Wide-field imaging demonstration
================================

This script makes a fake data set, fills it with a number of point
components, and then images it using a variety of algorithms. See
imaging-fits for a similar notebook that checks for errors in the
recovered properties of the images.

The measurement equation for a wide field of view interferometer is:

.. math:: V(u,v,w) =\int \frac{I(l,m)}{\sqrt{1-l^2-m^2}} e^{-2 \pi j (ul+um + w(\sqrt{1-l^2-m^2}-1))} dl dm

We will show various algorithms for computing approximations to this
integral. Calculation of the visibility V from the sky brightness I is
called predict, and the inverese is called invert.

.. code:: ipython3

    %matplotlib inline
    
    import os
    import sys
    
    sys.path.append(os.path.join('..', '..'))
    
    from data_models.parameters import arl_path
    
    results_dir = arl_path('test_results')
    
    from matplotlib import pylab
    
    import numpy
    
    from astropy.coordinates import SkyCoord
    from astropy import units as u
    from astropy.wcs.utils import pixel_to_skycoord
    
    from matplotlib import pyplot as plt
    from mpl_toolkits.mplot3d import Axes3D
    
    from data_models.polarisation import PolarisationFrame
    
    from processing_components.image import  image_raster_iter
    from processing_library.image import create_w_term_like
    
    # Use serial wrappers by default
    from processing_components.visibility import create_visibility, create_visibility, create_visibility_from_rows
    from processing_components.skycomponent import create_skycomponent
    from processing_components.image import show_image, export_image_to_fits
    from processing_components.visibility import  vis_timeslice_iter
    from processing_components.simulation.configurations import create_named_configuration
    from processing_components.imaging import invert_2d, create_image_from_visibility, \
        predict_skycomponent_visibility, advise_wide_field
    from processing_components.visibility import  vis_timeslice_iter
    from processing_components.imaging import  weight_visibility
    from processing_components.visibility import  vis_timeslices
    
    from processing_components.griddata import create_awterm_convolutionfunction
    from processing_components.griddata.convolution_functions import apply_bounding_box_convolutionfunction
    
    # Use arlexecute for imaging
    from wrappers.arlexecute.execution_support.arlexecute import arlexecute
    from workflows.arlexecute.imaging.imaging_arlexecute import invert_list_arlexecute_workflow
    
    import logging
    
    log = logging.getLogger()
    log.setLevel(logging.DEBUG)
    log.addHandler(logging.StreamHandler(sys.stdout))
    
    doplot = True



.. parsed-literal::

    Loaded backend module://ipykernel.pylab.backend_inline version unknown.


.. code:: ipython3

    pylab.rcParams['figure.figsize'] = (12.0, 12.0)
    pylab.rcParams['image.cmap'] = 'rainbow'

Construct the SKA1-LOW core configuration

.. code:: ipython3

    lowcore = create_named_configuration('LOWBD2-CORE')


.. parsed-literal::

    create_named_configuration: LOWBD2-CORE
    	(<Quantity -2565018.31203579 m>, <Quantity 5085711.90373391 m>, <Quantity -2861033.10788063 m>)
    	GeodeticLocation(lon=<Longitude 116.76444824 deg>, lat=<Latitude -26.82472208 deg>, height=<Quantity 300. m>)
    create_configuration_from_file: 166 antennas/stations


We create the visibility.

This just makes the uvw, time, antenna1, antenna2, weight columns in a
table

.. code:: ipython3

    times = numpy.array([-3.0, -2.0, -1.0, 0.0, 1.0, 2.0, 3.0]) * (numpy.pi / 12.0)
    frequency = numpy.array([1e8])
    channel_bandwidth = numpy.array([1e7])
    
    
    reffrequency = numpy.max(frequency)
    phasecentre = SkyCoord(ra=+15.0 * u.deg, dec=-45.0 * u.deg, frame='icrs', equinox='J2000')
    vt = create_visibility(lowcore, times, frequency, channel_bandwidth=channel_bandwidth,
                           weight=1.0, phasecentre=phasecentre, polarisation_frame=PolarisationFrame("stokesI"))


.. parsed-literal::

    create_visibility: 95865 rows, 0.010 GB
    create_visibility: flagged 0/95865 visibilities below elevation limit 0.261799 (rad)


Advise on wide field parameters. This returns a dictionary with all the
input and calculated variables.

.. code:: ipython3

    advice = advise_wide_field(vt, wprojection_planes=1)


.. parsed-literal::

    advise_wide_field: Maximum wavelength 2.998 (meters)
    advise_wide_field: Minimum wavelength 2.998 (meters)
    advise_wide_field: Maximum baseline 262.6 (wavelengths)
    advise_wide_field: Maximum w 169.4 (wavelengths)
    advise_wide_field: Station/dish diameter 35.0 (meters)
    advise_wide_field: Primary beam 0.0857 (rad) 4.91 (deg) 1.77e+04 (asec)
    advise_wide_field: Image field of view 0.514 (rad) 29.4 (deg) 1.06e+05 (asec)
    advise_wide_field: Synthesized beam 0.00381 (rad) 0.218 (deg) 785 (asec)
    advise_wide_field: Cellsize 0.00127 (rad) 0.0727 (deg) 262 (asec)
    advice_wide_field: Npixels per side = 405
    advice_wide_field: Npixels (power of 2) per side = 512
    advice_wide_field: Npixels (power of 2, 3) per side = 512
    advice_wide_field: Npixels (power of 2, 3, 4, 5) per side = 405
    advice_wide_field: W sampling for full image = 0.2 (wavelengths)
    advice_wide_field: W sampling for primary beam = 8.7 (wavelengths)
    advice_wide_field: Time sampling for full image = 25.2 (s)
    advice_wide_field: Time sampling for primary beam = 908.6 (s)
    advice_wide_field: Frequency sampling for full image = 29212.6 (Hz)
    advice_wide_field: Frequency sampling for primary beam = 1051653.8 (Hz)
    advice_wide_field: Number of planes in w stack 39 (primary beam)
    advice_wide_field: Number of planes in w projection 39 (primary beam)
    advice_wide_field: W support = 6 (pixels) (primary beam)


Plot the synthesized UV coverage.

