On the Existence of Epipolar Matrices
Sameer Agarwal, Hon-Leung Lee, Bernd Sturmfels, Rekha R. Thomas
OON THE EXISTENCE OF EPIPOLAR MATRICES
SAMEER AGARWAL, HON-LEUNG LEE, BERND STURMFELS, AND REKHA R. THOMAS
Abstract.
This paper considers the foundational question of the existence of a fundamental(resp. essential) matrix given m point correspondences in two views. We present a completeanswer for the existence of fundamental matrices for any value of m . Using examples we disprovethe widely held beliefs that fundamental matrices always exist whenever m ≤
7. At the sametime, we prove that they exist unconditionally when m ≤
5. Under a mild genericity condition,we show that an essential matrix always exists when m ≤
4. We also characterize the six andseven point configurations in two views for which all matrices satisfying the epipolar constrainthave rank at most one. Introduction
A set of point correspondences { ( x i , y i ) ∈ R × R , i = 1 , . . . , m } are the images of m pointsin R in two uncalibrated (resp. calibrated) cameras only if there exists a fundamental matrix F (resp. essential matrix E ) such that the ( x i , y i ) satisfy the epipolar constraints [11, Chapter 9].Under mild genericity conditions on the point correspondences, the existence of these matrices isalso sufficient for the correspondences ( x i , y i ) to be the images of a 3D scene [6, 13, 14, 18]. Thisbrings us to the following basic question in multiview geometry: Question 1.1.
Given a set of m point correspondences ( x i , y i ) ∈ R × R , when does there exist afundamental (essential) matrix relating them via the epipolar constraints? The answer to this question is known in several special cases [2, 12], but even in the minimallyconstrained and under-constrained cases ( m ≤ m ≤ m ≤
7, the popular statement of the so called sevenpoint algorithm will have you believe that there always exists a fundamental matrix [15, 27]. We willshow that this is not true. The problem is, that the matrix returned by the seven point algorithmis only guaranteed to be rank deficient, it is not guaranteed to have rank two.In the calibrated case, when m = 5, there exists up to 10 distinct complex essential matrices [5],but it is not known when we can be sure that one of them is real. Similarly, it is unknown whetherthere always exists a real essential matrix for m ≤ m and for essential matrices for m ≤
4. The problem of checking the existence of a fundamental(resp. essential) matrix for an arbitrary value of m reduces to one where m ≤
9. The situations of m = 8 , m ≤
7. We prove the following results:
Lee and Thomas were partially supported by NSF grant DMS-1418728, and Sturmfels by NSF grant DMS-1419018. a r X i v : . [ c s . C V ] O c t N THE EXISTENCE OF EPIPOLAR MATRICES 2 (1) For m ≤ m = 6 , m ≤ m ≤
4. We give a muchmore sophisticated proof to extend this result to m ≤ m ≤
3. The proof of (3) is much more complicated.A fundamental matrix can fail to exist in several ways. An important such case is when allmatrices that run for competition have rank at most one. We fully characterize this phenomenondirectly in terms of the geometry of the input point correspondences.The key technical task in all this is to establish conditions for the existence of a real point in theintersection of a subspace and a fixed set of 3 × Notation.
Capital roman letters (say
E, F, X, Y, Z ) denote matrices. For a matrix F , thecorresponding lower case letter f denotes the vector obtained by concatenating the rows of F .Upper case calligraphic letters denote sets of matrices (say E , F ).For a field F such as R or C , the projective space P n F is F n +1 \ { } in which we identify u and v if u = λv for some λ ∈ F \ { } . For example (1 , ,
3) and (4 , ,
12) are the same point in P R , denotedas (1 , , ∼ (4 , , m × n matrices with entries in F is denoted by F m × n , and by P m × n F if the matrices are only up to scale. For v ∈ R ,[ v ] × := − v v v − v − v v is a skew-symmetric matrix whose rank is two unless v = 0. Also, [ v ] × w = v × w , where × denotes the vector cross product. For A ∈ F m × n , we have ker F ( A ) = { u ∈ F n : Au = 0 } , andrank( A ) = n − dim(ker F ( A )). We use det( A ) to denote the determinant of A . Points x i and y i in F will be identified with their homogenizations ( x i , x i , (cid:62) and ( y i , y i , (cid:62) in P F . Also, y (cid:62) i ⊗ x (cid:62) i := (cid:0) y i x i y i x i y i y i x i y i x i y i x i x i (cid:1) ∈ F × .If P and Q are finite dimensional subspaces, then P ⊗ Q is the span of the pairwise Kroneckerproducts of the basis elements of P and Q .1.2. Linear algebra.
Below we list five facts from linear algebra that will be helpful in this paper.
Lemma 1.2. [24, pp. 399] If x , . . . , x n +1 and y , . . . , y n +1 are two sets of n + 2 points in R n +1 such that no n + 1 points in either set are linearly dependent. Then there is an invertible matrix H ∈ R ( n +1) × ( n +1) such that Hx i ∼ y i for any i = 0 , , . . . , n + 1 . Lemma 1.3. [22, Theorem 3]
Suppose V is a linear subspace of R n × n of dimension rn , such thatfor any A ∈ V , rank( A ) ≤ r . Then either V = W ⊗ R n or V = R n ⊗ W , for some r -dimensionalsubspace W ⊆ R n . Lemma 1.4. [8, Theorem 1]
Suppose V is a linear subspace of R m × n and r is the maximum rankof an element of V . Then dim( V ) ≤ r · max { m, n } . N THE EXISTENCE OF EPIPOLAR MATRICES 3
Lemma 1.5 (Matrix Determinant Lemma) . [16, Theorem 18.1.1] If A ∈ R n × n is invertible and u, v ∈ R n , then det( A + uv (cid:62) ) = (1 + v (cid:62) A − u ) det( A ) . In the following lemma we identify points in R with their homogenizations in P R as mentionedearlier. The proof of the lemma is in Appendix A. Lemma 1.6.
