Mathematical Analysis of the Multisolution Phenomenon in the P3P Problem

Abstract

The perspective 3-point problem, also known as pose estimation, has its origins in camera calibration and is of importance in many fields: for example, computer animation, automation, image analysis and robotics. One line of activity involves formulating it mathematically in terms of finding the solution to a quartic equation. However, in general, the equation does not have a unique solution, and in some situations there are no solutions at all. Here, we present a new approach to the solution of the problem; this involves closer scrutiny of the coefficients of the polynomial, in order to understand how many solutions there will be for a given set of problem parameters. We find that, if the control points are equally spaced, there are four positive solutions to the problem at 25 % of all available spatial locations for the control-point combinations, and two positive solutions at the remaining 75 %.

Appendix: \(\varDelta _{W},P,D,q\)

Substituting (43)–(47) into (49) gives, on factorizing,

$$\begin{aligned} \varDelta _{W}&=-16777216\left( p_{{2}}^{2}-p_{{3}}^{2}\right) ^{2} \left( p_{{1}}^{2}-p_{{3}}^{2}\right) ^{2}\left( p_{{1}}^{2}-p_{{2}}^{2}\right) ^2 \nonumber \\&\varDelta _T^{2}f_{1}\left( p_{1},p_{2},p_{3}\right) , \end{aligned}$$

(61)

where

$$\begin{aligned} \varDelta _T={p_{{1}}^{2}}+{p_{{2}}^{2}}+{p_{{3}}^{2}}-2p_{{1}}p_{{2}}p_{{3} }-1 \end{aligned}$$

and

$$\begin{aligned} f_{1}\left( p_{1},p_{2},p_{3}\right)&= -16{p_{{1}}^{4}p_{{2}}^{4}p_{{3} }^{4}}+32{p_{{1}}^{3}p_{{2}}^{3}p_{{3}}^{3}}\left( {p_{{1}}^{2}}+{p}_{2} ^{2}+p_{3}^{2}\right) \nonumber \\&-72{p_{{1}}^{2}p_{{2}}^{2}p_{{3}}^{2}}\left( {p_{{1}}^{2}}+{p}_{2} ^{2}+p_{3}^{2}\right) +184{p_{{1}}^{3}p_{{2}}^{3}p_{{3}}^{3}}\nonumber \\&+27\left( {p_{{1}}^{4}p_{{2}}^{4}}+{p_{{1}}^{4}p_{{3}}^{4}}+{p_{{2}} ^{4}p_{{3}}^{4}}\right) \nonumber \\&-6{p_{{1}}^{2}p_{{2}}^{2}p_{{3}}^{2}}\left( {p_{{1}}^{2}}+{p}_{2} ^{2}+p_{3}^{2}\right) \nonumber \\&-72p_{1}p_{2}p_{3}\left( {p_{{1}}^{2}p_{{2}}^{2}}+{p_{{1}}^{2}p_{{3}}^{2} }+{p_{{2}}^{2}p_{{3}}^{2}}\right) \nonumber \\&+84{p_{{1}}^{2}p_{{2}}^{2}p_{{3}}^{2}}+24p_{1}p_{2}p_{3}\left( {p_{{1} }^{2}}+{p_{{2}}^{2}}+{p_{{3}}^{2}}\right) \nonumber \\&-18\left( {p_{{1}}^{2}p_{{2}}^{2}}+{p_{{1}}^{2}p_{{3}}^{2}}+{p_{{2}} ^{2}p_{{3}}^{2}}\right) \nonumber \\&-8p_{{1}}p_{{2}}p_{{3}}+4\left( {p_{{1}}^{2}}+{p_{{2}}^{2}}+{p_{{3}}^{2} }\right) -1, \end{aligned}$$

(62)

as also obtained previously by Rieck [13]. Observe that $f_{1}\left( 0,0,0\right) =-1,$ which means that $\varDelta _{W}>0$ within the double crown-like shape shown in Fig. 8, and $\varDelta _{W}<0$ between it and the inflated tetrahedron. Note also that $f_1=0$ corresponds to what is commonly referred to as the danger cylinder [20, 21], as noted by [12].

Similarly, substituting (43)–(47) into (50) gives

$$\begin{aligned} P=-1024\varDelta _T^{2}f_{2}\left( p_{1},p_{2},p_{3}\right) , \end{aligned}$$

(63)

where

$$\begin{aligned}&f_{2}\left( p_{1},p_{2},p_{3}\right) \nonumber \\ =&12p_{1}^{4}p_{2}^{4}-36p_{1} ^{3}p_{2}^{3}p_{3}+27p_{1}^{2}p_{2}^{2}p_{3}^{2}\nonumber \\&\quad +4p_{1}p_{2}p_{3}\left( 2p_{1}^{2}+2p_{2}^{2}-p_{3}^{2}\right) \nonumber \\&\quad -6\left( p_{1}^{2}p_{2}^{2}+p_{1}^{2}p_{3}^{2}+p_{2}^{2}p_{3}^{2}\right) +p_{1}^{2}+p_{2}^{2}+p_{3}^{2}, \end{aligned}$$

(64)

whereas (52) leads to

$$\begin{aligned} D=-1048576\varDelta _T^{4}f_{3}\left( p_{1},p_{2},p_{3}\right) , \end{aligned}$$

