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Day x. 5-10 minute quiz SVD demo review of metric rectification algorithm build a 2x3 matrix A from orthog line pairs A = [L1M1, L1M2+L2M1, L2M2; L1’M1’, L1’M2’+L2’M1’, L2’M2’] find null vector s of A (using SVD) decode s into S Cholesky decompose S to find K: S = KK^t

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day x
Day x
  • 5-10 minute quiz
  • SVD demo
  • review of metric rectification algorithm
  • build a 2x3 matrix A from orthog line pairs

A = [L1M1, L1M2+L2M1, L2M2; L1’M1’, L1’M2’+L2’M1’, L2’M2’]

  • find null vector s of A (using SVD)
  • decode s into S
  • Cholesky decompose S to find K: S = KK^t
    • S and K are 2x2, so should be simple
  • metrically rectify image using affinity [K 0; 0 1]
    • image should have been affinely rectified already
slide2
Epipolar

geometry

HZ 9.1

Fundamental

matrix

HZ 9.2

normalized

8-pt alg

for F

HZ 11.2

7-point

algorithm

Camera

from F

Structure

computation

Alg 12.1, HZ318

linear constraints on f
Linear constraints on F
  • let x = (x,y,1) and x’ = (x’,y’,1) be a point correspondence between 2 images, and let F be the fundamental matrix of the images
  • we know that the point pair satisfies x’ F x = 0
  • this generates a linear constraint on the entries of F = (fij):
    • x’xf11 + x’yf12 + x’f13 + y’xf21 + ... + f33 = 0
    • (x’x, x’y, x’, y’x, y’y, y’, x, y, 1) f = 0
    • where f = (f11,12,...,f33) encodes F in row-major order
  • each point correspondence (xi,xi’) yields a row (xi’xi, xi’yi, xi’, yi’xi, yi’yi, yi’, xi, yi, 1)
  • collect n rows (x’x, x’y, x’, y’x, y’y, y’, x, y, 1) into nx9 matrix A
  • F is found as the solution of Af = 0
  • HZ279
solving af 0
Solving Af=0
  • if data is exact and rank(A) is exactly 8, then f is found as the null vector of A
    • in a perfect world, Af=0 has a solution
    • recall: use SVD (and last column of V) to compute the null vector
  • if data is noisy and rank(A) = 9, then f is found as a least squares solution
    • min ||Af||
    • SVD, use last column of V
    • just like in the DLT algorithm
  • HZ280
the singularity constraint
The singularity constraint