Given two lines l, m in R , and x ∈ l , y ∈ m , there is an invertible matrix H ∈ R × such that(1) Hx = y ; and(2) for any x ∈ l , Hx ∈ m . Projective varieties.
We recall some basic notions from algebraic geometry [3, 10, 25]. Let F [ u ] = F [ u , . . . , u n ] denote the ring of all polynomials with coefficients in the field F . Definition 1.7 (Homogeneous Polynomial) . A polynomial in F [ u ] is homogeneous (or called a form ) if all its monomials have the same total degree. For example, u u + u u is a form of degree three but u + u is not a form. Definition 1.8 (Projective Variety and Subvariety) . A subset
V ⊆ P n F is a projective variety if thereare homogeneous polynomials h , . . . , h t ∈ F [ u ] such that V = { u ∈ P n F : h ( u ) = . . . = h t ( u ) = 0 } .A variety V is a subvariety of V if V ⊆ V . Given homogeneous polynomials h , . . . , h t ∈ R [ u ], let V C := { u ∈ P n C : h i ( u ) = 0 for i =1 , . . . , t } be their projective variety over the complex numbers, and V R := V C ∩ P n R be the set of realpoints in V C . Definition 1.9 (Irreducibility) . A projective variety
V ⊆ P n F is irreducible if it is not the union oftwo nonempty proper subvarieties of P n F . We define the dimension of a projective variety over C in a form that is particularly suitable tothis paper. Definition 1.10 (Dimension) . [25, Corollary 1.6] The dimension dim( V ) of a projective variety V ⊆ P n C is d where n − d − is the maximum dimension of a linear subspace of P n C disjoint from V . As a special case, if L is a l -dimensional linear subspace in C n +1 , it can be viewed as an irreducibleprojective variety in P n C of dimension l − P n C . It is a consequence of the more general statementin [25, Theorem 1.24]. This result does not extend to varieties over R . Theorem 1.11.
Consider an irreducible projective variety V C ⊆ P C n of dimension d and a linearsubspace L ⊆ P C n of dimension l . If d + l = n then V must intersect L . If d + l > n then V intersects L at infinitely many points. Observe that the above theorem only applies over the complex numbers. As a simple illustrationthe curve x − y + z = 0 in P C is guaranteed to intersect the subspace y = 0 in two complex pointssince they have complementary dimensions in P C . However, neither of these intersection points isreal.If V ⊆ P n C is a projective variety, then it intersects any linear subspace of dimension n − dim( V )in P n C . If the subspace is general, then the cardinality of this intersection is a constant which is animportant invariant of the variety. N THE EXISTENCE OF EPIPOLAR MATRICES 4
Definition 1.12 (Degree) . [10, Definition 18.1] The degree of a projective variety
V ⊆ P n C , denotedby degree ( V ) , is the number of intersection points with a general linear subspace of dimension n − dim( V ) in P n C . Camera Matrices.
A general projective camera can be modeled by a matrix P ∈ P × R withrank( P ) = 3. Partitioning a camera as P = (cid:0) A b (cid:1) where A ∈ R × , we say that P is a finitecamera if A is nonsingular. In this paper we restrict ourselves to finite cameras.A finite camera P can be written as P = K (cid:0) R t (cid:1) , where t ∈ R , K is an upper triangularmatrix with positive diagonal entries known as the calibration matrix , and R ∈ SO(3) is a rotationmatrix that represents the orientation of the camera coordinate frame. If the calibration matrix K is known, then the camera is said to be calibrated , and otherwise the camera is uncalibrated . The normalization of a calibrated camera P = K (cid:0) R t (cid:1) is the camera K − P = (cid:0) R t (cid:1) .By dehomogenizing (i.e. scaling the last coordinate to be 1), we can view the image x = P w as apoint in R . If x is the image of w in the calibrated camera P , then K − x is called the normalizedimage of w , or equivalently, it is the image of w in the normalized camera K − P . This allows us toremove the effect of the calibration K by passing to the normalized camera K − P and normalizedimages ˜ x := K − x .1.5. Epipolar Matrices.
In this paper we use the name epipolar matrix to refer to either a fun-damental matrix or essential matrix derived from the epipolar geometry of a pair of cameras. Thesematrices are explained and studied in [11, Chapter 9].An essential matrix is any matrix in P × R of the form E = SR where S is a skew-symmetricmatrix and R ∈ SO(3). Essential matrices are characterized by the property that they have ranktwo (and hence one zero singular value) and two equal non-zero singular values. An essential matrixdepends on six parameters, three each from S and R , but since it is only defined up to scale, it hasfive degrees of freedom.The essential matrix of the two normalized cameras (cid:0) I (cid:1) and (cid:0) R t (cid:1) is E = [ t ] × R . For everypair of normalized images ˜ x and ˜ y in these cameras of a point w ∈ P R , the triple (˜ x, ˜ y, E ) satisfiesthe epipolar constraint ˜ y (cid:62) E ˜ x = 0 . (1.1)Further, any E = SR is the essential matrix of a pair of cameras as shown in [11, Section 9.6.2].If the calibrations K and K of the two cameras were unknown, then for a pair of correspondingimages ( x, y ) in the two cameras, the epipolar constraint becomes0 = ˜ y (cid:62) E ˜ x = y (cid:62) K −(cid:62) EK − x. (1.2)The matrix F := K −(cid:62) EK − is the fundamental matrix of the two uncalibrated cameras. This is arank two matrix but its two non-zero singular values are no longer equal. Conversely, any real 3 × F = [ b ] × H , where b is a non-zerovector and H is an invertible matrix 3 × N THE EXISTENCE OF EPIPOLAR MATRICES 5
X, Y and Z . Suppose we are given m point correspondences (normalized or not) { ( x i , y i ) , i =1 , . . . , m } ⊆ R × R . We homogenize this data and represent it by three matrices with m rows: X = x (cid:62) ... x (cid:62) m ∈ R m × , (1.3) Y = y (cid:62) ... y (cid:62) m ∈ R m × , and(1.4) Z = y (cid:62) ⊗ x (cid:62) ... y (cid:62) m ⊗ x (cid:62) m ∈ R m × . (1.5)The ranks of X and Y are related to the geometry of the point sets { x i } and { y i } . This is madeprecise by the following lemma which is stated in terms of X but obviously also applies to Y . Lemma 1.13. rank( X ) = If x i ’s, as points in R , are all equal. If all the x i ’s are collinear in R but not all equal. If the x i ’s are noncollinear in R . Notice that every row of X (resp. Y ) can be written as a linear combination of rank( X ) (resp.rank( Y )) rows of it. Using this and the bilinearity of Kronecker product, it is evident that: Lemma 1.14.
