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Characterization of the multiplicity of solutions for camera pose given two vertically-aligned landmarks and accelerometer

Alexander R. Pruss

Abstract

We consider the problem of recovering the position and orientation of a camera equipped with an accelerometer from sensor images of two labeled landmarks whose positions in a coordinate system aligned in a known way with gravity are known. This a variant on the much studied P$n$P problem of recovering camera position and orientation from $n$ points without any gravitational data. It is proved that in three types of singular cases there are infinitely many solutions, in another type of case there is one, and in a final type of case there are two. A precise characterization of each type of case. In particular, there is always a unique solution in the practically interesting case where the two landmarks are at the same altitude and the camera is at a different altitude. This case is studied by numerical simulation and an implementation on a consumer cellphone. It is also proved that if the two landmarks are unlabeled, then apart from the same singular cases, there are still always one or two solutions.

Characterization of the multiplicity of solutions for camera pose given two vertically-aligned landmarks and accelerometer

Abstract

We consider the problem of recovering the position and orientation of a camera equipped with an accelerometer from sensor images of two labeled landmarks whose positions in a coordinate system aligned in a known way with gravity are known. This a variant on the much studied PP problem of recovering camera position and orientation from points without any gravitational data. It is proved that in three types of singular cases there are infinitely many solutions, in another type of case there is one, and in a final type of case there are two. A precise characterization of each type of case. In particular, there is always a unique solution in the practically interesting case where the two landmarks are at the same altitude and the camera is at a different altitude. This case is studied by numerical simulation and an implementation on a consumer cellphone. It is also proved that if the two landmarks are unlabeled, then apart from the same singular cases, there are still always one or two solutions.

Paper Structure

This paper contains 8 sections, 5 theorems, 35 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Sensor data and singular cases
  3. Characterization of solutions
  4. Procedure
  5. Unlabeled landmarks
  6. Horizontally coplanar landmarks
  7. Physical implementation on smartphone
  8. Conclusions

Key Result

Theorem 1

Assume that the $z$-axis of the object coordinates is antiparallel to gravity and that we are given the upward vector $\mathbold u$ and the vectors $\mathbold v_i$ in camera coordinates from the camera to the respective landmarks, as well the object coordinates of the respective landmarks. In each o If none of (i)--(iii) obtain, there are at most two solutions for camera pose, and furthermore ther

Figures (8)

  • Figure 1: A pinhole camera coordinate system with the points corresponding to $(x_i,y_i)$ coordinates on the sensor.
  • Figure 2: The vertical tilt angles $\rho_i$ and the horizontal landmark angle $\beta$. The origin is the camera optical center.
  • Figure 3: Third singular case: The possible positions for the camera line on the interior of the solid segment $S$ between co-horizontal landmarks $A$ to $B$.
  • Figure 4: Region of unique solution for landmarks at different altitudes.
  • Figure 5: The tilt angle $\rho_i$ from camera to landmark, the height $h_i$ of the camera above landmark, and the horizontal distance $d_i$ to the landmark.
  • ...and 3 more figures

Theorems & Definitions (9)

  • Theorem 1
  • Lemma 1
  • proof : Proof of Lemma \ref{['lem:position-pose']}
  • Lemma 2
  • proof : Proof of Lemma \ref{['lem:hhdd']}
  • proof : Proof of Theorem \ref{['th:solutions']}
  • Lemma 3
  • Theorem 2
  • proof : Proof of Lemma \ref{['lem:identify']}