We present a new minimal problem for relative pose estimation mixing point features with lines incident at points observed in three views and its efficient homotopy continuation solver. We demonstrate the generality of the approach by analyzing and solving an additional problem with mixed point and line correspondences in three views. The minimal problems include correspondences of (i) three points and one line and (ii) three points and two lines through two of the points which is reported and analyzed here for the first time. These are difficult to solve, as they have 216 and -as shown here -312 solutions, but cover important practical situations when line and point features appear together, e.g., in urban scenes or when observing curves. We demonstrate that even such difficult problems can be solved robustly using a suitable homotopy continuation technique and we provide an implementation optimized for minimal problems that can be integrated into engineering applications. Our simulated and real experiments demonstrate our solvers in the camera geometry computation task in structure from motion. We show that new solvers allow for reconstructing challenging scenes where the standard two-view initialization of structure from motion fails.

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Systems of multivariate polynomial equations are ubiquitous throughout mathematics. They also appear prominently in scientific applications such as kinematics [20,22], computer vision [11,15], power flow engineering [18], and statistics [12]. Numerical homotopy continuation methods are a fundamental tool for both solving these systems and determining more refined information about their structure.In this article, we offer a brief glimpse of polynomial homotopy continuation methods: the general theory, a few applications, and some software packages that implement these methods. Our aim is to spark the reader's interest in this exciting and broad area of research. We invite those looking to learn more to join us at the AMS Short Course: Polynomial systems, homotopy continuation, and applications, to be held January 2-3 at the 2023 Joint Mathematics Meetings in Boston.