Parameter estimation class 5

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Parameter estimation class 5. Multiple View Geometry CPSC 689 Slides modified from Marc Pollefeys’ Comp 290-089. Projective 3D Geometry. Points, lines, planes and quadrics Transformations П ∞ , ω ∞ and Ω ∞. Singular Value Decomposition. Homogeneous least-squares. Parameter estimation. - PowerPoint PPT Presentation

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  • Parameter estimationclass 5Multiple View GeometryCPSC 689Slides modified from Marc Pollefeys Comp 290-089

  • Projective 3D GeometryPoints, lines, planes and quadrics

    Transformations

    , and

  • Singular Value DecompositionHomogeneous least-squares

  • Parameter estimation2D homographyGiven a set of (xi,xi), compute H (xi=Hxi)3D to 2D camera projectionGiven a set of (Xi,xi), compute P (xi=PXi)Fundamental matrixGiven a set of (xi,xi), compute F (xiTFxi=0)Trifocal tensor Given a set of (xi,xi,xi), compute T

  • Number of measurements requiredAt least as many independent equations as degrees of freedom requiredExample:

    2 independent equations / point8 degrees of freedom4x28

  • Approximate solutionsMinimal solution4 points yield an exact solution for HMore pointsNo exact solution, because measurements are inexact (noise)Search for best according to some cost functionAlgebraic or geometric/statistical cost

  • X, X : measured value from the first image and the second image, respectively -- Known values with noise : estimated value algorithm output : true value (unknown). Notations

  • Gold Standard algorithmCost function that is optimal for some assumptionsComputational algorithm that minimizes it is called Gold Standard algorithm Other algorithms can then be compared to it

  • Direct Linear Transformation(DLT)

  • Direct Linear Transformation(DLT)Equations are linear in hOnly 2 out of 3 are linearly independent (indeed, 2 eq/pt)(only drop third row if wi0)Holds for any homogeneous representation, e.g. (xi,yi,1)

  • Direct Linear Transformation(DLT)Solving for Hsize A is 8x9 or 12x9, but rank 8Trivial solution is h=09T is not interesting1-D null-space yields solution of interestpick for example the one with

  • Direct Linear Transformation(DLT)Over-determined solutionNo exact solution because of inexact measurementi.e. noiseFind approximate solution Additional constraint needed to avoid 0, e.g. not possible, so minimize

  • DLT algorithmObjectiveGiven n4 2D to 2D point correspondences {xixi}, determine the 2D homography matrix H such that xi=HxiAlgorithmFor each correspondence xi xi compute Ai. Usually only two first rows needed.Assemble n 2x9 matrices Ai into a single 2nx9 matrix AObtain SVD of A. Solution for h is last column of VDetermine H from h

  • Inhomogeneous solutionSince h can only be computed up to scale, pick hj=1, e.g. h9=1, and solve for 8-vectorSolve using Gaussian elimination (4 points) or using linear least-squares (more than 4 points)However, if h9=0 this approach fails also poor results if h9 close to zeroTherefore, not recommendedNote h9=H33=0 if origin is mapped to infinity

  • Degenerate configurationsx4x1x3x2x4x1x3x2H?H?x1x3x2x4Constraints:i=1,2,3,4H* is rank-1 matrix and thus not a homography(case A)(case B)If H* is unique solution, then no homography mapping xixi(case B)If further solution H exist, then also H*+H (case A) (2-D null-space in stead of 1-D null-space)

  • Solutions from lines, etc.2D homographies from 2D linesMinimum of 4 linesConic provides 5 constraintsMixed configurations?

  • Cost functionsAlgebraic distanceGeometric distanceReprojection error

    ComparisonGeometric interpretationSampson error

  • Algebraic distanceDLT minimizesresidual vectorpartial vector for each (xixi)algebraic error vectorNot geometrically/statistically meaningfull, but given good normalization it works fine and is very fast (use for initialization)

  • Geometric distancee.g. calibration patternd(.,.) Euclidean distance (in image)

  • Reprojection error

  • Comparison of geometric and algebraic distancesError in one imageFor affinities DLT can minimize geometric distancePossibility for iterative algorithm

  • represents 2 quadrics in R4 (quadratic in X)Geometric interpretation of reprojection errorEstimating homography~fit surface to points X=(x,y,x,y)T in R4

  • Sampson errorbetween algebraic and geometric error

  • Sampson errorbetween algebraic and geometric error(Sampson error)

  • Sampson approximationA few pointsFor a 2D homography X=(x,y,x,y) is the algebraic error vector is a 2x4 matrix, e.g.Similar to algebraic error in fact, same as Mahalanobis distanceSampson error independent of linear reparametrization (cancels out in between e and J)Must be summed for all pointsClose to geometric error, but much fewer unknowns

  • Statistical cost function and Maximum Likelihood EstimationOptimal cost function related to noise modelAssume zero-mean isotropic Gaussian noise (assume outliers removed)

    Error in one image

  • Statistical cost function and Maximum Likelihood EstimationOptimal cost function related to noise modelAssume zero-mean isotropic Gaussian noise (assume outliers removed)

    Error in both images

  • Mahalanobis distanceGeneral Gaussian caseMeasurement X with covariance matrix

    *Exact solutions are important for robust approaches*Other algorithms can have other advantages: faster, more reliable, etc.*row 3 = x.(row 1)+y.(row 2)*Nonlinear, except for line fitting = affine homographies (no quadratic terms in this case!) (x,x,y and x,y,y) => linear solution