Dense Correspondence and Its Applications - Computer Vision for Visual Effects - page 169

Π

P

1

2

I 1 I 2

(x',y')

(x,y)

(a) (b)

Figure 5.11. (a) A family of planes that tilts around the line connecting the two camera centers.
Every point in the scene has to lie on one of these planes. (b) Fixing one plane creates a pair of
conjugate epipolar lines in the two images. All of the epipolar lines in one image intersect at the
epipole, the projection of the camera center of the other image.

given . However, the epipolar lines are even more constrained than Figure 5.10

suggests. Figure 5.11 a illustrates a family of planes that tilts along a common “axis”—
the line connecting the two camera centers. Every point in the scene has to lie on one
of these planes. If we fix a plane

( x , y )

, as illustrated in Figure 5.11 b, it will intersect the

image planes in two lines, 2 . These two lines are exactly the epipolar lines

mentioned previously, and now we can see that they come in conjugate pairs . That
is, if a point on

1 and

2 , and vice versa . 9
Thus, in each image we have a one-dimensional family of epipolar lines. The epipolar
lines in each image all intersect at a special point called the epipole , which we can
see from Figure 5.11 b is the projection of the other camera. 10
In stereo, we reduce the optical flow problem to a one-dimensional correspon-
dence problem along each pair of epipolar lines. The next subsections discuss the
fundamentalmatrix , whichmathematically encapsulates the epipolar geometry and
can be estimated from feature matches in a similar manner to how we estimated a
projective transformation inSection 5.1 . It's conventional to rectify the images before
applying a stereo algorithm, which means that we transform each image with a pro-
jective transformation so that the epipolar lines coincide with rows of the resulting
pair of images. Section 5.4.3 discusses that process.

1 in I 1 has a correspondence in I 2 , it must lie on

5.4.1 The Fundamental Matrix

The epipolar line in one image corresponding to a point in the other image can be
computed from the fundamental matrix ,a3 3 matrix F that concisely expresses

the relationship between any two matching points. The key relationship is that any
correspondence

×

x , y ) ∈ { ( x , y ) ∈ I 1 , ( I 2 } must satisfy

x
y
1

x
y
1

=

F 0 (5.34)

9 Not all points on

1 may have a correspondence in I 2 due to occlusions; see Figure 5.16 .
10 In many real situations, the epipole is not visible in the captured image.