GUARNIERI, ALBERTO 
  
—1 . 
F'|g(o) |a] » 94 (a 9)5a(a. 9) (17) 
which, for reference convenience, is called angular phase correlation. The above mentioned hermitian symmetry of the 
Fourier transform makes possible estimating angle ¢ only within [-7/2, 7/2]. A simple way to extend the estimate 
beyond this limit is to backrotate the second image by « and qr and to subsequently use translational phase correlation 
in order to disambiguate the correct back rotation [7]. 
2 AN ANGULAR PHASE CORRELATION ALGORITHM 
An algorithm for estimating rotational displacement q can be structured as follows: 
1) compute magnitudes M;(k,, ky), i = 1, 2, of the Fourier transforms in cartesian coordinates of the two 
images fi(x); 
ii) interpolate magnitudes M;(k, k,) on a polar grid in order to ob tain M;(k,, ko); 
iii) compute m;(k,, &) by means of equation (12); 
iv) evaluate functions L(o) and normalized product O(c); 
v) compute angular phase correlation q(0.); 
vi) estimate rotational angle « as the location of the highest peak in q(0,). 
In practice image noise and border effects will not lead to a perfectly impulsive function q(o,q). However extensive 
experimentation has shown that angular phase correlation functions q(o,q) exhibit a distinctive peak in correspondence 
of the true rotational angle ¢. Fig.2 shows functions q(o0,q) relative to the images of Fig.l: the estimates are very 
accurate. This kind of performance is typical of the proposed algorithm. 
3 THE 3D CASE 
As for the estimate of planar rotations, the angular phase correlation technique can be profitable used also for the 
estimate of the 3D motion of free-form surfaces. This procedure is based on the Fourier transform of the 3-D intensity 
function, implicitly described by the registered time-sequences of range data. With respect to memory occupancy, the 
use of a time-sequence of a 3-D intensity function represents a considerable data reduction with respect to a pair of 
time-sequences of 2-D functions. . 
Let /,(x), xe R°, be a 3-D object data and let (x) be a rigidly rotated version of /,(x) (these data can be obtained from 
registered range and intensity data captured at different times). An example of /,(x) and /(x) is depicted in Fig. 3. 
It can be shown that /,(x) and /;(x) relate as: 
Lx) =L(R"x~1) (18) 
where Re SO(3) , te R?, and SO(3) is the group of the 3x3 special orthogonal matrices. According to (18) I(x) is first 
translated by the vector t and then rotated by the matrix R. Denote as 
Lk) - Fli,G)|k]- [[[ 71,696 7" *ax (19) 
with k-[k,, k,, k,]" the 3-D Fourier transform of /,(x), i= 1,2; it is straightforward to prove that, similarly to 2D case, 
the two transforms are related as 
js gia T 
Lk)sZ QU kem (20) 
From (20) one sees that the translation t affetcs only phases and not magnitudes. Magnitudes are related as 
IL, (&)| » | QR ^K) Q1) 
and (21) can be used in order to determine R. Therefore in the frequency domain the estimation of R and t can be 
decoupled and one can estimate first R from (21) and t from (20), according to the following two passes procedure. 
  
322 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000.