bundle block adjustment with 3d natural cubic splines
bundle block adjustment with 3d natural cubic splines bundle block adjustment with 3d natural cubic splines
Fourier transform has strength in signal and data compression, filtering and smoothing.However, Fourier transform has little strength in the representation of a curveaccurately. More terms to represent a free-form line exactly increase the time andthe computation complexity.2.3.3 Implicit polynomialsImplicit polynomials obtained by the computation of the least squares problemfor the coefficients of the polynomial have the mathematical form:X(t) = a 0 + a 1 t + a 2 t 2 + · · · + a n t nY (t) = b 0 + b 1 t + b 2 t 2 + · · · + b n t n (2.26)Z(t) = c 0 + c 1 t + c 2 t 2 + · · · + c n t nwhere a 0 , a 1 , · · · a n , b 0 , · · · b n , c 0 , · · · c n ∈ R 3 and the value of n is the degree of thepolynomial R n .The coefficients a 0 , b 0 , c 0 characterize a point in the object spaceR 3 and remaining parameters represent the direction.The number of parametersrequired to represent the n degree polynomial R n is 3(n + 1). Implicit polynomial ina degree of 3 is analogous to a natural cubic spline.A free-form line is formulated in a simple form geometrically independent accordingto a polynomial fit. However, a quality of the results is implemented in case ofhigh degrees. A prediction of the outcome of fitting with implicit polynomials is difficultsince coefficients are not geometrically correlated. The best solution is optimizedby the iterative try and error method. In the worst case, the implementation seemsto never end with a tight threshold. The number of trials to fit successful coefficientsdepends on the restriction of the search space. Implicit polynomials are originatedthrough constraints and various minimization criteria.32
For other polyline expressions, Ayache and Faugeras[4] described a line as theintersection of two planes that one plane is parallel to the X-axis and the other isparallel to the Y-axis with two constraints to extend this concept to general lines.Two constraints reduced line parameters six to four-dimensional vector.Since standard collinearity model can be extended to adopt parametric representationof spline, 3D natural cubic splines are employed for this research.Thecollinearity equations play an important role in photogrammetry since each controlpoint in the object space produces two collinearity equations for every photographin which the point appears. A natural 3D cubic spline allows the utilization of thecollinearity model for expressing orientation parameters and curve parameters.33
- 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: 2.3.2 Fourier transformFourier seri
- 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
- 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
For other polyline expressions, Ayache and Faugeras[4] described a line as theintersection of two planes that one plane is parallel to the X-axis and the other isparallel to the Y-axis <strong>with</strong> two constraints to extend this concept to general lines.Two constraints reduced line parameters six to four-dimensional vector.Since standard collinearity model can be extended to adopt parametric representationof spline, 3D <strong>natural</strong> <strong>cubic</strong> <strong>splines</strong> are employed for this research.Thecollinearity equations play an important role in photogrammetry since each controlpoint in the object space produces two collinearity equations for every photographin which the point appears. A <strong>natural</strong> 3D <strong>cubic</strong> spline allows the utilization of thecollinearity model for expressing orientation parameters and curve parameters.33
Change language