39.

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(3.41)

where d_XY denotes the distance between two cities X and Y, and A, B, C and D are the weighting factors determined by the user.

A complete energy function is constructed from Equations (3.38) to (3.41). The weights w_ij then can be derived from Equation (3.37) and become:

(3.42)

where δ_ab is defined as:

(3.43)

This example of energy function coding can be extended to solve the problem of automatic ground control point (GCP) matching in the process of the geometric correction of images (Chapter 1). Suppose there are m GCPs (denoted by a₁, a₂,…, a_m) in the reference image, and n (normally m<n) candidates (denoted by b₁, b₂,…, b_n) in the distorted image, then a Hopfield network similar to that shown in Figure 3.14 can be constructed, noting that the rows now represent the GCPs a_i (1≤i≤m) and the columns represent candidates b_j (1≤j≤n). Accordingly, three constraints similar to those used in the travelling salesman problem can be set up:

1 each GCP a_i can only be matched once; that requires the sum of each rows must be exactly 1,

2 only one GCP a_i can be matched at a given candidate b_j, which indicates that the sum of each column has to be no more than 1, and

3 all the GCP a_i must be matched, which means the total number of active units must be equal to the total number of GCPs (i.e. m).

Equations (3.38) to (3.40) can be adapted to meet these requirements. Regarding the objective function, one simple way for matching GCPs may be according to the absolute difference between the GCPs a_i and candidate b_j. Once the objective function has been defined, the energy function is fully specified. Equation (3.37) can then be used to initialise the Hopfield network weights and to search for the solution.

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