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 optimal unbiased estimate of the variance component can be obtained aŝσ 2 o = ẽT P ẽn − mẽ = y − A ̂ξ(4.4)where n is the number of equations and m is the number of parameters.If one or more of the three estimated parameter sets ξEOP k , ξSP i , ξt i are consideredas stochastic constraints, the reduction of the normal equation matrix can be applied.Control information is implemented as stochastic constraints in bundle blockadjustment. Distribution and quality of control features depend on the number andthe density of control features, the number of tie features and the degree of the overlapof tie features. If adding stochastic constraints removes the rank deficiency of theGauss-Markov model, bundle adjustment can be implemented employing only the extendedcollinearity equations for 3D natural cubic splines. Fixed exterior orientationparameters, control splines or control spline location parameters can be stochasticconstraints. In addition, splines in the object space can be divided into control featuresand tie features so that tie spline parameters can be recovered by bundle blockadjustment.Stochastic constraints assigned into ˆξ 2 , the integrated model can bewritten as62
[ ] [ ] [ ]N11 N 12 ˆξ1 c1N12T =N 22 ˆξ 2 c 2N 11 ˆξ1 + N 12 ˆξ2 = c 1N T 12 ˆξ 1 + N 22 ˆξ2 = c 2ˆξ 2 = N −122 c 2 − N −122 N T 12 ˆξ 2(N 11 − N 12 N −122 N T 12)ˆξ 1 = c 1 − N 12 N −122 c 2ˆξ 1 = (N 11 − N 12 N −122 N T 12) −1 (c 1 − N 12 N −122 c 2 )where the matrix of N, c, ξ depends on the stochastic constraints.D(ˆξ 1 ) = σ 2 oQ 11D(ˆξ 2 ) = σ 2 oQ 22D[ ] [ ] −1 [ ]ˆξ1= σ ˆξo2 N11 N 122 N12T = σoN 2 Q11 Q 1222 Q T 12 Q 22[ ] [ ] [N11 N 12 Q11 Q 12 I 0N12 T N 22 Q T =12 Q 22 0 I]N 11 Q 11 + N 12 Q T 12 = IN T 12Q 11 + N 22 Q T 12 = 0Q T 12 = −N −122 N T 12Q 11Q 12 = −Q 11 N 22 N −122(N 11 − N 12 N −122 N T 12)Q 11 = IQ 11 = (N 11 − N 12 N −122 N T 12) −163
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[ ] [ ] [ ]N11 N 12 ˆξ1 c1N12T =N 22 ˆξ 2 c 2N 11 ˆξ1 + N 12 ˆξ2 = c 1N T 12 ˆξ 1 + N 22 ˆξ2 = c 2ˆξ 2 = N −122 c 2 − N −122 N T 12 ˆξ 2(N 11 − N 12 N −122 N T 12)ˆξ 1 = c 1 − N 12 N −122 c 2ˆξ 1 = (N 11 − N 12 N −122 N T 12) −1 (c 1 − N 12 N −122 c 2 )where the matrix of N, c, ξ depends on the stochastic constraints.D(ˆξ 1 ) = σ 2 oQ 11D(ˆξ 2 ) = σ 2 oQ 22D[ ] [ ] −1 [ ]ˆξ1= σ ˆξo2 N11 N 122 N12T = σoN 2 Q11 Q 1222 Q T 12 Q 22[ ] [ ] [N11 N 12 Q11 Q 12 I 0N12 T N 22 Q T =12 Q 22 0 I]N 11 Q 11 + N 12 Q T 12 = IN T 12Q 11 + N 22 Q T 12 = 0Q T 12 = −N −122 N T 12Q 11Q 12 = −Q 11 N 22 N −122(N 11 − N 12 N −122 N T 12)Q 11 = IQ 11 = (N 11 − N 12 N −122 N T 12) −163
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