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The Euclidean distance degree of curves: from rational to line multiview varieties

Bella Finkel, Jose Israel Rodriguez

TL;DR

The paper derives a formula for the Euclidean distance degree of curves parameterized by rational functions within multiprojective spaces and applies it to line multiview varieties in computer vision. It establishes that, for a degree e rational world curve Y and generic n-camera configurations, the affine ED degree of the multiview image is 3en−2, under suitable dimensionality and genericity conditions. It then extends the framework to anchored multiview varieties via Grassmannians and wedge cameras, and uses this to resolve two conjectures on the ED degrees of line multiview varieties, showing affEDdeg = 6n−2 for h = 2,3. The results unify projective and affine perspectives and provide concrete, testable formulas for ED-degree computations in one-dimensional multiview geometry, with implications for triangulation and model optimization in vision systems.

Abstract

The Euclidean distance (ED) degree is an invariant that measures the algebraic complexity of optimizing the distance function of a point to a model. It has been studied in algebraic statistics, machine learning, and computer vision. In this article, we prove a formula for the ED degree of curves parameterized by rational functions with mild genericity assumptions. We apply our results to resolve conjectures on one-dimensional line multiview varieties from computer vision proposed by Duff and Rydell.

The Euclidean distance degree of curves: from rational to line multiview varieties

TL;DR

The paper derives a formula for the Euclidean distance degree of curves parameterized by rational functions within multiprojective spaces and applies it to line multiview varieties in computer vision. It establishes that, for a degree e rational world curve Y and generic n-camera configurations, the affine ED degree of the multiview image is 3en−2, under suitable dimensionality and genericity conditions. It then extends the framework to anchored multiview varieties via Grassmannians and wedge cameras, and uses this to resolve two conjectures on the ED degrees of line multiview varieties, showing affEDdeg = 6n−2 for h = 2,3. The results unify projective and affine perspectives and provide concrete, testable formulas for ED-degree computations in one-dimensional multiview geometry, with implications for triangulation and model optimization in vision systems.

Abstract

The Euclidean distance (ED) degree is an invariant that measures the algebraic complexity of optimizing the distance function of a point to a model. It has been studied in algebraic statistics, machine learning, and computer vision. In this article, we prove a formula for the ED degree of curves parameterized by rational functions with mild genericity assumptions. We apply our results to resolve conjectures on one-dimensional line multiview varieties from computer vision proposed by Duff and Rydell.

Paper Structure

This paper contains 13 sections, 11 theorems, 86 equations, 5 figures, 1 table.

Table of Contents

  1. Preliminaries
  2. Multiprojective varieties and multidegrees
  3. Multiprojective versus affine multiview varieties
  4. Euclidean distance degree
  5. The projective ED degree
  6. More multiview varieties and triangulation
  7. A Schubert variety in $\operatorname{Gr}(1,\mathbb{P}^3)$
  8. ED degree of multiview varieties of a rational curve
  9. Strengthening the ED-equivalence catalogue
  10. Anchoring the $n$-view variety
  11. Exterior algebra review and wedge cameras
  12. Resolving a pair of ED degree conjectures
  13. Conclusion and future directions

Key Result

Theorem 1.1

Suppose $X$ is an irreducible closed subvariety of $\mathbb{C}^{h}$. Then for general $\boldsymbol{\beta}:=(\beta_0,\dots,\beta_h)\in \mathbb{C}^{h+1}$ we have where $\operatorname{Eu}_X$ is the local Euler obstruction function on $X$ and

Figures (5)

  • Figure 1: A yellow cubic curve with its orange projection in the blue image plane (left). A yellow quartic curve with three orange projections in the blue image plane (right).
  • Figure 2: The red curve is $X_\mathbb{R}$ and the real ED critical points for the distance function to the orange points is the set of intersection points with the silver curve.
  • Figure 3: This commutative diagram shows two approaches for obtaining the $k$-plane multiview variety anchored at $\Lambda$.
  • Figure 4: For $h=2,3$, this shows the equality $\iota_{1,h,n}(\mathbf{C}^{h} \mathbin{\vcenter{\hbox{$\square$}}} {\texttt{L}^{\texttt{3}}})=\mathbf{D}^h \mathbin{\vcenter{\hbox{$\square$}}} \iota_{1,3,1}({\texttt{L}^{\texttt{3}}})$. The unlabeled horizontal arrows are inclusions and $\iota_{k,h,n}$ is as defined in \ref{['eq:khn-plucker-Gr-k-h']}.
  • Figure 5: The purple ruled surfaces consist of line segments connecting a pair of Bezier curves of degree one (left) and four (right). The lines spanned by the segments in a ruling give $\Lambda_{\mathbf{B}_1,\mathbf{B}_2} \subset\operatorname{Gr}(1,\mathbb{P}^3)$. The left surface has two rulings as discussed in \ref{['ss:schubert']}.

Theorems & Definitions (36)

  • Example 1
  • Example 2
  • Theorem 1.1: MRW-multiview
  • Theorem 1.2: MRW-multiview
  • Theorem 1.3: Theorem 1.3 MRW2021-edprojective
  • Theorem 1.4
  • Example 3: Standard grading giving discrepancy
  • Example 4: Projective closure discrepancy
  • Definition 1
  • Example 5: A one dimensional multiview variety
  • ...and 26 more