bundle block adjustment with 3d natural cubic splines
bundle block adjustment with 3d natural cubic splines bundle block adjustment with 3d natural cubic splines
(a, f(a)), ( a+b, f ( ))a+b2 2 , (b, f(b)). The three unknowns, a0 , a 1 , and a 2 , are obtainedfrom the following three equations:ff(a) = f 2 (a) = a 0 + a 1 a + a 1 a 2( ) ( )( ) ( ) 2a + b a + ba + b a + b= f 2 = a 0 + a 1 + a 1 (3.32)2222f(b) = f 2 (b) = a 0 + a 1 b + a 1 b 2Three unknowns areThusa 0 = a2 f(b) + abf(b) − 4abf ( )a+b2 + abf(a) + b 2 f(a)a 2 − 2ab + b 2a 1 = af(a) − 4af ( a+b2) (+ 3af(b) + 3bf(a)) − 4bfa+b2a 2 − 2ab + b 2)+ f(b)))+ bf(b)a 2 = 2 ( f(a) − 2f ( a+b2(3.33)a 2 − 2ab + b 2∫ abf(x)dx ∼ ===∫ ab∫ ab[Substituting (3.33) into (3.34) leads tof 2 (x)dx(a0 + a 1 x + a 2 x 2) dx (3.34)a 0 x + a 1x 22 + a 2x 3 ] b3a= a 0 (b − a) + a 1b 2 − a 22+ a 2b 3 − a 33∫ abf 2 (x)dx = b − a6[( ) ]a + bf(a) + 4f + f(b)2(3.35)In addition, Simpson’s rule can be derived by different methods such as the Lagrangepolynomial, the method of coefficients and the approximation by a second50
order polynomial using Newton’s divided difference polynomial. The error of Simpson’srule is known to be(b − a)5e = −2880 f 4 (ξ), a < ξ < b (3.36)The error of multiple segments using Simpson’s rule is the sum of each error ofSimpson’s rule as∑ n/2(b − a)5 i=1 f (4) (ξ i )e = −90n 4 n(3.37)The error is proportional to (b − a) 5 .Simpson’s rule is the numerical approximation of definite integrals. The geometricintegration of the arc-length in the image space can be calculated by Simpon’s ruleas followings.∫ t2Arc(t) =∫ t2t 1t 1f(t)dt ∼ =t 2 − t 16f(t)dt[f(t 1 ) + 4f( ) t2 + t 12]+ f(t 2 )f(t) = √ (x ′ p(t)) 2 + (y p(t)) ′ 2{(√= −f u(t) ) ′ } 2 {(+ −f v(t) ) ′ } 2(3.38)w(t)w(t)() √= −f u′ (t)w(t) − u(t)w ′ 2 ()(t)+ −f v′ (t)w(t) − v(t)w ′ 2(t)w 2 (t)w 2 (t)df(t) = 1 [12 (f(t))− 2 f 2x ′ p(t) w′w du − 2 2x′ p(t) 1 w du′ + 2y p(t) ′ w′w dv − 2 2y′ p(t) 1 w dv′{+ 2x ′ p(t) u′ w 2 − (u ′ w − uw ′ )2w− 2ywp(t) ′ v′ w 2 − (v ′ w − vw ′ })2wdw4 w 4+{2x ′ p(t) u w + 2 2y′ p(t) v } ]dw ′w 2Arc(t) − t0 2 − t 0 16[( tf(t 0 01) + 4f 2 + t 0 ) ]1+ f(t 022)(3.39)51
- 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 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: This equation can be replaced with
- 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
(a, f(a)), ( a+b, f ( ))a+b2 2 , (b, f(b)). The three unknowns, a0 , a 1 , and a 2 , are obtainedfrom the following three equations:ff(a) = f 2 (a) = a 0 + a 1 a + a 1 a 2( ) ( )( ) ( ) 2a + b a + ba + b a + b= f 2 = a 0 + a 1 + a 1 (3.32)2222f(b) = f 2 (b) = a 0 + a 1 b + a 1 b 2Three unknowns areThusa 0 = a2 f(b) + abf(b) − 4abf ( )a+b2 + abf(a) + b 2 f(a)a 2 − 2ab + b 2a 1 = af(a) − 4af ( a+b2) (+ 3af(b) + 3bf(a)) − 4bfa+b2a 2 − 2ab + b 2)+ f(b)))+ bf(b)a 2 = 2 ( f(a) − 2f ( a+b2(3.33)a 2 − 2ab + b 2∫ abf(x)dx ∼ ===∫ ab∫ ab[Substituting (3.33) into (3.34) leads tof 2 (x)dx(a0 + a 1 x + a 2 x 2) dx (3.34)a 0 x + a 1x 22 + a 2x 3 ] b3a= a 0 (b − a) + a 1b 2 − a 22+ a 2b 3 − a 33∫ abf 2 (x)dx = b − a6[( ) ]a + bf(a) + 4f + f(b)2(3.35)In addition, Simpson’s rule can be derived by different methods such as the Lagrangepolynomial, the method of coefficients and the approximation by a second50
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