astropy.convolution provides convolution functions and kernels that offer improvements compared to the SciPy scipy.ndimage convolution routines, including:
- Proper treatment of NaN values (ignoring them during convolution and replacing NaN pixels with interpolated values)
- Both direct and Fast Fourier Transform (FFT) versions
- Built-in kernels that are commonly used in Astronomy
The following thumbnails show the difference between SciPy and Astropy's convolve functions on an astronomical image that contains NaN values. Scipy's function returns NaN for all pixels that are within a kernel-sized region of any NaN value, which is often not the desired result.
.. plot::
:show-source-link:
import matplotlib.pyplot as plt
import numpy as np
from astropy.convolution import Gaussian2DKernel, convolve
from astropy.io import fits
from astropy.utils.data import get_pkg_data_filename
from scipy.ndimage import convolve as scipy_convolve
# Load the data from data.astropy.org
filename = get_pkg_data_filename('galactic_center/gc_msx_e.fits')
hdu = fits.open(filename)[0]
# Scale the file to have reasonable numbers
# (this is mostly so that colorbars do not have too many digits)
# Also, we crop it so you can see individual pixels
img = hdu.data[50:90, 60:100] * 1e5
# This example is intended to demonstrate how astropy.convolve and
# scipy.convolve handle missing data, so we start by setting the
# brightest pixels to NaN to simulate a "saturated" data set
img[img > 20] = np.nan
# We also create a copy of the data and set those NaNs to zero. We will
# use this for the scipy convolution
img_zerod = img.copy()
img_zerod[np.isnan(img)] = 0
# We smooth with a Gaussian kernel with x_stddev=1 (and y_stddev=1)
# It is a 9x9 array
kernel = Gaussian2DKernel(x_stddev=1)
# Convolution: scipy's direct convolution mode spreads out NaNs (see
# panel 2 below)
scipy_conv = scipy_convolve(img, kernel)
# scipy's direct convolution mode run on the 'zero'd' image will not
# have NaNs, but will have some very low value zones where the NaNs were
# (see panel 3 below)
scipy_conv_zerod = scipy_convolve(img_zerod, kernel)
# astropy's convolution replaces the NaN pixels with a kernel-weighted
# interpolation from their neighbors
astropy_conv = convolve(img, kernel)
# Now we do a bunch of plots. In the first two plots, the originally masked
# values are marked with red X's
fig, ax = plt.subplots(nrows=2, ncols=2, figsize=(8, 8), layout="tight")
fig.subplots_adjust(left=0.05, bottom=0.05, right=0.95, top=0.95,
wspace=0.3, hspace=0.3)
ax = ax.flatten()
for axis in ax:
axis.axis('off')
im = ax[0].imshow(img, vmin=-2.0, vmax=20.0, origin='lower',
interpolation='nearest', cmap='viridis')
y, x = np.where(np.isnan(img))
ax[0].plot(x, y, 'rx', markersize=4)
ax[0].set_title("Input Data")
ax[0].set_xticklabels([])
ax[0].set_yticklabels([])
im = ax[1].imshow(scipy_conv, vmin=-2.0, vmax=20.0, origin='lower',
interpolation='nearest', cmap='viridis')
ax[1].plot(x, y, 'rx', markersize=4)
ax[1].set_title("Scipy convolved")
ax[1].set_xticklabels([])
ax[1].set_yticklabels([])
im = ax[2].imshow(scipy_conv_zerod, vmin=-2.0, vmax=20.0, origin='lower',
interpolation='nearest', cmap='viridis')
ax[2].set_title("Scipy convolved (NaN to zero)")
ax[2].set_xticklabels([])
ax[2].set_yticklabels([])
im = ax[3].imshow(astropy_conv, vmin=-2.0, vmax=20.0, origin='lower',
interpolation='nearest', cmap='viridis')
ax[3].set_title("Astropy convolved")
ax[3].set_xticklabels([])
ax[3].set_yticklabels([])
# we make a second plot of the amplitudes vs offset position to more
# clearly illustrate the value differences
fig2, ax2 = plt.subplots(figsize=(8, 6))
ax2.plot(img[:, 25], label='Input data', drawstyle='steps-mid', linewidth=2,
alpha=0.5)
ax2.plot(scipy_conv[:, 25], label='SciPy convolved', drawstyle='steps-mid',
linewidth=2, alpha=0.5, marker='s')
ax2.plot(scipy_conv_zerod[:, 25], label='SciPy convolved (NaN to zero)',
drawstyle='steps-mid', linewidth=2, alpha=0.5, marker='s')
ax2.plot(astropy_conv[:, 25], label='Astropy convolved', drawstyle='steps-mid',
linewidth=2, alpha=0.5)
ax2.set(xlabel="Pixel", ylabel="Amplitude")
ax2.legend(loc='best')
plt.show()
Two convolution functions are provided. They are imported as:
from astropy.convolution import convolve, convolve_fft
and are both used as:
result = convolve(image, kernel) result = convolve_fft(image, kernel)
:func:`~astropy.convolution.convolve` is implemented as a direct convolution algorithm, while :func:`~astropy.convolution.convolve_fft` uses a Fast Fourier Transform (FFT). Thus, the former is better for small kernels, while the latter is much more efficient for larger kernels.