For any m , rank( Z ) ≤ rank( X ) rank( Y ) ≤ . In particular, if all points x i are collinear in R then rank( Z ) ≤
6. If all points x i are equal in R then rank( Z ) ≤ R ( Z ). Observe that for all m , ker R ( Z ) = ker R ( Z (cid:48) )for a supmatrix Z (cid:48) of Z consisting of rank( Z ) linearly independent rows. Therefore, we can replace Z with Z (cid:48) in order to study ker R ( Z ) which allows us to restrict our investigations to the values of m such that(1.6) 1 ≤ m = rank( Z ) ≤ . In light of the above discussion, it is useful to keep in mind that even though all our results arestated in terms of m ≤
9, we are in fact covering all values of m .2. Fundamental Matrices
Following Section 1.5, a fundamental matrix is any matrix in P × R of rank two [11, Section 9.2.4].In our notation, we denote the set of fundamental matrices as(2.1) F := { f ∈ P R : rank( F ) = 2 } , N THE EXISTENCE OF EPIPOLAR MATRICES 6 where the vector f is the concatenation of the rows of the matrix F . This notation allows us towrite the epipolar constraints (1.2) as(2.2) Zf = 0 . Hence a fundamental matrix F exists for the m given point correspondences if and only if thelinear subspace ker R ( Z ) intersects the set F , i.e.,ker R ( Z ) ∩ F (cid:54) = ∅ . (2.3)This geometric reformulation of the existence question for F is well-known in multiview geometry[11, 21].We now introduce two complex varieties that are closely related to F .Let R := { a ∈ P C : rank( A ) ≤ } be the set of matrices in P × C of rank one. It is an irreduciblevariety with dim( R ) = 4 and degree( R ) = 6.Let R := { a ∈ P C : rank( A ) ≤ } be the set of matrices in P × C of rank at most two. It is anirreducible variety with dim( R ) = 7 and degree( R ) = 3. Observe that(2.4) R = { a ∈ P C : det( A ) = 0 } . The set of fundamental matrices can now be written as F = ( R \R ) ∩ P R which is not a varietyover R .In this section we will give a complete answer to the question of existence of fundamental matricesfor any number m of point correspondences. Recall from Section 1.6 (1.6) that assuming m =rank( Z ), we only need to consider the cases 1 ≤ m ≤ Case: m = 9 . If m = 9, then ker R ( Z ) ⊆ P R is empty, and Z has no fundamental matrix.2.2. Case: m = 8 . If m = 8, then ker R ( Z ) ⊆ P R is a point a ∈ P R corresponding to the matrix A ∈ P × R . It is possible for A to have rank one, two or three. Clearly, Z has a fundamental matrixif and only if A has rank two.2.3. Case: m = 7 . The majority of the literature in computer vision deals with the case of m = 7which falls under the category of “minimal problems”; see for example [26, Chapter 3]. The namerefers to the fact that m = 7 is the smallest value of m for which ker C ( Z ) ∩ R is finite, making theproblem of estimating F well-posed (at least over C ).Indeed, when m = 7, ker C ( Z ) is a one-dimensional subspace of P C and hence by Theorem 1.11,generically it will intersect R in three points, of which at least one is real since det( A ) is a degreethree polynomial. Therefore, there is always a matrix of rank at most two in ker R ( Z ). This leadsto the common belief that when m = 7, there is always a fundamental matrix for Z .We first show an example for which ker R ( Z ) contains only matrices of ranks either one or three. Example 2.1.
Consider X = − − − − − −
12 1 − − and Y = −
14 7 15 −
16 9 1 . N THE EXISTENCE OF EPIPOLAR MATRICES 7 y y y y y y y x x x x x x x y y y y y y y x x x x x = x = x (a) (b) Figure 1.
Two examples where the conditions for Theorem 2.2 are satisfied andthere does not exist a fundamental matrix for m = 7 because ker R ( Z ) ⊆ R .Here, rank( Z ) = 7 and ker R ( Z ) is spanned by the rank three matrices I = and A = − −
15 3 11 . For any u , u ∈ R , one obtains det( Iu + A u ) = ( u + 5 u ) . If det( Iu + A u ) = 0, then u = − u and Iu + A u = u ( A − I ) = u − − − −
15 3 6 which has rank at most one. Thus for ( u , u ) (cid:54) = (0 , Iu + A u ) = (cid:40) u + 5 u = 03 if u + 5 u (cid:54) = 0 . Hence ker R ( Z ) consists of matrices of rank either one or three, and Z does not have a fundamentalmatrix.Another way for Z to not have a fundamental matrix is if ker R ( Z ) is entirely in R . The followingtheorem whose proof can be found in Appendix C, characterizes this situation. See Figure 1 forillustrations. Theorem 2.2. If m = 7 , ker R ( Z ) ⊆ R if and only if one of the following holds:(1) There is a nonempty proper subset τ ⊂ { , . . . , } such that as points in R , { y i : i ∈ τ } arecollinear and x i = x j for all i, j / ∈ τ .(2) There is a nonempty proper subset τ ⊂ { , . . . , } such that as points in R , { x i : i ∈ τ } arecollinear and y i = y j for all i, j / ∈ τ . Case: m = 6 . When m ≤
6, by Theorem 1.11, ker C ( Z ) ∩ R is infinite, and here the conven-tional wisdom is that there are infinitely many fundamental matrices for Z and thus these casesdeserve no study.Indeed, it is true that for six points in two views in general position, there exists a fundamentalmatrix relating them. To prove this, we first note the following fact which is a generalization of aresult of Chum et al. [2]. Its proof can be found in Appendix B. N THE EXISTENCE OF EPIPOLAR MATRICES 8
Lemma 2.3.