.. code:: ipython3

    if doplot:
        plt.clf()
        plt.plot(vt.data['uvw'][:, 0], vt.data['uvw'][:, 1], '.', color='b')
        plt.plot(-vt.data['uvw'][:, 0], -vt.data['uvw'][:, 1], '.', color='r')
        plt.xlabel('U (wavelengths)')
        plt.ylabel('V (wavelengths)')
        plt.show()
        
        plt.clf()
        plt.plot(vt.data['uvw'][:, 0], vt.data['uvw'][:, 2], '.', color='b')
        plt.xlabel('U (wavelengths)')
        plt.ylabel('W (wavelengths)')
        plt.show()
    
        plt.clf()
        plt.plot(vt.data['time'][vt.u>0.0], vt.data['uvw'][:, 2][vt.u>0.0], '.', color='b')
        plt.plot(vt.data['time'][vt.u<=0.0], vt.data['uvw'][:, 2][vt.u<=0.0], '.', color='r')
        plt.xlabel('U (wavelengths)')
        plt.ylabel('W (wavelengths)')
        plt.show()
    
        plt.clf()
        n, bins, patches = plt.hist(vt.w, 50, normed=1, facecolor='green', alpha=0.75)
        plt.xlabel('W (wavelengths)')
        plt.ylabel('Count')
        plt.show()


.. parsed-literal::

    update_title_pos
    findfont: Matching :family=sans-serif:style=normal:variant=normal:weight=normal:stretch=normal:size=10.0 to DejaVu Sans ('/home/jenkins-slave/workspace/ce-library_feature-improved-docs/_build/lib/python3.5/site-packages/matplotlib/mpl-data/fonts/ttf/DejaVuSans.ttf') with score of 0.050000.
    update_title_pos
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.. image:: imaging-wterm_arlexecute_files/imaging-wterm_arlexecute_11_1.png


.. parsed-literal::

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.. image:: imaging-wterm_arlexecute_files/imaging-wterm_arlexecute_11_3.png


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.. image:: imaging-wterm_arlexecute_files/imaging-wterm_arlexecute_11_5.png


.. parsed-literal::

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.. parsed-literal::

    /home/jenkins-slave/workspace/ce-library_feature-improved-docs/_build/lib/python3.5/site-packages/matplotlib/axes/_axes.py:6521: MatplotlibDeprecationWarning: 
    The 'normed' kwarg was deprecated in Matplotlib 2.1 and will be removed in 3.1. Use 'density' instead.
      alternative="'density'", removal="3.1")


.. parsed-literal::

    update_title_pos
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.. image:: imaging-wterm_arlexecute_files/imaging-wterm_arlexecute_11_9.png


Show the planar nature of the uvw sampling, rotating with hour angle

Create a grid of components and predict each in turn, using the full
phase term including w.

.. code:: ipython3

    npixel = 512
    cellsize=0.001
    facets = 4
    flux = numpy.array([[100.0]])
    vt.data['vis'] *= 0.0
    
    model = create_image_from_visibility(vt, npixel=512, cellsize=0.001, npol=1)
    spacing_pixels = npixel // facets
    log.info('Spacing in pixels = %s' % spacing_pixels)
    spacing = 180.0 * cellsize * spacing_pixels / numpy.pi
    centers = -1.5, -0.5, +0.5, +1.5
    comps=list()
    for iy in centers:
        for ix in centers:
            pra =  int(round(npixel // 2 + ix * spacing_pixels - 1))
            pdec = int(round(npixel // 2 + iy * spacing_pixels - 1))
            sc = pixel_to_skycoord(pra, pdec, model.wcs)
            log.info("Component at (%f, %f) %s" % (pra, pdec, str(sc)))
            comp = create_skycomponent(flux=flux, frequency=frequency, direction=sc, 
                                       polarisation_frame=PolarisationFrame("stokesI"))
            comps.append(comp)
    predict_skycomponent_visibility(vt, comps)


.. parsed-literal::

    create_image_from_visibility: Parsing parameters to get definition of WCS
    create_image_from_visibility: Defining single channel Image at <SkyCoord (ICRS): (ra, dec) in deg
        (15., -45.)>, starting frequency 100000000.0 Hz, and bandwidth 9999999.9999 Hz
    create_image_from_visibility: uvmax = 262.634709 wavelengths
    create_image_from_visibility: Critical cellsize = 0.001904 radians, 0.109079 degrees
    create_image_from_visibility: Cellsize          = 0.001 radians, 0.0572958 degrees
    create_image_from_visibility: image shape is [1, 1, 512, 512]
    Spacing in pixels = 128
    Component at (63.000000, 63.000000) <SkyCoord (ICRS): (ra, dec) in deg
        (34.54072794, -54.75874632)>
    Component at (191.000000, 63.000000) <SkyCoord (ICRS): (ra, dec) in deg
        (21.67016023, -55.97155392)>
    Component at (319.000000, 63.000000) <SkyCoord (ICRS): (ra, dec) in deg
        (8.53437599, -55.98101975)>
    Component at (447.000000, 63.000000) <SkyCoord (ICRS): (ra, dec) in deg
        (355.65593443, -54.78677607)>
    Component at (63.000000, 191.000000) <SkyCoord (ICRS): (ra, dec) in deg
        (31.62619218, -47.58256993)>
    Component at (191.000000, 191.000000) <SkyCoord (ICRS): (ra, dec) in deg
        (20.64032824, -48.59686272)>
    Component at (319.000000, 191.000000) <SkyCoord (ICRS): (ra, dec) in deg
        (9.53290242, -48.6047374)>
    Component at (447.000000, 191.000000) <SkyCoord (ICRS): (ra, dec) in deg
        (358.54340538, -47.60612847)>
    Component at (63.000000, 319.000000) <SkyCoord (ICRS): (ra, dec) in deg
        (29.67602513, -40.37841403)>
    Component at (191.000000, 319.000000) <SkyCoord (ICRS): (ra, dec) in deg
        (19.96147534, -41.27364866)>
    Component at (319.000000, 319.000000) <SkyCoord (ICRS): (ra, dec) in deg
        (10.19103897, -41.28057703)>
    Component at (447.000000, 319.000000) <SkyCoord (ICRS): (ra, dec) in deg
        (0.4748226, -40.3992699)>
    Component at (63.000000, 447.000000) <SkyCoord (ICRS): (ra, dec) in deg
        (28.31336506, -33.05766481)>
    Component at (191.000000, 447.000000) <SkyCoord (ICRS): (ra, dec) in deg
        (19.49068833, -33.88364257)>
    Component at (319.000000, 447.000000) <SkyCoord (ICRS): (ra, dec) in deg
        (10.64743131, -33.89002022)>
    Component at (447.000000, 447.000000) <SkyCoord (ICRS): (ra, dec) in deg
        (1.82415538, -33.07694985)>




