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a) original image ( 1600 × 1200 pixels)
b) projective transformation using
1 . 048 0 . 000 523 . 8
0 . 857 0 . 571 257 . 1
0 . 001 0 . 000
H =
1
c) projective transformation using
d) projective transformation using
0 . 794 0 . 000 397 . 0
0 . 000 0 . 476 142 . 9
0 . 001 0 . 001
2 . 072 0 . 414 911 . 9
0 . 933 1 . 306 1604
0 . 001 0 . 002
H =
H =
1
1
Fig. 2. A set of homographies which does not reflect a rigid-body transformation
toward the center of the image in different ways. The linear displacements are
represented by yellow lines in figure 2. Considering the initial rectangular shape
formed by the corners of the original image and those resulting from the shifted
corners, it is clear that the geometrical transformations imposed here cannot be
obtained by any rigid-body transformation. In figure 2‌b‌) the order of the corners
is changed by inverting the top-right and bottom-right corners, thus leading to
a kind of bow-tie shape. In figure 2‌c‌)and2‌d‌), the convexity of the initial shape
is modified and in figure 2‌d‌) the order of the top-left and bottom-left corners
is also inverted. However, even if it does not reflect a rigid-body transforma-
tion, a homography exists for any set of four correspondences. The beginnings
of an explanation for this matter lie in the fact that the projective plane
2 is
not the Euclidean plane of images and has a very different topology [3]. Even
if we start to work with Euclidean coordinates, the projective transformations
P
 
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