[Paper Review] Critical configurations for three projective views

Martin Bråtelund|arXiv (Cornell University)|Dec 10, 2021

Medical Imaging Techniques and Applications1 citations

TL;DR

This paper presents a novel algebraic approach to classifying critical configurations in three-projective-view structure from motion by analyzing intersections of multi-view varieties via blow-ups of camera centers. It proves that critical configurations of 8 or more points occur precisely when points lie on the strict transform of specific curves or surfaces—such as twisted cubics, rational quartics, or unions of conics and lines—while 7-point configurations are critical under a precise compatibility condition involving three ruled quadrics and their permissible lines.

ABSTRACT

The problem of structure from motion is concerned with recovering the 3-dimensional structure of an object from a set of 2-dimensional images taken by unknown cameras. Generally, all information can be uniquely recovered if enough images and point correspondences are provided, yet there are certain cases where unique recovery is impossible; these are called critical configurations. We use an algebraic approach to study the critical configurations for three projective cameras. We show that all critical configurations lie on the intersection of quadric surfaces, and classify exactly which intersections constitute a critical configuration.

Motivation & Objective

• To provide a complete classification of critical configurations for three projective cameras using algebraic geometry.
• To resolve a previously overlooked case in the literature where a twisted cubic with a secant line through two camera centers can be critical.
• To establish a general criterion for critical configurations beyond ad hoc examples, using multi-view varieties and blow-ups.
• To extend the theoretical understanding of instability in structure-from-motion algorithms near critical configurations.
• To offer a framework applicable to more complex camera configurations in future work.

Proposed method

• Use of blow-up of P³ at camera centers to resolve indeterminacies of camera projections, enabling morphism construction.
• Define a rational map ϕP: P³ ⇢ (P²)ⁿ that extends to a morphism on the blow-up BlP(P³), mapping configurations to multi-view varieties.
• Characterize critical configurations as those where two different camera-point configurations map to the same point in (P²)ⁿ via the extended morphism.
• Classify critical configurations by studying intersections of multi-view varieties, particularly intersections of three ruled quadrics.
• Apply techniques from algebraic geometry, including the use of line bundles and divisor classes (e.g., 2H − Ei − Ej), to constrain quadric surfaces.
• Use compatibility conditions between three quadrics and their associated permissible lines to determine criticality of 7-point configurations.

Experimental results

Research questions

• RQ1 Which configurations of three projective cameras and points in 3D space are critical, meaning that unique 3D structure recovery is impossible?
• RQ2 How do the intersections of multi-view varieties relate to the existence of critical configurations?
• RQ3 What is the precise condition under which a 7-point configuration with three cameras is critical?
• RQ4 Can the previously missed case—where a twisted cubic passes through one camera center and the line joining the other two is a secant—be shown to be critical?
• RQ5 How do degenerate configurations (e.g., unions of conics and lines) contribute to critical configurations?

Key findings

• All critical configurations for three projective cameras lie on the strict transform of curves or surfaces that are intersections of three ruled quadrics.
• The configuration of points on a twisted cubic passing through exactly one camera center is critical if the line joining the other two camera centers is a secant to the twisted cubic—correcting a prior error in the literature.
• For 8 or more points, a configuration is critical if and only if the points lie on the strict transform of one of the varieties shown in Figure 1 (e.g., rational quartics, twisted cubics, or unions of conics and lines), or form a subset of such a variety.
• For 6 or fewer points, every configuration is critical, but not maximal, as they are contained in larger critical configurations with at least 7 points.
• A 7-point configuration is critical if and only if there exists a compatible triple of quadrics containing the camera centers and the points, with the three planes spanned by permissible lines not intersecting any of the 7 points.
• The compatibility of the three quadrics and the geometric condition on the intersection of their associated planes fully determine criticality for 7-point configurations, as formalized in Proposition 6.19.