.. parsed-literal::

    <data_models.memory_data_models.Visibility at 0x7f2b719cfe10>



Make the dirty image and point spread function using the two-dimensional
approximation:

.. math:: V(u,v,w) =\int I(l,m) e^{2 \pi j (ul+um)} dl dm

Note that the shape of the sources vary with position in the image. This
space-variant property of the PSF arises from the w-term neglected in
the two-dimensional invert.

.. code:: ipython3

    arlexecute.set_client(use_dask=True)


.. parsed-literal::

    Using selector: EpollSelector
    Using selector: EpollSelector


.. code:: ipython3

    dirty = create_image_from_visibility(vt, npixel=512, cellsize=0.001, 
                                         polarisation_frame=PolarisationFrame("stokesI"))
    vt = weight_visibility(vt, dirty)
    
    future = invert_list_arlexecute_workflow([vt], [dirty], context='2d')
    dirty, sumwt = arlexecute.compute(future, sync=True)[0]
    
    if doplot:
        show_image(dirty)
    
    print("Max, min in dirty image = %.6f, %.6f, sumwt = %f" % (dirty.data.max(), dirty.data.min(), sumwt))
    
    export_image_to_fits(dirty, '%s/imaging-wterm_dirty.fits' % (results_dir))


.. parsed-literal::

    create_image_from_visibility: Parsing parameters to get definition of WCS
    create_image_from_visibility: Defining single channel Image at <SkyCoord (ICRS): (ra, dec) in deg
        (15., -45.)>, starting frequency 100000000.0 Hz, and bandwidth 9999999.9999 Hz
    create_image_from_visibility: uvmax = 262.634709 wavelengths
    create_image_from_visibility: Critical cellsize = 0.001904 radians, 0.109079 degrees
    create_image_from_visibility: Cellsize          = 0.001 radians, 0.0572958 degrees
    create_image_from_visibility: image shape is [1, 1, 512, 512]


.. parsed-literal::

    /home/jenkins-slave/workspace/ce-library_feature-improved-docs/_build/lib/python3.5/site-packages/distributed/worker.py:3235: UserWarning: Large object of size 2.10 MB detected in task graph: 
      ('getitem-0f26109ea1d3a935e74f14f4a5bf1dd8', <data ... -c71747eeb0c1')
    Consider scattering large objects ahead of time
    with client.scatter to reduce scheduler burden and 
    keep data on workers
    
        future = client.submit(func, big_data)    # bad
    
        big_future = client.scatter(big_data)     # good
        future = client.submit(func, big_future)  # good
      % (format_bytes(len(b)), s)


.. parsed-literal::

    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b903a1198>
    Using auto colorbar locator on colorbar
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b903a1198>
    Setting pcolormesh
    Max, min in dirty image = 49.220748, -8.719588, sumwt = 31701.000000
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    findfont: Matching :family=DejaVu Sans:style=normal:variant=normal:weight=normal:stretch=normal:size=10.0 to DejaVu Sans ('/home/jenkins-slave/workspace/ce-library_feature-improved-docs/_build/lib/python3.5/site-packages/matplotlib/mpl-data/fonts/ttf/DejaVuSans.ttf') with score of 0.050000.
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    findfont: Matching :family=DejaVu Sans:style=italic:variant=normal:weight=normal:stretch=normal:size=10.0 to DejaVu Sans ('/home/jenkins-slave/workspace/ce-library_feature-improved-docs/_build/lib/python3.5/site-packages/matplotlib/mpl-data/fonts/ttf/DejaVuSans-Oblique.ttf') with score of 0.150000.
    findfont: Matching :family=DejaVu Sans:style=normal:variant=normal:weight=bold:stretch=normal:size=10.0 to DejaVu Sans ('/home/jenkins-slave/workspace/ce-library_feature-improved-docs/_build/lib/python3.5/site-packages/matplotlib/mpl-data/fonts/ttf/DejaVuSans-Bold.ttf') with score of 0.000000.
    update_title_pos
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.. image:: imaging-wterm_arlexecute_files/imaging-wterm_arlexecute_17_3.png


This occurs because the Fourier transform relationship between sky
brightness and visibility is only accurate over small fields of view.

Hence we can make an accurate image by partitioning the image plane into
small regions, treating each separately and then glueing the resulting
partitions into one image. We call this image plane partitioning image
plane faceting.

