prrespondence 
| scheme is a 
| conventional 
ulation of the 
2, the surface 
irough a Hop- 
stage , the ob- 
natching. The 
ly singled out 
image and the 
)-fine strategy 
in 3-D object 
ks [8]. [9]. 
ject matching 
vides a more 
problem and a 
mentation. 
FOR 
er of neurons, 
unit to every 
ne weights on 
symmetrical. 
2 Hopfield net 
function E. 
n, an optimal 
ly reflected in 
ork. The ap- 
iltifarious. In 
graph match- 
formulated as 
gy function is 
is then solved 
1, a Hopfield 
on process to 
In [12]. the 
as an inexact 
formulated in 
a [13], ‘thé 
mputer vision 
es are the hy- 
hts. The net- 
work is then employed to select the optimal subset of 
hypotheses which satisfy the given constraints. 
In this paper, the Hopfield net for image matching 
is in the form of a two-dimensional array. The rows 
of the array represent the features of an input image; 
and the columns represent the features of an object 
model. The output of a neuron reflects the degree of 
similarity between two nodes, one from the image 
and the other from the object model. The matching 
process can be characterized as minimizing the follow- 
ing energy function [10]: 
E=— 212221 20 x 
+ E Da) + 3 - E (1) 
where Vj; is an output variable which converges to 
1. 0 if the ith node in the input image matches the kth 
node in the object model; otherwise, it converges to 
0. The first term in (1) is a compatibility constraint. 
The second and third terms are included for enforcing 
the uniqueness constraint so that each node in the ob- 
ject model eventually matches only one node in the 
input image and the summation of the outputs of the 
neurons in each row or column is equal to 1. The ma- 
jor component of the compatibility measure Cy; (or 
strength of interconnection) between a neuron in row 
i column k and a neuron in row j column / is ex- 
pressed in terms of a function F defined as follows: 
Fam = | 1; lead S e 
—], otherwise 
where 6 is a threshold value and z and y are features 
pertaining to row and column nodes , respectively. In 
this paper, two different sets of features and relations 
will be used for surface and vertex matching, and 
they will be detailedly described in Sections ll and 
IV , respectively. In general, Cy; can be expressed as 
Cap m S M, X Fn og (3) 
where x, is the ath feature of the node in row : col- 
umn k, y,is the nth feature of the node in row j col- 
ume /, and the summation of the weighting functions 
W,s is equal to I(i. e. , S Wl). Equation (1) 
can be simplified and fit into the energy function 
form of a Hopfield network [8] as follows: 
E=— i2 DES ZZ 
(4) 
where I,=2,and 
T ay = C ay "T ds; = i8u (5) 
where 8;;=1,if i— j, and ó;;7 0 otherwise. 
Matching can be considered as a constraint satis- 
faction process. The global information extracted 
from the image provides positive or negative supports 
for local feature matching. According to Hopfield and 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
Tank [10], the strength of the connection between 
each neuron pair can be derived from the energy 
function. Based on these connections, the equation of 
motion for state 44 of a neuron at position (?,k) can 
be derived as follows : 
dug/dt = SD) Cas à — 2.7 
i 253 
=D Va val s (6) 
where 
Va=9g wy) = [1+ exp(— 2uz/ug)]™* (M) 
Since the sum of V 4 on all the neurons at initializa- 
tion is constrained to be equal to the number of the fi- 
nal desired output, i.e. , 
2, » = N (8) 
where N is either the number of rows or the number 
of columns of the array depenting on which one is 
smaller, we can derive the initial condition for uj 
from (7) and (8) as follows: 
pq N (9) 
In order to prevent the system from being trapped 
in an unstable equilibrium in which the voltage of 
each neuron is equal, a certain amount of noise must 
be added to this initial value. We can rewrite the ini- 
tial conditions as follows : 
ub md) (10) 
and 
V3 = g (ui) (11) 
where ó is a random number uniformly distributed 
between — 0. lug, and +0. Tug. 
The algorithm for matching , based on the continu- 
ous Hopfield network model, is summarized as fol- 
lows. 
Algorithm 
Input: A set of neurons arranged in a two-dimen- 
sional array with initial values V3, where 0<Ci<< 
row A _maz — 1,0 k<column_maz — 1, and row _ 
maz and column_maz are the numbers of rows and 
columns in the array, respectively. 
Output: A set of stabilized neurons with output 
values V4, where 0. 0<CV <1 for 0<CeCrow _maz 
and 0<<t<column_maz. 
Method : 
1)Set the initial conditions using (10) and (11). 
2)Set index—]1 and limit —. 
3)Randomly pick up a node (i,k). 
4)U pdate the value of uy. 
5)Calculate the new output of neuron (t,k) as fol- 
lows : 
Va=g gy) 
6)Increment index by 1. 
If index <n, then go to (3), else stop and out- 
put the final values of all the neurons based on the 
1011 
TER —