bundle block adjustment with 3d natural cubic splines
bundle block adjustment with 3d natural cubic splines bundle block adjustment with 3d natural cubic splines
space features is the coplanarity approach. Projection plane defined by the perspectivecenter in the image space and the plane including the straight line in the objectspace are identical. The extended collinearity model using linear features comparingwith the coplanarity method was proposed. Zielinski[80] described another straightline expression with independent parameters.Zalmanson and Schenk[79] extended their algorithm to relative orientation usingline parameters. Epipolar lines were employed to adopt linear features in relativeorientation with the extended collinearity equations (2.10).Gauss-Markov modelwas partitioned into orientation parameters and line parameters as (2.13).y = [ A R A L] [ ξ Rξ L]+ e (2.13)where ξ R relative orientation parameters, ξ L line parameters, A R the partial derivativesof extended collinearity equations with respect to relative orientation parameters,A L the partial derivatives of extended collinearity equations with respect to lineparameters, and e the error vector. They demonstrated that parallel lines to epipolarlines contributed to relative orientation totally and vertical lines to epipolar linesrendered only surface reconstruction. In addition, they introduced a regularizationscheme with soft constraints for the general cases.Zalmanson[78] updated EOPs using the correspondence between the parametriccontrol free-form line in object space and the projected 2D free-form line in imagespace. The hierarchical approach, the modified iteratively close point (ICP) method,was developed to estimate curve parameters. The ray lies on the free-form line whoseparametric equation represented with l parameter is as following. Besl and McKay[7]employed the ICP algorithm to solve matching problem of point sets, free-form curves,20
surfaces and terrain models in 2D and 3D space. ICP algorithm is executed withoutthe prior knowledge of correspondence between points.The ICP method affectedZalmanson’s dissertation in the development of the recovery of EOPs using 3D freeformlines in photogrammetry. Euclidean 3D transformation is employed in the searchof the closest entity on the geometric data set. Rabbani et al.[52] utilized ICP methodin registration of Lidar point clouds to divide into four categories spheres, planes,cylinder and torus with direct and indirect method.Ξ(l) =⎡⎢⎣X(l)Y (l)Z(l)⎤⎥⎦ =⎡⎢⎣X k 0Y k0Z k 0⎤⎥⎦ +⎡⎢⎣⎤ρ 1⎥ρ 2ρ 3where X 0 , Y 0 , Z 0 , ω, ϕ, κ the EOPs and ρ 1 , ρ 2 , ρ 3 the direction vector.⎦ l (2.14)The parametric curve Γ(t) = [X(t) Y (t) Z(t)] Twas obtained by minimizing theEuclidian distance between two parametric curves.Φ(t, l) ≡ ‖Γ(t) − Ξ(l)‖ 2 = (X(t) − X 0 − ρ 1 l) 2 +(Y (t) − Y 0 − ρ 2 l) 2 +(Z(t) − Z 0 − ρ 3 l) 2(2.15)Φ(t, l) had a minimum value at ∂Φ/∂l = ∂Φ/∂t = 0 with two independent variablesl and t as (2.16).∂Φ/∂l = −2ρ 1 (X(t) − X 0 − ρ 1 l) − 2ρ 2 (Y (t) − Y 0 − ρ 2 l) − 2ρ 3 (Z(t) − Z 0 − ρ 3 l) = 0∂Φ/∂t = 2X ′ (t) (X(t) − X 0 − ρ 1 l) + 2Y ′ (t) (Y (t) − Y 0 − ρ 2 l) +2Z ′ (t) (Z(t) − Z 0 − ρ 3 l) = 0(2.16)Akav et al.[2] employed planar free form curves for aerial triangulation with the ICPmethod. Since the effect of Z parameter as compared with X and Y was large innormal plane equation aX +bY +cZ = 1, different plane representation was developed21
- Page 2: c○ Copyright byWon Hee Lee2008
