Related Solutions to the Perspective Three-Point Pose Problem

Rieck, Michael Q.

doi:10.1007/s10851-015-0572-1

“…(Note that the sign convention used for ∆ there is the opposite of the one used here.) Corollary 1 in [11] also indicates that when ∆ > 0, there are exact two P3P solution points (with real Cartesian coordinates). To establish that the proposed solution points (two of the four ξ j ) are the actual solution points, it suffices to show that the equation ζ L = ζ 2 − 2ζ has only two solutions.…”

Section: P3p Solutions In the Limit Casementioning

confidence: 93%

Understanding the Deltoid Phenomenon in the Perspective 3-Point Problem

Rieck¹

2022

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0

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<p>Concerning the Perspective 3-Point (P3P) Problem, Grunert's system of three quadratic equations has a repeated solution if and only if the cubic polynomial introduced by Finsterwalder has a repeated root. This polynomial is shown to be equivalent to a particularly simple cubic polynomial with complex coefficients that provides surprising geometric insight into the P3P problem. In particular, (1) its discriminant can be written using the formula for the standard deltoid, and (2) this discriminant vanishes on a surface that approaches a deltoid shape when the camera is moved infinitely far from the control points in a direction perpendicular to the control points plane (the ``limit case"). These two facts have been previously reported, but obscure reasoning was required to establish them. In contrast, the present article uses the newly discovered cubic polynomial to easily produce the first fact, which then provides a basis for better understanding the second fact. Geometric insight into the P3P solution points in the limit case is also provided.</p>

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“…We will now focus on the so-called "limit case," studied in [11]. The idea is to consider what happens as the third coordinate z of the camera's optical center becomes very large in relation to the other two coordinates and also to the distances between the control points.…”

Section: P3p Solutions In the Limit Casementioning

confidence: 99%

“…Unfortunately, the system typically produces multiple positive-valued solutions, as many as four of these. Understanding how many such solutions result for given values of the parameters a, b, c, α, β and γ has been the focus of much investigation over the past few decades (for example, [1], [2], [11], [12], [13], [15]). Similar to the situation for a single polynomial in a single variable, this question is closely related to determining which parameter values produce a repeated solution to the system, and this amounts to examining a quantity, the "discriminant," based on the parameter values.…”

Section: Introductionmentioning

confidence: 99%

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Understanding the Deltoid Phenomenon in the Perspective 3-Point Problem

Rieck¹

2023

Preprint

1

0

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<p>Concerning the Perspective 3-Point (P3P) Problem, Grunert's system of three quadratic equations has a repeated solution if and only if the cubic polynomial introduced by Finsterwalder has a repeated root. This polynomial is shown to be equivalent to a particularly simple cubic polynomial with complex coefficients that provides surprising geometric insight into the P3P problem. In particular, (1) its discriminant can be written using the formula for the standard deltoid, and (2) this discriminant vanishes on a surface that approaches a deltoid shape when the camera is moved infinitely far from the control points in a direction perpendicular to the control points plane (the ``limit case"). These two facts have been previously reported, but obscure reasoning was required to establish them. In contrast, the present article uses the newly discovered cubic polynomial to easily produce the first fact, which then provides a basis for better understanding the second fact. Geometric insight into the P3P solution points in the limit case is also provided.</p>

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“…Gao et al [14] have given a complete classification, although the procedure is lengthy, mainly as they try to solve two quadratic equations simultaneously, rather than the single quartic equation. Even more recent are the papers by Zhang and Hu [15][16][17], Hu et al [18], and Rieck [19][20][21][22]; in particular, Rieck [20] focused 2 Mathematical Problems in Engineering on the multisolution phenomenon. An alternative approach is that of Vynnycky and Kanev [23], who analyzed in detail the behaviour of coefficients of the quartic polynomial derived by Linnainmaa et al [7]; this enabled them to draw conclusions on the comparative likelihoods of obtaining different numbers of solutions, albeit only for the case of equidistant spacing between the vertices, constituting an isosceles triangle.…”

Section: Introductionmentioning

confidence: 99%

On the Effect of Control-Point Spacing on the Multisolution Phenomenon in the P3P Problem

Vynnycky

¹

,

Reddy

²

2018

Mathematical Problems in Engineering

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The perspective 3-point (P3P) 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 possibility is to formulate it mathematically in terms of finding the solution to a quartic equation. However, there is yet no quantitative knowledge as to how control-point spacing affects the solution structure—in particular, the multisolution phenomenon. Here, we consider this problem through an algebraic analysis of the quartic’s coefficients and its discriminant and find that there are significant variations in the likelihood of two or four solutions, depending on how the spacing is chosen. The analysis indicates that although it is never possible to remove the occurrence of the four-solution case completely, it could be possible to choose spacings that would maximize the occurrence of two real solutions. Moreover, control-point spacing is found to impact significantly on the reality conditions for the solution of the quartic equation.

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Related Solutions to the Perspective Three-Point Pose Problem

Cited by 9 publications

References 10 publications

Understanding the Deltoid Phenomenon in the Perspective 3-Point Problem

Understanding the Deltoid Phenomenon in the Perspective 3-Point Problem

Understanding the Deltoid Phenomenon in the Perspective 3-Point Problem

On the Effect of Control-Point Spacing on the Multisolution Phenomenon in the P3P Problem

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