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 lot in man-made structures and mobile mapping system in reality, they have focusedon implementation with straight linear features with geometric constraints.Since many man-made environments including buildings often have straight edgesand planar faces, it is advantageous to employ line photogrammetry instead of pointphotogrammetry when mapping polyhedral model objects.Mikhail[43] and Habib et al.[27] accomplished the geometrical modeling and theperspective transformation of linear features within a triangulation process. Linearfeatures were used to recover relative orientation parameters. Habib et al. proposeda free-form line in object space by a sequence of 3D points along the object spaceline.Lee and Bethel[38] proposed employing both points and linear features were moreaccurate than using only points in ortho-rectification of airborne hyperspectral imagery.EOPs were recovered accurately and serious distortions were removed by thecontribution of linear features.Schenk[59] extended the concept of aerial triangulation from point features tolinear features. The line equation of six dependent parameters replaced the pointbased collinearity equation.X = X A + t · aY = Y A + t · bZ = Z A + t · c(2.9)where a real variable t, the start point (X A , Y A , Z A ) and direction vector (a, b, c).Traditional point-based collinearity equation was extended to line features18
x p = −f (X A + t · a − X C )r 11 + (Y A + t · b − Y C )r 12 + (Z A + t · c − Z C )r 13(X A + t · a − X C )r 31 + (Y A + t · b − Y C )r 32 + (Z A + t · c − Z C )r 33y p = −f (X A + t · a − X C )r 21 + (Y A + t · b − Y C )r 22 + (Z A + t · c − Z C )r 23(X A + t · a − X C )r 31 + (Y A + t · b − Y C )r 32 + (Z A + t · c − Z C )r 33(2.10)with x p , y p photo coordinates, f the focal length, X C , Y C , Z C camera perspectivecenter, and r ij the elements of the 3D orthogonal rotation matrix. The extendedcollinearity equation with six parameters was derived as the line expression of fourparameters (φ, θ, x o , y o ) since a 3D straight line has only four independent parameters.Two constrains are required to solve a common form of the 3D straight equations usingsix parameters determined by two vectors.⎡⎢⎣XYZ⎤⎥⎦ =⎡⎢⎣cos θ cos φ · x o − sin φ · y o + sin θ cos φ · zcos θ sin φ · x o + cos φ · y o + sin θ sin φ · z− sin θ · x o + cos θ · z⎤⎥⎦ (2.11)where z is a real variable. The advantage of the 3D straight line using four independentparameters is that it reduces the computation and time complexity in theadjustment processes such as a bundle block adjustment. The collinearity equationas the straight line function of four parameters was developed.x p = −f (X − X C)r 11 + (Y − Y C )r 12 + (Z − Z C )r 13(X − X C )r 31 + (Y − Y C )r 32 + (Z − Z C )r 33y p = −f (X − X (2.12)C)r 21 + (Y − Y C )r 22 + (Z − Z C )r 23(X − X C )r 31 + (Y − Y C )r 32 + (Z − Z C )r 33where X, Y, and Z were defined in (2.11).The solution of the bundle block adjustment with linear features was implemented sothat the line-based aerial triangulation can provide a more robust and autonomousenvironment than the traditional point-based bundle block adjustment.Anothermathematical model of the perspective relationship between the image and the object19
- 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: ⎡⎢⎣x i − x py i − y p−f
- 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 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
a lot in man-made structures and mobile mapping system in reality, they have focusedon implementation <strong>with</strong> straight linear features <strong>with</strong> geometric constraints.Since many man-made environments including buildings often have straight edgesand planar faces, it is advantageous to employ line photogrammetry instead of pointphotogrammetry when mapping polyhedral model objects.Mikhail[43] and Habib et al.[27] accomplished the geometrical modeling and theperspective transformation of linear features <strong>with</strong>in a triangulation process. Linearfeatures were used to recover relative orientation parameters. Habib et al. proposeda free-form line in object space by a sequence of 3D points along the object spaceline.Lee and Bethel[38] proposed employing both points and linear features were moreaccurate than using only points in ortho-rectification of airborne hyperspectral imagery.EOPs were recovered accurately and serious distortions were removed by thecontribution of linear features.Schenk[59] extended the concept of aerial triangulation from point features tolinear features. The line equation of six dependent parameters replaced the pointbased collinearity equation.X = X A + t · aY = Y A + t · bZ = Z A + t · c(2.9)where a real variable t, the start point (X A , Y A , Z A ) and direction vector (a, b, c).Traditional point-based collinearity equation was extended to line features18
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