To convolve a 1D dataset with a user-specified kernel, you can do:
>>> from astropy.convolution import convolve >>> convolve([1, 4, 5, 6, 5, 7, 8], [0.2, 0.6, 0.2]) # doctest: +FLOAT_CMP array([1.4, 3.6, 5. , 5.6, 5.6, 6.8, 6.2])
The boundary keyword determines how the input array is extended
beyond its boundaries. The default value is 'fill', meaning values
outside of the array boundary are set to the input fill_value
(default is 0.0). Setting boundary='extend' causes values near the
edges to be extended using a constant extrapolation beyond the boundary.
The values at the end are computed assuming that any value below the
first point is 1, and any value above the last point is 8:
>>> from astropy.convolution import convolve >>> convolve([1, 4, 5, 6, 5, 7, 8], [0.2, 0.6, 0.2], boundary='extend') # doctest: +FLOAT_CMP array([1.6, 3.6, 5. , 5.6, 5.6, 6.8, 7.8])
For a more detailed discussion of boundary treatment, see :doc:`using`.
The convolution module also includes built-in kernels that can be imported as, for example:
>>> from astropy.convolution import Gaussian1DKernel
To use a kernel, first create a specific instance of the kernel:
>>> gauss = Gaussian1DKernel(stddev=2)
gauss is not an array, but a kernel object. The underlying array can be
retrieved with:
>>> gauss.array # doctest: +FLOAT_CMP
array([6.69162896e-05, 4.36349021e-04, 2.21596317e-03, 8.76430436e-03,
2.69959580e-02, 6.47599366e-02, 1.20987490e-01, 1.76035759e-01,
1.99474648e-01, 1.76035759e-01, 1.20987490e-01, 6.47599366e-02,
2.69959580e-02, 8.76430436e-03, 2.21596317e-03, 4.36349021e-04,
6.69162896e-05])
The kernel can then be used directly when calling :func:`~astropy.convolution.convolve`:
.. plot::
:show-source-link:
import numpy as np
import matplotlib.pyplot as plt
from astropy.convolution import Gaussian1DKernel, convolve
fig, ax = plt.subplots()
# Generate fake data
rng = np.random.default_rng(963)
x = np.arange(1000).astype(float)
y = np.sin(x / 100.) + rng.normal(0., 1., x.shape)
y[::3] = np.nan
# Create kernel
g = Gaussian1DKernel(stddev=50)
# Convolve data
z = convolve(y, g)
# Plot data before and after convolution
ax.plot(x, y, label='Data')
ax.plot(x, z, label='Convolved Data', linewidth=2)
ax.legend(loc='best')
plt.show()
astropy's convolution methods can be used to replace bad data with values
interpolated from their neighbors. Kernel-based interpolation is useful for
handling images with a few bad pixels or for interpolating sparsely sampled
images.
The interpolation tool is implemented and used as:
from astropy.convolution import interpolate_replace_nans result = interpolate_replace_nans(image, kernel)
Some contexts in which you might want to use kernel-based interpolation include:
- Images with saturated pixels. Generally, these are the highest-intensity regions in the imaged area, and the interpolated values are not reliable, but this can be useful for display purposes.
- Images with flagged pixels (e.g., a few small regions affected by cosmic rays or other spurious signals that require those pixels to be flagged out). If the affected region is small enough, the resulting interpolation will have a small effect on source statistics and may allow for robust source-finding algorithms to be run on the resulting data.
- Sparsely sampled images such as those constructed with single-pixel detectors. Such images will only have a few discrete points sampled across the imaged area, but an approximation of the extended sky emission can still be constructed.
