The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B3h. Beijing 2008 
Figure 1: Planar homography between two views of a target. 
Planar homography requires corresponding points across two 
views. 
and the Power Spectrum, then using the values of the Power 
Spectrum as the basis for the least squares adjustment to find 
the projectivity between the two Power Spectrums. Section 4 
shows the robust recognition results from the condition number 
and variance. 
2 PROJECTIVE GEOMETRY 
Methods, like similarity, affine and projective transforms, which 
exploit projective geometry require point correspondences 
across different views, such that any given point in one view 
corresponds to one and only one point in the other, and vice 
versa. We establish a relationship between the x, y coordinates 
of two contours of the same object on different sources with the 
planar homography matrix. If a planar object is imaged from 
multiple viewing positions, the result is a projective image-to- 
image homography (Hartley and Zisserman, 2000). Under 
planar homography, points in one image are transformed to the 
points in the other image as: 
x’ = Hx, (1) 
where x’ = (x\ y\ 1) and x = (x, y, 1) are corresponding points 
across images in homogeneous coordinates and H is the 3 x 3 
matrix known as the planar homography matrix: 
H= 
h 2 
h, 
l h 
\ 
h 3 
K 
h 9 
(2) 
For geometric features imaged from a distant viewpoint, the 
projective homography reduces to affine homography: 
H= 
( h x h 2 h 3 
K k 5 K 
o o 1 
(3) 
This approximation is considered adequate, since the projective 
portion is minimized compared to the affine (Shapiro and 
Zisserman, 1995). 
3 METHODOLOGY 
While establishing correspondences across images is feasible 
and has been a common practice among researchers, finding 
corresponding points between two contours is not easy and at 
times it is impossible. Due to this observation, a common 
tradition has been to represent the contours in parametric form 
prior to any processing. Parametric contour representation can 
be generated by methods including but not limited to: 
curvature/spline based, polar coordinate (geometric) based, 
trigonometric based, wavelet transform based, Fourier 
descriptor (FD) based. In contrast to others, FD and wavelet 
based parametric contour representations provide continuous 
functions. Compared to the FD based methods, wavelet based 
methods, however, involve intensive computation, and it is 
usually not clear which basis would be a better choice to 
represent the contour. (Zhang and Lu, 2001). Let two periodic 
functions x(t) and y(t) describe the contour, such that we treat x 
and y coordinates as independent dimensions (Zahn and Roskies, 
1972). We will be using periodic functions, since it is analogous 
to shifting the starting point on a contour. In order to objectively 
compare these functions to other x and y coordinates, we set a 
standard reference to the length of the contour, and select its 
period as 2 n. Let 1 be the arc length from an arbitrary starting 
point, to a point p, and let L be the entire length of the closed 
curve. In this form the arc-length is converted to its angular 
representation by: 
/ 
(o= 2n—- (4) 
L 
This suggests that L is the period of the x and y functions of a 
contour. Given a digital image, the contour of the geometric 
shape constitutes a dense set of discrete points. Expressing the 
parametric form of the discrete point set can be obtained by 
taking the Fourier transform of x and y independently: 
n=00 
F(x) = £ f(x)e~‘“ (n)t , (5) 
77=-GO 
77=GO 
F(y)=X , (6) 
77=—00 
where F is the Fourier transform of the f(x). For finite terms, n = 
(-N/2)...(N/2 - 1), where N is the number of points (even 
number). 
Using equations (5) and (6) and after some manipulations, 
which includes dividing each side with [e~ 1<0(1)t , e~'“ (2)t , ..., 
e- |l0(n)l ], it is easy to show that the homography transform in 
equation 1 becomes 
F(X0) = H F(X) (7) 
(3,n) (3,3) (3,n) 
where X = [x y scale] 
Let Gk be the k th Fourier coefficient. Then the coefficients in eq. 
7 can be expressed by: 
GOk = H Gk. (8) 
(3,n) (3,3) (3,n) 
Considering the case of a different starting point on the contour, 
according to Fourier theory a shift in the starting point of a 
function is the same as multiplying the coefficients by a rotation 
matrix (Schenk, 2001) 
GOk = Gk e i2nkn0/N (9) 
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