bundle block adjustment with 3d natural cubic splines
bundle block adjustment with 3d natural cubic splines bundle block adjustment with 3d natural cubic splines
Generally the larger noise level the more accurate approximations are required toachieve the ultimate convergence of the results. The worst case scenario for estimatingis that the large noise level leads the proposed model not to converge into thespecific estimates since the convergence radius is proportional to the noise level. Theparameter estimation is sensitive to the noise of the image measurement. Error propagationrelated with the noise error in the image space observation is one of the mostimportant elements in the estimation theory. The proposed bundle block adjustmentcan be evaluated statistically using the variances and the covariances of parameterssince a small variance represents that the estimated values have a small range anda large variance means that the estimated values are not calculated properly. Therange of the parameter variance is from zero in case of error free parameters to infinitein case of completely unknown parameters. The result of one spline segmentis expressed as table 5.4 with ξ 0 the initial values and ˆξ the estimates. The estimatedspline and spline location parameters along with their standard deviations areestablished without the knowledge of the point-to-point correspondence.If no random noise is added to image points, the estimates are converged to thetrue values. The quality of initial estimates is important in least squares system sinceit determines the iteration number of the system and the accuracy of the convergence.The assumption is that two points on one spline segment are measured ineach image so the total number of equations are 2 × 6(the number of images)×2(thenumber of points) +6(the number of the arc-length) and the total number of unknownsare 12(the number of spline parameters)+12(the number of spline locationparameters).The redundancy (= the number of equations - the number of parameters),the degrees of freedom, is 6. While some of geometric constraints such as80
Spline location parametersImage 1 Image 2 Image 3t 1 t 7 t 2 t 8 t 3 t 9ξ 0 0.02 0.33 0.09 0.41 0.16 0.47ˆξ 0.0415 0.3651 0.0917 0.4158 0.1412 0.4617±0.0046 ±0.0016 ±0.0017 ±0.0032 ±0.0043 ±0.0135Image 4 Image 5 Image 6t 4 t 10 t 5 t 11 t 6 t 12ξ 0 0.18 0.51 0.28 0.52 0.33 0.57ˆξ 0.2174 0.4974 0.2647 0.5472 0.3133 0.6157±0.0098 ±0.0079 ±0.0817 ±0.0317 ±0.0127 ±0.1115Spline parametersa 10 a 11 a 12 a 13 b 10 b 11ξ 0 3322.17 72.16 -45.14 27.15 4377.33 69.91ˆξ 3335.0080 70.4660 -48.8529 16.5634 4343.0712 63.0211±0.0004 ±0.0585 ±0.8310 ±1.2083 ±0.0004 ±0.0258b 12 b 13 c 10 c 11 c 12 c 13ξ 0 -17.49 13.68 48.82 10.15 -27.63 21.90ˆξ -28.7770 9.8893 51.9897 8.1009 -39.3702 13.3904±0.2193 ±0.2067 ±0.0006 ±0.0589 ±0.7139 ±1.0103Table 5.4: Spline parameter and spline location parameter recovery81
- 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
- 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: ξ kiSP = [ da i0 da i1 da i2 da i3
- Page 95 and 96: Spline location parametersImage 1 I
- Page 97 and 98: 5.3 Recovery of EOPs and spline par
- 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
Generally the larger noise level the more accurate approximations are required toachieve the ultimate convergence of the results. The worst case scenario for estimatingis that the large noise level leads the proposed model not to converge into thespecific estimates since the convergence radius is proportional to the noise level. Theparameter estimation is sensitive to the noise of the image measurement. Error propagationrelated <strong>with</strong> the noise error in the image space observation is one of the mostimportant elements in the estimation theory. The proposed <strong>bundle</strong> <strong>block</strong> <strong>adjustment</strong>can be evaluated statistically using the variances and the covariances of parameterssince a small variance represents that the estimated values have a small range anda large variance means that the estimated values are not calculated properly. Therange of the parameter variance is from zero in case of error free parameters to infinitein case of completely unknown parameters. The result of one spline segmentis expressed as table 5.4 <strong>with</strong> ξ 0 the initial values and ˆξ the estimates. The estimatedspline and spline location parameters along <strong>with</strong> their standard deviations areestablished <strong>with</strong>out the knowledge of the point-to-point correspondence.If no random noise is added to image points, the estimates are converged to thetrue values. The quality of initial estimates is important in least squares system sinceit determines the iteration number of the system and the accuracy of the convergence.The assumption is that two points on one spline segment are measured ineach image so the total number of equations are 2 × 6(the number of images)×2(thenumber of points) +6(the number of the arc-length) and the total number of unknownsare 12(the number of spline parameters)+12(the number of spline locationparameters).The redundancy (= the number of equations - the number of parameters),the degrees of freedom, is 6. While some of geometric constraints such as80
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