Introduction

The problem states that the simply connected binary image can be converted from one to another by interchanging between pixels, at the main time, the topology and the number of 1's of these images have no change.

Several difinition need to declear:

Topology:

The number of components of 1's and 0's. By convention, we imply 4-connected when we refer components of 1's, and 8-connected, when we refer components of 0's.

Let the red pixel represents 1 and the white pixel represents 0, which is also the background pixel. We keep this convention for the rest graphs.

one 1-Component
one 0-Component

One 1-Component
Two 0-Component

Obviously, a simply connected binary image has one 1-component and one 0-component, which is the topology this tutorial addressed on.

Interchangeble Pair:

A pair of pixels have opposite values and interchange them doesn't change the topology of the image.

The following graph shows one pair of interchangeble pair. As we can see, there are six interchangeble pairs in this graph.

Some binary images may have no interchangeble pair.

E.g.

Some simply connected 1 components may have no interchangeble pair.

E.g. the center pixel of the following graph has no interchangeble pair.

A simply connected binary image always has at least two interchangeble pair.

Please do not confused simply connected binary image with simply conected 1-component. The former is globally simple and the latter is locally simple.

IP-equiverlent:

Let I, J be two digital images.
If I has an interchangeble pair of pixels p, q, and J is obtained from I by interchanging p, q. I, J will be called directly IP_equivelent.
I, J are called IP-equivelent if I = I₀, I₁ ... I_n = J such that I_k is directly IP-equilevent.

If I and J are IP-equivelent, they must have the same topology and the same number of 1's.

The converse is not always true.

E.g.

The last statement is proved by the following four propositions. At the end, we can see that any simply connected binary image can always be transformed into a vertical bar by swapping the interchangeble pair. Therefore, these images can always be converted from one to another if they have the same number of 1's.