Twisted Cubic:Degeneracy Degree and Relationship with

General Degeneracy

Tian Lan, YiHong Wu, and Zhanyi Hu

National Laboratory of Pattern Recognition, Institute of Automation, ChineseAcademy 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 betweentwo images, plays an important role in 3-dimensional computer vision.Degenerate configurations of space points and two camera optical cen-ters affect stability of computation for fundamental matrix. In orderto robustly estimate fundamental matrix, it is necessary to study thesedegenerate configurations. We analyze all possible degenerate configu-rations caused by twisted cubic and give the corresponding degeneraterank for each case. Relationships with general degeneracies, the previousruled quadric degeneracy and the homography degeneracy, are also re-ported in theory, where some interesting results are obtained such as acomplete homography relation between two views. Based on the resultof the paper, by applying RANSAC for degenerate data, we could obtainmore robust estimations for fundamental matrix.

1 Introduction

Fundamental matrix describes geometric relation between two 2-dimensionalviews. It plays an important role in image matching, epipolar geometry, cameramotion determination, camera self-calibration and 3-dimensional reconstruction.Robust and accurate estimation for fundamental matrix has been the researchfocus 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 can-not be obtained, no matter how many correspondences are used. One of themain reasons is that the cameras and the scene lie on a degenerate or quasi-degenerate configuration. If a space configuration is degenerate mathematicallybut the noise from the measured image makes it non-degenerate, any estima-tion under such a configuration would be useless [9]. It follows that we shouldknow what configurations might cause degeneracy for estimating the fundamen-tal matrix. Moreover in order for a robust RANSAC like [10, 7], we still need toknow how great the degenerate degree is, namely, to know the degenerate rankof 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 equiva-lent to say the studies on the degenerate configurations are useless. At least,such studies can give more geometric intuition, which could be as guidance forplacing cameras to avoid degeneracy in practice. Furthermore, if we can judgethe degeneracy by applying geometric knowledge, RANSAC work will be mucheasier.

Due to the importance of degeneracy analysis, many of such works havebeen reported previously. The planar scene is a trivial degenerate configurationfor computing fundamental matrix, where the images can provide only six inde-pendent constraints [11, 7] but the general fundamental matrix has seven degreesof freedom. Degeneracy from twisted cubic configuration has also been discussed.In [12], Buchanan stated that camera calibration from known space points un-der a single view is not unique if the optical center and the space points lie ona twisted cubic. The corresponding detection as well as emendations includingother unreliability was given by Wu et al [13]. Then, under two views, Maybank[14] analyzed the characterizations of horopter curve and the relations betweenthe curve and the ambiguous case of reconstruction. The horopter curve is re-garded as a twisted cubic, which intersects the plane at infinity at three particularpoints. The ambiguous case of reconstruction implies ambiguity of fundamentalmatrix. Luong and Faugeras reported the stability for computing fundamentalmatrix caused by quadric critical surface in [15]. Hartley and Zisserman [11] alsogave systematic discussions for degeneracy of camera projection estimation fromtwisted cubic under a single view and for degeneracy from ruled quadric surfaceunder two views. Under three views, critical configurations are provided in [16],which is an extension of the critical surface under two views. Degeneracy undera 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 sixpoints when the six points and the camera optical centers lie on a hyperboloidof one sheet. In [17], Hartley and Kahl presented a classification of all possiblecritical configurations for any number of points from three images and showedthat in most cases, the ambiguity could extend to any number of cameras.

Relative to the above works on degenerate configurations, there are fewerdeep studies on degeneracy degrees of degenerate configurations. Torr et al [7]catalogued all two-view non-degenerate and degenerate cases in a logical way bydimensions of the right null space of equations on fundamental matrix and thenproposed a PLUNDER-DL method to detect degeneracy and outliers. Chum etal [10] also analyzed those dimensions when the two views or most of the pointcorrespondences are related by a homography and presented an algorithm to es-timate fundamental matrix through detecting the homography degeneracy. Theyall [7, 10] generalized the robust estimator RANSAC [19]. The plane degeneracyin [7, 10] is consistent with the ruled quadric degeneracy proposed by Hartleyand Zisserman [11] because a plane and two camera optical centers always lieon a degenerate ruled quadric. What are the degeneracy degrees when estimat-ing the fundamental matrix for other non-trivial degenerate configurations? Inthis paper, we discuss all possible degenerate situations caused by twisted cubic

