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A linear method to derive 3D projective invariants from 4 uncalibrated images.

Wang Y, Wang X, Zhang B, Wang Y - ScientificWorldJournal (2014)

Bottom Line: We show that this form of equations can be solved linearly and uniquely.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.

View Article: PubMed Central - PubMed

Affiliation: College of Information Science and Engineering, Northeastern University, Shenyang 110819, China.

ABSTRACT
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.

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Related in: MedlinePlus

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Mentions: We have validated the proposed method on the mathematica platform. The implementation is very simple. The code is given in Algorithm 1.


A linear method to derive 3D projective invariants from 4 uncalibrated images.

Wang Y, Wang X, Zhang B, Wang Y - ScientificWorldJournal (2014)

© Copyright Policy - open-access
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3926373&req=5

Mentions: We have validated the proposed method on the mathematica platform. The implementation is very simple. The code is given in Algorithm 1.

Bottom Line: We show that this form of equations can be solved linearly and uniquely.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.

View Article: PubMed Central - PubMed

Affiliation: College of Information Science and Engineering, Northeastern University, Shenyang 110819, China.

ABSTRACT
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.

Show MeSH
Related in: MedlinePlus