About the Contents: In places, this explanation bifurcates into streams for readers who are and readers who are not comfortable with equations.  These passages are marked as follows:  No Equations Equation Area Thus, if you are not familiar with the sorts of equations involved, you will still get the gist of the explanation. A bug in Netscape Navigator disarranges some of the equations and page formats.  Internet Explorer is recommended for them. If an equation looks scrambled even with Internet Explorer, widen its window.

Contents

1. The Problem
   Vision is based on massive parallel processing, far beyond our ability to reproduce directly.

2. Field Theory in Vision
   The fields of physics use infinite parallel processing, and there are good techniques to understand them.

3. Properties of a Vision Field
   Credible postulates about what sort of field might be relevant to vision give us guidelines from which to infer a field equation.

4. Laplace’s Equation
   Those postulates yield Laplace’s equation, which is easy to solve in two dimensions.

5. Source of the Field
   Laplace’s equation needs a source term to have an interesting solution.

6. Choice of the Kernel
   The kernel gives it the necessary source, driven by the image.

7. Equation for Linearity
   This yields a tractable expression for the linearity.

8. Emergent Properties of Linearity
   The resulting formula has remarkable properties, far beyond those that were built in during the derivation.

9. Methods to Evaluate Quadrupole Convolution
   Various methods of evaluating linearity give the answers we have seen, and tell us why.