- Page 7 and 8: ACKNOWLEDGMENTSThanks be to God, my
- Page 10 and 11: 3. BUNDLE BLOCK ADJUSTMENTWITH 3D N
- Page 13 and 14: CHAPTER 1INTRODUCTION1.1 OverviewOn
- Page 15 and 16: y an intersection employing more th
- Page 17 and 18: similarity of geometric properties
- Page 19 and 20: straight linear features or formula
- Page 21 and 22: • Bundle block adjustment by the
- Page 23 and 24: Hessian. Interest point operators w
- Page 25 and 26: [60], Ebner and Ohlhof(1994) [16],
- Page 27 and 28: a complicated problem. The developm
- Page 29 and 30: ⎡⎢⎣x i − x py i − y p−f
- Page 31: x p = −f (X A + t · a − X C )r
- Page 35 and 36: f(u) − e(u) = g(u)f(u) − e(u) =
- Page 37 and 38: Tankovich[69] used linear features
- Page 39 and 40: (a) 0th order continuity (b) 1st or
- Page 41 and 42: Cardinal splineA Cardinal spline is
- Page 43 and 44: 2.3.2 Fourier transformFourier seri
- Page 45 and 46: For other polyline expressions, Aya
- Page 47 and 48: Each segment of a natural cubic spl
- Page 49 and 50: ⎡⎢⎣2 11 4 11 4 1· · ·1 4 1
- Page 51 and 52: 3.2 Extended collinearity equation
- Page 53 and 54: R −1 = R T . The matrix R T (= R
- Page 55 and 56: dx p = M 1 dX C + M 2 dY C + M 3 dZ
- Page 57 and 58: In this research, the arc-length pa
- Page 59 and 60: =√∫ √√√ ()ti+1−f u′ (
- Page 61 and 62: This equation can be replaced with
- Page 63 and 64: order polynomial using Newton’s d
- Page 65 and 66: y collinearity equations, tangents
- Page 67 and 68: d tan(θ t ) = w′ (v ′ w − w
- Page 69 and 70: y each two points, which are four e
- Page 71 and 72: +M 14 db i3 + M 15 dc i0 + M 16 dc
- Page 73 and 74: collinearity model are described in
- Page 75 and 76: [ ] [ ] [ ]N11 N 12 ˆξ1 c1N12T =N
- Page 77 and 78: systematic errors in the image spac
- Page 79 and 80: interval based on the normal distri
- Page 81 and 82: 1 ∂Φ2 ∂l= (X C + d 1 l − a i
surfaces and terrain models in 2D and 3D space. ICP algorithm is executed <strong>with</strong>outthe prior knowledge of correspondence between points.The ICP method affectedZalmanson’s dissertation in the development of the recovery of EOPs using 3D freeformlines in photogrammetry. Euclidean 3D transformation is employed in the searchof the closest entity on the geometric data set. Rabbani et al.[52] utilized ICP methodin registration of Lidar point clouds to divide into four categories spheres, planes,cylinder and torus <strong>with</strong> direct and indirect method.Ξ(l) =⎡⎢⎣X(l)Y (l)Z(l)⎤⎥⎦ =⎡⎢⎣X k 0Y k0Z k 0⎤⎥⎦ +⎡⎢⎣⎤ρ 1⎥ρ 2ρ 3where X 0 , Y 0 , Z 0 , ω, ϕ, κ the EOPs and ρ 1 , ρ 2 , ρ 3 the direction vector.⎦ l (2.14)The parametric curve Γ(t) = [X(t) Y (t) Z(t)] Twas obtained by minimizing theEuclidian distance between two parametric curves.Φ(t, l) ≡ ‖Γ(t) − Ξ(l)‖ 2 = (X(t) − X 0 − ρ 1 l) 2 +(Y (t) − Y 0 − ρ 2 l) 2 +(Z(t) − Z 0 − ρ 3 l) 2(2.15)Φ(t, l) had a minimum value at ∂Φ/∂l = ∂Φ/∂t = 0 <strong>with</strong> two independent variablesl and t as (2.16).∂Φ/∂l = −2ρ 1 (X(t) − X 0 − ρ 1 l) − 2ρ 2 (Y (t) − Y 0 − ρ 2 l) − 2ρ 3 (Z(t) − Z 0 − ρ 3 l) = 0∂Φ/∂t = 2X ′ (t) (X(t) − X 0 − ρ 1 l) + 2Y ′ (t) (Y (t) − Y 0 − ρ 2 l) +2Z ′ (t) (Z(t) − Z 0 − ρ 3 l) = 0(2.16)Akav et al.[2] employed planar free form curves for aerial triangulation <strong>with</strong> the ICPmethod. Since the effect of Z parameter as compared <strong>with</strong> X and Y was large innormal plane equation aX +bY +cZ = 1, different plane representation was developed21
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