基本矩阵的5点和4点算法_英文_吴福朝

F = K (2) -T [ t] ×R K (1) -1
(2)
0 -t3 t 2
w here [ t ] × = t 3 0 -t 1 is the skew-symmet ric m at rix induced by the translation
-t2 t1
0
vect or t =(t 1 , t 2 , t 3)T .
176
ACTA A U T OM AT ICA SIN ICA
Vol .29
mat ri x and t a t ranslation vector .Suppose that the internal parameters of the tw o cameras are
respectively :
pressed as[ 6] :
H ≈ K (2)R
K (1)
-1
+
1 d
K
(2)tnT
K (1) -1
(4)
w here n is t he norm al of plane πunder the first camera frame , and d is the distance of the plane
dent of scene structure , and depends only on the camera internal parameters and their relative pose .T he fundamental mat rix encapsulates this epipolar geomet ry , and plays a very impo rtant role in 3D computer vision[ 1 , 2 , 3] , such as camera calibration , 3D reconstruction .The f undamental m at rix is a 3 ×3 matrix of rank 2 , and has 7 deg rees of freedom[ 4] .When the scene structure and camera' s motion are unknow n , the fundamental m at rix can be no n-linearly determined using at least 7 point co rrespondences[ 4] .In general at least 8 point correspondences are required to linearly det ermine the fundamental mat rix[ 5] .When there exist 4 co rrespondences induced from coplanar 3D point s (hereinaf ter , such a correspondence is called a coplanar one), only 6 point correspondences are required to linearly determine t he fundamental mat rix[ 6] .If there exist 4 coplanar correspondences , and if addi tional know ledge on camera mo tio n is available , t hen can the number of point co rrespondences be f urther reduced ?Our answer is af firmative .In this paper , the follow ing conclusions w ill be proved :If t he camera motion is of a pure translation , t hen given 5 point co rrespo ndences across tw o images , the fundamental matrix can be linearly determined if 4 of them are coplanar .In addit ion , we will show that if the distortion factor in t he pinhole camera model is null , then the fundamental mat rix can be linearly determined by only t hese 4 coplanar correspondences .
If {u(j 1) =(u(j 1), v(j 1), 1)T u(j 2) =(u(j 2), v(j 2), 1)T }is a set of point correspondences ,
then t he follow ing matrix equality holds :
Fu(j 1) = λe ×u(j 2)or equivalently , u(j2) T Fu(j1) = 0
第 29 卷 第 2 期 2003 年 3 月
自 动 化 学 报
ACT A A U TO MA T ICA SIN ICA
Vol.29 , N o.2 M ar ., 2003
5-point and 4-point Algorithm to Determine of the Fundamental Matrix1)
WU Fu-Chao HU Z han-Yi
(N ational Laboratory of Patt ern Recogni tion , Inst it ute of Automat ion , Chi nese Academy of S ciences , Bei ji ng 100080) (E-mail :{f cw u , huzy}@nlpr .ia.ac .cn)
T he paper is organized as follow s .In Sectio n 2 , some preliminaries on t he fundamental mat ri x and homog raphy are briefed .In Sect ions 3 and 4 , 5-point and 4-piont algori thm for the fundamental mat rix computation are elaborated.Some conclusions are given in Section 5 .
2 The fundamental matrix and homography
The fundamental matrix F Let (R , t)be the camera' s motion , that is , the t ransf ormation f rom t he f irst camera
frame (bef ore mo tion)t o the seco nd (af ter mot ion)is x(2) =Rx(1)+t , w here R is a rotation
1)Support ed by t he National Natu ral Science Foundat ion of P.R .C hi na(60075004 , 60033010)and M ultidisciplinary Research Program of CA S(K JCX1-07) Received November 26 , 2001 ;in revi sed f orm June 20 , 2002 收稿日期 2001-11-26 ;收修改稿日期 2002-06-20
Key words Fundamental matrix , homography , camera intrinsic parameters
1 Introduction T he epipolar geometry is the f undament al const raint between tw o images .It is indepen-
f
(j) u
s(j)
u
(j) 0
K (j) = 0
f
(j) v
v(0j) , j =1 , 2
(1)
001
T he f undament al matrix F corresponding to the tw o im ages(I(1), I(2))can be expressed as[ 6] :
8 point correspondences, it is possible to determine the fundamental mat rix by solving a set of
Байду номын сангаас
linear equations as show n in (3).
The homography H T he homog raphy H of the t wo images (I(1), I(2))from a same space plane πcan be ex-
(3)
w here λis a non-zero scale f actor , and e is the epipole in t he second image . T he fundamental mat rix is unique up to a non-zero scale facto r .In general , given at least
Abstract The fundamental matrix encapsulates all the information between tw o 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 g iven 5 point correspondences across tw o images , the fundamental matrix can be linearly determined if four correspondences of the 5 ones are from coplanar space points (called coplanar co rrespondences).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 cor respondences .T o our knowledge , such results have not been reported yet in the literature .