following rule: 
y.- 
n. if Va >01 (12) 
0, otherw ise 
where 601 is a threshold value. 
In the actual implementation, N is replaced by a 
value N , which is greater than N. This is to adjust 
the neutral positions of neurons. The coefficient 7 is 1 
and ug is 0. 002. The instability problem of the Hop- 
field neural network has been extensively studied 
[17]. It is well known that the optimal solution is 
not always found. In the algorithm, two termination 
strategies are used to handle different situations. The 
freeze strategy is adopted whenever the outputs of all 
the neurons in the network are convergent. When a 
small number of neurons are not convergent after a 
long period of time, the time-out strategy is adopted 
to force the system to stop. 
Il. HOPFIELD NETWORKS FOR SURFACE 
CORRESPONDENCE ESTABLISHMENT 
In this section, a Hopfield net is designed to estab- 
lish the surface correspondences between an unknown 
object and an object model in the database. It is as- 
sumed that the unknown object is in the form of line. 
drawings which are obtained by segmenting the origi- 
nal image and each object model is a 2-D projection of 
a 3-D object whose identity and pose are to be deter- 
mined. The features for surface matching are firstly 
described. This is followed by the introduction of 
row-column assignment. Then, the strength of in- 
terconnection Cy; is defined. A method for quantita- 
tively evaluating the degree of match between the in- 
put image and the object model is presented. Finally, 
the characteristics of the networks are discussed. 
Since regions (in 2-D) are the projections of surfaces 
(in 3-D), we use the two terms interchangeably in 
this paper. 
A. Feature Selection and Row-Column Assignments 
Before establishing surface correspondence, each 
object model or image has to be preprocessed. We 
first label all the regions in the image or model in or- 
der from left to right and top to bottom. This label- 
ing scheme provides the basis for subsequent row-col- 
umn assignment process. An example of labeled im- 
age is shown in Fig. 1(b). During the labeling pro- 
cess for each region, the area is calculated and the 
boundary traced for locating high curvature (or cor- 
ner) points [15]. The original image is then convert- 
ed to a set of polygons with vertices numbered in a 
certain order. The centroid of each polygon is then 
computed. For each polygon, two feature are ex- 
tracted for surface matching. One is a local feature, 
which is the area of the polygon. The other is a rela- 
tional feature, which is defined as a set of distances 
1012 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
originated from its centroid to all the centroids in oth- 
er polygons of the image. An example demonstrating 
both features of a polygon is shown in Fig. 1. Since 
these two features are not scale invariant, a normal- 
ization process is performed to compensate this ef- 
fect. This process starts with selecting the longest 
distance from the set of intercentroid distances in the 
input image. Then, based on the ratio between the 
longest, intercentroid distance in the input image and 
the longest intercentroid distance in the object model, 
all the distances in the input image are divided by this 
ratio for normalization. The area of each polygon in 
the input image is normalized by dividing it by the 
square of this ratio. 
  
  
  
  
  
  
  
(b) 
Fig. 1 (a)A aerial image of a building ; (b)Its poly- 
gons labeled and the feature set of polygon 2, local 
feature: area of polygon 2, relational features: 
{d20 »dz1 » d23 » 24 » d25 > d 26 » d27 } - 
The labels of the polygons in an input image or an 
object model is derived during the labeling process. 
In order to perform matching, each polygon in the 
input image is assigned a row index and each polygon 
in the object model is assigned a column index. An 
example is shown in Fig. 2. 
    
  
  
  
Fig. 2 
Sui 
B. Cj for 
At the 
ment. Cui 
Cog = 
where I, r 
polygon in 
polygon in 
distance be 
polygons i 
between tl 
in the obje 
portance o 
as the sum 
associated 
W 2). The 
more impx 
with highe 
C. Simdar 
After t 
matching 
active neu 
the degree 
ject model 
sists of th: 
Step 1 
maích to 1 
Step 2: 
there is n«