bundle block adjustment with 3d natural cubic splines
bundle block adjustment with 3d natural cubic splines bundle block adjustment with 3d natural cubic splines
B-splineThe equation of B-spline with m + 1 knot vector T , the degree p ≡ m − n − 1which must be satisfied from equality property of B-spline since a basis function isrequired for each control points and n + 1 control points P 0 , P 1 , · · · , P n is as [75]C(t) =n∑P i N i,p (t) (2.23)i=owithN i,0 (t) =N i,p (t) ={1 if ti ≤ t ≤ t i+1 and t i < t i+10 otherwiset − t iN i,p−1 (t) +t i+p+1 − tN i+1,p−1 (t)t i+p − t i t i+p+1 − t i+1Each point is defined by control points affected by segments and B-splines are apiecewise curve of degree p for each segment. Advantages of B-splines are to representcomplex shapes with lower degree polynomials while B-splines are drawn closer totheir original control polyline as the degree decreases. B-splines have an inside convexhull property that B-splines are contained in the convex hull presented by its polyline.With an inside convex hull property, the shape of B-splines can be controlled indetail. Although B-splines need more information and more complex algorithm thanother splines, a B-spline provides many important properties than other splines, suchas degree independence for control points.B-splines change the degree of curveswithout changing control points, and control points can be changed without affectingthe shape of whole B-splines. B-splines, however, cannot represent simple constrainedcurves such as straight lines and conic sections.30
2.3.2 Fourier transformFourier series and transform have been widely employed to represent 2D and 3Dcurves as well as 3D surfaces. Fourier series is useful for periodic curves and Fouriertransform is proper for non-periodic free-form lines. Parametric representations ofcurves have been established in many applications. Fourier transform is the generalizationof the complex Fourier series with the infinite limit. Let w be the frequencydomain, then time domain t is described as follow.X(w) = 1 √2π∫ ∞−∞x(t)e −iwt dt (2.24)Discrete Fourier transform (DFT) is commonly employed to solve a sampled signaland partial differential equations. Fast Fourier transform (FFT) which reduces thecomputation complexity for N points from 2N 2 to 2Nlog 2 N improves the efficiencyof DFT. DFT transforms N complex numbers x 0 , · · · , x N−1 into the sequence of Ncomplex number X 0 , · · · , X N−1 .X k =N−1 ∑k=0x n e − 2πiN kn k = 0, · · · , N − 1 (2.25)Shape information can be obtained with the first few coefficients of Fourier transformsince the low frequency covers most shape information. In addition, Fouriertransform can be developed in the smoothing of a 3D free-form line in the objectspace using the convolution theorem with the use of the FFT algorithm.Boundary conditions affect the application of Fourier related transforms especiallywhen solving partial differential equations. In some cases, discontinuities occur dueto the number of Fourier terms used.The fewer terms, the smoother a function.31
- 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
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- Page 21 and 22: • Bundle block adjustment by the
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- 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: Cardinal splineA Cardinal spline is
- 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 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
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- Page 75 and 76: [ ] [ ] [ ]N11 N 12 ˆξ1 c1N12T =N
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- Page 79 and 80: interval based on the normal distri
- Page 81 and 82: 1 ∂Φ2 ∂l= (X C + d 1 l − a i
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- Page 91 and 92: ξ kiSP = [ da i0 da i1 da i2 da i3
2.3.2 Fourier transformFourier series and transform have been widely employed to represent 2D and 3Dcurves as well as 3D surfaces. Fourier series is useful for periodic curves and Fouriertransform is proper for non-periodic free-form lines. Parametric representations ofcurves have been established in many applications. Fourier transform is the generalizationof the complex Fourier series <strong>with</strong> the infinite limit. Let w be the frequencydomain, then time domain t is described as follow.X(w) = 1 √2π∫ ∞−∞x(t)e −iwt dt (2.24)Discrete Fourier transform (DFT) is commonly employed to solve a sampled signaland partial differential equations. Fast Fourier transform (FFT) which reduces thecomputation complexity for N points from 2N 2 to 2Nlog 2 N improves the efficiencyof DFT. DFT transforms N complex numbers x 0 , · · · , x N−1 into the sequence of Ncomplex number X 0 , · · · , X N−1 .X k =N−1 ∑k=0x n e − 2πiN kn k = 0, · · · , N − 1 (2.25)Shape information can be obtained <strong>with</strong> the first few coefficients of Fourier transformsince the low frequency covers most shape information. In addition, Fouriertransform can be developed in the smoothing of a 3D free-form line in the objectspace using the convolution theorem <strong>with</strong> the use of the FFT algorithm.Boundary conditions affect the application of Fourier related transforms especiallywhen solving partial differential equations. In some cases, discontinuities occur dueto the number of Fourier terms used.The fewer terms, the smoother a function.31
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