(0 images. The 
'ased matching, 
rmined through 
2 more accurate 
sntation then is 
hape similarity 
features. All 
ed into a cost 
Hopfield-Tank 
imal matching. 
en images are 
daptive to the 
oises as well as 
scale. 
the / 
al 
of | 
features | 
rier 
feature | 
| 
Task 1: 
Determining 
approximation 
Task li: 
Point 
matching 
os ctr 
system 
rest points are 
nitial conjugate 
locations on the right image can be obtained based on 
the approximate orientation. More accurate conjugate 
locations then can be matched by using template 
matching techniques. This procedure can be speeded 
up by using coarse-to-fine matching when images are 
stored in multi-resolution format. 
2. MATCHING FEATURES 
Boundary features can be modeled with Elliptic Fourier 
descriptors [Lin and Hwang, 1987] [Zahn and Roskies, 
1972]. The best fit of shapes between features can be: 
obtained by matching their Fourier descriptors [Tseng 
and Schenk, 1992]. When the best fit of features is 
obtained, the shape difference (the mean square 
difference between two fitted features) can be 
calculated as well. 
2.1 Fourier Descriptors 
A two-dimensional closed line (Fig. 2a) can be 
expressed by two parametric functions as Eq. (1). 
pti nt. Deng TOR M rn alu e 
In which, the / is defined as a time period from 0 to 2m. 
It means tracing a closed line one cycle. The functions 
become periodic if a line is traced repeatedly. Fig. 2b 
shows the periodic functions of x(t) and y(t). 
y 
starting point 
  
X 
Figure 2a: A closed line feature. 
  
x(t) | y(t) 
| 
| A p, | A . 
0 + E00 = o 
Figure 2b: The periodic forms of x(t) and y(t). 
The periodic functions can be transformed into the 
frequency domain and expressed as Fourier series: 
x(f)| |20| ic 5, | cos 2 
PAM n 2) 
where 
] ga : a ] (27 : 
ag = 3; Jor Co — 5 Od 
1 027 . 1 027 ; : 
d, =— J x(coskt dt» h = Jasin kt dt^ 
- 1 [»tcosu dt» d, Lye in kt dt 
ur p Ds 
In which a,, b,, c,, andd, are Fourier descriptors of 
the Ath harmonic. The descriptors of Oth harmonic, a, 
and c,, represent the centroid of the line feature. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
  
  
  
   
  
   
  
    
   
    
    
   
   
    
   
   
  
  
  
    
        
   
   
  
  
  
     
      
2.2 Shape similarity 
It is well known that a linear transformation in the 
spatial domain can be easily modeled in the frequency 
domain [Dougherty and Giardina, 1988]. As a 
centroid-based similarity transformation in the spatial 
domain is expressed as follows: 
x cosh —smBllx—x [| |x | ] Ax (3) 
7 = S 3 + + 
y sn8. cos8 || y- Y, ] Ly.] LA 
Where (35; y,) are the coordinates of the centroid; 
Ax andAy represent the translation of the centroid; and 
S and 6 are scaling and rotation factors. The 
translation of the centroid can be modeled by using the 
0th harmonic of Fourier descriptors as Eq. (4), and the 
transformation of scaling, rotation and phase shift 
(shift of starting point), At, can also be modeled in the 
frequency domain as Eq. (5). 
ath m a, « Ax (4) 
et €, Ay 
a, b, S cost —sinb || a, b, || coskAt —smkAt (5) 
ed — |sinO cos c, d, || snkAt coskAt 
When two features are expressed by using Fourier 
descriptors, one can use Eq. (4) to calculate the location 
difference of their centroids. Computing the 
differences of scaling, rotation and phase shift is more 
complicated. First, we need to reorganize Eq. (5) to be: 
a, Var a b, -—e, -d,||cos0coskAr 
bi | vi |_o|B eu de ©, || cosOsin kAr (6) 
e] n e, d, a, 5b, ||smOcoskA 
di} | va: d, —c, b, —a, | | sin Osm KA# 
In which, v,,V,,V,. andv, are the residuals of 
k k Cr? k 
descriptors after transformation. They represent the 
shape differences between features. Based on the 
principle of least-squares solution, the transformation 
to obtain the best fit between features can be solved by 
minimizing the sum of the squares of residual. 
A unique expression of the shape difference can be 
formed by the residuals of matching Fourier descriptors. 
It is the mean square difference (MSD), or say the 
average discrepancy, between fitted features. MSD can 
be computed as: 
MSD = AG + +2 +42) (7) 
k=l 
Obviously, MSD can be used as an evaluation of the 
shape similarity between features. Therefore, it is used 
as the first criterion to find conjugate features in a pair 
of stereo images. 
2.3 Orientation consistency 
Based alone on shape similarity, it tends to fail in 
recognizing conjugate features when there are some 
similar features in an image. Under this circumstance,