7A-3-3 
has more practical use than sign function in some cases. In 
addition to its simplicity, it provides a smoother function in 
comparison to sign function^ 
c) Sigmoid function 
A sigmoid function 
f(x,y,k,t)=-lanh(k(\x-)\-t) (8) 
satisfies our decision function requirements appropriately. The 
shape of the curve near the threshold is approximately that of 
the linear function. The curve is generally smooth. Depending 
on the parameter k, the first derivative at t could also be 
smooth. 
Output function 
For neuron i, if its charge is Uj that is computed in the energy 
minimization, its neuron state output is represented as 
v ' = g(u,) = 
1 
\+e 
-HI, IT 
(9) 
T is the “temperature” (an annealing term) that determines the 
speed and quality of the final solution. A very large value of T 
will cause neuron values to be 1, while a very small a value 
will drive the network to a local minimum state, or a slow 
convergence. An annealing process keeps the value of T large 
at the beginning and reduces the T value as iteration 
progresses. This is important for achieving a global minimum 
and a fast convergence. 
Initialization 
The initial values of neuron states can be chosen randomly as 
described in Lin et al. (1991). The network may converge to a 
local minimum state. As stated above, an annealing process 
may overcome this problem. However, C jk may be calculated 
and used as a byproduct to set the initial neuron states as 
c l' Yf*’ > w andC) k > w 
jeS, jeS { 
K=\ C ifjc),>wandc‘ > >() 
0, ifCl< 0 
(10) 
r l N ' 
where £. is I Cj k > 0] and w= 
Combining matched features 
After iterations using homomorphism, each neuron reached its 
final state V jk ■ Those final states close to 1 yield matches 
between corresponding input image features with model 
features. However, there is still a need to put the matched 
features to form object(s). The following procedure combines 
the features under the assumption that there are N features 
forming an object. 
a) Establish N sets of ={fcll^»l}, i=l,...,/V. Each set 
contains all the input features that matched the 
corresponding model features. 
b) Establish an empty set Q. 
c) Set i to be 1. 
d) For each k^S i we get m ik =V ik ,keS i if Q is empty, 
/ otherwise m v Find the feature k„ that 
"Hk ~ ZJ ^¡kjl ^ n 
UJ*Q 
satisfies \/k p e S t , m iK - m ikp > add (i,kj t0 Q 
e) If i=N, one object is recognized and detected, go back to 
step a); otherwise, i =/+1 and go back to step b). 
3. MULTILAYER HOPFIELD NEURAL NETWORK 
The above object recognition is based on separate single layer 
neural networks. However, the interrelationship between line 
pattern recognition and region pattern recognition adds more 
constraints and thus achieves better results. Our objective is to 
utilize a two-layer network for truck recognition from aerial 
images (Figure 4). The interrelationship is based on the 
following realities: 
The top of a truck shown in the image is nearly rectangular in 
shape, while its shadow has a more complex shape due to the 
sun light direction and the truck head shape. 
Figure 3. Two-layer (line pattern and region pattern) 
Hopfield neural network 
The line pattern recognition could yield a match for the top, but 
may fail in shadow verification, which is a very important clue; 
and the region pattern recognition considers the top and 
shadow at the same time, but it does not take full advantages of 
line patterns. 
Connections among neurons in each single layer are fully 
dependent on geometric and photogrammetric constraints and 
are fixed before the initial iteration. During iterations the 
interconnections between the two layers vary. Let L, denote 
layer 1, which is a line pattern layer, and L, denote layer 2, 
which is a region pattern layer. We thus have an energy 
function