Source of the Field
If Laplace’s equation is satisfied for all (x, y), the field is constant. It can yield useful information only if points exist where it is not obeyed. Correspondingly, an analytic function is constant, unless it has a singularity for at least one value of z. Each point where Laplace’s equation is dissatisfied, or the analytic function used as its solution has a singularity, is a source of the field.
Say there is just one point source of the field, that it has unit strength, and that it is at point (x', y'). We denote the resulting field at point (x, y)
| K ((x, y), (x', y')), or, equivalently, K (z, z'). | (E3) |
We denote the image I (x', y'), or I (z'). In the accompanying examples, we have defined I to be 1 in a black area, 0 in a white area. This is just to represent black marks on white paper, but the opposite convention could be equally meaningful in other contexts. As we see in the Vortex example, the possibility of opposite conventions is a weakness of quadrupole convolution.
With this definition, the addition of the contribution from every point in the image is given by
|
(E4) | ||||||
or
|
(E5) | ||||||
This integral is the continuous equivalent of multiplication by a square matrix. Such an integral is called a “convolution”. The factor K (z, z') is called the “kernel” of the convolution.
If Laplace’s equation is satisfied at every point, the solution is the same everywhere. It then yields no useful information. This corresponds to the fact that, if a function has no singularity (discontinuity) anywhere, it is constant. Any point where Laplace’s equation is violated is a source of the field.
We postulate that each point in the image is a point source of the field. The intensity of the source per unit area, at each point, is the value of the image at that point. Because the field is linear, the total field is the sum of the fields that result from the infinite number of point sources. This type of sum is known as a “convolution”.
That sum contains a term that specifies the field that would be created by a point source of unit strength. That unit component field is called the “kernel”.