(65)

where

$$\begin{aligned} f_{3}\left( p_{1},p_{2},p_{3}\right)&= 48p_{1}^{8}p_{2}^{8}-288p_{1} ^{7}p_{2}^{7}p_{3}+648p_{1}^{6}p_{2}^{6}p_{3}^{2}\nonumber \\&+64p_{1}^{7}p_{2}^{5}p_{3}+64p_{1}^{5}p_{2}^{7}p_{3}-680p_{1}^{5}p_{2} ^{5}p_{3}^{3}\nonumber \\&-48p_{1}^{6}p_{2}^{6}-240p_{1}^{6}p_{2}^{4}p_{3}^{2}-240p_{1}^{4}p_{2} ^{6}p_{3}^{2}\nonumber \\&+339p_{1}^{4}p_{2}^{4}p_{3}^{4}+144p_{1}^{5}p_{2}^{5}p_{3}+288p_{1} ^{5}p_{2}^{3}p_{3}^{3}\nonumber \\&+288p_{1}^{3}p_{2}^{5}p_{3}^{3}-72p_{1}^{3}p_{2}^{3}p_{3}^{5}+8p_{1} ^{6}p_{2}^{4}\nonumber \\&+20p_{1}^{6}p_{2}^{2}p_{3}^{2}+8p_{1}^{4}p_{2}^{6}-56p_{1}^{4}p_{2} ^{4}p_{3}^{2}\nonumber \\&-128p_{1}^{4}p_{2}^{2}p_{3}^{4}+20p_{1}^{2}p_{2}^{6}p_{3}^{2}-128p_{1} ^{2}p_{2}^{4}p_{3}^{4}\nonumber \\&+4p_{1}^{2}p_{2}^{2}p_{3}^{6}-54p_{1}^{5}p_{2}^{3}p_{3}-30p_{1}^{5} p_{2}p_{3}^{3}\nonumber \\&-54p_{1}^{3}p_{2}^{5}p_{3}-84p_{1}^{3}p_{2}^{3}p_{3}^{3}+18p_{1}^{3} p_{2}p_{3}^{5}\nonumber \\&-30p_{1}p_{2}^{5}p_{3}^{3}+18p_{1}p_{2}^{3}p_{3}^{5}+9p_{1}^{4}p_{2} ^{4}\nonumber \\&+45p_{1}^{4}p_{2}^{2}p_{3}^{2}+9p_{1}^{4}p_{3}^{4}+45p_{1}^{2}p_{2} ^{4}p_{3}^{2}\nonumber \\&+45p_{1}^{2}p_{2}^{2}p_{3}^{4}+9p_{2}^{4}p_{3}^{4}+4p_{1}^{5}p_{2} p_{3}\nonumber \\&+12p_{1}^{3}p_{2}^{3}p_{3}+4p_{1}^{3}p_{2}p_{3}^{3}+4p_{1}p_{2}^{5} p_{3}\nonumber \\&+4p_{1}p_{2}^{3}p_{3}^{3}-4p_{1}p_{2}p_{3}^{5}-3p_{1}^{4}p_{2} ^{2}\nonumber \\&-3p_{1}^{4}p_{3}^{2}-3p_{1}^{2}p_{2}^{4}-18p_{1}^{2}p_{2}^{2}p_{3} ^{2}-3p_{1}^{2}p_{3}^{4}\nonumber \\&-3p_{2}^{4}p_{3}^{2}-3p_{2}^{2}p_{3}^{4}+p_{1}^{2}p_{2}^{2}+p_{1}^{2} p_{3}^{2}+p_{2}^{2}p_{3}^{2}.\nonumber \\ \end{aligned}$$

(66)

Also,

$$\begin{aligned} q=\frac{-4096f_{4}\left( p_{1},p_{2},p_{3}\right) }{\varDelta _T^{3}}, \end{aligned}$$

(67)

where

$$\begin{aligned} f_{4}\left( p_{1},p_{2},p_{3}\right)&= 8p_{1}^{6}p_{2}^{6}-36p_{1} ^{5}p_{2}^{5}p_{3}+54p_{1}^{4}p_{2}^{4}p_{3}^{2}\nonumber \\&+8p_{1}^{5}p_{2}^{3}p_{3}+8p_{1}^{3}p_{2}^{5}p_{3}-31p_{1}^{3}p_{2} ^{3}p_{3}^{3}\nonumber \\&-6p_{1}^{4}p_{2}^{4}-18p_{1}^{4}p_{2}^{2}p_{3}^{2}-18p_{1}^{2}p_{2} ^{4}p_{3}^{2}\nonumber \\&+6p_{1}^{2}p_{2}^{2}p_{3}^{4}+9p_{1}^{3}p_{2}^{3}p_{3}+9p_{1}^{3}p_{2} p_{3}^{3}\nonumber \\&+9p_{1}p_{2}^{3}p_{3}^{3}+p_{1}^{4}p_{2}^{2}+p_{1}^{4}p_{3}^{2}+p_{1} ^{2}p_{2}^{4}\nonumber \\&+4p_{1}^{2}p_{2}^{2}p_{3}^{2}-p_{1}^{2}p_{3}^{4}+p_{2}^{4}p_{3}^{2} -p_{2}^{2}p_{3}^{4}\nonumber \\&-3p_{1}^{3}p_{2}p_{3}-3p_{1}p_{2}^{3}p_{3}-3p_{1}p_{2}p_{3}^{3}+p_{1} p_{2}p_{3}.\nonumber \\ \end{aligned}$$

(68)