and give the corresponding degeneracy degrees. Let SO be a set of space pointsand the two camera optical centers. We find that if all the points of SO lieon a twisted cubic, the configuration is degenerate for estimating fundamentalmatrix and the corresponding rank of coefficient matrix is five; if all the pointsother than one lie on a twisted cubic, the corresponding rank is six; if all thepoints other than two lie on a twisted cubic, the corresponding rank is seven.The previous general degeneracies are ruled quadric degeneracy and homographydegeneracy. Few studies are given on relationships of twisted cubic degeneracywith them. We investigate the relationships in detail and then present our con-tribution relative to the general degeneracies.

The organization of the paper is as follows. Some preliminaries are listedin Section 2. The complete and unified degeneracy study from twisted cubic iselaborated in Section 3. Some experimental results are displayed in Section 4and Section 5 makes some conclusions.

2 Preliminaries

The camera model used is a perspective camera. A space point or its homoge-neous coordinates is denoted by M, an image point or its homogeneous coordi-nates is denoted by m, P denotes the camera projection matrix, and O denotesthe camera optical center. Under two views, P′ denotes the second camera pro-jection matrix, O′ denotes its optical center, and m′ denotes the correspondingimage point of m. Let F be the fundamental matrix between the two views.Other vectors or matrices are also denoted in boldface. The symbol ≈ meansequality 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 projec-tion matrix P is a 3 × 4 matrix such that mi ≈ PMi. For the camera opticalcenter O, we have the equation:

PO = 0 (1)

Fundamental matrix: Let m′i be the corresponding image points of the space

points Mi under another view. Then, mi and m′i are related by the fundamental

matrix F through:m′T

i Fmi = 0, i = 1 . . . N (2)

We denote F as

f1 f2 f3

f4 f5 f6

f7 f8 f9

If mi ≈

(ui vi wi

)and m′

i ≈(u′i v′i w′i

), we

expand (2) and have:

. . .u′iui u′ivi u′iwi v′iui v′ivi v′iwi w′iui w′ivi w′iwi

. . .

N×9

f = 0 (3)

where f =(f1 f2 f3 f4 f5 f6 f7 f8 f9

)T is the vector consisting of all elementsin F. The N × 9 coefficient matrix of f is denoted by G .Twisted cubic: The locus of points X =

(X Y Z T

)T in a 3-dimensionalprojective space satisfying the parametric equation:

(X Y Z T

)T ≈ H(θ3 θ2 θ 1

)T (4)

is a twisted cubic, where H is a 4×4 matrix and θ is the parameter[20]. Twistedcubic is an extension of a conic to 3-dimensional space by increasing the degreeof curve parameter from two to three. The properties of twisted cubic underliemany of the ambiguous cases that arise in 3-dimensional reconstruction.

3 Degeneracies from twisted cubic

The previously known degenerate configuration of two views for fundamentalmatrix or projective reconstruction is that two camera optical centers and allspace points lie on a ruled quadric. For such a general ruled quadric, the rightnull space of G in (3) is of dimension two as given in the section 2 of [7] and in theparagraph five of the introduction section of [16]. The more critically degenerateconfiguration is from a plane, of which the right null space of G in (3) is ofdimension three [10, 7]. This is not at the most since the nontrivial degenerateconfiguration—-twisted cubic can cause more critically degeneracy than a planeas 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 determineduniquely by linear 8-point algorithm. Otherwise, if the rank of G is 7, the solu-tion of f from (3) has one degree of freedom and the freedom can be removedby det(F) = 0 to obtain three or one solution. But if we only rely on the linearequations (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 deter-mined finitely. The configuration making the rank of G deficient is degeneratefor computing F. Due to noise of image data, generally we always can calcu-late a unique solution of f from (3) with 8 corresponding points. However, thedegenerate configurations or the configurations near to degeneracy will terriblyinfluence stability of the calculation. Therefore, in order for robust estimationof fundamental matrix, we need to know the degenerate configurations. The de-generate configurations from twisted cubic and the corresponding degeneracydegrees are provided in the following theorem.

