Degenerate configurations of space points and two camera optical cen- ... twisted cubic under a single view and for degeneracy from ruled quadric surface.
Twisted Cubic: Degeneracy Degree and Relationship with General Degeneracy Tian Lan, YiHong Wu, and Zhanyi Hu National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Science, P.O. Box 2728, 100190, Beijing, China {tlan, yhwu, zyhu}@nlpr.ia.ac.cn http://www.springer.com/lncs
Abstract. Fundamental matrix, drawing geometric relationship between two images, plays an important role in 3-dimensional computer vision. Degenerate configurations of space points and two camera optical centers affect stability of computation for fundamental matrix. In order to robustly estimate fundamental matrix, it is necessary to study these degenerate configurations. We analyze all possible degenerate configurations caused by twisted cubic and give the corresponding degenerate rank for each case. Relationships with general degeneracies, the previous ruled quadric degeneracy and the homography degeneracy, are also reported in theory, where some interesting results are obtained such as a complete homography relation between two views. Based on the result of the paper, by applying RANSAC for degenerate data, we could obtain more robust estimations for fundamental matrix.
1
Introduction
Fundamental matrix describes geometric relation between two 2-dimensional views. It plays an important role in image matching, epipolar geometry, camera motion determination, camera self-calibration and 3-dimensional reconstruction. Robust and accurate estimation for fundamental matrix has been the research focus of extensive researchers[1–8]. From at least seven pairs of point-point correspondences between two views, the fundamental matrix can be estimated. Sometimes, a reliable estimation cannot be obtained, no matter how many correspondences are used. One of the main reasons is that the cameras and the scene lie on a degenerate or quasidegenerate configuration. If a space configuration is degenerate mathematically but the noise from the measured image makes it non-degenerate, any estimation under such a configuration would be useless [9]. It follows that we should know what configurations might cause degeneracy for estimating the fundamental matrix. Moreover in order for a robust RANSAC like [10, 7], we still need to know how great the degenerate degree is, namely, to know the degenerate rank of the coefficient matrix of the equations for computing fundamental matrix.In [3], RANSAC loop to estimate relation from quasi-degenerate data is reported,
where the degenerate configurations need not be known. This is not equivalent to say the studies on the degenerate configurations are useless. At least, such studies can give more geometric intuition, which could be as guidance for placing cameras to avoid degeneracy in practice. Furthermore, if we can judge the degeneracy by applying geometric knowledge, RANSAC work will be much easier. Due to the importance of degeneracy analysis, many of such works have been reported previously. The planar scene is a trivial degenerate configuration for computing fundamental matrix, where the images can provide only six independent constraints [11, 7] but the general fundamental matrix has seven degrees of freedom. Degeneracy from twisted cubic configuration has also been discussed. In [12], Buchanan stated that camera calibration from known space points under a single view is not unique if the optical center and the space points lie on a twisted cubic. The corresponding detection as well as emendations including other unreliability was given by Wu et al [13]. Then, under two views, Maybank [14] analyzed the characterizations of horopter curve and the relations between the curve and the ambiguous case of reconstruction. The horopter curve is regarded as a twisted cubic, which intersects the plane at infinity at three particular points. The ambiguous case of reconstruction implies ambiguity of fundamental matrix. Luong and Faugeras reported the stability for computing fundamental matrix caused by quadric critical surface in [15]. Hartley and Zisserman [11] also gave systematic discussions for degeneracy of camera projection estimation from twisted cubic under a single view and for degeneracy from ruled quadric surface under two views. Under three views, critical configurations are provided in [16], which is an extension of the critical surface under two