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6.3.1.2 Edges in the spatial domain ¶

In purely ‘linear’ spatial filtering (convolution), there are problems with 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, the sum of the weights we use on the actual image pixels is not unity¹⁶⁶. 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 Brightness, Flux, Magnitude and Surface brightness) will not be equal, some of the flux is going to be ‘eaten’ by the edges.

If you run $ make check on the source files of Gnuastro, you can see 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 process¹⁶⁷. 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 do not have any more data on the edges) and not in modeling.