Degeneracy in geometry transfer

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The error influence on homography

0 50 100 150 200 250 300 350 400 450

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Distance d(point, plane)/d(camera, plane)

Error (pixels)

Figure 2-11 The error influence on homography.

2.4.1.4 Degeneracy in the homography estimation

Projective transformation, is also called as homography, may have the degeneracy problem. Hartely and Zisserman [38] discuss about two cases. Assume that a minimal solution needs four corresponding points. Case one is that if x1, x2, and x3 are collinear, and x1', x2', and x3' are not collinear, and there exists not any projective transformation. The second case is that if x1, x2, and x3 are collinear, and x1', x2', and x3' are also collinear, and there is not a unique solution.

2.4.2 Degeneracy in geometry transfer

In this section, we will discuss about the second category of the degeneracy. We discuss

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several common geometry transfers, such as the transfer by homography, epipolar transfer, and the transfer by trifocal tensor. We try to understand which situations will cause the degeneracy problems.

2.4.2.1 Transfer by homography

Hartely [38] showed that when a plane contains one of the camera centers in the 3D space, the transfer by its homohraphy is degenerate. It is because that the plane will be projected to a line in one of the cameras. The homography matrix does not have full rank and they are rank 2 or 1. All the points on the plane are respectively mapped to a line or a point.

Assume that when a plane contains the second camera center, this plane will intersect with the second image plane in a line l'. A point x in the first view is transferred to the intersection x' between its epipolar line and l' as the following equation.

  ' '

x l Fx.

2.4.2.2 Epipolar transfer

As shown in Figure 2-12 when the corresponding points in image I1 and I2 are known, the epipolar transfer can be used to transfer the corresponding points to I3, which are the intersections of the epipolar line from the points x1 in I1 and x2 in I2.

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Figure 2-12 The epipolar transfer.

We will discuss following three situations which will cause the degeneracy problems.

(1) Points on the trifocal plane

Hartely [38] discussed that when the 3D point is estimated from the corresponding points x1 and x2 is on the trifocal plane, the plane formed with three camera centers, the epipolar lines got from x1 and x2 are collinear. There is no intersection of the epipolar lines and it causes the degeneracy problem. Even if the 3D points are near the trifocal plane, the transfer points still have errors.

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C

0

C

1

C

2

e

01

e

02

e

10

e

₁₂

e

21

e

20

I

0

I

1

I

2

Figure 2-13 The degeneracy in the trifocal plane.

For this problem, there are two methods to process. First, judge if the corresponding point is close to the line between tow epipoles such as e13 and e12. Do not process the points close to the line. Second, the best method is to remove the bad camera configurations which the trifocal plane can be projected to the images. As shown in Figure 2-13, it will make each camera capture the object near the plane. However, when the camera configuration is as shown in Figure 2-14, the trifocal plane is vertical to the ground. The object captured will not be close to the plane, and the degeneracy problem will be avoided.

C

C' C''

X

Figure 2-14 The camera configuration for the degeneracy in epipolar transfer.

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We also consider if there are other conditions cause the degeneracy problems. As shown in Figure 2-15, when the corresponding points x in I1 and x' in I2 are known, we can draw the ray between the camera centers and the 3D point. There is an intersection X between the two lines. The two rays projected into the image I3 are the epipolar lines, and the intersection is the corresponding points, the projection point of the point X. We consider that when the C'' moves around the 3D space, which conditions will make the lines have no intersection?

C

C' x C''

x'

X

I₁

I₂

I₃

Figure 2-15 Move the third camera C'' in the 3D space.

Therefore, we show another two configurations which will cause the degeneracy problem as follows.

(2) The ray between the 3D point and the camera centers of the two views are collinear This condition is a special case of (1). As shown in Figure 2-16, when the lines CX and C X are collinear, there is not an intersection in I3. Usually we can denote the epipolar line l' in the I2 as the equation (2-26) where F is the fundamental matrix. It is also the line between

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epipole e' and corresponding point x'. As shown in Figure 2-17, the transfer point x'' in I3 is the intersection of the two epipolar line as the equation (2-27). Because the two epipolar lines in I3 are collinear, there is no solution by the epipolar transfer. However, only one point will cause this problem, so we can ignore it.

C

Figure 2-16 The ray between the 3D point and the camera centers of the two views are collinear.

Figure 2-17 The transfer point by using epipolar transfer.

= =

  

l e x Fx (2-26)

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31 32

31 31

ˆ ˆ

= ( ) ( )

ˆ ˆ

= ( ) ( )

= 0

  



x F x F x

F x F x (2-27)

(3) Three camera centers are collinear

This condition is also a special case of (1). As shown in Figure 2-18, when three cameras are collinear, the transfer of all corresponding points will cause the degeneracy problem. As shown in Figure 2-19, it is the same as the case (2). The relation can be shown by the equation (2-28). We can find that all epipolar lines from the corresponding points are collinear, so it is not a unique solution by the epipolar transfer.

C C' C''

x x'

X

I₁ I₂ I₃

Figure 2-18 Three camera centers are collinear.

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Figure 2-19 The transfer point by using epipolar transfer.

31 32

2.4.2.3 Transfer by the trifocal tensor

As shown in Figure 2-20, the trifocal tensor is similar to epipolar transfer. It can transfer the corresponding points x in I1 and x' in I2 to x'' in I3. However, the probability of causing the degeneracy in the transfer by the trifocal tensor is lower. It occurs when the 3D points close to the baseline, the line between two camera centers. But, it is not easy to capture the objects near the baseline. Therefore, it may be used to replace the epipolar transfer.

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C

0

C

1

C

2

e

01

e

10

I

0

I

1

I

2

l

X  

B

01

B

₁₂

x x 

Figure 2-20 Degeneracy for point transfer by trifocal tensor.

在文檔中 多視角影像中的退化問題與參數估測 - 政大學術集成 (頁 49-57)