without

with

  • force F to be singular
    • we know the correct F is singular (see above)
    • we need F to be singular, since the epipoles are computed using its null space
    • due to noise, the computed F will not be singular
  • how? use another SVD!
  • let F = UDV^t where D = diag(r,s,t) with r>=s>=t
  • let F’ = UD’V^t where D’ = diag(r,s,0)
    • this is guaranteed to be singular since det(F’) = 0
  • HZ280-281
preconditioning
Preconditioning
  • recall preconditioning in DLT algorithm
  • apply it in 8 point algorithm for F too
  • before creating A (as in Af=0) from the point correspondences, normalize the points
    • centroid to origin translation + scaling to \sqrt 2 average distance
    • also called root mean square (RMS) scaling
  • let xi Txi and xi’  T’xi’ be the transformation matrices
  • denormalize via F  T’t F T after finding F (i.e., after enforcing the singularity constraint)
    • shampoo?
  • HZ282
normalized 8 point algorithm
Normalized 8 point algorithm
  • basically 2 SVD’s, the first to find F and the second to refine it
  • the fodder for the SVD’s is a matrix built from the point correspondences
  • SIFT: find a set of at least 8 point correspondences (xi,xi’)
  • precondition each pointset {xi} and {xi’} independently (with T and T’)
  • [solve for F as a null vector f]
    • build nx9 matrix A from x^t F x = 0 constraints, n>=8
    • compute SVD of A: A = U1 D V1^t
    • f = row-major encoding of F = null vector of A = last column of V1
  • [impose singularity constraint]
    • compute SVD of F: F = U2 D V2t
    • replace F by U2 diag(d11,d22,0) V2t
  • postcondition: replace F by T’t F T
  • SPASSP (SIFT, precondition, A, SVD for f, SVD for singularity, postcondition)
  • HZ282, Alg 11.1
7 point algorithm
7-point algorithm
  • F has 8dof so we need 8 constraints
  • but the singularity constraint (det F = 0) can act as one of the 8 constraints
  • hence, only need 7 constraints from point correspondences
  • build 7x9 matrix A (as above)
  • the 2D null space of Af=0 is spanned by the last 2 columns of V in the SVD of A
    • let v1 and v2 be the last 2 columns of V
    • the vector family f(t) = (1-t)v1 + tv2 yields a matrix family F(t) = (1-t)F1 + tF2
  • now solve det (F(t)) = 0 for t
  • since det(F(t)) is a cubic polynomial in t, it has 1 or 3 real roots
  • hence, this algorithm yields 1 or 3 solutions for F
  • HZ281
more than one solution for f
More than one solution for F
  • degeneracy: 3 solutions if the camera centers and 3D points Xi (from which the point pairs (xi,xi’)) all lie on a ‘critical surface’
    • critical surface can be two planes, cone, cylinder or hyperboloid of one sheet (N.B. always a ruled quadric)
  • this is quite likely in the 7-point algorithm: 7 points and 2 camera centers always lie on some quadric, since 9 points define a quadric
    • the only question is whether the quadric is ruled or not (ellipsoid, hyperboloid of one sheet)
  • beware of all points in the same plane, or identical camera centers
    • this will yield a 2d family of family of fundamental matrices!
  • HZ296
iterative optimization
Iterative optimization
  • an even better approximation to F can be found using an iterative method, bootstrapping from the 7-point or 8-point solutions
  • based on the idea of searching only for singular matrices that minimize ||Af||
    • rather than separating out the singularity constraint in a later stage
  • requires the epipole, which we do not know, so it leads to an iterative process
    • guess F, compute associated e, compute new F, iterate
  • this goes beyond our scope here: we’ll talk about it in seminar
  • techniques: 2 SVD’s (HZ595 minimizations) and Levenberg-Marquardt algorithm (HZ600): Algorithm 11.2, HZ284
  • there is also a gold standard algorithm that involves even more computation: Algorithm 11.3, HZ285
  • HZ282-285; HZ595,600
which algorithm is best
Which algorithm is best?
  • an algorithm for computing F may be evaluated by computing the distance of a point xi’ from the computed epipolar line Fxi
  • conclusions from HZ experiments (Figure 11.3):
    • normalized 8-pt algorithm performs just fine
    • use more than 8 points: error is reduced up to about 15 points, then starts leveling off
    • 8-point with 15 points is better than more elaborate algorithms on 8 points
  • add RANSAC estimation for added robustness (Alg 11.4)
  • note that we are computing F with no user interaction
  • HZ288-291
adding ransac
Adding RANSAC

without

with

  • avoids outlier contamination
  • SIFT will have plenty of outliers
  • HZ290-291
don t use line correspondences
Don’t use line correspondences
  • fascinating: line correspondences offer no information in 2 images!
  • reason: 2 points backproject to 2 lines, which must intersect if the points are corresponding
    • this adds a constraint, since 2 lines in 3-space do not intersect in general
  • however, 2 lines backproject to 2 planes, and 2 planes always intersect, so there is no added constraint
  • line correspondences do become useful in 3 views: see trifocal tensors
  • HZ294
but curve correspondences are useful
But curve correspondences are useful
  • shared tangency at epipolar lines can be leveraged to add a constraint
    • corresponding curves in the 2 images must be tangent to the corresponding epipolar lines
  • see Exercises (ii) and (iii) at end of Chapter 11
  • tangency again: we will explore later
  • HZ295, 309
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