Theorem 1 Let SO be a set of space points and two camera optical centers forcapturing these points. If all the points of SO are on a twisted cubic, then therank of the coefficient matrix G for computing F is five. If all the points otherthan one of SO are on a twisted cubic, the rank of G is six. If all the pointsother 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 isH

(θ3 θ2 θ 1

)T , where H is a 4 × 4 matrix. Let the parameter of the spacepoint Mi be θi and the parameters of the two camera optical centers be θ0, θ′0.Then,Mi = H

(θ3

i θ2i θi 1

)T , O = H(θ30 θ2

0 θ0 1)T , O′ = H

(θ′30 θ′20 θ′0 1

)T .By (1), we have:

0 = PO = PH(θ30 θ2

0 θ0 1)T

, 0 = P′O′ = P′H(θ′30 θ′20 θ′0 1

)T (5)

where P, P′ are the two camera projection matrices. So we also have:

mi ≈ PMi = PH(θ3

i θ2i θi 1

)T, m′

i ≈ P′Mi = P′H(θ′3i θ′2i θ′i 1

)T (6)

Do subtraction from both sides for (5) and (6), we obtain:

mi ≈ PH(θ3

i θ2i θi 1

)T −PH(θ30 θ2

0 θ0 1)T

≈ PH(θi − θ0)(θ2

i + θ20 + θ0θi θi + θ0 1 0

)T

≈ PH(θ2

i + θ20 + θ0θi θi + θ0 1 0

)T

(7)

Denote PH as Q =

q1 q2 q3 q4

q5 q6 q7 q8

q9 q10 q11 q12

, and P′H as Q′ =

q′1 q′2 q′3 q′4q′5 q′6 q′7 q′8q′9 q′10 q′11 q′12

.

Then, (7) is changed into:

mi ≈

q1

q5

q9

(θ2

i + θ20 + θ0θi) +

q2

q6

q10

(θi + θ0) +

q3

q7

q11

(8)

Similarly, do subtraction from both sides for (5) and (6), there is:

m′i ≈

q′1q′5q′9

(θ2

i + θ′02 + θ′0θi) +

q′2q′6q′10

(θi + θ′0) +

q′3q′7q′11

(9)

θi varies with(mi m′

i

), while qk, q′k, θ0, θ′0 are unchanged.

Substitute (8) and (9) into (3), we get the coefficient matrix G with eachelement of the i-th row being a four-order polynomial in θi as c1θ

4i + c2θ

3i +

c3θ2i + c4θi + c5. The coefficients cs of θi in these four-order polynomials are

functions on while qk, q′k, θ0, θ′0. Since while qk, q′k, θ0, θ′0 are not varying withimage pair varying, cs are also not varying with the row number varying. Itfollows 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) = c1jθ4 + c2jθ

3 + c3jθ2 + c4jθ + c5j , j = 1 . . . N . We equivalently

change G into: G =

θ41 θ3

1 θ21 θ1 1

. . .θ4

i θ3i θ2

i θi 1. . .

N×5

c11 . . . c1j . . . c19

c21 . . . c2j . . . c29

c31 . . . c3j . . . c39

c41 . . . c4j . . . c49

c51 . . . c5j . . . c59

5×9

From the expression, we know the rank of G is generally five. By now, weproved that if all the space points and the optical centers of the two camerasare 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 byanother camera optical center and the space points, assumed to be O, thenthe degree of θi for representing mi in (6) can not decrease to two but thedegree for representing m′

i can do, i.e. m′i 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 thepoint not lying on the twisted cubic is one of the space points other than oneof the camera optical center, assumed to be Mi0 , then the row in G fromthe image pair mi0 , m′

i0is not in the polynomial form of some θ. It follows

that this row is not linearly related to other rows in general. Thus, the rankof G increases from five to six.Similarly, if all the points other than two of SO are in a twisted cubic, therank of G is seven. The theorem is proved.

In the above theorem, we analyze all possible degenerate configurations forcomputing F caused from twisted cubic. In all the cases, F can not be determinedfinitely by linear 8-point algorithm and the dimensions of the right null space ofG in (3) are respectively 4, 3, 2. By 7-point algorithm, F still can not be solvedin 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 aruled quadric [11]. The theorem in Section 3.1 is consistent with the ruled quadricdegeneracy. In this subsection, we at first give two lemmas about twisted cubicand ruled quadric for the consistency. Then, the contribution of our work isdiscussed.

