C++ Neural Networks and Fuzzy Logic: Preface

Another Example for the Hopfield Network

You will see in Chapter 12 an application of Kohonen’s feature map for pattern recognition. Here we give an

example of pattern association using a Hopfield network. The patterns are some characters. A pattern

representing a character becomes an input to a Hopfield network through a bipolar vector. This bipolar vector

is generated from the pixel (picture element) grid for the character, with an assignment of a 1 to a black pixel

and a −1 to a pixel that is white. A grid size such as 5x7 or higher is usually employed in these approaches.

The number of pixels involved will then be 35 or more, which determines the dimension of a bipolar vector

for the character pattern.

We will use, for simplicity, a 3x3 grid for character patterns in our example. This means the Hopfield network

has 9 neurons in the only layer in the network. Again for simplicity, we use two exemplar patterns, or

reference patterns, which are given in Figure 1.5. Consider the pattern on the left as a representation of the

character “plus”, +, and the one on the right that of “minus”, − .

Figure 1.5

  The “plus” pattern and “minus” pattern.

The bipolar vectors that represent the characters in the figure, reading the character pixel patterns row by row,

left to right, and top to bottom, with a 1 for black and −1 for white pixels, are C+ = (−1, 1, −1, 1, 1, 1, −1, 1,

−1), and C− = (−1, −1, −1, 1, 1, 1, −1, −1, −1). The weight matrix W is:

      0  0  2 −2 −2 −2  2  0  2

      0  0  0  0  0  0  0  2  0

C++ Neural Networks and Fuzzy Logic:Preface

Binary and Bipolar Inputs

27

---

      2  0  0 −2 −2 −2  2  0  2

      2  0 −2  0  2  2 −2  0 −2

W=    2  0 −2  2  0  2 −2  0 −2

      2  0 −2  2  2  0 −2  0 −2

      2  0  2 −2 −2 −2  0  0  2

      0  2  0  0  0  0  0  0  0

      2  0  2 −2 −2 −2  2  0  0

The activations with input C+ are given by the vector (−12, 2, −12, 12, 12, 12, −12, 2, −12). With input C−,

the activations vector is (−12, −2, −12, 12, 12, 12, −12, −2, −12).

When this Hopfield network uses the threshold function

             1  if x >= 0

f(x) = 

{

            −1  if x [le] 0

the corresponding outputs will be C+ and C−, respectively, showing the stable recall of the exemplar vectors,

and establishing an autoassociation for them. When the output vectors are used to construct the corresponding

characters, you get the original character patterns.

Let us now input the character pattern in Figure 1.6.