bundle block adjustment with 3d natural cubic splines

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total number of equations is 2 × 6(the number of images)×3(the number of points) =36 and the total number of unknowns is 18(the number of spline location parameters)so the redundancy is 18. Since spline location parameters are independent of eachother, the arc-length parameterization is not required.Spline location parametersImage 1 Image 2t 1 t 7 t 13 t 2 t 8 t 14ξ 0 0.01 0.37 0.63 0.09 0.44 0.71ˆξ 0.0589 0.3570 0.6712 0.1134 0.4175 0.7069±0.0015 ±0.0076 ±0.0197 ±0.0072 ±0.0054 ±0.0080Image 3 Image 4t 3 t 9 t 15 t 4 t 10 t 16ξ 0 0.17 0.46 0.74 0.21 0.49 0.81ˆξ 0.1757 0.4784 0.7631 0.2039 0.4869 0.8122±0.0031 ±0.0071 ±0.0095 ±0.0102 ±0.0030 ±0.0044Image 5 Image 6t 5 t 11 t 17 t 6 t 12 t 18ξ 0 0.26 0.53 0.84 0.29 0.61 0.89ˆξ 0.2544 0.5554 0.8597 0.3151 0.6284 0.9013±0.0050 ±0.0069 ±0.0089 ±0.0095 ±0.0052 ±0.0086Table 5.6: Spline location parameter recoveryThe result represents that a convergence of spline location parameters has beenachieved with fixed spline parameters considered as stochastic constraints. The proposedmodel is robust with respect to the initial approximations of spline parameters.The uncertain information related to the representation of a natural cubic spline isdescribed in the dispersion matrix.84
5.3 Recovery of EOPs and spline parametersThe object space knowledge about splines is available to recover the exterior orientationparameters in bundle block adjustment.Control spline and partial controlspline are applied to verify the feasibility of control information with splines.In both cases, equations of the arc-length parameterization are not necessary if wehave enough equations to solve the system since spline parameters are independentof each other. In the experiment of full control spline, the total number of equationsare 2 × 6(the number of images)×4(the number of points) +3(the number ofarc-length)×6(the number of images) = 66 and the total number of unknowns are36(the number of EOPs) +24(the number of spline location parameters) = 60. Theredundancy is 6.In the case of the partial control spline with one spline segment, the total numberof equations are 2 × 6(the number of images)×4(the number of points) +3(the numberof arc-length)×6(the number of images) = 66 and the total number of unknownsare 36(the number of EOPs)+9(the number of partial spline parameters)+24(thenumber of spline location parameters) = 69. Thus one more segment is required tosolve the underdetermined system. The total number of equations using two splinesegments are 2 × 6(the number of images)×4(the number of points)×2(the numberof spline segment) +3(the number of arc-length)×6(the number of images)×2(thenumber of spline segment) = 132 and the total number of unknowns are 36(the numberof EOPs)+9(the number of partial spline parameters)×2(the number of splinesegment)+24(the number of spline location parameters)×2(the number of spline segment)= 102. The redundancy is 30. A convergence of EOPs of an image block andspline parameters have been achieved in both experiments.85
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c○ Copyright byWon Hee Lee2008
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ACKNOWLEDGMENTSThanks be to God, my
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3. BUNDLE BLOCK ADJUSTMENTWITH 3D N
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CHAPTER 1INTRODUCTION1.1 OverviewOn
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y an intersection employing more th
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similarity of geometric properties
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straight linear features or formula
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• Bundle block adjustment by the
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Hessian. Interest point operators w
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[60], Ebner and Ohlhof(1994) [16],
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a complicated problem. The developm
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⎡⎢⎣x i − x py i − y p−f
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x p = −f (X A + t · a − X C )r
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surfaces and terrain models in 2D a
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f(u) − e(u) = g(u)f(u) − e(u) =
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Tankovich[69] used linear features
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(a) 0th order continuity (b) 1st or
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Cardinal splineA Cardinal spline is
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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
Page 83 and 84: about splines, their relationships,
Page 85 and 86: cubic spline in the image and the o
Page 87 and 88: The redundancy budget of a tie poin
Page 89 and 90: of bundle block adjustment is requi
Page 91 and 92: ξ kiSP = [ da i0 da i1 da i2 da i3
Page 93 and 94: Spline location parametersImage 1 I
Page 95: Spline location parametersImage 1 I
Page 99 and 100: Table 5.7 expressed the convergence
Page 101 and 102: Iteration with an incorrect spline
Page 103 and 104: Vertical aerial photographData 9 Ju
Page 105 and 106: All locations are assumed as on the
Page 107 and 108: of the Gauss-Markov model correspon
Page 109 and 110: estimation is obstacled by the corr
Page 111 and 112: Interior orientation defines a tran
Page 113 and 114: + fu ( w2 31 (X i (t) − X C ) + s
Page 115 and 116: A.2 Derivation of arc-length parame
Page 117 and 118: +2f( t [1 + t 2) − 1 22s 12 (Y i
Page 119 and 120: +Du ′ ( t 1 + t 22)2r 11 t + Dv
Page 121 and 122: A 17 = t [2 − t 1 16 2 f(t 1) −
Page 123 and 124: 1−u ′ w − w ′ u {w′ [s 21
Page 125 and 126: BIBLIOGRAPHY[1] Ackerman, F., and V
Page 127 and 128: [24] Haala, N., and G. Vosselman. 1
Page 129 and 130: [49] Parian, J.A., and A. Gruen. 20
Page 131: [73] Vosselman, G., and H. Veldhuis
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bundle block adjustment with 3d natural cubic splines

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