The Projective Invariants of Six 3D Points from Three 2D Uncalibrated Images

2010 ◽  
Author(s):  
Yuanbin Wang ◽  
Bin Zhang ◽  
Fenghua Hou
Keyword(s):  
Projective Invariants ◽  
Uncalibrated Images

Algorithm for the estimation of projective invariants of six points from uncalibrated images

1995 ◽  
Vol 31 (25) ◽  
pp. 2162-2163
Author(s):  
Yan Xiong ◽  
Jia-xiong Peng ◽  
Ming-yue Ding ◽  
Dong-hui Xue
Keyword(s):  
Projective Invariants ◽  
Uncalibrated Images

The unique solution of projective invariants of six points from four uncalibrated images

1997 ◽  
Vol 30 (3) ◽  
pp. 513-517 ◽  
Author(s):  
Xiong Yan ◽  
Peng Jia-Xiong ◽  
Ding Ming-Yue ◽  
Xue Dong-Hui
Keyword(s):  
Unique Solution ◽  
Projective Invariants ◽  
Uncalibrated Images

scholarly journals A Linear Method to Derive 3D Projective Invariants from 4 Uncalibrated Images

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
YuanBin Wang ◽  
XingWei Wang ◽  
Bin Zhang ◽  
Ying Wang
Keyword(s):  
Linear Method ◽  
Projective Reconstruction ◽  
Reconstruction Problem ◽  
Linear Transformations ◽  
Explicit Formulas ◽  
Projective Invariants ◽  
Projective Invariant ◽  
Uncalibrated Images ◽  
Bilinear Equations ◽  
2D 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 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.

Distributions of projective invariants and model-based machine vision

1996 ◽  
Vol 28 (03) ◽  
pp. 641-661 ◽  
Author(s):  
K. V. Mardia ◽  
Colin Goodall ◽  
Alistair Walder
Keyword(s):  
Machine Vision ◽  
Projective Transformation ◽  
Likelihood Estimation ◽  
Cauchy Distribution ◽  
Projective Line ◽  
Cross Ratio ◽  
Projective Invariants ◽  
Inference Problems ◽  
The Cross ◽  
Insight Into

In machine vision, objects are observed subject to an unknown projective transformation, and it is usual to use projective invariants for either testing for a false alarm or for classifying an object. For four collinear points, the cross-ratio is the simplest statistic which is invariant under projective transformations. We obtain the distribution of the cross-ratio under the Gaussian error model with different means. The case of identical means, which has appeared previously in the literature, is derived as a particular case. Various alternative forms of the cross-ratio density are obtained, e.g. under the Casey arccos transformation, and under an arctan transformation from the real projective line of cross-ratios to the unit circle. The cross-ratio distributions are novel to the probability literature; surprisingly various types of Cauchy distribution appear. To gain some analytical insight into the distribution, a simple linear-ratio is also introduced. We also give some results for the projective invariants of five coplanar points. We discuss the general moment properties of the cross-ratio, and consider some inference problems, including maximum likelihood estimation of the parameters.

IDENTIFICATION AND MATCHING OF PLANES IN A PAIR OF UNCALIBRATED IMAGES

2003 ◽  
Vol 17 (07) ◽  
pp. 1127-1143 ◽  
Author(s):  
B. S. BOUFAMA ◽  
D. J. O'CONNELL
Keyword(s):  
Prior Knowledge ◽  
Seed Region ◽  
Uncalibrated Images ◽  
Dense Matching ◽  
Key Points ◽  
Current Region ◽  
Correlation Techniques ◽  
Surface Discontinuity ◽  
And Performance ◽  
Plane Homography

In this paper, we propose a new method to simultaneously achieve segmentation and dense matching in a pair of stereo images. In contrast to conventional methods that are based on similarity or correlation techniques, this method is based on geometry, and uses correlations only on a limited number of key points. Stemming from the observation that our environment is abundant in planes, this method focuses on segmentation and matching of planes in an observed scene. Neither prior knowledge about the scene nor camera calibration are needed. Using two uncalibrated images as inputs, the method starts with a rough identification of a potential plane, defined by three points only. Based on these three points, a plane homography is then calculated and, used for validation. Starting from a seed region defined by the original three points, the method grows the current region by successive move/confirmation steps until occlusions and/or surface discontinuity occur. In this case, the homography-based mapping of points between the two images will not be valid anymore. This condition is detected by the correlation, used in the confirmation process. In particular, this method grows a region even across different colors as long as the region is planar. Experiments on real images validated our method and showed its capability and performance.

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