.. math::

   V(u,v,w) = \sum_{i,j} \frac{1}{\sqrt{1- l_{i,j}^2- m_{i,j}^2}} e^{-2 \pi j (ul_{i,j}+um_{i,j} + w(\sqrt{1-l_{i,j}^2-m_{i,j}^2}-1))}
   \int  I(\Delta l, \Delta m) e^{-2 \pi j (u\Delta l_{i,j}+u \Delta m_{i,j})} dl dm

.. code:: ipython3

    dirtyFacet = create_image_from_visibility(vt, npixel=512, cellsize=0.001, npol=1)
    future = invert_list_arlexecute_workflow([vt], [dirtyFacet], facets=4, context='facets')
    dirtyFacet, sumwt = arlexecute.compute(future, sync=True)[0]
    
    if doplot:
        show_image(dirtyFacet)
    
    print("Max, min in dirty image = %.6f, %.6f, sumwt = %f" % (dirtyFacet.data.max(), dirtyFacet.data.min(), sumwt))
    export_image_to_fits(dirtyFacet, '%s/imaging-wterm_dirtyFacet.fits' % (results_dir))


.. parsed-literal::

    create_image_from_visibility: Parsing parameters to get definition of WCS
    create_image_from_visibility: Defining single channel Image at <SkyCoord (ICRS): (ra, dec) in deg
        (15., -45.)>, starting frequency 100000000.0 Hz, and bandwidth 9999999.9999 Hz
    create_image_from_visibility: uvmax = 262.634709 wavelengths
    create_image_from_visibility: Critical cellsize = 0.001904 radians, 0.109079 degrees
    create_image_from_visibility: Cellsize          = 0.001 radians, 0.0572958 degrees
    create_image_from_visibility: image shape is [1, 1, 512, 512]
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b901c7b00>
    Using auto colorbar locator on colorbar
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b901c7b00>
    Setting pcolormesh
    Max, min in dirty image = 102.477593, -11.772359, sumwt = 507216.000000
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos



.. image:: imaging-wterm_arlexecute_files/imaging-wterm_arlexecute_19_1.png


That was the best case. This time, we will not arrange for the
partitions to be centred on the sources.

.. code:: ipython3

    dirtyFacet2 = create_image_from_visibility(vt, npixel=512, cellsize=0.001, npol=1)
    future = invert_list_arlexecute_workflow([vt], [dirtyFacet2], facets=2, context='facets')
    dirtyFacet2, sumwt = arlexecute.compute(future, sync=True)[0]
    
    
    if doplot:
        show_image(dirtyFacet2)
    
    print("Max, min in dirty image = %.6f, %.6f, sumwt = %f" % (dirtyFacet2.data.max(), dirtyFacet2.data.min(), sumwt))
    export_image_to_fits(dirtyFacet2, '%s/imaging-wterm_dirtyFacet2.fits' % (results_dir))


.. parsed-literal::

    create_image_from_visibility: Parsing parameters to get definition of WCS
    create_image_from_visibility: Defining single channel Image at <SkyCoord (ICRS): (ra, dec) in deg
        (15., -45.)>, starting frequency 100000000.0 Hz, and bandwidth 9999999.9999 Hz
    create_image_from_visibility: uvmax = 262.634709 wavelengths
    create_image_from_visibility: Critical cellsize = 0.001904 radians, 0.109079 degrees
    create_image_from_visibility: Cellsize          = 0.001 radians, 0.0572958 degrees
    create_image_from_visibility: image shape is [1, 1, 512, 512]
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b9012e4e0>
    Using auto colorbar locator on colorbar
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b9012e4e0>
    Setting pcolormesh
    Max, min in dirty image = 51.663404, -9.843857, sumwt = 126804.000000
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos



.. image:: imaging-wterm_arlexecute_files/imaging-wterm_arlexecute_21_1.png


Another approach is to partition the visibility data by slices in w. The
measurement equation is approximated as:

.. math:: V(u,v,w) =\sum_i \int   \frac{ I(l,m) e^{-2 \pi j (w_i(\sqrt{1-l^2-m^2}-1))})}{\sqrt{1-l^2-m^2}} e^{-2 \pi j (ul+um)} dl dm

If images constructed from slices in w are added after applying a
w-dependent image plane correction, the w term will be corrected.

The w-dependent w-beam is:

.. code:: ipython3

    if doplot:
        wterm = create_w_term_like(model, phasecentre=vt.phasecentre, w=numpy.max(vt.w))
        show_image(wterm)
        plt.show()


.. parsed-literal::

    create_w_term_image: For w = 169.4, field of view = 0.512000, Fresnel number = 11.10
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b90394b38>
    Using auto colorbar locator on colorbar
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b90394b38>
    Setting pcolormesh
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos



.. image:: imaging-wterm_arlexecute_files/imaging-wterm_arlexecute_24_1.png


.. code:: ipython3

    dirtywstack = create_image_from_visibility(vt, npixel=512, cellsize=0.001, npol=1)
    future = invert_list_arlexecute_workflow([vt], [dirtywstack], vis_slices=101, context='wstack')
    dirtywstack, sumwt = arlexecute.compute(future, sync=True)[0]
    
    show_image(dirtywstack)
    plt.show()
    
    print("Max, min in dirty image = %.6f, %.6f, sumwt = %f" % 
          (dirtywstack.data.max(), dirtywstack.data.min(), sumwt))
    
    export_image_to_fits(dirtywstack, '%s/imaging-wterm_dirty_wstack.fits' % (results_dir))


.. parsed-literal::

    create_image_from_visibility: Parsing parameters to get definition of WCS
    create_image_from_visibility: Defining single channel Image at <SkyCoord (ICRS): (ra, dec) in deg
        (15., -45.)>, starting frequency 100000000.0 Hz, and bandwidth 9999999.9999 Hz
    create_image_from_visibility: uvmax = 262.634709 wavelengths
    create_image_from_visibility: Critical cellsize = 0.001904 radians, 0.109079 degrees
    create_image_from_visibility: Cellsize          = 0.001 radians, 0.0572958 degrees
    create_image_from_visibility: image shape is [1, 1, 512, 512]
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b3ccb4128>
    Using auto colorbar locator on colorbar
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b3ccb4128>
    Setting pcolormesh
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos



