Emergent Properties of Linearity. 

1. L (x, y) is dimensionless.

   This means:

   • If the image is magnified or diminished by any non-zero amount, the value of L (x, y) at every point in the image is not changed. 

   • If L (x, y) dominates the perception of shape, then the perception of shape is independent of the perception of size. 

   • If L (x, y) dominates the perception of shape, then an object is perceived with the same shape, no matter what is its distance from the observer.

2. Anywhere in an ideal line, a lone black stripe of infinite length but finite width, the linearity has a magnitude of π and an argument that gives the orientation per equation (E9)

   This means that a line yields a clear, readily detected indication of its location and direction.

3. As a line, initially ideal, is degraded by

   • becoming stubby,
   • taking on an irregular outline, or
   • breaking up into isolated pieces,
   the orientation indication in it remains the same, and the magnitude decreases slowly.

   This means that an indistinct line yields as clear an indication as a distinct one, with a clear measure of its degree of distinctness.  A pointillist image is perceived as clearly as a solid one.

4. Anywhere outside an ideal line, the linearity is zero.

   Since the linearities from all points of an image add, this property is essential.  It means that the perception of a line is not affected by other lines.

5. As a line, initially ideal, is degraded as above, the contribution outside it starts to grow.

   This characteristic is as important as the previous one.  If a line is not very distinct, then maybe it is a component of some other line at a different orientation.  The line has to be rather indistinct before its contribution to the linearity outside it becomes important.

6. If a constant value is added to the image, it makes no contribution to the linearity.

   This means that the perception of an image is not affected by meaningless changes of gain and bias in the channel that conveys the image.

These characteristics explain the remarkable succeses that linearity analysis by quadrupole convolution shows in the examples.