Assume there is a real × invertible matrix H such that for at least m − of theindices i ∈ { , . . . , m } , y i ∼ Hx i . Then Z has a fundamental matrix. An immediate consequence of Lemma 2.3 with m = 6 and Lemma 1.2 with n = 2 is the following. Theorem 2.4. If m = 6 and τ is a subset of , . . . , with four elements such that no set of threepoints in either { x i : i ∈ τ } or { y i : i ∈ τ } is collinear in R , then Z has a fundamental matrix. Note that in [12, Theorem 1.2], Hartley shows that a fundamental matrix associated with sixpoint correspondences is uniquely determined under certain geometric assumptions on the pointcorrespondences and world points. One of Hartley’s assumptions is the assumption of Theorem 2.4.Theorem 2.4 hints at the possibility that collinearity of points in any one of the two views mayprevent a fundamental matrix from existing. The following theorem, whose proof can be found inAppendix D, characterizes the conditions under which ker R ( Z ) ⊆ R when m = 6. No fundamentalmatrix can exist in this case. Theorem 2.5. If m = 6 , ker R ( Z ) ⊆ R if and only if either all points x i are collinear in R or allpoints y i are collinear in R . We remark that when m = 6, it is impossible that as points in R , all x i are collinear and all y i are collinear. If this were the case, then by Lemmas 1.13 and 1.14, rank( Z ) ≤ < m whichviolates our assumption (1.6).2.5. Existence of fundamental matrices in general.
In the previous two sections, we havedemonstrated that dimension counting is not enough to argue for the existence of a fundamentalmatrix for m = 6 and m = 7. We have also described particular configurations in two views whichguarantee the existence and non-existence of a fundamental matrix. We are now ready to tacklethe general existence question for fundamental matrices for m ≤
8. To do this, we first need thefollowing key structural lemma. It provides a sufficient condition for ker R ( Z ) to have a matrix ofrank two. Lemma 2.6.
Let L be a positive dimensional subspace in P × R that contains a matrix of rank three.If the determinant restricted to L is not a power of a linear form, then L contains a real matrix ofrank two. The proof of this lemma can be found in Appendix E, but we elaborate on its statement. If { A , . . . , A t } is a basis of a subspace L in P × R , then any matrix in L is of the form A = u A + · · · + u t A t for scalars u , . . . , u t ∈ R , and det( A ) is a polynomial in u , . . . , u t of degree at mostthree. Lemma 2.6 says that if det( A ) is not a power of a linear form a u + · · · + a t u t where a , . . . , a t ∈ R , then L contains a matrix of rank two.It is worth noting that Lemma 2.6 is only a sufficient condition and not necessary for a subspace L ⊆ P × R to have a rank two matrix. This is illustrated by the following example: Example 2.7.
For X = − − and Y = − − − − − , N THE EXISTENCE OF EPIPOLAR MATRICES 9 ker R ( Z ) is spanned by A = and A = , and det( A u + A u ) = u . Since rank( A ) = 2 , A is a fundamental matrix of Z . We now present a general theorem that characterizes the existence of a fundamental matrix for m ≤ Theorem 2.8.
For a basis { A , . . . , A t } of ker R ( Z ) , define M ( u ) := (cid:80) ti =1 A i u i , and set d ( u ) :=det( M ( u )) .(1) If d ( u ) is the zero polynomial then Z has a fundamental matrix if and only if some × minor of M ( u ) is nonzero.(2) If d ( u ) is a nonzero polynomial that is not a power of a linear form in u then Z has afundamental matrix.(3) If d ( u ) = ( b (cid:62) u ) k for some k ≥ and non-zero vector b ∈ R t , then Z has a fundamentalmatrix if and only if some × minor of M (cid:0) u − b (cid:62) ub (cid:62) b b (cid:1) is nonzero.Proof. Note that M ( u ) is a parametrization of ker R ( Z ) and d ( u ) is a polynomial in u of degree atmost three.(1) If d ( u ) is the zero polynomial, then all matrices in ker R ( Z ) have rank at most two. In thiscase, Z has a fundamental matrix if and only if some 2 × M ( u ) is a nonzeropolynomial in u .(2) If d ( u ) is a nonzero polynomial in u , then we factor d ( u ) and see if it is the cube of a linearform. If it is not, then by Lemma 2.6, Z has a fundamental matrix.(3) Suppose d ( u ) = ( b (cid:62) u ) k for some k ≥ b . Then the set of rankdeficient matrices in ker R ( Z ) is M := (cid:8) M ( u ) : u ∈ b ⊥ (cid:9) where, b ⊥ := { u ∈ R t : b (cid:62) u = 0 } .The hyperplane b ⊥ consists of all vectors u − b (cid:62) ub (cid:62) b b where u ∈ R t . Therefore, M = (cid:110) M (cid:0) u − b (cid:62) ub (cid:62) b b (cid:1) : u ∈ R t (cid:111) . As a result, Z has a fundamental matrix if and only if some 2 × M (cid:0) u − b (cid:62) ub (cid:62) b b (cid:1) is nonzero. (cid:3) Cases: m ≤ . While Theorem 2.8 provides a general existence condition for fundamentalmatrices for m ≤
8, we now show that for m ≤ Theorem 2.9.