.. image:: imaging-wterm_arlexecute_files/imaging-wterm_arlexecute_25_1.png


.. parsed-literal::

    Max, min in dirty image = 728001.342797, -283031.542936, sumwt = 31701.000000


The w-term can also be viewed as a time-variable distortion.
Approximating the array as instantaneously co-planar, we have that w can
be expressed in terms of :math:`u,v`

.. math:: w = a u + b v

Transforming to a new coordinate system:

.. math::  l' = l + a (\sqrt{1-l^2-m^2}-1))

.. math::  m' = m + b (\sqrt{1-l^2-m^2}-1))

Ignoring changes in the normalisation term, we have:

.. math:: V(u,v,w) =\int \frac{I(l',m')}{\sqrt{1-l'^2-m'^2}} e^{-2 \pi j (ul'+um')} dl' dm'

To illustrate this, we will construct images as a function of time. For
comparison, we show difference of each time slice from the best facet
image. Instantaneously the sources are un-distorted but do lie in the
wrong location.

.. code:: ipython3

    for rows in vis_timeslice_iter(vt):
        visslice = create_visibility_from_rows(vt, rows)
        dirtySnapshot = create_image_from_visibility(visslice, npixel=512, cellsize=0.001, npol=1, compress_factor=0.0)
        future = invert_list_arlexecute_workflow([visslice], [dirtySnapshot], context='2d')
        dirtySnapshot, sumwt = arlexecute.compute(future, sync=True)[0]
        
        print("Max, min in dirty image = %.6f, %.6f, sumwt = %f" % 
              (dirtySnapshot.data.max(), dirtySnapshot.data.min(), sumwt))
        if doplot:
            dirtySnapshot.data -= dirtyFacet.data
            show_image(dirtySnapshot)
            plt.title("Hour angle %.2f hours" % (numpy.average(visslice.time) * 12.0 / 43200.0))
            plt.show()


.. parsed-literal::

    create_image_from_visibility: Parsing parameters to get definition of WCS
    create_image_from_visibility: Defining single channel Image at <SkyCoord (ICRS): (ra, dec) in deg
        (15., -45.)>, starting frequency 100000000.0 Hz, and bandwidth 9999999.9999 Hz
    create_image_from_visibility: uvmax = 193.168337 wavelengths
    create_image_from_visibility: Critical cellsize = 0.002588 radians, 0.148305 degrees
    create_image_from_visibility: Cellsize          = 0.001 radians, 0.0572958 degrees
    create_image_from_visibility: image shape is [1, 1, 512, 512]
    Max, min in dirty image = 95.641822, -16.727650, sumwt = 4216.610661
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b3cc60320>
    Using auto colorbar locator on colorbar
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b3cc60320>
    Setting pcolormesh
    update_title_pos
    findfont: Matching :family=sans-serif:style=normal:variant=normal:weight=normal:stretch=normal:size=12.0 to DejaVu Sans ('/home/jenkins-slave/workspace/ce-library_feature-improved-docs/_build/lib/python3.5/site-packages/matplotlib/mpl-data/fonts/ttf/DejaVuSans.ttf') with score of 0.050000.
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos



.. image:: imaging-wterm_arlexecute_files/imaging-wterm_arlexecute_27_1.png


.. parsed-literal::

    create_image_from_visibility: Parsing parameters to get definition of WCS
    create_image_from_visibility: Defining single channel Image at <SkyCoord (ICRS): (ra, dec) in deg
        (15., -45.)>, starting frequency 100000000.0 Hz, and bandwidth 9999999.9999 Hz
    create_image_from_visibility: uvmax = 231.753249 wavelengths
    create_image_from_visibility: Critical cellsize = 0.002157 radians, 0.123614 degrees
    create_image_from_visibility: Cellsize          = 0.001 radians, 0.0572958 degrees
    create_image_from_visibility: image shape is [1, 1, 512, 512]
    Max, min in dirty image = 100.519607, -17.807714, sumwt = 4508.422216
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b3cbf9550>
    Using auto colorbar locator on colorbar
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b3cbf9550>
    Setting pcolormesh
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos



.. image:: imaging-wterm_arlexecute_files/imaging-wterm_arlexecute_27_3.png


.. parsed-literal::

    create_image_from_visibility: Parsing parameters to get definition of WCS
    create_image_from_visibility: Defining single channel Image at <SkyCoord (ICRS): (ra, dec) in deg
        (15., -45.)>, starting frequency 100000000.0 Hz, and bandwidth 9999999.9999 Hz
    create_image_from_visibility: uvmax = 255.914039 wavelengths
    create_image_from_visibility: Critical cellsize = 0.001954 radians, 0.111943 degrees
    create_image_from_visibility: Cellsize          = 0.001 radians, 0.0572958 degrees
    create_image_from_visibility: image shape is [1, 1, 512, 512]
    Max, min in dirty image = 99.690697, -17.608646, sumwt = 4735.435580
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b3cb05940>
    Using auto colorbar locator on colorbar
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b3cb05940>
    Setting pcolormesh
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos



.. image:: imaging-wterm_arlexecute_files/imaging-wterm_arlexecute_27_5.png


.. parsed-literal::

    create_image_from_visibility: Parsing parameters to get definition of WCS
    create_image_from_visibility: Defining single channel Image at <SkyCoord (ICRS): (ra, dec) in deg
        (15., -45.)>, starting frequency 100000000.0 Hz, and bandwidth 9999999.9999 Hz
    create_image_from_visibility: uvmax = 262.634709 wavelengths
    create_image_from_visibility: Critical cellsize = 0.001904 radians, 0.109079 degrees
    create_image_from_visibility: Cellsize          = 0.001 radians, 0.0572958 degrees
    create_image_from_visibility: image shape is [1, 1, 512, 512]
    Max, min in dirty image = 102.909868, -19.624505, sumwt = 4806.769444
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b3caa5dd8>
    Using auto colorbar locator on colorbar
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b3caa5dd8>
    Setting pcolormesh
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos



