bundle block adjustment with 3d natural cubic splines

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CHAPTER 3BUNDLE BLOCK ADJUSTMENTWITH 3D NATURAL CUBIC SPLINES3.1 3D natural cubic splinesIn this chapter, the correspondence between the 3D curve in the object spacecoordinate system and its projected 2D curve in the image coordinate system isimplemented using a natural cubic spline accommodating curve feature because of itsboundary conditions the zero second derivatives at the end points. A natural cubicspline is composed of a sequence of cubic polynomial segments as figure 3.1 withx 0 , x 1 , · · · , x n the n + 1 control points and X 0 , X 1 , · · · , X n−1 the ground coordinate ofn segments.Figure 3.1: Natural cubic spline34
Each segment of a natural cubic spline is expressed with parametric representationsas (3.1).X 0 (t) = a 00 + a 01 t + a 02 t 2 + a 03 t 3 , t ∈ [0, 1]X 1 (t) = a 10 + a 11 t + a 12 t 2 + a 13 t 3 , t ∈ [0, 1]· · · (3.1)X i (t) = a i0 + a i1 t + a i2 t 2 + a i3 t 3 , t ∈ [0, 1]Y i (t) = b i0 + b i1 t + b i2 t 2 + b i3 t 3 , t ∈ [0, 1]Z i (t) = c i0 + c i1 t + c i2 t 2 + c i3 t 3 , t ∈ [0, 1]Bartels et al.[5] represented the solution of a natural cubic spline with given controlpoints.Since four coefficients are required for each interval, the number of totalparameters is 4n to define the spline X i .X i (t) = a i0 + a i1 t + a i2 t 2 + a i3 t 3X (1)i (t) = a i1 + 2a i2 t + 3a i3 t 2 (3.2)X (2)i (t) = 2a i2 + 6a i3 tThe first and second order derivatives are equal at the joints of a spline and therelationship between the control points and the segments leads (3.3).X i−1 (1) = x iX i (0) = x iX (1)i−1(1) = X (1)i (0) (3.3)X (2)i−1(1) = X (2)i (0) [1]X 0 (0) = x 0X n−1 (1) = x n35
Page 2: c○ Copyright byWon Hee Lee2008
Page 7 and 8: ACKNOWLEDGMENTSThanks be to God, my
Page 10 and 11: 3. BUNDLE BLOCK ADJUSTMENTWITH 3D N
Page 13 and 14: CHAPTER 1INTRODUCTION1.1 OverviewOn
Page 15 and 16: y an intersection employing more th
Page 17 and 18: similarity of geometric properties
Page 19 and 20: straight linear features or formula
Page 21 and 22: • Bundle block adjustment by the
Page 23 and 24: Hessian. Interest point operators w
Page 25 and 26: [60], Ebner and Ohlhof(1994) [16],
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: For other polyline expressions, Aya
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 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
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+ 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
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+Du ′ ( t 1 + t 22)2r 11 t + Dv
Page 121 and 122:
A 17 = t [2 − t 1 16 2 f(t 1) −
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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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