views. Degeneracy under a sequence of images is also investigated [17, 18]. Maybank and Shashua [18] pointed out there is a three-way ambiguity for reconstruction from images of six points when the six points and the camera optical centers lie on a hyperboloid of one sheet. In [17], Hartley and Kahl presented a classification of all possible critical configurations for any number of points from three images and showed that in most cases, the ambiguity could extend to any number of cameras. Relative to the above works on degenerate configurations, there are fewer deep studies on degeneracy degrees of degenerate configurations. Torr et al [7] catalogued all two-view non-degenerate and degenerate cases in a logical way by dimensions of the right null space of equations on fundamental matrix and then proposed a PLUNDER-DL method to detect degeneracy and outliers. Chum et al [10] also analyzed those dimensions when the two views or most of the point correspondences are related by a homography and presented an algorithm to estimate fundamental matrix through detecting the homography degeneracy. They all [7, 10] generalized the robust estimator RANSAC [19]. The plane degeneracy in [7, 10] is consistent with the ruled quadric degeneracy proposed by Hartley and Zisserman [11] because a plane and two camera optical centers always lie on a degenerate ruled quadric. What are the degeneracy degrees when estimating the fundamental matrix for other non-trivial degenerate configurations? In this paper, we discuss all possible degenerate situations caused by twisted cubic
and give the corresponding degeneracy degrees. Let SO be a set of space points and the two camera optical centers. We find that if all the points of SO lie on a twisted cubic, the configuration is degenerate for estimating fundamental matrix and the corresponding rank of coefficient matrix is five; if all the points other than one lie on a twisted cubic, the corresponding rank is six; if all the points other than two lie on a twisted cubic, the corresponding rank is seven. The previous general degeneracies are ruled quadric degeneracy and homography degeneracy. Few studies are given on relationships of twisted cubic degeneracy with them. We investigate the relationships in detail and then present our contribution relative to the general degeneracies. The organization of the paper is as follows. Some preliminaries are listed in Section 2. The complete and unified degeneracy study from twisted cubic is elaborated in Section 3. Some experimental results are displayed in Section 4 and Section 5 makes some conclusions.
2
Preliminaries
The camera model used is a perspective camera. A space point or its homogeneous coordinates is denoted by M, an image point or its homogeneous coordinates is denoted by m, P denotes the camera projection matrix, and O denotes the camera optical center. Under two views, P0 denotes the second camera projection matrix, O0 denotes its optical center, and m0 denotes the corresponding image point of m. Let F be the fundamental matrix between the two views. Other vectors or matrices are also denoted in boldface. The symbol ≈ means equality up to a scale. Camera Projection Matrix: Mi , i = 1 . . . N are 3-dimensional space points. And their corresponding image points are mi , i = 1 . . . N . The camera projection matrix P is a 3 × 4 matrix such that mi ≈ PMi . For the camera optical center O, we have the equation: PO = 0
(1)
Fundamental matrix: Let m0i be the corresponding image points of the space points Mi under another view. Then, mi and m0i are related by the fundamental matrix F through: m0T (2) i Fmi = 0, i = 1 . . . N f1 f2 f3 ¡ ¢ ¡ ¢ We denote F as f4 f5 f6 If mi ≈ ui vi wi and m0i ≈ u0i vi0 wi0 , we f7 f8 f9 expand (2) and have:
... u0i ui u0i vi u0i wi vi0 ui vi0 vi vi0 wi wi0 ui wi0 vi wi0 wi f =0 ... N ×9
(3)
¡ ¢T where f = f1 f2 f3 f4 f5 f6 f7 f8 f9 is the vector consisting of all elements in F. The N × 9 coefficient matrix of f is denoted by G . ¡ ¢T Twisted cubic: The locus of points X = X Y Z T in a 3-dimensional projective space satisfying the parametric equation: ¡ ¢T ¡ ¢T XY ZT ≈ H θ3 θ2 θ 1 (4) is a twisted cubic, where H is a 4 × 4 matrix and θ is the parameter[20]. Twisted cubic is an extension of a conic to 3-dimensional space by increasing the degree of curve parameter from two to three. The properties of twisted cubic underlie many of the ambiguous cases that arise in 3-dimensional reconstruction.