Note
Care must be taken to ensure that the kernel is large enough to completely
cover potential contiguous regions of NaN values.
An AstropyUserWarning is raised if NaN values are detected
post-convolution, in which case the kernel size should be increased.
The script below shows an example of kernel interpolation to fill in flagged-out pixels:
.. plot::
:context:
:show-source-link:
import numpy as np
import matplotlib.pyplot as plt
from astropy.io import fits
from astropy.utils.data import get_pkg_data_filename
from astropy.convolution import Gaussian2DKernel, interpolate_replace_nans
# Load the data from data.astropy.org
filename = get_pkg_data_filename('galactic_center/gc_msx_e.fits')
hdu = fits.open(filename)[0]
img = hdu.data[50:90, 60:100] * 1e5
# This example is intended to demonstrate how astropy.convolve and
# scipy.convolve handle missing data, so we start by setting the brightest
# pixels to NaN to simulate a "saturated" data set
img[img > 2e1] = np.nan
# We smooth with a Gaussian kernel with x_stddev=1 (and y_stddev=1)
# It is a 9x9 array
kernel = Gaussian2DKernel(x_stddev=1)
# create a "fixed" image with NaNs replaced by interpolated values
fixed_image = interpolate_replace_nans(img, kernel)
# Now we do a bunch of plots. In the first two plots, the originally masked
# values are marked with red X's
fig, axs = plt.subplots(ncols=2, figsize=(12, 6))
ax1 = axs[0]
im = ax1.imshow(img, vmin=-2., vmax=2.e1, origin='lower',
interpolation='nearest', cmap='viridis')
y, x = np.where(np.isnan(img))
ax1.plot(x, y, 'rx', markersize=4)
ax1.set(
autoscale_on=False,
title="Original",
xticklabels=[],
yticklabels=[],
)
ax2 = axs[1]
im = ax2.imshow(fixed_image, vmin=-2., vmax=2.e1, origin='lower',
interpolation='nearest', cmap='viridis')
ax2.set(
title="Fixed",
xticklabels=[],
yticklabels=[],
)
This script shows the power of this technique for reconstructing images from sparse sampling. Note that the image is not perfect: the pointlike sources are sometimes missed, but the extended structure is very well recovered by eye.
.. plot::
:context:
:show-source-link:
import numpy as np
import matplotlib.pyplot as plt
from astropy.io import fits
from astropy.utils.data import get_pkg_data_filename
from astropy.convolution import Gaussian2DKernel, interpolate_replace_nans
# Load the data from data.astropy.org
filename = get_pkg_data_filename('galactic_center/gc_msx_e.fits')
hdu = fits.open(filename)[0]
img = hdu.data[50:90, 60:100] * 1e5
rng = np.random.default_rng(1379)
indices = rng.integers(low=0, high=img.size, size=300)
sampled_data = img.flat[indices]
# Build a new, sparsely sampled version of the original image
new_img = np.tile(np.nan, img.shape)
new_img.flat[indices] = sampled_data
# We smooth with a Gaussian kernel with x_stddev=1 (and y_stddev=1)
# It is a 9x9 array
kernel = Gaussian2DKernel(x_stddev=1)
# create a "reconstructed" image with NaNs replaced by interpolated values
reconstructed_image = interpolate_replace_nans(new_img, kernel)
# Now we do a bunch of plots. In the first two plots, the originally masked
# values are marked with red X's
fig, axs = plt.subplots(ncols=3, figsize=(12, 6))
ax1 = axs[0]
im = ax1.imshow(img, vmin=-2., vmax=2.e1, origin='lower',
interpolation='nearest', cmap='viridis')
y, x = np.where(np.isnan(img))
ax1.set(
autoscale_on=False,
title="Original",
xticklabels=[],
yticklabels=[],
)
ax2 = axs[1]
im = ax2.imshow(new_img, vmin=-2., vmax=2.e1, origin='lower',
interpolation='nearest', cmap='viridis')
ax2.set(
title="Sparsely Sampled",
xticklabels=[],
yticklabels=[],
)
ax3 = axs[2]
im = ax3.imshow(reconstructed_image, vmin=-2., vmax=2.e1, origin='lower',
interpolation='nearest', cmap='viridis')
ax3.set(
title="Reconstructed",
xticklabels=[],
yticklabels=[],
)
.. toctree:: :maxdepth: 2 using.rst kernels.rst non_normalized_kernels.rst
.. toctree:: :maxdepth: 2 ref_api