In projective space, quadrics are classified into ruled and unruled ones. Quadricswith positive index of inertia 2 are ruled quadrics and the degenerate quadricsexcept one point case are all ruled ones [11]. Here the positive index of inertiameans 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 canalways be embedded on a ruled quadric, conversely, any quadric containing aproper 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 propertwisted 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 oneor two points can be embedded on a ruled quadric. We take seven points on thetwisted cubic and combine the additional one or two points to generate a quadric.This is reasonable because generally nine space points uniquely determine aquadric. Since this quadric contains seven points of the twisted cubic, by Lemma2, we know it contains the whole twisted cubic. Furthermore by Lemma 1, weknow the generated quadric is ruled. It follows that the theorem in Section 3.1is consistent with the previous ruled quadric degeneracy.

Remark 2. The contribution of Theorem 1 is that it gives more intuitive de-generacy and the degeneracy degrees for all possible cases caused by twistedcubic. For the general ruled quadric degeneracy, there are a finite number of so-lutions for the fundamental matrix by combining with the additional constraintof det(F) = 0. This degeneracy degree is the same as the rank 7 case in thetheorem. For rank 5, 6 cases in the theorem, the degeneracy is more criticalwhich 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. Thesedetails are not discussed in the previous ruled quadric degeneracy. Usually, sixpoints determine a unique twisted cubic and nine points determine a uniquequadric. A twisted cubic is not a class in the ruled quadrics. Therefore, fromfewer non-incidence points to make F computations, quadric degeneracy maynot come to mind, which also could ignore the twisted cubic degeneracy. How-ever indeed the twisted cubic can make the F computation degenerate severelyas 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 etal. [10], where the H-degeneracy means the degeneracy caused by a 3×3 homog-raphy between two views. They also discussed the degeneracy degrees for the Fcomputation and mentioned the twisted cubic degeneracy. There are differencesbetween our work and theirs. In this subsection, we discuss the contribution ofour work relative to the study [10].

Firstly, we give complete cases that two views are related by a 3× 3 homog-raphy.

Lemma 3 If the image point correspondences (mi,m′i) between two views are

related by a homography H , that is m′i = 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 alanalyzed degrees of the H-degeneracy on three cases: i) two views are relatedby a homography; ii) all image point pairs other than one pair are related by ahomography; iii) all image point pairs other than two pairs are related by a ho-mography. Then based on the degrees, they developed a DEGENSAC algorithmto compute F unaffected by a dominant plane by detecting H-degeneracy.

The relationship and differences between our work and Chum et al’s [10] areas follows. The cases in the theorem of Section 3.1 related to the H-degeneracyare: (a1) The two camera optical centers and all the space points lie on a twistedcubic. (a2) The two camera optical centers and all other than one the space pointslie on a twisted cubic. (a3) The two camera optical centers and all other thantwo 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, andthe 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 casesare the same as Chum et al’s, the degeneracy degrees are different. Here in ourwork, the degeneracy is more critical. For case (a1), since the coefficient matrixhas rank 5, the linear space of F has dimension 4 while in [10] for two viewsrelated with a homography the dimension is 3. For case (a2), the correspondingdimension is 3 while that in [10] is 2. For case (a3), the corresponding dimensionis 2 while that in [10] could be 1 if linear 8-point algorithm is applied. It followsthat the twisted cubic cases could cause more critical degeneracy than the planecases, 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 thespace points and one of the camera optical centers lie on a twisted cubic. Theother camera optical center is not on this twisted cubic. (b2) All other than oneof 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 spacepoints 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 leastone of the optical centers does not lie on the twisted cubic and the space pointsare also not coplanar. Thus according to Lemma 3 all or most of the image pointpairs 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 somenew contribution in theory. The aim of [10] is to stably estimate F unaffected bya dominant plane. We also will explore a detection method on the degeneracycaused from twisted cubic and then apply the RANSAC on degenerate datain [13] to robustly compute fundamental matrix. Detection on the degeneracydeserves studies also because usually computations of matrix rank or its singularvalues are very sensitive to noise and presetting a threshold to discriminate thedegeneracy 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 verifythe established theorem. One group of the experiments is reported below.

