If m=4, choosing one row from each of four different views gives a quadrilinear four-view constraint, expressed by the quadrifocal tensor.

If we assume for a moment that the projective depths ζij are known, then matrix W is known too and we can compute its singular value decomposition:

This equation can be used to recover T (likewise we did for F). The coefficient matrix is a 9×27 matrix; its rank is four, being the Kronecker product of a vector by a rank-2 matrix by a rank-2 matrix.

Epipolar transfer fails when the three optical rays are coplanar, because the epipolar lines are coincident. This happens:

Writing Eq. (45) for each camera pair (taking the centre of the third camera as the point M) results in three epipolar constraints:

A third equivalent formulation of the trifocal constraint is derived if we look at the vector T⁢m1 in Eq. (73) as the vectorization of a suitable matrix. This is easy to write thanks to the vector transposition2The vector transposition operator Apgeneralizes the transpose of a matrix A by operating on vectors of p entries at a time.:

This geometry could be described in terms of fundamental matrices linking pairs of cameras, but a more compact and elegant description is provided by a suitable trilinear form, in the same way as the epipolar (bifocal) geometry is described by a bilinear form.

Geometric perspective definition

This is the point transfer equation: if m1 and m2 are conjugate points in the first and second view respectively, the position of the conjugate point m3 in the third view is computed by means of the trifocal matrix.

If the intrinsic parameters of the cameras are known, we can obtain a Euclidean reconstruction, that differs from the true reconstruction by a similarity transformation. This is composed by a rigid displacement (due to the arbitrary choice of the world reference frame) plus a a uniform change of scale (due to the well-known depth-speed ambiguity).

This indicates that no interesting constraints can be written for more than four views5Actually, it can be proven that also the quadrifocal constraints are not independent (Ma et al.,2003)..

The Kronecker notation and the tensorial notation are deeply related, as both represents multilinear forms. To draw this relationship in the case of the trifocal geometry, let us expand the trifocal matrix into its columns T=t1⁢t2⁢t3 and m1 into its components m1=u,v,wT. Then, thanks to the linearity of the vector transposition:

The plane containing the three optical centres is called the trifocal plane. It intersects each image plane along a line which contains the two epipoles.

This implies that T⁢m13 can be seen as the linear combination of the matrices t13,t23,t33 with the components of m1 as coefficients. Therefore, the action of the trilinear form Eq. (81) is to first combine matrices t13,t23,t33 according to m1, then combine the columns of the resulting matrix according to s3 and finally to combine the elements of the resulting vector according to s2, to obtain a scalar.

This is the trifocal constraint, that links (via a trilinear form) m1, s2 (any line through m2) and s3 (any line through m3).

We outline here an alternative and elegant way to derive all the meaningful multi-linear constraints on N views, based on determinants, described in (Heyden,1998). Consider one image point viewed by m cameras:

This implies that the transpose of the leftmost term in parentheses (which is a 3-D vector) belongs to the kernel of m3×, which is equal to m3 (up to a scale factor) by construction. Hence

Please note that given two (arbitrary) lines in two images, they can be always seen as the image of a 3-D line L, because two planes always define a line, in projective space (this is why there is no such thing as the epipolar constraint between lines.)

A line s2 in the second view defines (by back-projection) a 3-D plane, which induces a homography H between the first and the third view.

In presence of noise, σ5 will not be zero. By forcing D=diag⁢σ1,σ2,σ3,σ4,0,…⁢0 one computes the solution that minimizes the following error:

Geometric perspective Renaissance

Starting from an initial guess for ζij (typically ζij=1), the following iterative procedure4Whilst this procedure captures the main idea of Sturm and Triggs, it is not exactly the algorithm proposed in (Sturm and Triggs,1996). To start with, the original algorithm (Sturm and Triggs,1996) was not iterative and used the epipolar constraint (Eq.45) to fix the ratio of the projective depths of one point in successive images. It was Triggs (1996) who made the scheme iterative. Moreover in (Sturm and Triggs,1996) the normalization of W is performed by normalizing rows and columns of W. The Frobenius norm was used by (Oliensis,1999). A similar scheme was also proposed by Heyden (1997). is used:

Denoting the cameras by 1,2,3, there are now three fundamental matrices, F1,2, F1,3, F2,3, and six epipoles, ei,j, as in Figure 11. The three fundamental matrices describe completely the trifocal geometry (Faugeras and Robert,1994).

If the trifocal geometry is known, given two conjugate points m1 and m2 in view 1 and 2 respectively, the position of the conjugate point m3 in view 3 is completely determined (Figure 12).

In this section we study the relationship that links three or more views of the same 3-D scene, known in the three-view case as trifocal geometry.

View in geometryexamples

In addition, by generalizing the case of two views, one might conjecture that the trifocal geometry should be represented by a trilinear form in the coordinates of three conjugate points.

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This is the trifocal constraint for lines, which also allows direct line transfer: if s3 and s2 are two lines in the third and second view respectively, the image s1 in the first view of the line in space determined by s2 and s3 is obtained by means of the trifocal matrix.

