# Using Sparse Elimination for Solving Minimal Problems in Computer Vision

^{1}

##### Citations

15 citations

### Cites methods from "Using Sparse Elimination for Solvin..."

...Successful applications of the resultant method in computer vision can be found in [36], [37]....

[...]

7 citations

### Cites methods from "Using Sparse Elimination for Solvin..."

...Polynomial eigenvalue methods have been successfully used for many minimal problems in computer vision, such as the 9-point one-parameter radial distortion problem [11], the 5- and 6-point relative pose problems [21], the 6point one unknown focal length problem [4], and the selfcalibration problems [17]....

[...]

6 citations

### Cites methods from "Using Sparse Elimination for Solvin..."

...This was recently extended by Heikkila [86] using techniques for constructing sparse resultants [205, 56]....

[...]

5 citations

### Cites background or methods from "Using Sparse Elimination for Solvin..."

...The most promising results in this direction were proposed by Emiris [12] and Heikkilä [18], where methods based on sparse resultants were proposed and applied to camera geometry problems....

[...]

...The most promising results in this direction were proposed by Emiris [12] and Heikkilä [18], where methods based on sparse resultants were proposed and applied to camera geometry problems....

[...]

...The augmented polynomial system is solved by hiding λ and reducing a constraint similar to (2) into a regular eigenvalue problem that leads to smaller solvers than [12, 18]....

[...]

...Our algorithm is inspired by the ideas explored in [18, 12], but thanks to the special form of added equation and by solving the resultant as a small eigenvalue problem, in contrast to a polynomial eigenvalue problem in [18], the new approach achieves significant improvements over [18, 12] in terms of efficiency of the generated solvers....

[...]

...The algorithm by Heikkilä [18] basically computes the Minkowski sum of the Newton polytopes of a subset of input polynomials, Q = ΣiNP(fi(x))....

[...]

5 citations

##### References

11,465 citations

### "Using Sparse Elimination for Solvin..." refers methods in this paper

...Well-known Zhang’s calibration method [23] provides a closed form solution to the calibration problem from images of a known planar target....

[...]

1,834 citations

### "Using Sparse Elimination for Solvin..." refers methods in this paper

...Nistér [17] converted the resulting system of polynomial equations to a tenth degree univariate polynomial that can be efficiently solved using standard numerical techniques....

[...]

545 citations

535 citations

### "Using Sparse Elimination for Solvin..." refers background or methods in this paper

...In [8] the minimal problem of computing the radial distortion coefficient was expressed as a quadratic polynomial eigenvalue problem and later it was extended in [16] to include an additional constraint....

[...]

...Fitzgibbon [8] augmented the fundamental matrix estimation to include one term of radial lens distortion, and solved them from 9 point correspondences....

[...]

335 citations

### "Using Sparse Elimination for Solvin..." refers background in this paper

...Such minimal problems include for example, the classical P3P (Perspective-ThreePoint) problem for a calibrated camera where an image of three points with known distances is sufficient to compute the camera pose, but it requires solving a system of three quadratic equations in three variables [9]....

[...]