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Fig.2. Row-column assignment for 
surface correspondence establishment 
B. Cj for Surface Matching 
At the stage of surface correspondence establish- 
ment. Cj; is expressed as follows : 
Cup = W1 X F;,M + W 9 X F(;,M,) 
+ Wy X Fy sduym) (13) 
where I, represents the (normalized) area of the zth 
polygon in the input image, M, the area of the yth 
polygon in the object model, dy, the (normalized ) 
distance between the centroids of the ith and the jth 
polygons in the input image, and dum, the distance 
between the centroids of the kth and the Ith polygons 
in the object model. The values of W ;s reflect the im- 
portance of each term. They can be adjusted as long 
as the sum equals to 1. For the symmetric terms, the 
associated weights should be set equal (e.g. , W,;= 
W,). The weight of the relational feature (W3) in 
more important than the other two and is thus set 
with higher value. 
C. Similarity Measure from Surface Matching 
After the states of the Hopfield net for surface 
matching are stabilized, we can count the number of 
active neurons in the network and use it to measure 
the degree of match (or similarity) between the ob- 
ject model and the input image. The procedure con- 
sists of the following four steps. 
Step 1: Initialize both. row. match and column... 
match to be 0. 
Step 2; Count the number of l's in each row. If 
there is nol in a row, skip to the next row and leave 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
row _ match unchanged. If there is only one lina 
row , add 1 to row match. If there are n 1s (421) in 
a row, then add 1/n to row match. Repeat this for 
all the rows. 
Step 3: Do the same ealculation for all the columns 
and updaté column. match. 
Step 4; Pick up the larger one from row _ match 
and column .maích , divide it by the number of rows, 
and take the result as the similarity measure. 
Given an input image and a large number of object 
models (which is usually the case in a multiple-view 
approach), the degree of match between the input 
image and different object models can be derived by 
comparing the input image and each of the object 
models in the final state of the Hopfield net. Ideally, 
there should be at most one active neuron in each row 
or column. However, due to the influence of the first 
term in the right-hand side of (1),'it is possible to 
have more than one candidate in the same row or col- 
umn in the final state of the network. As far as 
matching is concerned, this situation should be con- 
sidered as unfavorable and hence decreases the degree 
of match. This is the reason why 1/» is added to the 
degree of match (row or column) instead of ] when 
there are 4 1s simultaneously existing in the same 
row or column. When there is no 1 in a row (or col- 
umn), it means a surface in the input image (or ob- 
ject model) does not have a corresponding surface in 
the object model (or input image). This does not 
contribute to the degree of match between the input 
image and the object model and thus the degree of 
match is left unchanged. For a model-based 3-D ob- 
ject recognition system using multiple-view approach , 
a set of 2-D object models are in the model database. 
To derive their degrees of match with the input im- 
age in the Hopfield net, we associate the input image 
with row indexes and object models column indexes. 
This arrangement allows us to compare all the object 
models simultaneously with the input image, provid- 
ed that the dimension of the neuron array is large e- 
nough. This also explains why the number of rows is 
used as the denominator in the derivation of similarity 
measure. 
D . Discussion 
The proposed Hopfield net for surface matching 
has a flexible structure and is able to solve the surface 
correspondence problem even if the numbers of poly- 
gons in the input image and the object model are dif- 
ferent. in other words, the two-dimensional neuron 
array may have different numbers of rows and 
columns and an inexact matching [18] can be per- 
formed in this net. Furthermore, based on the out- 
puts of the neurons in the network , a similarity mea- 
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