Every three-dimensional subspace of P × R contains a rank two matrix. In particular,if m ≤ , then Z has a fundamental matrix.Proof. Suppose L is a three-dimensional subspace in P × R generated by the basis { A , . . . , A } , andsuppose L does not contain a rank two matrix. Since the dimension of L as a linear subspace isfour, by applying Lemma 1.4 with m = n = 3 and r = 1, we see that L cannot be contained inthe variety of rank one matrices. Therefore, we may assume that A has rank three. Since L isassumed to have no matrices of rank two, by Lemma 2.6 we also have thatdet( λ A + λ A + λ A + λ A ) = ( a λ + a λ + a λ + a λ ) (2.5)where λ , · · · , λ are variables. Note that a (cid:54) = 0 since otherwise, choosing λ = λ = λ = 0 and λ = 1 we get det( A ) = 0 which is impossible. N THE EXISTENCE OF EPIPOLAR MATRICES 10
By a change of coordinates, we may assume thatdet( λ A + λ A + λ A + λ A ) = λ ,(2.6)and in particular, det( A ) = 1. Indeed, consider (cid:101) A := A − a a A , (cid:101) A := A − a a A , (cid:101) A := A − a a A , (cid:101) A := 1 a A (2.7)which also form a basis of L . Then using (2.5) with the variables η , η , η , η , we obtaindet( η (cid:101) A + η (cid:101) A + η (cid:101) A + η (cid:101) A )= det (cid:18) η A + η A + η A + ( η − η a − η a − η a ) a A (cid:19) = ( a η + a η + a η + ( η − η a − η a − η a )) = η , which is the desired conclusion.Setting λ = 0 in (2.6) we get det( λ A + λ A + λ A ) = 0. Hence, span { A , A , A } consistsonly of rank one matrices since there are no rank two matrices in L . Therefore, by Lemma 1.3 with n = 3 and r = 1, up to taking transposes of all A i , there are column vectors u, v , v , v ∈ P R suchthat A j = uv (cid:62) j for all j = 1 , , λ = 1, by the Matrix Determinant Lemma, we have1 = det( λ A + λ A + λ A + A )= det( A + u ( λ v (cid:62) + λ v (cid:62) + λ v (cid:62) ))= 1 + ( λ v (cid:62) + λ v (cid:62) + λ v (cid:62) ) A − u. Hence ( λ v (cid:62) + λ v (cid:62) + λ v (cid:62) ) A − u is the zero polynomial, and so A − u is a non-zero vector orthog-onal to span { v , v , v } . This means that v , v , v are linearly dependent, and so are A , A , A ,which is impossible. This completes the proof of the first statement.If m ≤
5, then rank( Z ) ≤ R ( Z ) is a subspace in P R of dimension at leastthree. By the first statement of the theorem, Z has a fundamental matrix. (cid:3) Note that when m ≤ m = 5.2.7. Comments.
As far as we know, the seven and eight point algorithms are the only generalmethods for checking the existence of a fundamental matrix. They work by first computing thematrices in ker C ( Z ) ∩ R and then checking if there is a real matrix of rank two in this collection.While such an approach might decide the existence of a fundamental matrix for a given input X and Y , it does not shed light on the structural requirements of X and Y to have a fundamentalmatrix. The goal of this paper is to understand the existence of epipolar matrices in terms of theinput data.When the input points x i and y i are rational, the results in this section also certify the existenceof a fundamental matrix by exact rational arithmetic in polynomial time. The only calculation thatscales with m is the computation of a basis for ker R ( Z ), which can be done in polynomial timeusing Gaussian elimination. N THE EXISTENCE OF EPIPOLAR MATRICES 11 Essential Matrices
We now turn our attention to calibrated cameras and essential matrices. The set of essentialmatrices is the set of real 3 × R \ R . Wedenote the set of essential matrices by(3.1) E R = { e ∈ P R : σ ( E ) = σ ( E ) and σ ( E ) = 0 } , where σ i ( E ) denotes the i th singular value of the matrix E . Demazure [5] showed that E R = (cid:8) e ∈ P R : p j ( e ) = 0 for j = 1 , . . . , (cid:9) , (3.2)where the p j ’s are homogeneous polynomials of degree three defined as p p p p p p p p p := 2 EE (cid:62) E − Tr( EE (cid:62) ) E, and(3.3) p := det( E ) . (3.4)Therefore, E R is a real projective variety in P R .Passing to the common complex roots of the cubics p , . . . , p , we get E C := { e ∈ P C : p j ( e ) = 0 , ∀ j = 1 , . . . , } . (3.5)This is an irreducible projective variety with dim( E C ) = 5 and degree( E C ) = 10 (see [5]), and E R = E C ∩ P R . See [21] for many interesting facts about E C and E R and their role in reconstructionproblems in multiview geometry.As before, our data consists of m point correspondences, which are now normalized image co-ordinates. For simplicity we will denote them as { ( x i , y i ) , i = 1 , . . . , m } instead of { (˜ x i , ˜ y i ) , i =1 , . . . , m } .As in the uncalibrated case, we can write the epipolar constraints (cf. (1.1)) as Ze = 0 where e ∈ E R , and Z has an essential matrix if and only ifker R ( Z ) ∩ E R (cid:54) = ∅ . (3.6)Hence the existence of an essential matrix for a given Z is equivalent to the intersection of a subspacewith a fixed real projective variety being non-empty. This formulation can also be found in [21,Section 5.2].3.1. Cases: m = 8 , . As in the previous section it is easy to settle the existence of E for m = 8 , m = 8, then the subspace ker R ( Z ) ⊆ P R is a point a in P R , and Z has an essential matrix if andonly if A satisfies the conditions of (3.1) or (3.2). If m = 9, then ker R ( Z ) ⊆ P R is empty, and Z has no essential matrix.3.2. Cases: ≤ m ≤ . The “minimal problem” for essential matrices is the case of m = 5 where,by Definition 1.12, E C ∩ ker C ( Z ) is a finite set of points. Since degree( E C ) = 10, generically weexpect ten distinct complex points in this intersection. An essential matrix exists for Z if and onlyif one of these points is real. There can be ten distinct real points in E C ∩ ker C ( Z ) as shown in [21,Theorem 5.14]. On the other extreme, it can also be that no point in E C ∩ ker C ( Z ) is real as weshow below. N THE EXISTENCE OF EPIPOLAR MATRICES 12
Example 3.1.
