Locating Perspective Three-Point Problem Solutions In Spatial Regions

Rieck, Michael Q.; Wang, Bo

doi:10.1007/s10851-021-01040-4

“…An important quantity for analyzing the P3P Problem is the Gramian determinant associated with the unit vectors pointing from the camera's optical center to the control points. Following [12], [13] and [15], this will be denoted by η 2 . It is equal to…”

Section: A Special Cubic Polynomialmentioning

confidence: 99%

Understanding the Deltoid Phenomenon in the Perspective 3-Point Problem

Rieck¹

2022

Preprint

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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“…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%

“…As explained in [10], [12], [13], [15] and [16], Grunert's system has a repeated positive solution if and only if one of the possible positions for the camera's optical center is on the circular cylinder based on the circle containing the three control points. This is the so-called "danger cylinder."…”

Section: Introductionmentioning

confidence: 99%

“…Together with three special toroids, it separates space into regions having differing numbers of positive solutions to the system (1) (with corresponding parameter values). (See [13]. )…”

Section: Introductionmentioning

confidence: 99%

“…Following the practice adopted in [11], [12], [13] and [15], a special rectangular coordinate system will be used, though this in no way restricts the nature of the problem. The three coordinates are called x, y and z.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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Geometric Conditions for the Existence or Non-existence of a Solution to the Perspective 3-Point Problem

Rieck

2023

J Math Imaging Vis

0

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Locating Perspective Three-Point Problem Solutions In Spatial Regions

Cited by 6 publications

References 8 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

Geometric Conditions for the Existence or Non-existence of a Solution to the Perspective 3-Point Problem

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