bundle block adjustment with 3d natural cubic splines
bundle block adjustment with 3d natural cubic splines bundle block adjustment with 3d natural cubic splines
Calculate the maximum distance between the approximated spline and the originalinput dataWhile(the maximum distance > the threshold of the maximum distance)Add break point to the break point set at the location of the maximum distanceCompare the maximum distance with the thresholdThe larger threshold makes the more break points with more accurate spline to theoriginal input data.N piecewise cubic polynomial functions between two adjacent break points aredefined from the N + 1 break points. There is a separate cubic polynomial for eachsegment with its own coefficients.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]· · · (2.21)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]The strength is that segmented lines represent a free-form line with analytical parameters.The number of break points is reduced and the input error should be absorbedby a mathematical model especially in the expression of points on a straight line. Anatural cubic spline is data independent curve fitting. The disadvantage is that theentire curve shape depends on all of the passing points. Changing any one of themchanges the entire curve.28
Cardinal splineA Cardinal spline is defined by a cubic Hermite spline, a third-degree spline.Hermite form defined by two control points and two control tangents is implied byeach polynomial of a Hermite form. The basic scheme of solving the system of a cubicHermit spline is continuity C 1 that the first derivatives are equal at break points. Adisadvantage of a cubic Hermite spline is that tangents are always required while thisinformation is not available for all curves.A Cardinal spline consists of three points before and after a control point and atension parameter. A tangent m i of a cardinal spline is as following with given n + 1points, p 0 , · · · , p n .m i = 1 2 (1 − c)(p i+1 − p i−1 ) (2.22)where c is a tension parameter which contributes the length of the tangent. A tensionparameter is between 0 and 1.If tension is 0, a Cardinal spline is referred to asa Catmull-Rom spline[11]. A Catmull-Rom spline is frequently used to interpolatesmoothly between point data in mathematics and between key-frames in computergraphics. A cardinal spline represents the curve from the second point to the lastsecond point of the input point set. The input points are the control points thatdefine a Cardinal spline.A Cardinal spline produces a C 1 continuous curve, nota C 2 continuous curve, and the second derivatives are linearly interpolated withineach segment. The advantage is no need for tangents but the disadvantage is theimprecision of the tangent approximation.29
- 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
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- 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
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- 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 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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Calculate the maximum distance between the approximated spline and the originalinput dataWhile(the maximum distance > the threshold of the maximum distance)Add break point to the break point set at the location of the maximum distanceCompare the maximum distance <strong>with</strong> the thresholdThe larger threshold makes the more break points <strong>with</strong> more accurate spline to theoriginal input data.N piecewise <strong>cubic</strong> polynomial functions between two adjacent break points aredefined from the N + 1 break points. There is a separate <strong>cubic</strong> polynomial for eachsegment <strong>with</strong> its own coefficients.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]· · · (2.21)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]The strength is that segmented lines represent a free-form line <strong>with</strong> analytical parameters.The number of break points is reduced and the input error should be absorbedby a mathematical model especially in the expression of points on a straight line. A<strong>natural</strong> <strong>cubic</strong> spline is data independent curve fitting. The disadvantage is that theentire curve shape depends on all of the passing points. Changing any one of themchanges the entire curve.28
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