3
Degeneracies from twisted cubic
The previously known degenerate configuration of two views for fundamental matrix or projective reconstruction is that two camera optical centers and all space points lie on a ruled quadric. For such a general ruled quadric, the right null space of G in (3) is of dimension two as given in the section 2 of [7] and in the paragraph five of the introduction section of [16]. The more critically degenerate configuration is from a plane, of which the right null space of G in (3) is of dimension three [10, 7]. This is not at the most since the nontrivial degenerate configuration—-twisted cubic can cause more critically degeneracy than a plane as shown below. 3.1
Degeneracy degree from twisted cubic
In (3), if the rank of the coefficient matrix G is 8, then F can be determined uniquely by linear 8-point algorithm. Otherwise, if the rank of G is 7, the solution of f from (3) has one degree of freedom and the freedom can be removed by det(F) = 0 to obtain three or one solution. But if we only rely on the linear equations (3), the freedom cannot be removed. If the rank is 6 or less than 6, solutions of f has two or more degrees of freedom and so F cannot be determined finitely. The configuration making the rank of G deficient is degenerate for computing F. Due to noise of image data, generally we always can calculate a unique solution of f from (3) with 8 corresponding points. However, the degenerate configurations or the configurations near to degeneracy will terribly influence stability of the calculation. Therefore, in order for robust estimation of fundamental matrix, we need to know the degenerate configurations. The degenerate configurations from twisted cubic and the corresponding degeneracy degrees are provided in the following theorem. Theorem 1 Let SO be a set of space points and two camera optical centers for capturing these points. If all the points of SO are on a twisted cubic, then the rank of the coefficient matrix G for computing F is five. If all the points other than one of SO are on a twisted cubic, the rank of G is six. If all the points other than two of SO are on a twisted cubic, the rank of G is seven.
Proof: Firstly, we give the proof when SO are all on a twisted cubic. According to (4), assume the parametric equation of this twisted cubic is ¡ ¢T H θ3 θ2 θ 1 , where H is a 4 × 4 matrix. Let the parameter of the space point Mi be θi and the parameters of the two camera optical centers be θ0 , θ00 . ¡ ¢T ¡ ¢T ¡ ¢T Then,Mi = H θi3 θi2 θi 1 , O = H θ03 θ02 θ0 1 , O0 = H θ003 θ002 θ00 1 . By (1), we have: ¢T ¡ 0 = PO = PH θ03 θ02 θ0 1 ,
¢T ¡ 0 = P0 O0 = P0 H θ003 θ002 θ00 1
(5)
where P, P0 are the two camera projection matrices. So we also have: ¡ ¢T mi ≈ PMi = PH θi3 θi2 θi 1 ,
¡ ¢T m0i ≈ P0 Mi = P0 H θi03 θi02 θi0 1 (6)
Do subtraction from both sides for (5) and (6), we obtain: ¡ ¢T ¡ ¢T mi ≈ PH θi3 θi2 θi 1 − PH θ03 θ02 θ0 1 ¡ ¢T ≈ PH(θi − θ0 ) θi2 + θ02 + θ0 θi θi + θ0 1 0 ¡ ¢T ≈ PH θi2 + θ02 + θ0 θi θi + θ0 1 0
(7)
0 0 0 0 q1 q2 q3 q4 q1 q2 q3 q4 Denote PH as Q = q5 q6 q7 q8 , and P0 H as Q0 = q50 q60 q70 q80 . 0 0 0 q12 q9 q10 q11 q12 q90 q10 q11 Then, (7) is changed into: q1 q2 q3 mi ≈ q5 (θi2 + θ02 + θ0 θi ) + q6 (θi + θ0 ) + q7 (8) q9 q10 q11 Similarly, do subtraction from both sides for (5) and (6), there is: 0 0 0 q1 q2 q3 m0i ≈ q50 (θi2 + θ00 2 + θ00 θi ) + q60 (θi + θ00 ) + q70 0 0 q90 q10 q11
(9)
¡ ¢ θi varies with mi m0i , while qk , qk0 , θ0 , θ00 are unchanged. Substitute (8) and (9) into (3), we get the coefficient matrix G with each element of the i-th row being a four-order polynomial in θi as c1 θi4 + c2 θi3 + c3 θi2 + c4 θi + c5 . The coefficients cs of θi in these four-order polynomials are functions on while qk , qk0 , θ0 , θ00 . Since while qk , qk0 , θ0 , θ00 are not varying with image pair varying, cs are also not varying with the row number varying. It follows that G is in this form: ... G = g1 (θi ) g2 (θi ) g3 (θi ) g4 (θi ) g5 (θi ) g6 (θi ) g7 (θi ) g8 (θi ) g9 (θi ) (10) ...