4.1 Simulations

The parametric equation of a space twisted cubic is:

M ≈

2 5 −3 2.51 −1 12 16 −15 −2 3−7 5 3 2

θ3

θ2

θ1

(11)

Ten points Mi on this twisted cubic are taken, of which the parameters arerespectively −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 thespace points lie on the same twisted cubic. Let the two points of the twisted cubicwith parameters 1.25, 1.5 be the two optical centers O, O′. The space distributionis 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 * denotesthe space points, and o denotes the camera optical centers.

with the optical centers are set as follows:P =

1000 0 512 431980 900 384 954840 0 1 −103

,

P′ =

−529 648.1 287.4 −4321.3338.6 −295.4 748.7 −1810.1−0.7 −0.1 0.7 −3.9

. Projected by P, P′, we generated two

simulated images of the ten space points and established the equations on thefundamental matrix. Under the noise level of zero, the rank of the coefficientmatrix 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 thetwisted cubic any more. Let Q2 =

(3 7.3 2 1

)T which is not on this twisted

cubic, the corresponding camera projection matrix is set as:

P2 =

1000 0 512 −40240 900 384 −73380 0 1 −2

. By this camera projection matrix, another new

image is generated. From this image and that of P′, we established equations onthe fundamental matrix and then computed the rank of the coefficient matrixunder noise level of zero. The result is six that is consistent with the proposedtheorem.

Finally, we give the experimental result of the case when the two opticalcenters do not lie on the twisted cubic (11). Another optical center is set asQ′

2 =(0.67 −1.49 −2.8 1

)T which is also far away from the twisted cubic (11).From the two images generated, we computed rank of the coefficient matrix ofthe 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 twistedcubic determined by other space points and the optical centers, the same resultsare 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 rankcomputation by the singular values is only correct in the absence of noise. Whenwe add noise to the image, the rank of the coefficient matrix becomes to 8 andthe computation becomes very unstable. Therefore, in order to robustly estimatethe fundamental matrix, it is necessary to develop a method of detection on thedegenerate configuration. We will explore a detection method on the degeneracycaused 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 morepoints on real data need to be performed after the detection on degenerate dataand the corresponding RANSAC are proposed.

We took the images of six space points at different viewpoints. Four of themwith a size of 640× 480 pixels are shown in Fig. 2, where the dot points denotethe used image points.

In order to know whether the six space points and the corresponding opticalcenter lie on a twisted cubic or not, we measured the space coordinates of the sixpoints 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 respectively1.0655, 1.0504, 2.2934, and 2.5091. Then, by the method in [13], we know thatthe six points and the two corresponding optical centers of Fig. 2 (a)(b) are ona same twisted cubic, while the six points and the two corresponding opticalcenters of (c)(d) are not. We also computed the singular values of the coefficientmatrix 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 inFig. 2 (c)(d) is:508796.41, 138904.18, 33040.13, 9883.42, 112.31, 37.68. We seethat 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 degeneracyby using the condition number because the singular values are very sensitive tonoise and presetting a threshold to discriminate the degeneracy from the non-degeneracy is not easy, as pointed out in [21]. We found sometimes the conditionnumber of the degeneracy is yet smaller than that of the non-degeneracy. Thisis why we would like to pursue a detection method for the degeneracy from twoimage data in the future.

5 Conclusion

This paper provides all the possible degenerate configurations caused by twistedcubic and the corresponding degeneracy degrees for estimating fundamental ma-trix. Relationships with the ruled quadric degeneracy and the homograghy de-generacy are also given. The result is helpful to improving the accuracy of theestimations. Indeed, for a robust RANSAC, initial samples with worse estima-tions should be removed or mended. These initial samples not only are thoseincluding mismatching pairs but also are those that are degenerate. The lattercase 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 studiedthoroughly. We give some research on the degeneracy in this work and furtherrobust detection on the twisted cubic configurations will be developed.

Acknowledgement This work was supported by the National Natural ScienceFoundation of China under grant No. 60633070, 60773039.

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