bundle block adjustment with 3d natural cubic splines

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EOPsParameter X c [m] Y c [m] Z c [m] ω [deg] ϕ [deg] κ [deg]ξ 0 3007.84 4001.17 501.81 8.7090 -9.7976 -12.5845Image 1 ˆξ 3001.5852 4001.2238 503.2550 -0.8909 0.3252 6.0148±0.0154 ±0.0215 ±0.1386 ±0.3895 ±0.1351 ±0.8142ξ 0 3308.17 4001.17 497.52 10.2300 8.3144 -5.5004Image 2 ˆξ 3305.1962 4004.9827 501.2641 -0.1247 -0.5497 -5.2858±0.3804 ±0.1785 ±0.2489 ±0.0308 ±0.0798 ±0.4690ξ 0 3612.68 3993.37 506.32 5.2731 7.2581 -10.135Image 3 ˆξ 3611.8996 3995.7891 505.1299 0.1486 0.1192 2.3372±0.1226 ±0.0695 ±0.0337 ±0.4467 ±0.0168 ±0.0794ξ 0 3619.75 4612.78 506.88 6.2571 -5.3482 183.6600Image 4 ˆξ 3612.7128 4613.0145 507.0654 -0.0921 -0.152 184.5016±0.0258 ±0.1895 ±0.0251 ±0.7485 ±0.4505 ±0.2289ξ 0 3301.84 4618.63 497.61 -6.1731 7.5182 187.7145Image 5 ˆξ 3302.8942 4617.0538 492.9424 -0.6347 0.2662 171.9808±0.0467 ±0.0817 ±0.0704 ±0.1413 ±0.8006 ±0.6445ξ 0 2999.59 4615.74 508.49 -7.1651 -4.8427 185.1057Image 6 ˆξ 2997.9827 4610.1432 509.2952 -0.1360 -0.1279 183.1789±0.0513 ±0.0249 ±0.0401 ±0.5659 ±0.6225 ±0.2271Spline location parametersImage 1 Image 2t 1 t 7 t 13 t 19 t 2 t 8 t 14 t 20ξ 0 0.04 0.28 0.52 0.76 0.08 0.32 0.56 0.80ˆξ 0.0432 0.2980 0.5176 0.7705 0.0813 0.3338 0.5715 0.8136±0.0033 ±0.0012 ±0.0039 ±0.0077 ±0.0082 ±0.0041 ±0.0039 ±0.0069Image 3 Image 4t 3 t 9 t 15 t 21 t 4 t 10 t 16 t 22ξ 0 0.12 0.36 0.60 0.84 0.16 0.40 0.64 0.88ˆξ 0.1294 0.3578 0.6024 0.8437 0.1594 0.4112 0.6418 0.9783±0.0036 ±0.0092 ±0.0046 ±0.0079 ±0.0115 ±0.0057 ±0.0029 ±0.0037Image 5 Image 6t 5 t 11 t 17 t 23 t 6 t 12 t 18 t 24ξ 0 0.20 0.44 0.68 0.92 0.24 0.48 0.72 0.96ˆξ 0.2039 0.4461 0.6713 0.9264 0.2483 0.4860 0.7181 0.9613±0.0057 ±0.0125 ±0.0080 ±0.0061 ±0.0085 ±0.0073 ±0.0084 ±0.0079Table 5.7: EOP and spline location parameter recovery86
Table 5.7 expressed the convergence achievement of EOPs and spline location parameters.The correlation coefficient between parameter X c and ϕ is high (ρ ≈ 1) inthe dispersion matrix, that is two parameters are highly correlated among EOPs. Thecorrelation coefficient between parameter Y c and ω is approximately 0.85. Lee[39] representedthat parameter X c &ϕ and parameter Y c &ω are highly correlated in the linecamera orientation parameters. In general, the correlation coefficient between parameterX c and ϕ is higher than between parameter Y c and ω. In addition, Lee demonstratedthat image coordinates in x direction are related with parameter X c , Z c , ω, ϕand κ and image coordinates in y direction are the function of parameter Y c , Z c , ω, ϕand κ in a line camera.Since a control spline provides the object space information about the coordinatesystem having datum defects of seven, tie spline parameters and EOPs can be recoveredsimultaneously. In the experiment of combined splines, the total number ofequations are 2 × 6(the number of images)×3(the number of points) ×2(the numberof splines)+12(the number of the arc-length)×2(the number of splines) = 96 andthe total number of unknowns are 36(the number of EOPs) +12(the number of tiespline parameters)+18(the number of tie spline location parameters)+18(the numberof control spline location parameters) = 84. Since the knowledge of the object spaceinformation about spline referred to as a full control spline is available prior to aerialtriangulation. A control spline is considered as a stochastic constraint in the proposedadjustment model and the representation of a control spline is the same with that oftie spline. The result of the combined splines which demonstrates the feasibility oftie spline and control spline for bundle block adjustment is described in table 5.8.87
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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
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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 and 96: Spline location parametersImage 1 I
Page 97: 5.3 Recovery of EOPs and spline par
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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