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The Trifocal Tensor Multiple View Geometry

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Scene planes and homographies plane induces homography between two views

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6-point algorithm x 1,x 2,x 3,x 4 in plane, x 5,x 6 out of plane Compute H from x 1,x 2,x 3,x 4

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Three-view geometry

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The trifocal tensor Three back-projected lines have to meet in a single line Incidence relation provides constraint on lines Let us derive the corresponding algebraic constraint…

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Notations

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Incidence e.g. is part of bundle formed by ’ and ”

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Incidence relation

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The Trifocal Tensor Trifocal Tensor = {T 1,T 2,T 3 } Only depends on image coordinates and is thus independent of 3D projective basis Also and but no simple relation General expression not as simple as DOF T: 3x3x3=27 elements, 26 up to scale 3-view relations: 11x3-15=18 dof 8(=26-18) independent algebraic constraints on T (compare to 1 for F, i.e. rank-2)

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Homographies induced by a plane

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Line-line-line relation Eliminate scale factor: (up to scale)

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Point-line-line relation

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Point-line-point relation note: valid for any line through x”, e.g. l”=[x”] x x” arbitrary

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Point-point-point relation note: valid for any line through x’, e.g. l’=[x’] x x’ arbitrary

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Overview incidence relations

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Non-incident configuration incidence in image does not guarantee incidence in space

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Epipolar lines if l’ is epipolar line, then satisfied for arbitrary l” inversely, epipolar lines are right and left null-space of

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Epipoles With points becomes respectively Epipoles are intersection of right resp. left null-space of (e=P’C and e”=P”C)

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Algebraic properties of T i matrices

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Extracting F good choice for l” is e” (V 3 T e”=0)

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Computing P,P‘,P“ ? ok, but not specifically, (no derivation)

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matrix notation is impractical Use tensor notation instead

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Definition affine tensor Collection of numbers, related to coordinate choice, indexed by one or more indices Valency = ( n+m ) Indices can be any value between 1 and the dimension of space ( d (n+m) coefficients)

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Conventions Einstein’s summation: (once above, once below) Index rule: Contravariant indices Covariant indices

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More on tensors Transformations (covariant) (contravariant)

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Some special tensors Kronecker delta Levi-Cevita epsilon (valency 2 tensor) (valency 3 tensor)

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Trilinearities

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Compute F and P from T

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matrix notation is impractical Use tensor notation instead

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Definition affine tensor Collection of numbers, related to coordinate choice, indexed by one or more indices Valency = ( n+m ) Indices can be any value between 1 and the dimension of space ( d (n+m) coefficients)

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Conventions Contraction: (once above, once below) Index rule:

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More on tensors Transformations (covariant) (contravariant)

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Some special tensors Kronecker delta Levi-Cevita epsilon (valency 2 tensor) (valency 3 tensor)

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Trilinearities

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Transfer: epipolar transfer

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Transfer: trifocal transfer Avoid l’=epipolar line

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Transfer: trifocal transfer point transfer line transfer degenerate when known lines are corresponding epipolar lines

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Computation of Trifocal Tensor Linear method (7-point) Minimal method (6-point) Geometric error minimization method RANSAC method

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Basic equations Three points Correspondence Relation #lin. indep.Eq. 4 Two points, one line One points, two line 2 1 2 Three lines At=0 (26 equations) (more equations) min||At|| with ||t||=1

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Normalized linear algorithm At=0 Points Lines or Normalization: normalize image coordinates to ~1

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Normalized linear algorithm Objective Given n 7 image point correspondences across 3 images, or a least 13 lines, or a mixture of point and line corresp., compute the trifocal tensor. Algorithm (i)Find transformation matrices H,H’,H” to normalize 3 images (ii)Transform points with H and lines with H -1 (iii)Compute trifocal tensor T from At=0 (using SVD) (iv)Denormalize trifocal tensor

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Internal constraints 27coefficients 1 free scale 18 parameters 8 internal consistency constraints (not every 3x3x3 tensor is a valid trifocal tensor!) (constraints not easily expressed explicitly) Trifocal Tensor satisfies all intrinsic constraints if it corresponds to three cameras {P,P’,P”}

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Maximum Likelihood Estimation data cost function parameterization (24 parameters+3N) also possibility to use Sampson error (24 parameters)

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Objective Compute the trifocal tensor between two images Algorithm (i)Interest points: Compute interest points in each image (ii)Putative correspondences: Compute interest correspondences (and F) between 1&2 and 2&3 (iii)RANSAC robust estimation: Repeat for N samples (a) Select at random 6 correspondences and compute T (b) Calculate the distance d for each putative match (c) Compute the number of inliers consistent with T (d

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108 putative matches 18 outliers 88 inliers 95 final inliers (26 samples) (0.43) (0.23) (0.19)

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additional line matches

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Matrix formulation for m-View Consider one object point X and its m images: i x i =P i X i, i=1, ….,m: i.e. rank(M) < m+4.

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Laplace expansions The rank condition on M implies that all (m+4)*(m+4) minors of M are equal to 0. These can be written as sums of products of camera matrix parameters and image coordinates.

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Matrix formulation for non-trivially zero minors, one row has to be taken from each image (m). 4 additional rows left to choose

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only interesting if 2 or 3 rows from view

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The three different types 1.Take the 2 remaining rows from one image block and the other two from another image block, gives the 2-view constraints. 2.Take the 2 remaining rows from one image block 1 from another and 1 from a third, gives the 3-view constraints. 3.Take 1 row from each of four different image blocks, gives the 4-view constraints.

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