基本矩阵的5点和4点算法英文

dent of scene st ruct ure , and depends only on t he camera internal parameters and t heir relative pose. The f undamental mat rix encapsulates t his epipolar geomet ry , and plays a very important role in 3D computer vision[1 ,2 ,3 ] , such as camera calibration , 3D reconst ruction. The f unda2 mental mat rix is a 3 ×3 mat rix of rank 2 , and has 7 degrees of f reedom[4 ] . When t he scene st ruct ure and camera’s motion are unknown , t he f undamental mat rix can be non2linearly deter2 mined using at least 7 point correspondences[4 ] . In general at least 8 point correspondences are required to linearly determine t he f undamental mat rix[5 ] . When t here exist 4 correspondences induced f rom coplanar 3D point s (hereinafter , such a correspondence is called a coplanar one) , only 6 point correspondences are required to linearly determine t he f undamental mat rix[6 ] . If t here exist 4 coplanar correspondences , and if additional knowledge on camera motion is avail2 able , t hen can t he number of point correspondences be f urt her reduced ? Our answer is affirma2 tive. In t his paper , t he following conclusions will be proved : If t he camera motion is of a pure t ranslation , t hen given 5 point correspondences across two images , t he f undamental mat rix can be linearly determined if 4 of t hem are coplanar. In addition , we will show t hat if t he distortion factor in t he pinhole camera model is null , t hen t he f undamental mat rix can be linearly deter2 mined by only t hese 4 coplanar correspondences.
Key words Fundamental matrix , homography , camera intrinsic parameters
1 Introduction The epipolar geomet ry is t he f undamental const raint between two images. It is indepen2
F = K (2) - T[ t ] ×R K (1) - 1
(2)
0 - t3 t2
where [ t ] × = t3 0 - t1 is t he skew2symmet ric mat rix induced by t he t ranslation
- t2 t1
0
vector t = ( t1 , t2 , t3) T .
176
ACTA AU TOMA TICA SIN ICA
Vol. 29
mat rix and t a t ranslation vector. Suppose t hat t he internal parameters of t he two cameras are
respectively :
f
8 point correspondences , it is possible to determine t he f undamental mat rix by solving a set of
linear equations as shown in (3) .
The homography H The homograp hy H of t he two images ( I (1) , I (2) ) f rom a same space plane πcan be ex2
2 The f undamental matrix and homography
The f undamental matrix F Let ( R , t) be t he camera’s motion , t hat is , t he t ransformation f rom t he first camera
第 29 卷 第 2 期 2003 年 3 月
自 动 化 学 报
ACTA AU TOMA TICA SIN ICA
Vol129 ,No12 Mar. ,2003
52point and 42point Algorithm to Determine of the Fundamental Matrix1)
If
{
u
(1) j
=
(
u
(1) j
,
v
(1) j
,1)
T
∴
u
(2) j
=
(
u
(2) j
,
v
(2) j
,1) T}
is
a
set
of
point
correspondences ,
t hen t he following mat rix equality holds :
Fu
(1) j
=
λe
×
u
(2) j
or
Abstract The fundamental matrix encapsulates all the information between two images , and plays a very important role in camera calibration and 3D reconstruction. In this paper , the following conclusions have been rigorously proved : If the camera motion is of a pure translation , then given 5 point correspon2 dences across two images , the fundamental matrix can be linearly determined if four correspondences of the 5 ones are from coplanar space points (called coplanar correspondences) . In addition , we show that if the distortion factor in the pinhole camera model is null , then the fundamental matrix can be linearly determined by only these 4 coplanar correspondences. To our knowledge , such results have not been re2 ported yet in the literature.
The paper is organized as follows. In Section 2 , some preliminaries on t he f undamental mat rix and homograp hy are briefed. In Sections 3 and 4 , 52point and 42piont algorit hm for the fundamental matrix computation are elaborated. Some conclusions are given in Section 5.
πto t he origin.
The homograp hy
H is uniquely defined up to a non2zero scale factor.
If
{
u
(1) j
∴
u
(2) j
}
is
a
set of coplanar correspondences , t hen t he following equality holds :
WU Fu2Chao HU Zhan2Yi
( N ational L aboratory of Pattern Recognition , I nstit ute of A utom ation , Chi nese A cademy of Sciences , Beiji ng 100080) ( E2mail : {fcwu , huzy} @nlpr. ia. ac. cn)