We verified using
Maple that the following set of five point correspondences has noessential matrix. X = , Y = . None of the ten points in ker C ( Z ) ∩ E C are real. As we have mentioned earlier, the existence of an essential matrix is equivalent to existence of areal point in the intersection ker C ( Z ) ∩ E C . In general, this is a hard question which falls under theumbrella of real algebraic geometry .The reason we were able to give a general existence result for fundamental matrices is becausewe were able to exploit the structure of the set of rank 2 matrices (Lemma 2.6). We believe thata general existence result for essential matrices would require a similar result about the variety ofessential matrices. One that still eludes us, and therefore, we are unable to say more about theexistence of essential matrices for the case 5 ≤ m ≤ real Nullstellensatz [20] and checked degree by degree via semidefinite programming [28].Or given a Z we could solve the Demazure cubics together with the linear equations cutting outker R ( Z ) and check if there is a real solution among the finitely many complex solutions [23, 26]. Inboth of these approaches, it is a case by case computation for each instance of Z and will not yielda characterization of those Z ’s for which there is an essential matrix.3.3. Cases: m ≤ . We now consider the cases of m ≤ E C ∩ ker C ( Z ) is infinite and theconventional wisdom is that an essential matrix always exists. It turns out that an essential matrixdoes indeed exist when m ≤
4. Again, such a result does not follow from dimension counting forcomplex varieties since we have to exhibit the existence of a real matrix in E C ∩ ker C ( Z ).When m ≤
3, there is a short proof that Z always has an essential matrix using the fact thatan essential matrix can be written in the form E = [ t ] × R where t is a nonzero vector in R and R ∈ SO(3).
Theorem 3.2. If m ≤ then Z has an essential matrix.Proof. Without loss of generality we assume m = 3. Choose a rotation matrix R so that y ∼ Rx .Then consider t ∈ R \ { } which is orthogonal to both y × Rx and y × Rx . Now check thatfor each i = 1 , , y (cid:62) i [ t ] × Rx i = 0 and hence [ t ] × R is an essential matrix for Z . It helps to recallthat y (cid:62) i [ t ] × Rx i = t (cid:62) ( y i × Rx i ). (cid:3) The above argument does not extend to m = 4. Our main result in this section is Theorem 3.4which proves the existence of E when m = 4 under the mild assumption that all the x i ’s (respec-tively, y i ’s) are distinct. This result will need the following key lemma which is a consequence ofTheorems 5.19 and 5.21 in [19]. Lemma 3.3.
If there is a matrix H ∈ R × of rank at least two such that for each i , either y i ∼ Hx i or Hx i = 0 , then Z has an essential matrix. Theorem 3.4. If m = 4 and all the x i ’s are distinct points and all the y i ’s are distinct points for i = 1 , . . . , , then Z has an essential matrix. N THE EXISTENCE OF EPIPOLAR MATRICES 13 x x x x y y y y x x x x y y y y Case 1: No three points in the two images arecollinear. Case 2: No three points in x , x , x , x arecollinear, and y , y , y are collinear. y y y y x x x x x x x x y y y y Case 3: x , x , x are collinear, while x is not onthe line, and y , y , y are collinear while y is noton the line. Case 4: x , x , x are collinear, and y , y , y arecollinear. Figure 2.
The four point configurations (up to rearranging indices and swapping x with y ) for m = 4. Theorem 3.4 proves the existence of an E matrix for m = 4by treating each of these cases separately. Proof.
We divide the proof into several cases; see Figure 2. The first is the generic situation inwhich the x i ’s and y i ’s are in general position. In the remaining cases the input data satisfy specialnon-generic conditions. Together, these cases exhaust all possibilities, up to rearranging indicesand swapping x with y .In the first three cases, the proof proceeds by exhibiting an explicit matrix H that satisfies theassumption of Lemma 3.3. Cases 1 and 2 are easy to check. The H in case 3 is quite a bit moreinvolved, although it only suffices to verify that it satisfies Lemma 3.3, which is mechanical. Thelast case uses a different argument to construct an essential matrix associated to Z .(1) No three of the x i ’s are collinear in R , and no three of the y i ’s are collinear in R ; seeFigure 2. In this case, there is an invertible matrix H ∈ R × such that y i ∼ Hx i by Lemma 1.2with n = 2. The conclusion now follows from Lemma 3.3.(2) No three points in x , x , x , x are collinear in R , and the points y , y , y are collinear in R ; see Figure 2. By Lemma 1.2, we can choose an invertible matrix H ∈ R × such that H x ∼ (1 , , (cid:62) , H x ∼ (0 , , (cid:62) , H x ∼ (0 , , (cid:62) and H x ∼ (1 , , (cid:62) . On the other hand, by Lemma 1.6, there is an invertible matrix H ∈ R × such that H y = (0 , , (cid:62) , H y = (0 , α, (cid:62) , H y = (0 , β, (cid:62) N THE EXISTENCE OF EPIPOLAR MATRICES 14 for some non-zero distinct real numbers α and β . Consider the rank two matrix H := − αβ αβ − α β . Then we obtain H (1 , , (cid:62) ∼ H y , H (0 , , (cid:62) ∼ H y ,H (0 , , (cid:62) ∼ H y and H (1 , , (cid:62) = (0 , , (cid:62) . Consequently, if we consider the rank two matrix H := H − H H , then y i ∼ Hx i for i = 1 , , Hx = 0. Thus, the result follows from Lemma 3.3.