bundle block adjustment with 3d natural cubic splines

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with x pi , y pi the photo coordinates of the ith segment, f the focal length, X C , Y C , Z Cthe camera perspective center, and r ij the elements of the 3D orthogonal rotationmatrix R T by angular elements (ω, ϕ, κ) of EOPs.Figure 3.2: The projection of a point on a splineThe 3D orthogonal rotation matrix R T rotates the object coordinate system parallelto the photo coordinate system employing three sequential rotations: the primaryrotation angle ω around the X-axis (X ω Y ω Z ω ), the secondary rotation angle ϕ aroundthe once rotated Y -axis (X ωϕ Y ωϕ Z ωϕ ) and the tertiary rotation angle κ around thetwice rotated Z-axis (X ωϕκ Y ωϕκ Z ωϕκ ). Most often the consecutive rotations with thesequence R ω , R ϕ , R κ or R ϕ , R ω , R κ are used in photogrammetry.Since each threesequential rotations R ω , R ϕ , R κ are orthogonal matrices, the matrix R is orthogonal40
R −1 = R T . The matrix R T (= R −1 ) rotates the object coordinate system parallel tothe photo coordinate system and the matrix R rotates the photo coordinate systemparallel to the object coordinate system.R = R ω R ϕ R κ ==⎡⎢⎣⎡⎢⎣1 0 00 cos ω − sin ω0 sin ω cos ω⎤ ⎡⎥ ⎢⎦ ⎣cos ϕ 0 sin ϕ0 1 0− sin ϕ 0 cos ϕ⎤ ⎡⎥ ⎢⎦ ⎣cos κ − sin κ 0sin κ cos κ 00 0 1cos ϕ cos κ − cos ϕ sin κ sin ϕcos ω sin κ + sin ω sin ϕ cos κ cos ω cos κ − sin ω sin ϕ sin κ − sin ω cos ϕsin ω sin κ − cos ω sin ϕ cos κ sin ω cos κ + cos ω sin ϕ sin κ cos ω cos ϕThe extended collinearity equations can be written as follows:⎤⎥⎦⎤⎥⎦(3.13)x p = −f u w ,y p = −f v w(3.14)⎡⎢⎣uvw⎤⎡⎥⎦ = R T ⎢(ω, ϕ, κ) ⎣X i (t) − X CY i (t) − Y CZ i (t) − Z C⎤⎥⎦To recover the 3D natural cubic spline parameters and the exterior orientation parametersin a bundle block adjustment, a non-linear mathematical model of the extendedcollinearity equation is differentiated. The models of exterior orientation recoveryare classified into linear and non-liner methods. While linear methods decreasethe computation load, accuracy and robustness of linear algorithms are not excellent.Otherwise non-linear methods are more accurate and robust. However, non-linearmethods require initial estimates and increase the computational complexity. Therelationship between a point in the image space and a corresponding point in theobject space is established by the extended collinearity equation. Prior knowledge41
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 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: 3.2 Extended collinearity equation
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
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
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[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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