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Simple Case

If we know the 3D position of the points and know they are on an ellipsoid Q, we could directly use the equation: $X^TQX=0$, where Q is the 4x4 matrix.

Question

I'm wondering is it possible to build a constraint only use the information on image planes. If we know the projected points coordinates $[u_1, u_2, ...]$ of points $[X_1, X_2, ...]$, is there an equation that: $f(u_1, u_2, ...) = y(Q, P)$, where P is the projection Matrix.

Maybe we could use the distance between two points, crossratios between several distances and so on?

Thank you so much.

Z-crow
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  • You cannot do miracles... You cannot retrieve the $n+1$ Dimensional information knowing only the $n$ dimensional data. which is the projection. Nevertheless, here is a question to which I have indirectly contributed : https://math.stackexchange.com/q/2431159 ; it is shown there that the connection between say an ellipsoid and its projection (an ellipse) can be rather easily understood and treated using the technique of Schur's complements. – Jean Marie Sep 08 '19 at 21:47
  • Is this a subproblem of yours : knowing an ellipse on xOy plane, find a generic implicit equation for all ellipsoids (in 3D) that will be (orthogonally) projected onto this ellipse ? – Jean Marie Sep 09 '19 at 10:22
  • (Ctd). A way to "know" an ellipse (more generally a conic curve) is to give 5 points on it. – Jean Marie Sep 09 '19 at 10:25
  • Thanks for your comments. It seems that indeed we can not reconstruct the ellipsoid using only one image. How about several images? If we have an image sequence with projected points, and known camera poses, we can first construct the 3D points, then constrain the ellipsoid using X^TQX=0. However, the 3D point estimation will bring uncertainty. Is it possible to construct the ellipsoid using 2d points observation directly? Which means to emit the step of estimating the 3D position of points. Thank you. – Z-crow Sep 12 '19 at 06:37
  • Maybe, you should be more explicit about your data. What can be said is that the objective is to attempt to "retrieve" the 10 coefficients $a,b,c,d,...$ (10 coefficients up to a constant), precisely : $ax^2+by^2+cz^2+2dxy+\cdots=0$ ; the idea is to have at least 10 equations in unknowns $a,b,c,d...$ may be much more and treat this issue as a least square issue. – Jean Marie Sep 12 '19 at 09:59

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