View in geometrypdf

If m=2 choosing two rows from one view and two rows from another view gives a bilinear two-view constraint, expressed by the bifocal tensor i.e., the fundamental matrix.

If m=3, choosing two rows from one view, one row from another view and one row from a third view gives a trilinear three-view constraint, expressed by the trifocal tensor.

An elegant method for multi-image reconstruction was described in Sturm and Triggs (1996), based on the same idea of the factorization method of Tomasi and Kanade (1992).

The trifocal constraint represents the trifocal geometry (nearly) without singularities. It only fails is when the cameras are collinear and the 3-D point is on the same line.

Geometrically, the trifocal constraint imposes that the optical rays of m1 intersect the 3-D line L that projects onto s2 in the second image and s3 in the third image.

Perspectiveinmath

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If one applies the method of Section 4.4.2 to view pairs 1-2, 1-3 and 2-3 one obtains three displacements R12,t⌃12,R13,t⌃13 and R23,t⌃23 known up a scale factor, as the norm of translation cannot be recovered, (the symbol ⌃ indicates a unit vector).

Mathematical perspectiveinart

Hence we can write a total of nine constraints similar to Eq. (72), only four of which are independent (two for each point):

Hence, at least one row has to be taken from each view to obtain a meaningful constraint, plus another row from each camera to prevent the constraint to be trivially factorized.

If a minor contains only one row from some view, the image coordinate corresponding to this row can be factored out (using Laplace expansion along the corresponding column).

In both cases, the solution is not a straightforward generalization of the two view case, as the problem of global consistency comes into play (i.e., how to relate each other the local reconstructions that can be obtained from view pairs).

Consider a line L in space projecting to s1, s2 and s3 in the three cameras. The trifocal constraint must hold for any point m1 contained in the line s1:

If one applies the method of Section 4.5.3 to consecutive pairs of views, she would obtain, in general, a set of reconstructions linked to each other by an unknown projective transformation (because each camera pair defines its own projective frame).

Three fundamental matrices include 21 free parameters, less the 3 constraints above; the trifocal geometry is therefore determined by 18 parameters.

This reconstruction is unique up to a (unknown) projective transformation. Indeed, for any non singular projective transformation T, T⁢P and T-1⁢M is an equally valid factorization of the data into projective motion and structure.

Epipolargeometry inComputer vision

In the noise-free case, D=diag⁢σ1,σ2,σ3,σ4,0,…⁢0, thus, only the first 4 columns of U (V) contribute to this matrix product. Let U3×m×4 (Vn×4) the matrix of the first 4 columns of U (V). Then:

Projectivegeometry

This description of the trifocal geometry fails when the three cameras are collinear, and the trifocal plane reduces to a line.

Therefore, every triplet {m1, m2, m3} of corresponding points gives four linear independent equations. Seven triplets determine the 27 entries of T.

This allows for point transfer or prediction. Indeed, m3 belongs simultaneously to the epipolar line of m1 and to the epipolar line of m2, hence:

Hence, s3=HT⁢s1 since s1 and s3 are both projection of the same line, that belongs to the plane 3The reader can verify that if H is the homography induced by a plane between two views, such that conjugate points are related by m2=H⁢m1, conjugate lines are related by s2=HT⁢s1..

Please note that Eq. (95) can be also used to triangulate one point M in multiple views, by solving the homogeneous linear system for M,-ζ1,-ζ2,⋯,-ζmT.

If m>4, there is no way to avoid that the minors contain only one row from some views. Hence, constraints involving more than 4 cameras can be factorised as product of the two-, three-, or four-views constraints and image point coordinates.

An equivalent formulation of the trifocal constraint that generalizes the expression of a bilinear form (Cfr. pg. 4.5.2) is obtained by applying once again the property v⁢e⁢c⁢A⁢X⁢B=BT⊗A⁢v⁢e⁢c⁢X:

The minors that does not contain at least one row from each camera are identically zero, since they contain a zero column.

In the weakly calibrated case, i.e., when point correspondences are the only information available, a projective reconstruction can be obtained.

This technique is fast, requires no initialization, and gives good results in practice, although there is no guarantee that the iterative process will converge. A provably convergent iterative method has been presented by Mahamud et al. (2001).

We also discover that three views are all we need, in the sense that additional views do not allow us to compute anything we could not already compute (Section 5.4).

In this formula the mij are known, but all the other quantities are unknown, including the projective depths ζij. Equation (89) tells us that W can be factored into the product of a 3×m×4 matrix P and a 4×n matrix M. This also means that W has rank four.

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View synthesisLaveau and Faugeras (1994); Avidan and Shashua (1997); Boufama (2000), exploit the trifocal geometry to generate novel (synthetic) images starting from two reference views. A related topic is image-based rendering(Lengyel,1998; Zhang and Chen,2003; Isgrò et al.,2004).

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This implies that the 3×m×m+4 matrix L is rank-deficient, i.e., r⁢a⁢n⁢k⁢L

The trifocal geometry could be used to link together consistently triplets of views. In Section 4.5.3 we saw how a camera pair can be extracted from the fundamental matrix. Likewise, a triplet of consistent cameras can extracted from the trifocal matrix (or tensor). The procedure is more tricky, though.