(3) The points x , x , x are collinear in R while x is not on the line, and the points y , y , y are collinear in R while y is not on the line; see Figure 2. Using Lemma 1.6, by multiplying two invertible matrices from the left to x i ’s and y i ’s ifnecessary, we may assume x = (0 , x = (0 , α ), x = (0 , β ), x = ( x , x ), y = (0 , y = (0 , γ ), y = (0 , δ ) and y = ( y , y ), where x , α, β, γ, δ are non-zero real numbers, α (cid:54) = β and γ (cid:54) = δ . Then there is a matrix H ∈ R × such that Hx = αβx ( γ − δ ) y Hx = αδx ( α − x ) y Hx = βγx ( α − x ) y Hx = x [ βγ ( α − x ) − αδ ( β − x )] y , given by H := H H H H H where H = ( α − x ) βγy − ( β − x ) αδy H = − αx γδ + βx γδ + αβγy − βx γy − αβδy + αx δy H = ( α − β ) x γδH = ( αδ − βγ ) x H = ( γ − δ ) x αβ. Notice that H H (cid:54) = 0, which implies rank( H ) ≥
2. Then, the result follows usingLemma 3.3.(4)
The points x , x , x are collinear in R , and the points y , y , y are collinear in R ; seeFigure 2. Let P X be the plane in R containing (0 , ,
0) and the common line l X joining x , x , x .Let P Y be the plane in R containing (0 , ,
0) and the common line joining y , y , y . Takea unit vector u ∈ P X so that u (cid:62) x = 0. Let U be the orthogonal matrix given by U := (cid:18) x (cid:107) x (cid:107) , u, x (cid:107) x (cid:107) × u (cid:19) N THE EXISTENCE OF EPIPOLAR MATRICES 15
Let w ∈ P Y be a unit vector so that w (cid:62) y = 0. We consider the orthogonal matrix W := (cid:18) y (cid:107) y (cid:107) , w, y (cid:107) y (cid:107) × w (cid:19) . Let R be an orthogonal matrix so that RU = W , namely, R := W U (cid:62) . Then, R x (cid:107) x (cid:107) = y (cid:107) y (cid:107) and Ru = w . If x ∈ l X , then x = α x (cid:107) x (cid:107) + βu for some real numbers α, β . Thus we have Rx = αR x (cid:107) x (cid:107) + βRu = α y (cid:107) y (cid:107) + βw ∈ P Y . Consider the essential matrix E = [ y ] × R . One has y (cid:62) Ex = y (cid:62) [ y ] × Rx = 0 (cid:62) Rx = 0 and y (cid:62) Ex = y (cid:62) [ y ] × Rx ∼ y (cid:62) [ y ] × y = 0 . For i = 2 ,
3, since Rx i ∈ P Y = span { y i , y } , one obtains y (cid:62) i Ex i ∼ [ y i × y ] (cid:62) Rx i = 0 . Hence E is an essential matrix of Z . (cid:3) Corollary 3.5.
An essential matrix always exists when m ≤ provided all the x i ’s are distinctand all the y i ’s are distinct. Discussion
In this paper, we have settled the existence problem for fundamental matrices for all values of m and essential matrices for m ≤ m for which rank( Z ) ≤ ≤ m ≤ m = 5 ,
6. For m = 7, we did find a testfor the existence of an essential matrix. This uses the classical theory of Chow forms [4, 9]. Wehave not included it in this paper since we felt that it deserves further attention and can possiblybe simplified. The interested reader can find the details in [1, Section 3.3]. Chow forms also providea test for whether ker C ( Z ) ∩ E C (cid:54) = ∅ when m = 6 [1, Section 3.1]. Again, we have left this out ofthe current paper since it does not answer the question of existence of a real essential matrix when m = 6.Even though our results are phrased in terms of the matrix Z , we have shown that they can bereinterpreted in terms of the input X and Y in most cases. We are curious about the set of six andseven point correspondences in two views for which there is no fundamental matrix. Theorems 2.2and 2.5 characterized the point configurations for which there is no fundamental matrix becauseker R ( Z ) ⊆ R . It would also be interesting to understand the configurations for which ker R ( Z )contains only matrices of ranks one and three as in Example 2.1.The results in this paper show that reasoning over real numbers is both a source of surprises andcomplications. We believe that similar surprises and complications lurk in other existence problemsin multiview geometry and are worthy of study. N THE EXISTENCE OF EPIPOLAR MATRICES 16
Appendix A. Proof of Lemma 1.6
Since l − x and m − y are lines in R passing through the origin, one can choose an orthogonalmatrix W ∈ R × such that m − y = W ( l − x ). It follows that m = W ( l − x ) + y = W l − W x + y = W l + z where z := y − W x is a point in R . Then, for the 3 × H := ( W z ), one has ( m ) = H ( l )which verifies the statement (2). In addition, H ( x ) = ( y ), and thus the assertion (1) alsoholds. (cid:3) Appendix B. Proof of Lemma 2.3
Recall that a fundamental matrix can be written in the form F = [ b ] × H where b is a nonzerovector in R and H ∈ R × is an invertible matrix. Then the epipolar constraints can be rewrittenas y (cid:62) i F x i = 0 , ∀ i = 1 , . . . , m. ⇐⇒ y (cid:62) i [ b ] × Hx i = 0 , ∀ i = 1 , . . . , m. ⇐⇒ y (cid:62) i ( b × Hx i ) = 0 , ∀ i = 1 , . . . , m. (B.1) ⇐⇒ b (cid:62) ( y i × Hx i ) = 0 , ∀ i = 1 , . . . , m. (B.2) ⇐⇒ b (cid:62) (cid:0) · · · y i × Hx i · · · (cid:1) = 0 . A non-zero b exists in the expression for F if and only ifrank (cid:0) · · · y i × Hx i · · · (cid:1) < . (B.3)The equivalence of (B.1) and (B.2) follows from the fact that p (cid:62) ( q × r ) = − q (cid:62) ( p × r ). The matrixin (B.3) is of size 3 × m . A sufficient condition for it to have rank less than 3 is for m − H = A given in the assumption. (cid:3) The observation about the scalar triple product and the resulting rank constraint has also beenused by Kneip et al. [17] but only in the calibrated case.
