bundle block adjustment with 3d natural cubic splines
bundle block adjustment with 3d natural cubic splines bundle block adjustment with 3d natural cubic splines
unknowns. The redundancy is 2nm-m-12 for one spline segment so that if two images(m=2) are used for bundle block adjustment, the redundancy is 4n-14. Four points arerequired to determine spline and spline location parameters in case one spline segmentand one degree of freedom to the overall redundancy budget is solved by each pointmeasurement with the extended collinearity equation. Arc-length parameterizationalso contributes one degree of freedom to the overall redundancy budget. The fifthpoint does not provide additional information to reduce the overall deficiency butonly makes spline parameters robust, which means it increases the overall precisionof the estimated parameters.This fact is an advantage of adopting splines which the number of degrees offreedom is four since in tie straight lines; only two points per line are independent.Independent information, the number of degrees of freedom, of a straight line is twofrom two points or a point with its tangent direction. A redundancy is r=2m-4 witha line expression of four parameters since equations are 2nm collinearity equationsand unknowns are 4+nm [59]. Only two points (n=2) are available to determine fourline parameters with two images (m=2) so at least three images must contain a tieline. The information content of t tie lines on m images is t(2m-4). One straight lineincreases two degrees of freedom to the redundancy budget and at least three linesare required in the space resection. An additional point on a straight line does notprovide additional information to reduce the rank deficiency of the recovery of EOPsbut only contributes image line coefficients. If spline location parameters or splineparameters enter the integrated adjustment model through stochastic constraints,employing extended collinearity equations is enough for solving the system withoutthe arc-length parameterization.74
The redundancy budget of a tie point is r=2m-3 so tie points provide one more independentequation than tie lines. However, using tie points requires semi-automaticmatching procedure to identify tie points on all images and employing linear featuresis more robust and accurate than point features for object recognition, posedetermination, and other higher photogrammetric activities.5.1 Synthetic data descriptionThe standard block configuration is generated by strips of images with approximately60% overlap in the flight direction and 20%-30% overlap in the neighboredflight strips. In bundle block adjustment, a ground feature is required at least intwo images to determine three dimensional coordinates. The simulation of aerial imageblock with six images, a forward overlap of 60% and a side overlap of 20%, isperformed to verify the feasibility of the proposed model in bundle block adjustmentwith synthetic data. EOPs of the simulation data set with six images are described intable 5.3 and figure 5.2 with 0.15m focal length and zero offsets from a fiducial-basedorigin to a perspective center origin of a camera. This means that interior orientationparameters are known and fixed. EOPs of six images are generated under theassumption of the vertical viewing condition.To evaluate the new bundle block adjustment model using natural cubic splines,the analysis of sensitivity and robustness of the model is required. Verification of themodel suitability can be assessed by the estimated parameters with the dispersionmatrix including standard deviations and correlations. The accuracy of bundle blockadjustment is determined by the geometry of a block of all images and the quality ofthe position and attitude information of a camera. For novel approaches, a simulation75
- 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
- Page 83 and 84: about splines, their relationships,
- Page 85: cubic spline in the image and the o
- 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 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
The redundancy budget of a tie point is r=2m-3 so tie points provide one more independentequation than tie lines. However, using tie points requires semi-automaticmatching procedure to identify tie points on all images and employing linear featuresis more robust and accurate than point features for object recognition, posedetermination, and other higher photogrammetric activities.5.1 Synthetic data descriptionThe standard <strong>block</strong> configuration is generated by strips of images <strong>with</strong> approximately60% overlap in the flight direction and 20%-30% overlap in the neighboredflight strips. In <strong>bundle</strong> <strong>block</strong> <strong>adjustment</strong>, a ground feature is required at least intwo images to determine three dimensional coordinates. The simulation of aerial image<strong>block</strong> <strong>with</strong> six images, a forward overlap of 60% and a side overlap of 20%, isperformed to verify the feasibility of the proposed model in <strong>bundle</strong> <strong>block</strong> <strong>adjustment</strong><strong>with</strong> synthetic data. EOPs of the simulation data set <strong>with</strong> six images are described intable 5.3 and figure 5.2 <strong>with</strong> 0.15m focal length and zero offsets from a fiducial-basedorigin to a perspective center origin of a camera. This means that interior orientationparameters are known and fixed. EOPs of six images are generated under theassumption of the vertical viewing condition.To evaluate the new <strong>bundle</strong> <strong>block</strong> <strong>adjustment</strong> model using <strong>natural</strong> <strong>cubic</strong> <strong>splines</strong>,the analysis of sensitivity and robustness of the model is required. Verification of themodel suitability can be assessed by the estimated parameters <strong>with</strong> the dispersionmatrix including standard deviations and correlations. The accuracy of <strong>bundle</strong> <strong>block</strong><strong>adjustment</strong> is determined by the geometry of a <strong>block</strong> of all images and the quality ofthe position and attitude information of a camera. For novel approaches, a simulation75
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