bundle block adjustment with 3d natural cubic splines
bundle block adjustment with 3d natural cubic splines bundle block adjustment with 3d natural cubic splines
to avoid numerical problems as⎡⎢⎣⎤n 1⎥n 2n 3⎦ =⎡⎢⎣sin θ cos ϕsin θ sin ϕcos ϕn 1 (X − X o ) + n 2 (Y − Y o ) + n 3 (Z − Z o ) = 0n 1 X + n 2 Y + n 3 Z = D⎤⎥⎦(2.17)with θ angle from XY plane, ϕ angle around Z axis, n unit vector of plane normal andD the distance between the plane and the origin. Five relative orientation parametersand three planar parameters were obtained by using the homography mapping systemwhich searched the conjugate point in an image corresponding to a point in the otherimage.Lin[40] proposed the method of the autonomous recovery of exterior orientationparameters by the extension of the traditional point-based Modified Iterated HoughTransform (MIHT) to the 3D free-form linear feature based MIHT. Straight polylineswere generalized for matching primitives in the pose estimation since the mathematicalrepresentation of straight lines are much clearer than the algebraic expression ofconic sections and splines.Gruen and Akca[21] matched 3D curves whose forms were defined by a cubicspline using the least squares matching. Subpixels were localized by the least squaresmatching and the quality of the localization was decided by the geometry of imagepatches. Two free-form lines were defined as (2.18).f(u) = [x(u) y(u) z(u)] T = a 0 + a 1 u + a 2 u 2 + a 3 u 3g(u) = [x ′ (u) y ′ (u) z ′ (u)] T = b 0 + b 1 u + b 2 u 2 + b 3 u 3 (2.18)where u ∈ [0, 1], a 0 , a 1 , a 2 , a 3 , b 0 , b 1 , b 2 , b 3 variables and f(u), g(u) ∈ R 3Taylor expansion was employed to adopt Gauss-Markov model as (2.19).22
f(u) − e(u) = g(u)f(u) − e(u) = g 0 (u) + ∂g0 (u)du ∂u(2.19)f(u) − e(u) = g 0 (u) + ∂g0 (u) ∂udx + ∂g0 (u) ∂udy + ∂g0 (u) ∂udz ∂u ∂x ∂u ∂y ∂u ∂z2.3 Parametric representations of curvesWhile progress for the automatic detection, segmentation and recognition of 3Dlines and objects consisting of free-form lines has become sophisticated by significantadvances in computer technology, considerable techniques such as the developmentin segmentation and classification for digital photogrammetry have been developedduring the last few decades, such as; geospatial image processing software, digitalorthophoto generation software, and softcopy workstations. Digital photogrammetryis closely related to the field of computer graphics and computer vision. Free-formlines and objects are an important element of many applications in computer graphicsand computer vision. A number of researchers in computer vision and artificial intelligencehave used suitable constraints or assumptions to reduce the solution space insegmentation and have extracted constrained features such as contours [64], convexoutlines [35], rectangles [54] and ellipses [53].Free-form lines are one of three linear features and other linear features are straightlinear features and linear features described by unique mathematical equations. Anumber of researchers have preferred straight lines for photogrammetric applicationssince straight lines have no singularity problem and straight lines are easily detectedin man-made environments. A list of curves is described as follows.23
- 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: surfaces and terrain models in 2D a
- 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 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,
f(u) − e(u) = g(u)f(u) − e(u) = g 0 (u) + ∂g0 (u)du ∂u(2.19)f(u) − e(u) = g 0 (u) + ∂g0 (u) ∂udx + ∂g0 (u) ∂udy + ∂g0 (u) ∂udz ∂u ∂x ∂u ∂y ∂u ∂z2.3 Parametric representations of curvesWhile progress for the automatic detection, segmentation and recognition of 3Dlines and objects consisting of free-form lines has become sophisticated by significantadvances in computer technology, considerable techniques such as the developmentin segmentation and classification for digital photogrammetry have been developedduring the last few decades, such as; geospatial image processing software, digitalorthophoto generation software, and softcopy workstations. Digital photogrammetryis closely related to the field of computer graphics and computer vision. Free-formlines and objects are an important element of many applications in computer graphicsand computer vision. A number of researchers in computer vision and artificial intelligencehave used suitable constraints or assumptions to reduce the solution space insegmentation and have extracted constrained features such as contours [64], convexoutlines [35], rectangles [54] and ellipses [53].Free-form lines are one of three linear features and other linear features are straightlinear features and linear features described by unique mathematical equations. Anumber of researchers have preferred straight lines for photogrammetric applicationssince straight lines have no singularity problem and straight lines are easily detectedin man-made environments. A list of curves is described as follows.23
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