r"""
======================
Image denoising by FFT
======================

Denoise an image (:download:`../../../../data/moonlanding.png`) by
implementing a blur with an FFT.

Implements, via FFT, the following convolution:

.. math::

    f_1(t) = \int dt'\, K(t-t') f_0(t')

.. math::

    \tilde{f}_1(\omega) = \tilde{K}(\omega) \tilde{f}_0(\omega)

"""

############################################################
# Read and plot the image
############################################################
import numpy as np
import matplotlib.pyplot as plt

im = plt.imread('../../../../data/moonlanding.png').astype(float)

plt.figure()
plt.imshow(im, plt.cm.gray)
plt.title('Original image')


############################################################
# Compute the 2d FFT of the input image
############################################################
import scipy as sp
im_fft = sp.fftpack.fft2(im)

# Show the results

def plot_spectrum(im_fft):
    from matplotlib.colors import LogNorm
    # A logarithmic colormap
    plt.imshow(np.abs(im_fft), norm=LogNorm(vmin=5))
    plt.colorbar()

plt.figure()
plot_spectrum(im_fft)
plt.title('Fourier transform')

############################################################
# Filter in FFT
############################################################

# In the lines following, we'll make a copy of the original spectrum and
# truncate coefficients.

# Define the fraction of coefficients (in each direction) we keep
keep_fraction = 0.1

# Call ff a copy of the original transform. NumPy arrays have a copy
# method for this purpose.
im_fft2 = im_fft.copy()

# Set r and c to be the number of rows and columns of the array.
r, c = im_fft2.shape

# Set to zero all rows with indices between r*keep_fraction and
# r*(1-keep_fraction):
im_fft2[int(r*keep_fraction):int(r*(1-keep_fraction))] = 0

# Similarly with the columns:
im_fft2[:, int(c*keep_fraction):int(c*(1-keep_fraction))] = 0

plt.figure()
plot_spectrum(im_fft2)
plt.title('Filtered Spectrum')


############################################################
# Reconstruct the final image
############################################################

# Reconstruct the denoised image from the filtered spectrum, keep only the
# real part for display.
im_new = sp.fftpack.ifft2(im_fft2).real

plt.figure()
plt.imshow(im_new, plt.cm.gray)
plt.title('Reconstructed Image')


############################################################
# Easier and better: :func:`scipy.ndimage.gaussian_filter`
############################################################
#
# Implementing filtering directly with FFTs is tricky and time consuming.
# We can use the Gaussian filter from :mod:`scipy.ndimage`

im_blur = sp.ndimage.gaussian_filter(im, 4)

plt.figure()
plt.imshow(im_blur, plt.cm.gray)
plt.title('Blurred image')

plt.show()