Experimental and Theoretical Scrutiny of the Geometric Derivation of the Fundamental Matrix

被引:0
作者
Basta, Tayeb [1 ]
机构
[1] Al Ghurair Univ, Coll Engn & Comp, Dubai, U Arab Emirates
来源
AIPR 2020: 2020 3RD INTERNATIONAL CONFERENCE ON ARTIFICIAL INTELLIGENCE AND PATTERN RECOGNITION | 2020年
关键词
Cross product; skew-matrix; essential matrix; fundamental matrix; epipolar geometry; stereo vision; 3D reconstruction;
D O I
10.1145/3430199.3430227
中图分类号
TP18 [人工智能理论];
学科分类号
081104 ; 0812 ; 0835 ; 1405 ;
摘要
In this paper, we prove mathematically that the geometric derivation of the fundamental matrix.. of the two-view reconstruction problem is flawed. Although the fundamental matrix approach is quite classic, it is still taught in universities around the world. Thus, analyzing the derivation of F now is a non-trivial subject. The geometric derivation of E is based on the cross product of vectors in R-3. The cross product (or vector product) of two vectors is x x y where.. = < x(1), x(2), x(3)> and y = < y(1), y(2), y(3)> in R-3. The relationship between the skew-matrix of a vector.. in R-3 and the cross product is [t](x) y = t x y for any vector t in R3. In the derivation of the essential matrix we have E = [t](x).. which is the result of replacing t x R by [t](x) R, the cross product of a vector t and a 3x3 matrix R. This is an undefined operation and therefore the essential matrix derivation is flawed. The derivation of F, is based on the assertion that the set of all points in the first image and their corresponding points in the second image are protectively equivalent and therefore there exists a homography H-pi between the two images. An assertion that does not hold for 3D non-planar scenes.
引用
收藏
页码:25 / 29
页数:5
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