# Reconstruction of Quadrics from two views

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.

## 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.

## 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.

# 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:

## 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.

## 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:

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

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## 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.