Essential Matrix: Pure Rotation

Question

I am trying to estimate an Essential Matrix from point correspondences. I set my init values to [rx,ry,rz,tx,ty,tz] to 0. But the first problem I encounter is, the Essential Matrix will be zero if all translation vector entries are zero. Am I right that Essential Matrix can not be used if I got pure rotation?

Math:

E = R [R^T t]x

Hartley,Zisserman: Multiple View Geometry p.257

Should I use a linear Algorithm like 'normalized 8-point' for calculating init values?

score 3 · Answer 1 · answered Mar 31 '19 at 06:28

If you try to estimate the essential matrix from a pure rotation (no motion) then here is what should happen. Suppose that x'_i = R x_i. We are looking for an essential matrix E = [t]x R' (possibly not the same as R, so I write it as R'). However, if t is anything at all and R' = R, then you find that x_i'^t [t]x R x_i = x_i'^t [t]x x_i', which will always be 0, because [t]x is skew-symmetric. Since this holds for any t, it follows that you cannot compute translation t from purely rotational motion. You should get the right value of R, however.

score 0 · Accepted Answer · answered Jul 14 '15 at 13:15

0

Under a pure rotation images are related by a homography.

answered Jul 14 '15 at 13:15

Francesco Callari

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I agree with that, does the translation vector refer on the camera centre? – crazyduck Jul 14 '15 at 16:00
Yes, in your expression above t is the vector from one camera center to the other's. – Francesco Callari Jul 15 '15 at 14:01

Essential Matrix: Pure Rotation

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