A Linear Method to Derive 3D Projective Invariants from 4 Uncalibrated Images
A well-known method proposed by Quan to compute projective invariants of 3D points uses six points in three 2D images. The method is nonlinear and complicated. It usually produces three possible solutions. It is noted previously that the problem can be solved directly and linearly using six points i...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | The Scientific World Journal |
| Online Access: | http://dx.doi.org/10.1155/2014/109318 |
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| author | YuanBin Wang XingWei Wang Bin Zhang Ying Wang |
| author_facet | YuanBin Wang XingWei Wang Bin Zhang Ying Wang |
| author_sort | YuanBin Wang |
| collection | DOAJ |
| description | A well-known method proposed by Quan to compute projective invariants of 3D points uses six points in three 2D images. The method is nonlinear and complicated. It usually produces three possible solutions. It is noted previously that the problem can be solved directly and linearly using six points in five images. This paper presents a method to compute projective invariants of 3D points from four uncalibrated images directly. For a set of six 3D points in general position, we choose four of them as the reference basis and represent the other two points under this basis. It is known that the cross ratios of the coefficients of these representations are projective invariant. After a series of linear transformations, a system of four bilinear equations in the three unknown projective invariants is derived. Systems of nonlinear multivariable equations are usually hard to solve. We show that this form of equations can be solved linearly and uniquely. This finding is remarkable. It means that the natural configuration of the projective reconstruction problem might be six points and four images. The solutions are given in explicit formulas. |
| format | Article |
| id | doaj-art-ec62c604c96a4de1b28fde03baa825de |
| institution | Kabale University |
| issn | 2356-6140 1537-744X |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | The Scientific World Journal |
| spelling | doaj-art-ec62c604c96a4de1b28fde03baa825de2025-02-03T01:21:30ZengWileyThe Scientific World Journal2356-61401537-744X2014-01-01201410.1155/2014/109318109318A Linear Method to Derive 3D Projective Invariants from 4 Uncalibrated ImagesYuanBin Wang0XingWei Wang1Bin Zhang2Ying Wang3College of Information Science and Engineering, Northeastern University, Shenyang 110819, ChinaCollege of Information Science and Engineering, Northeastern University, Shenyang 110819, ChinaCollege of Information Science and Engineering, Northeastern University, Shenyang 110819, ChinaDepartment of Computer Science, Worcester Polytechnic Institute, Worcester, MA 01609, USAA well-known method proposed by Quan to compute projective invariants of 3D points uses six points in three 2D images. The method is nonlinear and complicated. It usually produces three possible solutions. It is noted previously that the problem can be solved directly and linearly using six points in five images. This paper presents a method to compute projective invariants of 3D points from four uncalibrated images directly. For a set of six 3D points in general position, we choose four of them as the reference basis and represent the other two points under this basis. It is known that the cross ratios of the coefficients of these representations are projective invariant. After a series of linear transformations, a system of four bilinear equations in the three unknown projective invariants is derived. Systems of nonlinear multivariable equations are usually hard to solve. We show that this form of equations can be solved linearly and uniquely. This finding is remarkable. It means that the natural configuration of the projective reconstruction problem might be six points and four images. The solutions are given in explicit formulas.http://dx.doi.org/10.1155/2014/109318 |
| spellingShingle | YuanBin Wang XingWei Wang Bin Zhang Ying Wang A Linear Method to Derive 3D Projective Invariants from 4 Uncalibrated Images The Scientific World Journal |
| title | A Linear Method to Derive 3D Projective Invariants from 4 Uncalibrated Images |
| title_full | A Linear Method to Derive 3D Projective Invariants from 4 Uncalibrated Images |
| title_fullStr | A Linear Method to Derive 3D Projective Invariants from 4 Uncalibrated Images |
| title_full_unstemmed | A Linear Method to Derive 3D Projective Invariants from 4 Uncalibrated Images |
| title_short | A Linear Method to Derive 3D Projective Invariants from 4 Uncalibrated Images |
| title_sort | linear method to derive 3d projective invariants from 4 uncalibrated images |
| url | http://dx.doi.org/10.1155/2014/109318 |
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