.. image:: imaging-wterm_arlexecute_files/imaging-wterm_arlexecute_27_7.png


.. parsed-literal::

    create_image_from_visibility: Parsing parameters to get definition of WCS
    create_image_from_visibility: Defining single channel Image at <SkyCoord (ICRS): (ra, dec) in deg
        (15., -45.)>, starting frequency 100000000.0 Hz, and bandwidth 9999999.9999 Hz
    create_image_from_visibility: uvmax = 253.664251 wavelengths
    create_image_from_visibility: Critical cellsize = 0.001971 radians, 0.112936 degrees
    create_image_from_visibility: Cellsize          = 0.001 radians, 0.0572958 degrees
    create_image_from_visibility: image shape is [1, 1, 512, 512]
    Max, min in dirty image = 101.931847, -18.677038, sumwt = 4732.772801
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b3cb12748>
    Using auto colorbar locator on colorbar
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b3cb12748>
    Setting pcolormesh
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos



.. image:: imaging-wterm_arlexecute_files/imaging-wterm_arlexecute_27_9.png


.. parsed-literal::

    create_image_from_visibility: Parsing parameters to get definition of WCS
    create_image_from_visibility: Defining single channel Image at <SkyCoord (ICRS): (ra, dec) in deg
        (15., -45.)>, starting frequency 100000000.0 Hz, and bandwidth 9999999.9999 Hz
    create_image_from_visibility: uvmax = 236.819632 wavelengths
    create_image_from_visibility: Critical cellsize = 0.002111 radians, 0.120969 degrees
    create_image_from_visibility: Cellsize          = 0.001 radians, 0.0572958 degrees
    create_image_from_visibility: image shape is [1, 1, 512, 512]
    Max, min in dirty image = 102.692951, -18.823002, sumwt = 4524.935947
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b3ccb4198>
    Using auto colorbar locator on colorbar
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b3ccb4198>
    Setting pcolormesh
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos



.. image:: imaging-wterm_arlexecute_files/imaging-wterm_arlexecute_27_11.png


.. parsed-literal::

    create_image_from_visibility: Parsing parameters to get definition of WCS
    create_image_from_visibility: Defining single channel Image at <SkyCoord (ICRS): (ra, dec) in deg
        (15., -45.)>, starting frequency 100000000.0 Hz, and bandwidth 9999999.9999 Hz
    create_image_from_visibility: uvmax = 204.762355 wavelengths
    create_image_from_visibility: Critical cellsize = 0.002442 radians, 0.139908 degrees
    create_image_from_visibility: Cellsize          = 0.001 radians, 0.0572958 degrees
    create_image_from_visibility: image shape is [1, 1, 512, 512]
    Max, min in dirty image = 98.916621, -17.862308, sumwt = 4176.053350
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b901a3ef0>
    Using auto colorbar locator on colorbar
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b901a3ef0>
    Setting pcolormesh
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos



.. image:: imaging-wterm_arlexecute_files/imaging-wterm_arlexecute_27_13.png


This timeslice imaging leads to a straightforward algorithm in which we
correct each time slice and then sum the resulting timeslices.

.. code:: ipython3

    dirtyTimeslice = create_image_from_visibility(vt, npixel=512, cellsize=0.001, npol=1)
    future = invert_list_arlexecute_workflow([vt], [dirtyTimeslice], vis_slices=vis_timeslices(vt, 'auto'),
                                           padding=2, context='timeslice')
    dirtyTimeslice, sumwt = arlexecute.compute(future, sync=True)[0]
    
    
    show_image(dirtyTimeslice)
    plt.show()
    
    print("Max, min in dirty image = %.6f, %.6f, sumwt = %f" % 
          (dirtyTimeslice.data.max(), dirtyTimeslice.data.min(), sumwt))
    
    export_image_to_fits(dirtyTimeslice, '%s/imaging-wterm_dirty_Timeslice.fits' % (results_dir))


.. parsed-literal::

    create_image_from_visibility: Parsing parameters to get definition of WCS
    create_image_from_visibility: Defining single channel Image at <SkyCoord (ICRS): (ra, dec) in deg
        (15., -45.)>, starting frequency 100000000.0 Hz, and bandwidth 9999999.9999 Hz
    create_image_from_visibility: uvmax = 262.634709 wavelengths
    create_image_from_visibility: Critical cellsize = 0.001904 radians, 0.109079 degrees
    create_image_from_visibility: Cellsize          = 0.001 radians, 0.0572958 degrees
    create_image_from_visibility: image shape is [1, 1, 512, 512]
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b3ca31668>
    Using auto colorbar locator on colorbar
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b3ca31668>
    Setting pcolormesh
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos



.. image:: imaging-wterm_arlexecute_files/imaging-wterm_arlexecute_29_1.png


.. parsed-literal::

    Max, min in dirty image = 3967923.507510, -689431.950928, sumwt = 31701.000000


Finally we try w-projection. For a fixed w, the measurement equation can
be stated as as a convolution in Fourier space.

.. math:: V(u,v,w) =G_w(u,v) \ast \int \frac{I(l,m)}{\sqrt{1-l^2-m^2}} e^{-2 \pi j (ul+um)} dl dm

where the convolution function is:

.. math:: G_w(u,v) = \int \frac{1}{\sqrt{1-l^2-m^2}} e^{-2 \pi j (ul+um + w(\sqrt{1-l^2-m^2}-1))} dl dm

Hence when gridding, we can use the transform of the w beam to correct
this effect while gridding.