where gj (θ) = c1j θ4 + c2j θ3 + c3j θ2 + c4j θ +c5j , j = 1 . . . N . Weequivalently 4 3 2 c11 . . . c1j . . . c19 θ1 θ1 θ1 θ1 1 c21 . . . c2j . . . c29 ... c31 . . . c3j . . . c39 change G into: G = θi4 θi3 θi2 θi 1 c41 . . . c4j . . . c49 ... N ×5 c51 . . . c5j . . . c59 5×9 From the expression, we know the rank of G is generally five. By now, we proved that if all the space points and the optical centers of the two cameras are on a twisted cubic, the rank of the coefficient matrix G is five. If a camera optical center does not lie on the twisted cubic determined by another camera optical center and the space points, assumed to be O, then the degree of θi for representing mi in (6) can not decrease to two but the degree for representing m0i can do, i.e. m0i is still in the form (9). Thus, the degrees of θi in the obtained coefficient matrix G of (10) become into 5. Then by the same reason as above, we have the corresponding rank 6. If the point not lying on the twisted cubic is one of the space points other than one of the camera optical center, assumed to be Mi0 , then the row in G from the image pair mi0 , m0i0 is not in the polynomial form of some θ. It follows that this row is not linearly related to other rows in general. Thus, the rank of G increases from five to six. Similarly, if all the points other than two of SO are in a twisted cubic, the rank of G is seven. The theorem is proved. In the above theorem, we analyze all possible degenerate configurations for computing F caused from twisted cubic. In all the cases, F can not be determined finitely by linear 8-point algorithm and the dimensions of the right null space of G in (3) are respectively 4, 3, 2. By 7-point algorithm, F still can not be solved in rank 5, 6 cases but can be solved in rank 7 case. 3.2
Relationship with ruled quadric degeneracy
The degenerate configuration of two views for reconstruction is well known as a ruled quadric [11]. The theorem in Section 3.1 is consistent with the ruled quadric degeneracy. In this subsection, we at first give two lemmas about twisted cubic and ruled quadric for the consistency. Then, the contribution of our work is discussed. In projective space, quadrics are classified into ruled and unruled ones. Quadrics with positive index of inertia 2 are ruled quadrics and the degenerate quadrics except one point case are all ruled ones [11]. Here the positive index of inertia means the number of positive entries in the canonical form for a quadric. Lemma 1 In a 3-dimensional projective space, a proper real twisted cubic can always be embedded on a ruled quadric, conversely, any quadric containing a proper real twisted cubic is a ruled one. Due to space limit, the proof is omitted. It is similar for the following lemmas.
Lemma 2 In a 3-dimensional projective space, if seven points of a real proper twisted cubic lie on a quadric, then the whole twisted cubic lies on the quadric. Remark 1. By Lemma 1 and Lemma 2, we conclude that a twisted cubic plus one or two points can be embedded on a ruled quadric. We take seven points on the twisted cubic and combine the additional one or two points to generate a quadric. This is reasonable because generally nine space points uniquely determine a quadric. Since this quadric contains seven points of the twisted cubic, by Lemma 2, we know it contains the whole twisted cubic. Furthermore by Lemma 1, we know the generated quadric is ruled. It follows that the theorem in Section 3.1 is consistent with the previous ruled quadric degeneracy. Remark 2. The contribution of Theorem 1 is that it gives more intuitive degeneracy and the degeneracy degrees for all possible cases caused by twisted cubic. For the general ruled quadric degeneracy, there are a finite number of solutions for the fundamental matrix by combining with the additional constraint of det(F) = 0. This degeneracy degree is the same as the rank 7 case in the theorem. For rank 5, 6 cases in the theorem, the degeneracy is more critical which makes the fundamental matrix free in a four- or three-dimensional space. Even though by the additional constraint det(F) = 0, it cannot be solved. These details are not discussed in the previous ruled quadric degeneracy. Usually, six points determine a unique twisted cubic and nine points determine a unique quadric. A twisted cubic is not a class in the ruled quadrics. Therefore, from fewer non-incidence points to make F computations, quadric degeneracy may not come to mind, which also could ignore the twisted cubic degeneracy. However indeed the twisted cubic can make the F computation degenerate severely as shown in the theorem in Section 3.1. 3.3
Relationship with H-degeneracy
One previous work closely related to ours is the H-degeneracy studied by Chum et al. [10], where the H-degeneracy means the degeneracy caused by a 3 × 3 homography between two views. They also discussed the degeneracy degrees for the F computation and mentioned the twisted cubic degeneracy. There are differences between our work and theirs. In this subsection, we discuss the contribution of our work relative to the study [10]. Firstly, we give complete cases that two views are related by a 3 × 3 homography. Lemma 3 If the image point correspondences (mi , m0i ) between two views are related by a homography H , that is m0i = Hmi , then generally there are the following complete three situations: 1)The camera performs a pure rotation; 2)The space points are coplanar; 3)The space points and the two camera optical centers lie on a twisted cubic.
