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We study the Perspective-n-Point (PNP) problem, which is fundamental in 3D vision, for the recovery of camera translation and rotation. A common solution applies polynomial sum-of-squares (SOS) relaxation techniques via semidefinite programming. Our main result is that the polynomials which should be optimized can be non-negative but not SOS, hence the resulting convex relaxation is not tight; specifically, we present an example of real-life configurations for which the convex relaxation in the Lasserre Hierarchy fails, in both the second and third levels. In addition to the theoretical contribution, the conclusion for practitioners is that this commonly-used approach can fail; our experiments suggest that using higher levels of the Lasserre Hierarchy reduces the probability of failure. The methods we use are mostly drawn from the area of polynomial optimization and convex relaxation; we also use some results from real algebraic geometry, as well as Matlab optimization packages for PNP.

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The empirical entropy is a key statistical measure of data frequency vectors, enabling one to estimate how diverse the data are. From the computational point of view, it is important to quickly compute, approximate, or bound the entropy. In a distributed system, the representative (“global”) frequency vector is the average of the “local” frequency vectors, each residing in a distinct node. Typically, the trivial solution of aggregating the local vectors and computing their average incurs a huge communication overhead. Hence, the challenge is to approximate, or bound, the entropy of the global vector, while reducing communication overhead. In this paper, we develop algorithms which achieve this goal.

The Perspective-n-Point problem aims to estimate the relative pose between a calibrated monocular camera and a known 3D model, by aligning pairs of 2D captured image points to their corresponding 3D points in the model. We suggest an algorithm that runs on weak IoT devices in realtime but still provides provable theoretical guarantees for both running time and correctness. Existing solvers provide only one of these requirements. Our main motivation was to turn the popular DJI's Tello Drone (<90gr, <$100) into an autonomous drone that navigates in an indoor environment with no external human/laptop/sensor, by simply attaching a Raspberry PI Zero (<9gr, <$25) to it. This tiny micro-processor takes as input a real-time video from a tiny RGB camera, and runs our PnP solver on-board. Extensive experimental results, open source code, and a demonstration video are included.