Choice of the Kernel

Many phenomena satisfy Laplace’s equation, (E1); the choice of the source defines the meaning of the field.  The kernel, in turn, defines the source of the field.

One characteristic of the kernel follows from a very general postulate about the space in which the field exists.  This is the postulate that the space is everywhere the same.  This postulate means that a point source of the field will have the same effect on its environment, no matter where it is. 

In other words, the kernel depends only on the relative displacement between the location of the point source and the location at which the resulting field is considered.

The kernel is thus a function of only one vector, not two; the most general form is

K (x − x', y − y'), or K (z − z'). (E6)

The specific choice of the kernel defines the specific meaning of the field.  In this work, we choose as our task the detection of linearity, because of the overwhelming importance of lines in human visual experience.

The defining characteristic of a line, recognizable both by artists and by mathematicians, is its symmetry.  If one chooses a point anywhere in a line, and rotates the image 180° or 360° about that point, the line, at least in that immediate neighborhood, is unchanged.  Any other rotation about that point changes the line.

Point sources of Laplace’s equation can be classified into a set of possible forms, and only one of them has this rotational symmetry. 

Although we speak of the field as a quantity defined at each point, it is a “complex” number; that is, a number with two components.  The kernel has two corresponding parts, which would result from the two source configurations 


   -
 +   + 
   -
, and  

 +   - 

 -   +
 .

The four “poles” in each part give the “quadrupole” its name.  These configurations have the same rotational symmetry as a line has.

The kernel must be a solution of Laplace’s equation everywhere except at the source.  Therefore, its only singularity must be at the source, where z = z'.  Therefore, any acceptable kernel is expressible as a linear combination of the terms

1  /  (z − z')n,
(E7)

in which n is a positive integer.  If we express z − z' in terms of r and θ according to the definition

z − z' = r (cos (θ) + i sin (θ)), (E8)

then

1  /  (z − z')n = (cos (n θ) − i sin (n θ))  /  rn.
(E9)

This expression is unchanged when the image is rotated about point z' through an angle of 2 π / n radians, but it is changed by any other rotation.  Thus the term with n = 2, the “quadrupole” term, alone yields the correct rotational symmetry.

The background image on this and other pages is a plot of this kernel, represented by chromaticity.  The hue represents its argument (the angle formed by the real and imaginary parts in a right triangle), and the saturation represents its absolute value.

Back to Source of the Field. Forward to Equation for Linearity.
Up to How Quadrupole Convolution Works.