GUARNIERI, ALBERTO 
It can be easily proved [8] that functions Mi(kp, ke) are related as images f(x), ie. 
M,(k,,k,) =M,(k,,k, — 0) (11) 
However there is a difference between relationships (7) and (11) concerning the periodicity of the functions: functions 
fi(p,0) have 2 as periodicity whereas functions Mi(kp, ke) have m as periodicity, owing to the hermitian syrnmetry of the 
Fourier transform. 
Let mi(k,, 01), i= 1, 2, be two auxiliary functions, along a generic circumference of radius ky, defined as 
m, (ky) = [M,(k, ky )e "gk, (12) 
-— 
Notice that functions m; K», 0t) are the partial angular Fourier transforms of Mi(kp» Ke) , i.e. mi(k,, o) = F[M(. , kg) | o]. 
From the translational theorem of the Fouriei transform we obtain 
m, (k,,00) — m, (k, ,or)e "?7* (13) 
In order to be independent from the specific circumference , one can build two further auxiliary functions 
R 
u,(@) = [km (k,,00dk, i=12 (14) 
0 
where R is a fixed radius. 
  
  
  
  
  
  
  
  
  
  
  
Fig. 2. Angular phase correlation q(a,®) relative to functions pairs a)-b) and a)-c) of Fig.1. ;the true rotational angles 
and their estimates are respectively (from left to right): ¢ =1° and ¢' =1°, 9 =20° and ¢' =20° . 
Functions (a) are reciprocally related similarly to functions m;(k,, c) In fact 
H,(0)=[k,m,(k, 0)dk, = [k,m,(k, 00) e dk, = ome [my (k, 00d, = 4, (a)e > (15) 
0 0 0 
Rotational angle ¢ (in the exponent of equation (1 5)) can be computed from normalized product 
Ox) = u, ()u, (x) Lg mee 
= (16) 
[44 (0)44, (0)| 
Indeed inverse Fourier transform of Q(a) gives 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 321