bundle block adjustment with 3d natural cubic splines
bundle block adjustment with 3d natural cubic splines bundle block adjustment with 3d natural cubic splines
Substituting (3.2) into (3.3 [1]) leads toand substituting (3.2) into (3.3) leads to2a (i−1)2 + 6a (i−1)3 = 2a i2 (3.4)a i0 = x ia i1 = D ia i2 = 3(x i+1 − x i ) − 2D i − D i+1 (3.5)where D is the derivative.Substituting (3.5) into (3.4) leads towith 1 ≤ i ≤ n − 1.a i3 = 2(x i − x i+1 ) + D i + D i+1D i−1 + 4D i + D i+1 = 3(x i+1 − x i−1 ) (3.6)Since the total number of D i unknowns is n and the total number of equationsis n − 2, two more equations are required to solve the underdetermined system. Ina natural cubic spline, two boundary conditions are given to complete the system ofn − 2 equations. The second derivatives at the end points are set to zero as (3.7).X (2)0 (0) = 0X (2)n−1(0) = 0 (3.7)Otherwise, the first and the last segment can be set to the 2nd order polynomial toreduce two unknowns, for which two boundary conditions are not required.Substituting (3.5) into (3.7) can be described as followings.2D 0 + D 1 = 3(x 1 − x 0 )D n−1 + 2D n = 3(x n − x n−1 ) (3.8)36
⎡⎢⎣2 11 4 11 4 1· · ·1 4 11 2⎤ ⎡⎥⎦ ⎢⎣D 0D 1D 2.D n−1D n⎤⎡=⎥ ⎢⎦ ⎣3(x 1 − x 0 )3(x 2 − x 0 )3(x 3 − x 1 ).3(x n − x n−2 )3(x n − x n−1 )⎤⎥⎦(3.9)D i is obtained in the case of close curves as⎡⎤ ⎡4 11 4 11 4 1⎥⎢⎢⎣· · ·1 4 11 1 4⎥⎦ ⎢⎣D 0D 1D 2.D n−1D n⎤⎡=⎥ ⎢⎦ ⎣3(x 1 − x 0 )3(x 2 − x 0 )3(x 3 − x 1 ).3(x n − x n−2 )3(x n − x n−1 )⎤⎥⎦(3.10)The normalized spline system is⎡⎢⎣b 11 4 11 4 1· · ·1 4 11 1 b⎤ ⎡⎥⎦ ⎢⎣D 0D 1D 2.D n−1D n⎤⎡= k⎥ ⎢⎦ ⎣(x 1 − x 0 )(x 2 − x 0 )(x 3 − x 1 ).(x n − x n−2 )(x n − x n−1 )⎤⎥⎦(3.11)where the value of b and k depends on the boundary conditions of a spline and kdepends on the type of a spline. Normally the value of b and k in a natural cubicare 2 and 3 in an unclosed curve case and 4 and 3 in a closed curve case respectively.The values of b and k in B-Splines are 5 and 6 respectively [13].In case of two parameters, the corresponding relationship between two parameterscan be calculated without an intermediate t parameter. n + 1 point pairs,(x 0 , y 0 ), (x 1 , y 1 ), · · ·, (x n , y n ), have 4n unknown spline parameters in n segments. 2nequations from 0th continuity condition, n − 1 equations from 1st continuity condition,n − 1 equations from 2nd continuity condition, 2 equations from boundaryconditions that the second derivatives at the end points are set to zero.37
- 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: Each segment of a natural cubic spl
- 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
⎡⎢⎣2 11 4 11 4 1· · ·1 4 11 2⎤ ⎡⎥⎦ ⎢⎣D 0D 1D 2.D n−1D n⎤⎡=⎥ ⎢⎦ ⎣3(x 1 − x 0 )3(x 2 − x 0 )3(x 3 − x 1 ).3(x n − x n−2 )3(x n − x n−1 )⎤⎥⎦(3.9)D i is obtained in the case of close curves as⎡⎤ ⎡4 11 4 11 4 1⎥⎢⎢⎣· · ·1 4 11 1 4⎥⎦ ⎢⎣D 0D 1D 2.D n−1D n⎤⎡=⎥ ⎢⎦ ⎣3(x 1 − x 0 )3(x 2 − x 0 )3(x 3 − x 1 ).3(x n − x n−2 )3(x n − x n−1 )⎤⎥⎦(3.10)The normalized spline system is⎡⎢⎣b 11 4 11 4 1· · ·1 4 11 1 b⎤ ⎡⎥⎦ ⎢⎣D 0D 1D 2.D n−1D n⎤⎡= k⎥ ⎢⎦ ⎣(x 1 − x 0 )(x 2 − x 0 )(x 3 − x 1 ).(x n − x n−2 )(x n − x n−1 )⎤⎥⎦(3.11)where the value of b and k depends on the boundary conditions of a spline and kdepends on the type of a spline. Normally the value of b and k in a <strong>natural</strong> <strong>cubic</strong>are 2 and 3 in an unclosed curve case and 4 and 3 in a closed curve case respectively.The values of b and k in B-Splines are 5 and 6 respectively [13].In case of two parameters, the corresponding relationship between two parameterscan be calculated <strong>with</strong>out an intermediate t parameter. n + 1 point pairs,(x 0 , y 0 ), (x 1 , y 1 ), · · ·, (x n , y n ), have 4n unknown spline parameters in n segments. 2nequations from 0th continuity condition, n − 1 equations from 1st continuity condition,n − 1 equations from 2nd continuity condition, 2 equations from boundaryconditions that the second derivatives at the end points are set to zero.37
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