7A-3-2 
Each dot in the matrix represents a neuron that stands for 
similarity between one input feature and one feature of the 
candidate model. Its state (1 meaning absolutely similar and 0 
meaning totally different,) can be determined when the 
minimization of the energy function is reached. 
Energy function 
The following is a detailed discussion on the single layer 
Hopfield neural network. Let Q denote similarity/disparity 
between a model feature pair Qj) and an input image feature 
pair (&,/)• It is then represented as: 
C ~C' +C 1 +C 2 • (2) 
We use a top-down strategy to achieve object recognition. The 
problem is treated as an optimization problem, where the 
correct answer is given when a global minimized energy state 
is reached. Let C\k and C 2 ikji be unary and binary similarity 
measure respectively. The energy function is 
E =-¿XXXXw*+flX<> - X 1 '. > : + 
I k i I i k 
C XXX 1 '. xV > * D X<' - X v .) ! +*xxx*i xV i‘■ 
Ì k l*k k i k i j*i 
(1) 
The neuron state, V ik , converges to 1.0 if the model feature i 
matches the input image feature k perfectly, otherwise, it is 
equal or close to 0. Thus, the first term measures similarity 
between the model and image features. The second term 
implies that the final states of neurons in the 
same row 
XX2X xV 'i 
add up to 1, and the third term 
confirms that there is at most one neuron that 
has a value greater than 0 in each row. This means that only 
one input image feature matches with each model feature. The 
forth term Y(i-^Tv;..) 2 * m Pl ies that the final states of neurons 
k /' 
in the same column add up to 1, and the fifth term 
confirms that there is at most one neuron that 
XX» 
k i jt> 
ik*Vjk 
has a value greater than 0 in each column. That means that each 
input image feature matches with only one model feature. 
Combining the second term Vo-Yl/ ) 2 with third term 
/' k 
XXXK» xV 5 
i k tek 
gives a solution that forces each model feature to 
match only one input image feature. Similarly, combining the 
forth term y with the fifth term yyyy x v; §’ ves 
k i k i jm 
a solution that guarantees each input image feature will match 
only one model feature. The determination of coefficients A, B, 
C, D and E depends on how strictly the unique matching 
conditions should be implemented. Different values of in 
Equation (1) apply to various cases of our tasks. For 
monomorphism, coefficients B, C, D and E are assigned with 
high values based on the assumption that one model feature 
will uniquely match one input feature. The final solution yields 
a one-to-one mapping. In the case of homomorphism, 
coefficients B and C are assigned with low values (even zero) 
based on the assumption that one model feature will match 
several image input features. 
where 
C L (3) 
n=l 
md cf„=fy./„ J (v;y*»)- <4) 
/1=1 
In the above equations C] k and C% represent unary and binary 
similarity respectively. C] k encodes compatibility between 
model feature i and input feature k, and Cf kjl encodes 
compatibility between the correspondence of the model feature 
pair (ij) and that of the input feature pair (k,l)■ f is a 
similarity-measuring function and weighted by w that meets the 
condition 
V, n 2 
2^\v n + 
«=1 71=1 
YW 2 T1 = 1 • 
(5) 
Measuring functions 
(a) Sign function 
Figure 2. 
(b) Linear function (c) Sigmoid function 
Three types of measuring functions 
a) Sign function 
Sign function is a simple function with one parameter 0. 
otherwise 
(6) 
x and y are similarity measures (such as length of a line) of an 
input feature and model feature, respectively. The parameter 0 
is sometimes difficult to determine in case the measure selected 
is sensitive. A small change of 9 may alter the recognition 
result. A training is usually necessary. 
b) Linear function 
A linear function 
/ = 
1, 
2j.v - )| - (a + b) I (a - b), 
-1, 
ij\x->\ <a 
if a<\x-)\<b 
tf\ x ~y\ >b 
(7)