bundle block adjustment with 3d natural cubic splines
bundle block adjustment with 3d natural cubic splines bundle block adjustment with 3d natural cubic splines
The extended collinearity equations are a mathematical model for bundle blockadjustment.The mathematical model of bundle block adjustment consists of twomodels, a functional model and a stochastic model. The functional model representsgeometrical properties and the stochastic model describes statistical properties. Therepeated measurements at the same location in the image space are represented withrespect to the functional model and the redundant observations of image locationsin the image space are expressed with respect to the stochastic model. While theGauss-Markov model uses indirect observations, condition equations such as coordinatetransformations, and the coplanarity condition can be employed in the adjustment.The Gauss-Markov model and the condition equation can be combined intothe Gauss-Helmert model. In addition, functional constraints such as points havingthe same height or straight railroad segments can be added into the block adjustment.The difference between condition equations and constraint equations is that conditionequations consist of observations and parameters, and constraint equationsconsist of only parameters. With the advances of technology, the input data in photogrammetryhas been increased so adequate formulation of adjustment is required.All variables are involved in the mathematical equations and the weight matrix ofvariables is changed from zero to infinity depending on variances. Variables with thenear to zero weight are considered as unknown parameters and variables with thenear to infinite weight are considered as constants. Most actual observations are inexistence between two boundary cases. Assessment of adjustment, post-adjustmentanalysis, is important in photogrammetry to analyze the results. One of the assessmentmethods is to compare the estimated variance with the two-tailed confidence66
interval based on the normal distribution. The two-tailed confidence interval is computedby a reference variance σ 2 o with χ 2 distribution asr̂σ 2 oχ 2 r,α/2< σ 2 o < r̂σ2 oχ 2 r,1−α/2(4.6)where r is degrees of freedom and α is a confidence coefficient (or a confidence level).If σo 2 has the value outside of the interval, we can assume the mathematical modelof adjustment is incorrect such as the wrong formulation, the wrong linearization,blunders or systematic errors.4.3 Pose estimation with ICP algorithmUnlike the previous case with spline segments which the correspondence betweenspline segments in the image and the object space are assumed, now it is unknownwhich image points belong to which spline segment. ICP algorithm can be utilized forthe recovery of EOPs since the initial estimated parameters of the relative pose canbe obtained from orientation data in general photogrammetric tasks. The originalICP algorithm steps are as follows. The closest point operators search the associatepoint by the nearest neighboring algorithm and then the transformation parametersare estimated using mean square cost function.The point is transformed by theestimated parameters and this step is iteratively established until converging into alocal minimum of the mean square distance. The transformation including translationand rotation between two clouds of points is estimated iteratively converging into aglobal minimum. In other words, the iterative calculation of the mean square errorsis terminated when a local minimum falls below a predefined threshold. The smallerglobal minimum or the fluctuated curve requires more memory intensive and timeconsuming computation. In every iteration step, a local minimum is calculated with67
- Page 27 and 28: a complicated problem. The developm
- Page 29 and 30: ⎡⎢⎣x i − x py i − y p−f
- Page 31 and 32: x p = −f (X A + t · a − X C )r
- Page 33 and 34: surfaces and terrain models in 2D a
- 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: systematic errors in the image spac
- 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 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
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- 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
The extended collinearity equations are a mathematical model for <strong>bundle</strong> <strong>block</strong><strong>adjustment</strong>.The mathematical model of <strong>bundle</strong> <strong>block</strong> <strong>adjustment</strong> consists of twomodels, a functional model and a stochastic model. The functional model representsgeometrical properties and the stochastic model describes statistical properties. Therepeated measurements at the same location in the image space are represented <strong>with</strong>respect to the functional model and the redundant observations of image locationsin the image space are expressed <strong>with</strong> respect to the stochastic model. While theGauss-Markov model uses indirect observations, condition equations such as coordinatetransformations, and the coplanarity condition can be employed in the <strong>adjustment</strong>.The Gauss-Markov model and the condition equation can be combined intothe Gauss-Helmert model. In addition, functional constraints such as points havingthe same height or straight railroad segments can be added into the <strong>block</strong> <strong>adjustment</strong>.The difference between condition equations and constraint equations is that conditionequations consist of observations and parameters, and constraint equationsconsist of only parameters. With the advances of technology, the input data in photogrammetryhas been increased so adequate formulation of <strong>adjustment</strong> is required.All variables are involved in the mathematical equations and the weight matrix ofvariables is changed from zero to infinity depending on variances. Variables <strong>with</strong> thenear to zero weight are considered as unknown parameters and variables <strong>with</strong> thenear to infinite weight are considered as constants. Most actual observations are inexistence between two boundary cases. Assessment of <strong>adjustment</strong>, post-<strong>adjustment</strong>analysis, is important in photogrammetry to analyze the results. One of the assessmentmethods is to compare the estimated variance <strong>with</strong> the two-tailed confidence66
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