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2.1 Geometric primitives and transformations 51

p = (X,Y,Z,1)
^
n 0·p+c 0= 0
~
~ ~ x 1 = (x 1,y 1,1)
x 1 = (x 1,y 1,1,d 1)
x 0 = (x 0,y 0,1)
~
x 0 = (x 0,y 0,1,d 0)
H 10
M 10 .
(a) (b)

Figure 2.12 A point is projected into two images: (a) relationship between the 3D point coordinate (X, Y, Z, 1)
and the 2D projected point (x, y, 1,d); (b) planar homography induced by points all lying on a common plane
ˆ n 0 · p + c 0 =0.

The other special case where we do not need to know depth to perform inter-camera
mapping is when the camera is undergoing pure rotation (Section 9.1.3), i.e., when t 0 = t 1 .
In this case, we can write

˜ x 1 ∼ K 1 R 1 R −1 K −1 ˜ x 0 = K 1 R 10 K −1 ˜ x 0 , (2.72)
0 0 0
which again can be represented with a 3 × 3 homography. If we assume that the calibration
matrices have known aspect ratios and centers of projection (2.59), this homography can be
parameterized by the rotation amount and the two unknown focal lengths. This particular
formulation is commonly used in image-stitching applications (Section 9.1.3).

Object-centered projection
When working with long focal length lenses, it often becomes difficult to reliably estimate
the focal length from image measurements alone. This is because the focal length and the
distance to the object are highly correlated and it becomes difficult to tease these two effects
apart. For example, the change in scale of an object viewed through a zoom telephoto lens
can either be due to a zoom change or a motion towards the user. (This effect was put to
dramatic use in some of Alfred Hitchcock’s film Vertigo, where the simultaneous change of
zoom and camera motion produces a disquieting effect.)
This ambiguity becomes clearer if we write out the projection equation corresponding to
the simple calibration matrix K (2.59),

r x · p + t x
x s = f + c x (2.73)
r z · p + t z
r y · p + t y
= f + c y , (2.74)
y s
r z · p + t z
where r x , r y , and r z are the three rows of R. If the distance to the object center t z ð p
(the size of the object), the denominator is approximately t z and the overall scale of the
projected object depends on the ratio of f to t z . It therefore becomes difficult to disentangle
these two quantities.