Appendix C. Proof of Theorem 2.2 “If” part : Suppose (1) holds and let τ be the set given in (1). Then there is a u ∈ P R suchthat u (cid:62) y i = 0 for any i ∈ τ . Let x k be the single element in the set { x i } i/ ∈ τ . Consider a basis { v , v } ⊆ P R of the orthogonal complement of x k . For j = 1 ,
2, define A j = uv (cid:62) j ∈ P × R and let a j ∈ P R be its vectorization. Then { a , a } is a linearly independent set spanning a subset of R .Moreover for any i = 1 , . . . , j = 1 , y (cid:62) i A j x i = ( y (cid:62) i u )( v (cid:62) j x i ) = 0. Hence a j ∈ ker R ( Z ) for j = 1 ,
2. As rank( Z ) = 7 (cf. (1.6)), ker R ( Z ) = span { a , a } ⊆ R . The same idea of proof worksif (2) holds. “Only if” part : Consider a basis { a , a } ⊆ P R of ker R ( Z ), which is inside R , and assume a j is the vectorization of A j ∈ P × R for j = 1 ,
2. For any j , rank( A j ) = 1, so A j = u j v (cid:62) j for some u j , v j ∈ P R . Since rank( A + A ) = 1, a simple check shows that either { u , u } or { v , v } islinearly dependent. Thus, up to scaling, we may assume either u = u or v = v . If u = u ,then { v , v } is linearly independent. In addition, 0 = y (cid:62) i A j x i = ( y (cid:62) i u )( v (cid:62) j x i ) for each i = 1 , . . . , j = 1 ,
2. Thus, either y (cid:62) i u = 0 or x i ∈ span { v , v } ⊥ . Notice that span { v , v } ⊥ is a singletonin P R . As rank( Z ) = 7, by the paragraph after Lemma 1.14, neither “ y (cid:62) i u = 0 for all i ” nor N THE EXISTENCE OF EPIPOLAR MATRICES 17 “ x i ∈ span { v , v } ⊥ for all i ” can happen. Hence (1) holds with the nonempty proper subset τ := { i : y (cid:62) i u = 0 } of { , . . . , } . If v = v , by the same idea one sees that (2) holds. (cid:3) Appendix D. Proof of Theorem 2.5
Recall that we are assuming that Z has full row rank, i.e., m = rank( Z ) = 6. By Lemma 1.14,this can only be true for m = 6 if x i and y i are not simultaneously collinear, i.e. one of X or Y hasto have full row rank. “If” part : If all points y i are collinear in R , then there is u ∈ P R such that u (cid:62) y i = 0 for any i = 1 , . . . ,
6. Let e = (1 , , (cid:62) , e = (0 , , (cid:62) , e = (0 , , (cid:62) . Consider the 3 × A j = ue (cid:62) j for j = 1 , , a j ∈ P R . Then, { a , a , a } is a linearly independent set spanning asubset of R . Moreover, for any i = 1 , . . . , j = 1 , , y (cid:62) i A j x i = ( y (cid:62) i u )( x (cid:62) i e j ) = 0. Hence a j ∈ ker R ( Z ). As rank( Z ) = 6 (cf. (1.6)), ker R ( Z ) = span { a , a , a } ⊆ R . The same idea of proofworks if all points x i are collinear in R . “Only if” part : Consider a basis { a , a , a } ⊆ P R of ker R ( Z ), which is inside R , and assume a j is the vectorization of A j ∈ P × R for j = 1 , ,
3. Then, by Lemma 1.3 with n = 3 and r = 1, upto taking transpose of all A j , there are nonzero vectors u, v , v , v ∈ P R such that A j = uv (cid:62) j for j = 1 , ,
3. The vectors v j are linearly independent as A j are. Moreover 0 = y (cid:62) i A j x i = ( y (cid:62) i u )( x (cid:62) i v j )for any i = 1 , . . . , j = 1 , ,
3. We fix i ∈ { , . . . , } and claim that y (cid:62) i u = 0. Indeed, if y (cid:62) i u (cid:54) = 0,then x (cid:62) i v j = 0 for any j = 1 , ,
3. As vectors v j are linearly independent we have x i = 0. This isimpossible because x i as a point in P R has nonzero third coordinate. Hence our claim is true andthus all points y i are collinear in R . If it is necessary to replace A j by A (cid:62) j , it follows that all points x i are collinear in R . (cid:3) Appendix E. Proof of Lemma 2.6
We first consider the case when L is a projective line, i.e., L = { Aµ + Bη : µ, η ∈ R } for some A, B ∈ R × , with B invertible. Then B − L = { B − Aµ + Iη : µ, η ∈ R } is an isomorphicimage of L and contains a matrix of rank two if and only if L does. Hence we can assume L = { M µ − Iη : µ, η ∈ R } for some M ∈ R × . The homogeneous cubic polynomial det( M µ − Iη ) isnot identically zero on L . When dehomogenized by setting µ = 1, it is the characteristic polynomialof M . Hence the three roots of det( M µ − Iη ) = 0 in P are ( µ , η ) ∼ (1 , λ ) , ( µ , η ) ∼ (1 , λ ) and( µ , η ) ∼ (1 , λ ) where λ , λ , λ are the eigenvalues of M . At least one of these roots is real sincedet( M µ − Iη ) is a cubic. Suppose ( µ , η ) is real. If rank( M µ − Iη ) = rank( M − Iλ ) = 2, then L contains a rank two matrix. Otherwise, rank( M − Iλ ) = 1. Then λ is a double eigenvalue of M and hence equals one of λ or λ . Suppose λ = λ . This implies that ( µ , η ) is a real root aswell. If it is different from ( µ , η ), then it is a simple real root. Hence, rank( M µ − Iη ) = 2, and L has a rank two matrix. So suppose ( µ , η ) ∼ ( µ , η ) ∼ ( µ , η ) ∼ (1 , λ ) where λ is the uniqueeigenvalue of M . In that case, det( M µ − Iη ) = α · ( η − λµ ) for some constant α . This finishes thecase dim( L ) = 1.Now suppose dim( L ) ≥
2. If det restricted to L is not a power of a homogeneous linear polyno-mial, then there exists a projective line L (cid:48) in L such that det restricted to L (cid:48) is also not the powerof a homogeneous linear polynomial. The projective line L (cid:48) contains a matrix of rank two by theabove argument. (cid:3) N THE EXISTENCE OF EPIPOLAR MATRICES 18
Appendix F. A proof for the existence of a fundamental matrix when m ≤ Theorem F.1. If m ≤ , then Z has a fundamental matrix.Proof. If m ≤
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