+m).to_global_data()
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 v4 **more or less stable!**
......
......@@ -52,8 +52,7 @@ if __name__ == "__main__":
tol_abs_gradnorm=0.1)
inverter = ift.ConjugateGradient(controller=IC)
D = (ift.SandwichOperator(R, N.inverse) + Sh.inverse).inverse
# MR FIXME: we can/should provide a preconditioner here as well!
D = ift.InversionEnabler(D, inverter)
D = ift.InversionEnabler(D, inverter, approximation=Sh)
m = D(j)
# Plotting
......
......@@ -236,12 +236,24 @@ class data_object(object):
def __ipow__(self, other):
return self._binary_helper(other, op='__ipow__')
def __eq__(self, other):
return self._binary_helper(other, op='__eq__')
def __lt__(self, other):
return self._binary_helper(other, op='__lt__')
def __le__(self, other):
return self._binary_helper(other, op='__le__')
def __ne__(self, other):
return self._binary_helper(other, op='__ne__')
def __eq__(self, other):
return self._binary_helper(other, op='__eq__')
def __ge__(self, other):
return self._binary_helper(other, op='__ge__')
def __gt__(self, other):
return self._binary_helper(other, op='__gt__')
def __neg__(self):
return data_object(self._shape, -self._data, self._distaxis)
......
from .operator_tests import consistency_check
from .energy_tests import check_value_gradient_consistency
from .energy_tests import *
......@@ -19,35 +19,63 @@
import numpy as np
from ..field import Field
__all__ = ["check_value_gradient_consistency"]
__all__ = ["check_value_gradient_consistency",
"check_value_gradient_curvature_consistency"]
def check_value_gradient_consistency(E, tol=1e-6, ntries=100):
def _get_acceptable_energy(E):
if not np.isfinite(E.value):
raise ValueError
dir = Field.from_random("normal", E.position.domain)
# find a step length that leads to a "reasonable" energy
for i in range(50):
try:
E2 = E.at(E.position+dir)
if np.isfinite(E2.value) and abs(E2.value) < 1e20:
break
except FloatingPointError:
pass
dir *= 0.5
else:
raise ValueError("could not find a reasonable initial step")
return E2
def check_value_gradient_consistency(E, tol=1e-6, ntries=100):
for _ in range(ntries):
dir = Field.from_random("normal", E.position.domain)
# find a step length that leads to a "reasonable" energy
E2 = _get_acceptable_energy(E)
dir = E2.position - E.position
Enext = E2
dirnorm = dir.norm()
dirder = E.gradient.vdot(dir)/dirnorm
for i in range(50):
try:
E2 = E.at(E.position+dir)
if np.isfinite(E2.value) and abs(E2.value) < 1e20:
break
except FloatingPointError:
pass
print(abs((E2.value-E.value)/dirnorm-dirder))
if abs((E2.value-E.value)/dirnorm-dirder) < tol:
break
dir *= 0.5
dirnorm *= 0.5
E2 = E2.at(E.position+dir)
else:
raise ValueError("could not find a reasonable initial step")
raise ValueError("gradient and value seem inconsistent")
# E = Enext