The above classification of the three cases are complete. In [10], Chum et al analyzed degrees of the H-degeneracy on three cases: i) two views are related by a homography; ii) all image point pairs other than one pair are related by a homography; iii) all image point pairs other than two pairs are related by a homography. Then based on the degrees, they developed a DEGENSAC algorithm to compute F unaffected by a dominant plane by detecting H-degeneracy. The relationship and differences between our work and Chum et al’s [10] are as follows. The cases in the theorem of Section 3.1 related to the H-degeneracy are: (a1) The two camera optical centers and all the space points lie on a twisted cubic. (a2) The two camera optical centers and all other than one the space points lie on a twisted cubic. (a3) The two camera optical centers and all other than two the space points lie on a twisted cubic. According to Lemma 3, the two views in (a1) are related by a homography, in (a2) the image point pairs except for one pair are related by a homography, and the image point pairs except for two pairs are related by a homography in (a3). Although these geometric relations between the two views in the three cases are the same as Chum et al’s, the degeneracy degrees are different. Here in our work, the degeneracy is more critical. For case (a1), since the coefficient matrix has rank 5, the linear space of F has dimension 4 while in [10] for two views related with a homography the dimension is 3. For case (a2), the corresponding dimension is 3 while that in [10] is 2. For case (a3), the corresponding dimension is 2 while that in [10] could be 1 if linear 8-point algorithm is applied. It follows that the twisted cubic cases could cause more critical degeneracy than the plane cases, though they have the same geometric H-relations between the two views. The cases in the theorem of Section 3.1 not involved in [10] are: (b1) All the space points and one of the camera optical centers lie on a twisted cubic. The other camera optical center is not on this twisted cubic. (b2) All other than one of the space points and one of the camera optical centers lie on a twisted cubic. The other camera optical center is not on this twisted cubic. (b3) All the space points but the two camera optical centers lie on a twisted cubic. The three cases do not fall into the work of [10]. In the three cases at least one of the optical centers does not lie on the twisted cubic and the space points are also not coplanar. Thus according to Lemma 3 all or most of the image point pairs in each case (b1), (b2), (b3) do not agree to a homography relation. Therefore, our work not only develops the work in [10] but also makes some new contribution in theory. The aim of [10] is to stably estimate F unaffected by a dominant plane. We also will explore a detection method on the degeneracy caused from twisted cubic and then apply the RANSAC on degenerate data in [13] to robustly compute fundamental matrix. Detection on the degeneracy deserves studies also because usually computations of matrix rank or its singular values are very sensitive to noise and presetting a threshold to discriminate the degeneracy from the non-degeneracy is not easy, as pointed out in [21].
4
Experiment
We performed both simulations and experiments on real data. The results verify the established theorem. One group of the experiments is reported below. 4.1
Simulations
The parametric equation of a space twisted cubic is: 3 2 5 −3 2.5 θ 1 −1 12 1 θ2 M≈ 6 −15 −2 3 θ −7 5 3 2 1
(11)
Ten points Mi on this twisted cubic are taken, of which the parameters are respectively −1.1, −0.35, −0.75, −0.22, −0.6, 0.1, −0.1, 0.2, 1.9, −2. At first, we consider the case of that both the two optical centers and the space points lie on the same twisted cubic. Let the two points of the twisted cubic with parameters 1.25, 1.5 be the two optical centers O, O0 . The space distribution is shown as Fig.1. Then, the corresponding camera projection matrices consistent
Fig. 1. Space points and two optical centers lie on a twisted cubic, where * denotes the space points, and o denotes the camera optical centers.
