Reconstruction of Quadrics from two views

Example setup of Quadric Reconstruction. The quadric Q is projected onto two planes using a perspective map to obtain the two elliptical conics C1 and C2. It is our goal to reconstruct Q from observed C1 and C2.

Quadrics

A quadric surface in 3d space is defined by a polynomial function of degree two with a total of 9 unknown coefficients. The most common quadric describes an ellipsoidal object such as an ellipsoid or a sphere.

Definition of quadric as 4x4 symmetric matrix. The null space of the second equation defines a quadric surface. Due to the homogenous representation, the parameter c can be omitted, which gives a total of 9 unknowns.

Conics

A conic is the 2d-analogous of a quadric. Typical conics include ellipses, hyperbolas and parabolas. The general conic equation is a polynomial with 5 independant coefficients.

Quadric Projection: Quadric Q projected along projection matrix P (3x4 matrix) yields view-conic C.

Reconstruction from two views

Cross, Zisserman (1998) describe a method of retriving the Quadric Q from two observed view-conics C1 and C2. To do so, the above formula for quadric projection is vectorized.

Vectoriation of quadric projection.
The 12x12 matrix M defines the constraints imposed on the quadric by two projection conics c and c’.

Ambiguity of two-view reconstruction

The paper continues to state that the resulting solution of a two-view equation system will be a family of quadrics given by:

Solution space for two-view reconstruction.

Choosing free parameter using the positive-definite constraint

To obtain a unique reconstruction from the parameter family, we can assess what happens for different choice of lambda and impose constraints on it. A simple constraint would be, that the upper left triangle matrix A must be positive definite if we know the quadric is an ellipsoid.

All upper left sub-matrices of A must have a positive determinant.

Example

We define an ellipsoid which has its longest axis along the y = x line. We proceed as described above to obtain a solution family. For the projection matrices we use a z-axis and y-axis projection for simplicity:

Ground truth Quadric and result of reconstruction. We see that if we set b=-1 and c=0, we obtain the original quadric.
Parameter space c and b. The positive definite constraint yields that b<0 and f(c, b) > 0. This yields the 2d parameter solution space where the blue function is positive and b negative.
Changing shape of reconstructed quadric wrt choice of free parameter c.

Conclusion

Using the constraint that the quadric must be positive definite, we can define bounds for the free parameters. However, the resulting family of quadrics does not seem to approximate the input ellipsoid well. Further work has to be dedicated to searching the optimal solution.

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Max Rohleder

Max Rohleder

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Data Analytics Student from Munich, Germany. Interested in web apps, machine learning systems and medical AI norms and regulations.