Perspective-n-points Problem

发表于 2018-12-15 更新于 2020-12-11 分类于算法

This article briefly discusses the main idea of PnP problem and set up the question model of P3P problem. No solutions are given.

Perspective n Points Problem - P3P

Problem Definition

Given a set of n 3D points in a world reference frame and their corresponding 2D image projections as well as the calibrated intrinsic camera parameters, determine the 6 DOF pose of the camera in the form of its rotation and translation with respect to the world. This follows the perspective project model for cameras:

\[s\ p_c = K[R|T]p_w\]

$p_w=[x\ y\ z\ 1]^T$is the homogeneous world point, $p_c = [u\ v\ 1]^T$is the corresponding homogeneous image point. $K$ denotes the intrinsic parameters matrix, where $\gamma$ is the skew parameter. $s$ is a scale factor for the image point. and $R$ and $T$ are the desired 3D rotation and 3D translation of the camera (extrinsic parameters) that are being calculated.

Aassumptions

The key assumption is that the intrinsic matrix $K$ should be given. For each solution to PnP, the chosen point correspondences cannot be coplanar.

Method Description

When n = 3, the PnP problem is in the minimal form of P3P and can be solved with three point correspondences. Let $P$ be the Center of Perspective, and $A,B,C$ the control points. Let$|PA|=X,\ |PB| = Y,\ |PC|=Z$, $\alpha = \angle BPC,\ \beta = \angle APC, \ \gamma = \angle APB$ , $p = 2\cos \alpha,\ q = 2\cos \beta, \ r = \cos \gamma$ , $|AB|=c',\ |BC| = a',|AC| = b'$.From triangles $PBC,PAC,PAB$ ，we obtain the equation system.

A set of solutions for $X,Y,Z$ is call a set of physical solutions if the following "reality conditions" are satisfied. These conditions are assumed through out the article.

To simplify the question, let $X = xZ,Y = yZ,|AB| = \sqrt{v}Z$ , $|BC| = \sqrt{av}Z$ , $|AC| = \sqrt{bv}Z$. Since $Z = |PC| $, we obtain the following equivalent equation system:

Since $|r|<2$, we have $v = x^2+y^2-xyr>0$. Thus $Z$ can be uniquely determined by $Z = |AB|/\sqrt{v}$. Eliminating $v$ from the equations above, we have

Now the P3P problem is reduced to finding the positive solutions of two quadratic equations. As a consequence, we obtain the following result:

The P3P problem has either an infinite number of solutions or at most four physical solutions

The following solutions to these two equations are not given in this article, please refer to the paper.¹

Complete solution classification for the perspective-three-point problem, Gao, Xiao ShanHou, Xiao RongTang, JianliangCheng, Hang Fei↩︎