1000 0 512 43198 with the optical centers are set as follows:P = 0 900 384 95484 , 0 0 1 −103 −529 648.1 287.4 −4321.3 P0 = 338.6 −295.4 748.7 −1810.1 . Projected by P, P0 , we generated two −0.7 −0.1 0.7 −3.9 simulated images of the ten space points and established the equations on the fundamental matrix. Under the noise level of zero, the rank of the coefficient matrix of these equations could be computed out and the result is as five. We also tested the case when one of the optical centers does not lie on the ¡ ¢T which is not on this twisted twisted cubic any more. Let Q2 = 3 7.3 2 1
cubic, the corresponding camera projection matrix is set as: 1000 0 512 −4024 P2 = 0 900 384 −7338 . By this camera projection matrix, another new 0 0 1 −2 image is generated. From this image and that of P0 , we established equations on the fundamental matrix and then computed the rank of the coefficient matrix under noise level of zero. The result is six that is consistent with the proposed theorem. Finally, we give the experimental result of the case when the two optical centers do not lie on the twisted cubic (11). Another optical center is set as ¡ ¢T which is also far away from the twisted cubic (11). Q02 = 0.67 −1.49 −2.8 1 From the two images generated, we computed rank of the coefficient matrix of the equations on the fundamental matrix and the result is seven. If there are one or two of the space points that do not lie on the twisted cubic determined by other space points and the optical centers, the same results are obtained. All the experimental results validate the theorem in Section 3.1. However, we find that the direct computation on the matrix rank or the rank computation by the singular values is only correct in the absence of noise. When we add noise to the image, the rank of the coefficient matrix becomes to 8 and the computation becomes very unstable. Therefore, in order to robustly estimate the fundamental matrix, it is necessary to develop a method of detection on the degenerate configuration. We will explore a detection method on the degeneracy caused from twisted cubic and apply the RANSAC on degenerate data in [19, 13] to robustly compute the fundamental matrix. 4.2
Experiments on real data
We tested the degeneracy of six points from real data. The experiments of more points on real data need to be performed after the detection on degenerate data and the corresponding RANSAC are proposed. We took the images of six space points at different viewpoints. Four of them with a size of 640 × 480 pixels are shown in Fig. 2, where the dot points denote the used image points. In order to know whether the six space points and the corresponding optical center lie on a twisted cubic or not, we measured the space coordinates of the six points and then by the criterion function proposed in [13] detected the situation. The values of the criterion function on the four images in Fig. 2 are respectively 1.0655, 1.0504, 2.2934, and 2.5091. Then, by the method in [13], we know that the six points and the two corresponding optical centers of Fig. 2 (a)(b) are on a same twisted cubic, while the six points and the two corresponding optical centers of (c)(d) are not. We also computed the singular values of the coefficient matrix G in (3). The result from the two images in Fig. 2 (a)(b) is: 623427.73, 156095.86, 41657.74, 6772.02, 79.53,9.81. And the result from the two images in Fig. 2 (c)(d) is:508796.41, 138904.18, 33040.13, 9883.42, 112.31, 37.68. We see that the condition number of coefficient matrix G from (a) (b)in Fig. 2 is larger
(a)
(b)
(c)
(d)
Fig. 2. Images of six points, where the space points and the two camera optical centers (a)(b): are on a same twisted cubic; (c)(d): are not on a same twisted cubic
than that from (c)(d). However, usually it is difficult to detect the degeneracy by using the condition number because the singular values are very sensitive to noise and presetting a threshold to discriminate the degeneracy from the nondegeneracy is not easy, as pointed out in [21]. We found sometimes the condition number of the degeneracy is yet smaller than that of the non-degeneracy. This is why we would like to pursue a detection method for the degeneracy from two image data in the future.
5
Conclusion
This paper provides all the possible degenerate configurations caused by twisted cubic and the corresponding degeneracy degrees for estimating fundamental matrix. Relationships with the ruled quadric degeneracy and the homograghy degeneracy are also given. The result is helpful to improving the accuracy of the estimations. Indeed, for a robust RANSAC, initial samples with worse estimations should be removed or mended. These initial samples not only are those including mismatching pairs but also are those that are degenerate. The latter case usually is ignored by people but really affects stability of the computations. The reason of the ignorance may be that the degeneracy has not been studied thoroughly. We give some research on the degeneracy in this work and further robust detection on the twisted cubic configurations will be developed. Acknowledgement This work was supported by the National Natural Science Foundation of China under grant No. 60633070, 60773039.
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