2 research outputs found
New insights on Multi-Solution Distribution of the P3P Problem
Traditionally, the P3P problem is solved by firstly transforming its 3
quadratic equations into a quartic one, then by locating the roots of the
resulting quartic equation and verifying whether a root does really correspond
to a true solution of the P3P problem itself. However, a root of the quartic
equation does not always correspond to a solution of the P3P problem. In this
work, we show that when the optical center is outside of all the 6 toroids
defined by the control point triangle, each positive root of the Grunert's
quartic equation must correspond to a true solution of the P3P problem, and the
corresponding P3P problem cannot have a unique solution, it must have either 2
positive solutions or 4 positive solutions. In addition, we show that when the
optical center passes through any one of the 3 toroids among these 6 toroids (
except possibly for two concentric circles) , the number of the solutions of
the corresponding P3P problem always changes by 1, either increased by 1 or
decreased by 1.Furthermore we show that such changed solutions always locate in
a small neighborhood of control points, hence the 3 toroids are critical
surfaces of the P3P problem and the 3 control points are 3 singular points of
solutions. A notable example is that when the optical center passes through the
outer surface of the union of the 6 toroids from the outside to inside, the
number of the solutions must always decrease by 1. Our results are the first to
give an explicit and geometrically intuitive relationship between the P3P
solutions and the roots of its quartic equation. It could act as some
theoretical guidance for P3P practitioners to properly arrange their control
points to avoid undesirable solutions
Companion Surface of Danger Cylinder and its Role in Solution Variation of P3P Problem
Traditionally the danger cylinder is intimately related to the solution
stability in P3P problem. In this work, we show that the danger cylinder is
also closely related to the multiple-solution phenomenon. More specifically, we
show when the optical center lies on the danger cylinder, of the 3 possible P3P
solutions, i.e., one double solution, and two other solutions, the optical
center of the double solution still lies on the danger cylinder, but the
optical centers of the other two solutions no longer lie on the danger
cylinder. And when the optical center moves on the danger cylinder, accordingly
the optical centers of the two other solutions of the corresponding P3P problem
form a new surface, characterized by a polynomial equation of degree 12 in the
optical center coordinates, called the Companion Surface of Danger Cylinder
(CSDC). That means the danger cylinder always has a companion surface. For the
significance of CSDC, we show that when the optical center passes through the
CSDC, the number of solutions of P3P problem must change by 2. That means CSDC
acts as a delimitating surface of the P3P solution space. These new findings
shed some new lights on the P3P multi-solution phenomenon, an important issue
in PnP study