.. code:: ipython3

    dirtyWProjection = create_image_from_visibility(vt, npixel=512, cellsize=0.001, npol=1)
    
    gcfcf = create_awterm_convolutionfunction(model, nw=101, wstep=800.0/101, oversampling=8, 
                                                        support=60,
                                                        use_aaf=True)
        
    future = invert_list_arlexecute_workflow([vt], [dirtyWProjection], context='2d', gcfcf=[gcfcf])
    
    dirtyWProjection, sumwt = arlexecute.compute(future, sync=True)[0]
    
    if doplot:
        show_image(dirtyWProjection)
    
    print("Max, min in dirty image = %.6f, %.6f, sumwt = %f" % (dirtyWProjection.data.max(), 
                                                                 dirtyWProjection.data.min(), sumwt))
    export_image_to_fits(dirtyWProjection, '%s/imaging-wterm_dirty_WProjection.fits' % (results_dir))



.. parsed-literal::

    create_image_from_visibility: Parsing parameters to get definition of WCS
    create_image_from_visibility: Defining single channel Image at <SkyCoord (ICRS): (ra, dec) in deg
        (15., -45.)>, starting frequency 100000000.0 Hz, and bandwidth 9999999.9999 Hz
    create_image_from_visibility: uvmax = 262.634709 wavelengths
    create_image_from_visibility: Critical cellsize = 0.001904 radians, 0.109079 degrees
    create_image_from_visibility: Cellsize          = 0.001 radians, 0.0572958 degrees
    create_image_from_visibility: image shape is [1, 1, 512, 512]
    create_w_term_image: For w = -396.0, field of view = 0.512000, Fresnel number = 25.95
    create_w_term_image: For w = -388.1, field of view = 0.512000, Fresnel number = 25.44
    create_w_term_image: For w = -380.2, field of view = 0.512000, Fresnel number = 24.92
    create_w_term_image: For w = -372.3, field of view = 0.512000, Fresnel number = 24.40
    create_w_term_image: For w = -364.4, field of view = 0.512000, Fresnel number = 23.88
    create_w_term_image: For w = -356.4, field of view = 0.512000, Fresnel number = 23.36
    create_w_term_image: For w = -348.5, field of view = 0.512000, Fresnel number = 22.84
    create_w_term_image: For w = -340.6, field of view = 0.512000, Fresnel number = 22.32
    create_w_term_image: For w = -332.7, field of view = 0.512000, Fresnel number = 21.80
    create_w_term_image: For w = -324.8, field of view = 0.512000, Fresnel number = 21.28
    create_w_term_image: For w = -316.8, field of view = 0.512000, Fresnel number = 20.76
    create_w_term_image: For w = -308.9, field of view = 0.512000, Fresnel number = 20.24
    create_w_term_image: For w = -301.0, field of view = 0.512000, Fresnel number = 19.73
    create_w_term_image: For w = -293.1, field of view = 0.512000, Fresnel number = 19.21
    create_w_term_image: For w = -285.1, field of view = 0.512000, Fresnel number = 18.69
    create_w_term_image: For w = -277.2, field of view = 0.512000, Fresnel number = 18.17
    create_w_term_image: For w = -269.3, field of view = 0.512000, Fresnel number = 17.65
    create_w_term_image: For w = -261.4, field of view = 0.512000, Fresnel number = 17.13
    create_w_term_image: For w = -253.5, field of view = 0.512000, Fresnel number = 16.61
    create_w_term_image: For w = -245.5, field of view = 0.512000, Fresnel number = 16.09
    create_w_term_image: For w = -237.6, field of view = 0.512000, Fresnel number = 15.57
    create_w_term_image: For w = -229.7, field of view = 0.512000, Fresnel number = 15.05
    create_w_term_image: For w = -221.8, field of view = 0.512000, Fresnel number = 14.53
    create_w_term_image: For w = -213.9, field of view = 0.512000, Fresnel number = 14.02
    create_w_term_image: For w = -205.9, field of view = 0.512000, Fresnel number = 13.50
    create_w_term_image: For w = -198.0, field of view = 0.512000, Fresnel number = 12.98
    create_w_term_image: For w = -190.1, field of view = 0.512000, Fresnel number = 12.46
    create_w_term_image: For w = -182.2, field of view = 0.512000, Fresnel number = 11.94
    create_w_term_image: For w = -174.3, field of view = 0.512000, Fresnel number = 11.42
    create_w_term_image: For w = -166.3, field of view = 0.512000, Fresnel number = 10.90
    create_w_term_image: For w = -158.4, field of view = 0.512000, Fresnel number = 10.38
    create_w_term_image: For w = -150.5, field of view = 0.512000, Fresnel number = 9.86
    create_w_term_image: For w = -142.6, field of view = 0.512000, Fresnel number = 9.34
    create_w_term_image: For w = -134.7, field of view = 0.512000, Fresnel number = 8.82
    create_w_term_image: For w = -126.7, field of view = 0.512000, Fresnel number = 8.31
    create_w_term_image: For w = -118.8, field of view = 0.512000, Fresnel number = 7.79
    create_w_term_image: For w = -110.9, field of view = 0.512000, Fresnel number = 7.27
    create_w_term_image: For w = -103.0, field of view = 0.512000, Fresnel number = 6.75
    create_w_term_image: For w = -95.0, field of view = 0.512000, Fresnel number = 6.23
    create_w_term_image: For w = -87.1, field of view = 0.512000, Fresnel number = 5.71
    create_w_term_image: For w = -79.2, field of view = 0.512000, Fresnel number = 5.19
    create_w_term_image: For w = -71.3, field of view = 0.512000, Fresnel number = 4.67
    create_w_term_image: For w = -63.4, field of view = 0.512000, Fresnel number = 4.15
    create_w_term_image: For w = -55.4, field of view = 0.512000, Fresnel number = 3.63
