B3b. Beijing 2008 
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B3b. Beijing 2008 
ition, a common 
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of points (even 
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it on the contour, 
arting point of a 
ents by a rotation 
where nO is the shift in starting point (number of points), k is 
the harmonic coefficient, N is the total number of points on the 
contour. In this formulation, the phase shift is different for every 
k harmonic coefficient. This observation presents several 
problems, the most major being that shifting the starting point 
of an object is directly proportional to the harmonic number. In 
order to eliminate this effect, we use the Power Spectrum (PS) 
of the Fourier transform. Since the change of the starting point 
corresponds to a rotation matrix, 
quality of the projective relation depicted in equation (13). 
Particularly, the quality of an equation system can be computed 
by the condition number of A or the empirical estimate of the 
residual variance. 
Among these two measures, the condition number evaluates the 
residual error calculated from the least squares solution and the 
values of the PS. In order to calculate the condition number, let 
e be the error in y. Then the error in the solution A 'y * s A 'e. 
R= 
^COS 6 
y sin 6 
-sin 6 N 
COS(9 y 
(10) 
Multiplying this matrix with its transpose would result in an 
identity matrix (since a rotation matrix is orthogonal). For 
complex numbers, this means multiplying the number with its 
complex conjugate: 
PS = £ ||f(x)e H “ (n)t || 2 =FF (11) 
n 
where F is the Fourier transform of f(x), and F is its complex 
conjugate. The Power Spectrum of an object is the sum of the 
square of the magnitude of the x, y Fourier descriptors. In 
equation (8), if Gkx and Gky is the k th complex Fourier 
descriptor for x and y, and GOkx and GOky is the kth complex 
Fourier descriptor for xO and yO, then the power spectrum is 
(||Gkx||) 2 + (||Gky||) 2 and (||G’kx||)2 + (||G’ky||) 2 for the k th 
harmonic. This value is constant and independent of rotation or 
starting point, for any k. The result of this operation is a 
function where the only variable is the magnitude of each 
harmonic. 
3.1 Matching with the Power Spectrum 
We hypothesize that projectivity between two contours is 
sufficient for the existence of projectivity between their 
respective PSs. Hence, checking for the existence of projectivity 
between PSs suggests that two contours are projectively 
equivalent. Introducing the power spectrum into the harmonics 
in equation (7) and developing these equations establish a set of 
equations as in the following: 
||G0kx|| 2 + ||G0ky|| 2 = (hi) 2 ||(Gkx)|| 2 + 
(h2) 2 ||(Gkx)|| 2 + 2||(Gkx)|| ||(Gky)|| hlh2 + 
(h3) 2 ||(Gky)|| 2 + (h4) 2 ||(Gky)|| 2 + 
2||(Gky)|| ||(Gkx)||h3h4.(12) 
Rearranging this equation and putting the unknowns into a 
vector form results in: 
(||G’kx|| 2 + ||G’ky|| 2 ) = 
( ||Gkx|| 2 IIGkyll 2 2||Gkx|| ||Gky|| ) 
( hlhl + h2h2 ^ 
h3h3 + h4h4 
hlh2 + h3h4 
(13) 
This is in the well known form of y = Ax, where y and A are the 
PS coefficients of the two contours, and x in this case is the 
unknown homography coefficients between the PSs. Since there 
are more equations than unknowns, we can use a least squares 
solution to calculate the homography coefficients. The matching 
between two contours can be expressed by evaluating the 
The ratio of the relative error in the solution to the relative error 
in y is 
k = (IIA 'ell IIA-’yll) / (Hell llyll)- O 4 ) 
A lower condition number suggests a better parameter 
estimation; hence a better matching between the contours. The 
variance, on the other hand, is the measure of statistical 
dispersion of the residual. In other words, it is a measure of how 
spread out a distribution is and how much variability there is in 
the distribution: 
Var(X) = E((X - p) 2 ), (15) 
where p is the average of the variables contained in X. In this 
paper, in order to compute the matching between two contours, 
we use a combination of both the condition number and the 
variance combination by weighting the variance with the 
inverse condition number: 
MatchingScore =1 /( k Var(X)). (16) 
(a) 
(b) 
(c) 
(a) original shape (F15), (b) x coordinate, (c) y coordinate 
Figure 2: Analysis of an object by separating it into its x and y 
coordinates. Starting point is at nose of FI5 plane, going 
clockwise. 
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