GUARNIERI, ALBERTO 
  
COSQ — Sin 
Rp) C O89. 7 m0 71 essOQ) 5) 
sinp  cosQ 
and SO(2) is the group of 2 x 2 special orthogonal matrices. Some examples of images pairs f(x) and f(x) are shown in 
Fig. 1. (“Madonna con bambino" of Donatello, St Anthony Basilic in Padua). 
aj 
  
    
50 100 150 200 250 
Xx 
Fig. 1.a) Image f(x); b) , c), d) images f(x) , versions of fi(x) rotated respectively by q-1? , (-20^, ¢p=40°. 
  
Equation (4) can be conveniently rewritten in polar coordinates 
x = pcos60 pao 
(6) 
y-psinü 0<#<27 
as 
f(p,9)7 fi(p,0 — 9) () 
where 
f.(p,0)7 f.(p cos?, p sin?) r2 (8) 
Let Fi(k) = F[fi(x) | k] be the 2-D cartesian Fourier transform of f, (x), i ^ 1, 2; denote polar coordinates in frequency 
domain as 
k, =k, cosk, k, 20 
k,=k, sink,  O<k,<27 
and define the magnitudes of F(k) in this reference system as 
M,(k,.k,) =|F,(k, cosk,, k sink, (10) 
  
320 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000.