    create_w_term_image: For w = -47.5, field of view = 0.512000, Fresnel number = 3.11
    create_w_term_image: For w = -39.6, field of view = 0.512000, Fresnel number = 2.60
    create_w_term_image: For w = -31.7, field of view = 0.512000, Fresnel number = 2.08
    create_w_term_image: For w = -23.8, field of view = 0.512000, Fresnel number = 1.56
    create_w_term_image: For w = -15.8, field of view = 0.512000, Fresnel number = 1.04
    create_w_term_image: For w = -7.9, field of view = 0.512000, Fresnel number = 0.52
    create_w_term_image: For w = 0.0, field of view = 0.512000, Fresnel number = 0.00
    create_w_term_image: For w = 7.9, field of view = 0.512000, Fresnel number = 0.52
    create_w_term_image: For w = 15.8, field of view = 0.512000, Fresnel number = 1.04
    create_w_term_image: For w = 23.8, field of view = 0.512000, Fresnel number = 1.56
    create_w_term_image: For w = 31.7, field of view = 0.512000, Fresnel number = 2.08
    create_w_term_image: For w = 39.6, field of view = 0.512000, Fresnel number = 2.60
    create_w_term_image: For w = 47.5, field of view = 0.512000, Fresnel number = 3.11
    create_w_term_image: For w = 55.4, field of view = 0.512000, Fresnel number = 3.63
    create_w_term_image: For w = 63.4, field of view = 0.512000, Fresnel number = 4.15
    create_w_term_image: For w = 71.3, field of view = 0.512000, Fresnel number = 4.67
    create_w_term_image: For w = 79.2, field of view = 0.512000, Fresnel number = 5.19
    create_w_term_image: For w = 87.1, field of view = 0.512000, Fresnel number = 5.71
    create_w_term_image: For w = 95.0, field of view = 0.512000, Fresnel number = 6.23
    create_w_term_image: For w = 103.0, field of view = 0.512000, Fresnel number = 6.75
    create_w_term_image: For w = 110.9, field of view = 0.512000, Fresnel number = 7.27
    create_w_term_image: For w = 118.8, field of view = 0.512000, Fresnel number = 7.79
    create_w_term_image: For w = 126.7, field of view = 0.512000, Fresnel number = 8.31
    create_w_term_image: For w = 134.7, field of view = 0.512000, Fresnel number = 8.82
    create_w_term_image: For w = 142.6, field of view = 0.512000, Fresnel number = 9.34
    create_w_term_image: For w = 150.5, field of view = 0.512000, Fresnel number = 9.86
    create_w_term_image: For w = 158.4, field of view = 0.512000, Fresnel number = 10.38
    create_w_term_image: For w = 166.3, field of view = 0.512000, Fresnel number = 10.90
    create_w_term_image: For w = 174.3, field of view = 0.512000, Fresnel number = 11.42
    create_w_term_image: For w = 182.2, field of view = 0.512000, Fresnel number = 11.94
    create_w_term_image: For w = 190.1, field of view = 0.512000, Fresnel number = 12.46
    create_w_term_image: For w = 198.0, field of view = 0.512000, Fresnel number = 12.98
    create_w_term_image: For w = 205.9, field of view = 0.512000, Fresnel number = 13.50
    create_w_term_image: For w = 213.9, field of view = 0.512000, Fresnel number = 14.02
    create_w_term_image: For w = 221.8, field of view = 0.512000, Fresnel number = 14.53
    create_w_term_image: For w = 229.7, field of view = 0.512000, Fresnel number = 15.05
    create_w_term_image: For w = 237.6, field of view = 0.512000, Fresnel number = 15.57
    create_w_term_image: For w = 245.5, field of view = 0.512000, Fresnel number = 16.09
    create_w_term_image: For w = 253.5, field of view = 0.512000, Fresnel number = 16.61
    create_w_term_image: For w = 261.4, field of view = 0.512000, Fresnel number = 17.13
    create_w_term_image: For w = 269.3, field of view = 0.512000, Fresnel number = 17.65
    create_w_term_image: For w = 277.2, field of view = 0.512000, Fresnel number = 18.17
    create_w_term_image: For w = 285.1, field of view = 0.512000, Fresnel number = 18.69
    create_w_term_image: For w = 293.1, field of view = 0.512000, Fresnel number = 19.21
    create_w_term_image: For w = 301.0, field of view = 0.512000, Fresnel number = 19.73
    create_w_term_image: For w = 308.9, field of view = 0.512000, Fresnel number = 20.24
    create_w_term_image: For w = 316.8, field of view = 0.512000, Fresnel number = 20.76
    create_w_term_image: For w = 324.8, field of view = 0.512000, Fresnel number = 21.28
    create_w_term_image: For w = 332.7, field of view = 0.512000, Fresnel number = 21.80
    create_w_term_image: For w = 340.6, field of view = 0.512000, Fresnel number = 22.32
    create_w_term_image: For w = 348.5, field of view = 0.512000, Fresnel number = 22.84
    create_w_term_image: For w = 356.4, field of view = 0.512000, Fresnel number = 23.36
    create_w_term_image: For w = 364.4, field of view = 0.512000, Fresnel number = 23.88
    create_w_term_image: For w = 372.3, field of view = 0.512000, Fresnel number = 24.40
    create_w_term_image: For w = 380.2, field of view = 0.512000, Fresnel number = 24.92
    create_w_term_image: For w = 388.1, field of view = 0.512000, Fresnel number = 25.44
    create_w_term_image: For w = 396.0, field of view = 0.512000, Fresnel number = 25.95
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b3c9e0400>
    Using auto colorbar locator on colorbar
    locator: <matplotlib.colorbar._ColorbarAutoLocator object at 0x7f2b3c9e0400>
    Setting pcolormesh
    Max, min in dirty image = 100.225951, -11.016452, sumwt = 31701.000000
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos
    update_title_pos



.. image:: imaging-wterm_arlexecute_files/imaging-wterm_arlexecute_31_1.png