# Code example: Wiener filter

## Introduction
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 (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.

## Wiener filter on one-dimensional fields

### 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:

$$\mathcal P (s|d) \propto P(s,d) = \mathcal G(d-Rs,N) \,\mathcal G(s,S) \propto \mathcal G (s-m,D) $$

where
$$m = Dj$$
with
$$D = (S^{-1} +R^\dagger N^{-1} R)^{-1} , \quad j = R^\dagger N^{-1} d.$$

Let us implement this in NIFTy!

### In NIFTy

- 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:
$$R = FFT_{\text{harmonic} \rightarrow \text{position}}$$



```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))
```

### Implementation

#### Import Modules


```python
%matplotlib inline
import numpy as np
import nifty8 as ift
import matplotlib.pyplot as plt

plt.rcParams['figure.dpi'] = 100
```

#### Implement Propagator


```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.WienerFilterCurvature(R,N,Sh,iteration_controller=IC,iteration_controller_sampling=IC)
```

#### 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

$$T = \mathcal F^\dagger S_h^{-1} \mathcal F.$$
-->

#### Generate Mock data

- Generate a field $s$ and $n$ with given covariances.
- Calculate $d$.


```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, sampling_dtype=float)
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(s_space, noise_amplitude**2, float)

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
```

#### Run Wiener Filter


```python
m = D(j)
```

#### Results


```python
# Get signal data and reconstruction data
s_data = HT(sh).val
m_data = HT(m).val
d_data = d.val

plt.plot(s_data, 'r', label="Signal", linewidth=2)
plt.plot(d_data, 'k.', label="Data")
plt.plot(m_data, 'k', label="Reconstruction",linewidth=2)
plt.title("Reconstruction")
plt.legend()
plt.show()
```


    
![png](old_nifty_getting_started_0_files/old_nifty_getting_started_0_16_0.png)
    



```python
plt.plot(s_data - s_data, 'r', label="Signal", linewidth=2)
plt.plot(d_data - s_data, 'k.', label="Data")
plt.plot(m_data - s_data, 'k', label="Reconstruction",linewidth=2)
plt.axhspan(-noise_amplitude,noise_amplitude, facecolor='0.9', alpha=.5)
plt.title("Residuals")
plt.legend()
plt.show()
```


    
![png](old_nifty_getting_started_0_files/old_nifty_getting_started_0_17_0.png)
    


#### Power Spectrum


```python
s_power_data = ift.power_analyze(sh).val
m_power_data = ift.power_analyze(m).val
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()
```


    
![png](old_nifty_getting_started_0_files/old_nifty_getting_started_0_19_0.png)
    


## Wiener Filter on Incomplete Data


```python
# Operators
Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec, sampling_dtype=float)
N = ift.ScalingOperator(s_space, noise_amplitude**2, sampling_dtype=float)
# 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)
```

### Partially Lose Data


```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_raw(s_space, mask)

R = ift.DiagonalOperator(mask) @ HT
n = n.val_rw()
n[l:h] = 0
n = ift.Field.from_raw(s_space, n)

d = R(sh) + n
```


```python
curv = Curvature(R=R, N=N, Sh=Sh)
D = curv.inverse
j = R.adjoint_times(N.inverse_times(d))
m = D(j)
```

### Compute Uncertainty



```python
m_mean, m_var = ift.probe_with_posterior_samples(curv, HT, 200, np.float64)
```

### Get data


```python
# Get signal data and reconstruction data
s_data = s.val
m_data = HT(m).val
m_var_data = m_var.val
uncertainty = np.sqrt(m_var_data)
d_data = d.val_rw()

# Set lost data to NaN for proper plotting
d_data[d_data == 0] = np.nan
```


```python
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=2)
plt.plot(d_data, 'k.', label="Data")
plt.plot(m_data, 'k', label="Reconstruction", linewidth=2)
plt.title("Reconstruction of incomplete data")
plt.legend()
```




    <matplotlib.legend.Legend at 0x7fe34f52df10>




    
![png](old_nifty_getting_started_0_files/old_nifty_getting_started_0_29_1.png)
    


## Wiener filter on two-dimensional field


```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])
```


```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, sampling_dtype=float)
N = ift.ScalingOperator(s_space, sigma2, sampling_dtype=float)

# 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_raw(s_space, mask)

R = ift.DiagonalOperator(mask)(HT)
n = n.val_rw()
n[l:h, l:h] = 0
n = ift.Field.from_raw(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, np.float64)

# Get data
s_data = HT(sh).val
m_data = HT(m).val
m_var_data = m_var.val
d_data = d.val
uncertainty = np.sqrt(np.abs(m_var_data))
```


```python
cmap = ['magma', 'inferno', 'plasma', 'viridis'][1]

mi = np.min(s_data)
ma = np.max(s_data)

fig, axes = plt.subplots(1, 2)

data = [s_data, d_data]
caption = ["Signal", "Data"]

for ax in axes.flat:
    im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cmap, 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)
```




    <matplotlib.colorbar.Colorbar at 0x7fe34dc05b90>




    
![png](old_nifty_getting_started_0_files/old_nifty_getting_started_0_33_1.png)
    



```python
mi = np.min(s_data)
ma = np.max(s_data)

fig, axes = plt.subplots(3, 2, figsize=(10, 15))
sample = HT(curv.draw_sample(from_inverse=True)+m).val
post_mean = (m_mean + HT(m)).val

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=cmap, 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)
```




    <matplotlib.colorbar.Colorbar at 0x7fe34f59bdd0>




    
![png](old_nifty_getting_started_0_files/old_nifty_getting_started_0_34_1.png)
    


### Is the uncertainty map reliable?


```python
precise = (np.abs(s_data-m_data) < uncertainty)
print("Error within uncertainty map bounds: " + str(np.sum(precise) * 100 / N_pixels**2) + "%")

plt.imshow(precise.astype(float), cmap="brg")
plt.colorbar()
```

    Error within uncertainty map bounds: 67.71392822265625%





    <matplotlib.colorbar.Colorbar at 0x7fe34d4122d0>




    
![png](old_nifty_getting_started_0_files/old_nifty_getting_started_0_36_2.png)