A Fast and Reliable Solution to PnP, Using Polynomial Homogeneity and a Theorem of Hilbert
One of the most-extensively studied problems in three-dimensional Computer Vision is “Perspective-n-Point” (PnP), which concerns estimating the pose of a calibrated camera, given a set of 3D points in the world and their corresponding 2D projections in an image captured by the camera. One solution m...
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MDPI AG
2023-06-01
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| Online Access: | https://www.mdpi.com/1424-8220/23/12/5585 |
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| author | Daniel Keren Margarita Osadchy Amit Shahar |
| author_facet | Daniel Keren Margarita Osadchy Amit Shahar |
| author_sort | Daniel Keren |
| collection | DOAJ |
| description | One of the most-extensively studied problems in three-dimensional Computer Vision is “Perspective-n-Point” (PnP), which concerns estimating the pose of a calibrated camera, given a set of 3D points in the world and their corresponding 2D projections in an image captured by the camera. One solution method that ranks as very accurate and robust proceeds by reducing PnP to the minimization of a fourth-degree polynomial over the three-dimensional sphere <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>S</mi><mn>3</mn></msup></semantics></math></inline-formula>. Despite a great deal of effort, there is no known fast method to obtain this goal. A very common approach is solving a convex relaxation of the problem, using “Sum Of Squares” (SOS) techniques. We offer two contributions in this paper: a faster (by a factor of roughly 10) solution with respect to the state-of-the-art, which relies on the polynomial’s homogeneity; and a fast, guaranteed, easily parallelizable approximation, which makes use of a famous result of Hilbert. |
| first_indexed | 2024-03-11T01:57:20Z |
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| issn | 1424-8220 |
| language | English |
| last_indexed | 2024-03-11T01:57:20Z |
| publishDate | 2023-06-01 |
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| spelling | doaj.art-bba161d3ee8447eb95030232b8ba09bb2023-11-18T12:33:11ZengMDPI AGSensors1424-82202023-06-012312558510.3390/s23125585A Fast and Reliable Solution to PnP, Using Polynomial Homogeneity and a Theorem of HilbertDaniel Keren0Margarita Osadchy1Amit Shahar2Department of Computer Science, University of Haifa, Haifa 3498838, IsraelDepartment of Computer Science, University of Haifa, Haifa 3498838, IsraelDepartment of Computer Science, University of Haifa, Haifa 3498838, IsraelOne of the most-extensively studied problems in three-dimensional Computer Vision is “Perspective-n-Point” (PnP), which concerns estimating the pose of a calibrated camera, given a set of 3D points in the world and their corresponding 2D projections in an image captured by the camera. One solution method that ranks as very accurate and robust proceeds by reducing PnP to the minimization of a fourth-degree polynomial over the three-dimensional sphere <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>S</mi><mn>3</mn></msup></semantics></math></inline-formula>. Despite a great deal of effort, there is no known fast method to obtain this goal. A very common approach is solving a convex relaxation of the problem, using “Sum Of Squares” (SOS) techniques. We offer two contributions in this paper: a faster (by a factor of roughly 10) solution with respect to the state-of-the-art, which relies on the polynomial’s homogeneity; and a fast, guaranteed, easily parallelizable approximation, which makes use of a famous result of Hilbert.https://www.mdpi.com/1424-8220/23/12/5585the PnP problempolynomial optimization |
| spellingShingle | Daniel Keren Margarita Osadchy Amit Shahar A Fast and Reliable Solution to PnP, Using Polynomial Homogeneity and a Theorem of Hilbert Sensors the PnP problem polynomial optimization |
| title | A Fast and Reliable Solution to PnP, Using Polynomial Homogeneity and a Theorem of Hilbert |
| title_full | A Fast and Reliable Solution to PnP, Using Polynomial Homogeneity and a Theorem of Hilbert |
| title_fullStr | A Fast and Reliable Solution to PnP, Using Polynomial Homogeneity and a Theorem of Hilbert |
| title_full_unstemmed | A Fast and Reliable Solution to PnP, Using Polynomial Homogeneity and a Theorem of Hilbert |
| title_short | A Fast and Reliable Solution to PnP, Using Polynomial Homogeneity and a Theorem of Hilbert |
| title_sort | fast and reliable solution to pnp using polynomial homogeneity and a theorem of hilbert |
| topic | the PnP problem polynomial optimization |
| url | https://www.mdpi.com/1424-8220/23/12/5585 |
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