Question about Lecture Shape from Texture

Question about Lecture Shape from Texture

by Ruihang Jiang -
Number of replies: 1

Hi,

I want to ask a question about the equation on P13 of Lecture Shape from Texture as following shown:


It seems that the left side is a scaled value of sin(θ) (θ is the angle between p1 and p2). And the right side is a constant for the plane with ω=60° to the image plane. It indicates that for all perpendicular lines in this plane, the angle between p1 and p2 will be the same but it is not the case.

Is there anything I mis-understand?

Thanks in advance!

In reply to Ruihang Jiang

Re: Question about Lecture Shape from Texture

by Benoît Guillard -
You can forget about the scaling factors l_1 and l_2. With p1 and p2 the projection of the axis vectors (ie. as measured on the image), you have
||p1 x p2|| / (||p1||^2 + ||p2||^2) = cos(omega) / (1 + cos(omega)^2)

If you are interested, the full proof and details are in Sec. 2.2 and Appendix A of this paper. This is outside the scope of the course. This slide was only meant to show that you can recover the orientation of the plane under specific assumptions (regular pattern, orthographic projection…).