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In purely ‘linear’ spatial filtering (convolution), there are problems on
the edges of the input image. Here we will explain the problem in the
spatial domain. For a discussion of this problem from the frequency domain
perspective, see Edges in the frequency domain. The problem
originates from the fact that on the edges, in practice^{108}, the sum of the weights we use on the actual image pixels is not
unity. For example, as discussed above, a profile in the center of an image
will have the same brightness before and after convolution. However, for
partially imaged profile on the edge of the image, the brightness (sum of
its pixel fluxes within the image, see Flux Brightness and magnitude)
will not be equal, some of the flux is going to be ‘eaten’ by the edges.

If you ran `$ make check`

on the source files of Gnuastro, you can
see the this effect by comparing the `convolve_frequency.fits` with
`convolve_spatial.fits` in the `./tests/` directory. In the
spatial domain, by default, no assumption will be made about pixels outside
of the image or any blank pixels in the image. The problem explained above
will also occur on the sides of blank regions (see Blank pixels). The
solution to this edge effect problem is only possible in the spatial
domain. For pixels near the edge, we have to abandon the assumption that
the sum of the kernel pixels is unity during the convolution
process^{109}. So taking \(W\) as the sum of the kernel pixels
that overlapped with non-blank and in-image pixels, the equation in
Convolution process will become:

$$C_{x,y}= { \sum_{s=-a}^{a}\sum_{t=-b}^{b}K_{s,t}\times{}I_{x+s,y+t} \over W}.$$

In this manner, objects which are near the edges of the image or blank
pixels will also have the same brightness (within the image) before and
after convolution. This correction is applied by default in Convolve when
convolving in the spatial domain. To disable it, you can use the
`--noedgecorrection` option. In the frequency domain, there is no
way to avoid this loss of flux near the edges of the image, see Edges in the frequency domain for an interpretation from the frequency domain
perspective.

Note that the edge effect discussed here is different from the one in If convolving afterwards. In making mock images we want to simulate a real observation. In a real observation the images of the galaxies on the sides of the CCD are first blurred by the atmosphere and instrument, then imaged. So light from the parts of a galaxy which are immediately outside the CCD will affect the parts of the galaxy which are covered by the CCD. Therefore in modeling the observation, we have to convolve an image that is larger than the input image by exactly half of the convolution kernel. We can hence conclude that this correction for the edges is only useful when working on actual observed images (where we don’t have any more data on the edges) and not in modeling.

Because we assumed the overlapping pixels outside the input image have a value of zero.

ofcourse the sum of the kernel pixels still have to be unity in general.

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GNU Astronomy Utilities 0.9 manual, April 2019.