The Chow form of the essential variety in computer vision

Abstract

The Chow form of the essential variety in computer vision is calculated. Our derivation uses secant varieties, Ulrich sheaves and representation theory. Numerical experiments show that our formula can detect noisy point correspondences

Complexity of model testing for dynamical systems with toric steady states

“…Applying (7) gives Remark 3.4. Theorem 3.3 is rather remarkable as it tells us that for any parameter vector and any measurement we will always find exactly one local minimum for the Euclidean distance problem associated to the sequestration network (22). Using Proposition 2.8 we can guarantee that all coordinates of this minimum point are positive real numbers; in particular this minimum is the desired model point.…”

Distortion Varieties

“…Example 2.2 (Essential Matrices). We now write E for the essential variety [10,16]. It has dimension 5 and degree 10 in P 8 .…”

“…The following proof, and the subsequent development in this section, assumes familiarity with two tools from computational algebraic geometry: the construction of initial ideals with respect to weight vectors, as in [34], and the Chow form of a projective variety [9,16,17,23].…”

“…Here the α i,j are fixed generic complex numbers and λ is an unknown that parametrizes the curve. If we take the Grassmannian in its Plücker embedding then the degree of our curve is u c + u c+1 + · · · + u n , which is the largest degree in λ of any maximal minor of (16).…”

“…We now consider the intersection of our curve with the hypersurface defined by Ch X . Equivalently, we substitute the maximal minors of (16) into Ch X and we examine the resulting polynomial in λ. Since the matrix entries α i,j in (15) are generic, the curve intersects the hypersurface of the Chow form Ch X outside its singular locus.…”