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Foundations for Strip Adjustment of Airborne Laserscanning Data with Conformal Geometric Algebra


Typically, airborne laserscanning includes a laser mounted on an airplane or drone (its pulsed beam direction can scan in flight direction and perpendicular to it) an intertial positioning system of gyroscopes, and a global navigation satellite system. The data, relative orientation and relative distance of these three systems are combined in computing strips of ground surface point locations in an earth fixed coordinate system. Finally, all laserscanning strips are combined via iterative closest point methods to an interactive three-dimensional terrain map. In this work we describe the mathematical framework for how to use the iterative closest point method for the adjustment of the airborne laserscanning data strips in the framework of conformal geometric algebra.

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Fig. 1


  1. Notation: In geometric algebra \(Cl^k(p,q), 0\le k \le n=p+q\), denotes the vector space of grade-k elements, e.g., \(Cl^2(3,0)\) is the three-dimensional space of bivectors in Cl(3, 0).

  2. Traditionally, null basis vectors \({\mathbf {e}}_{\infty } = {\mathbf {e}}_{+1} + {\mathbf {e}}_{-1}\), \({\mathbf {e}}_{0} = \frac{1}{2} ({\mathbf {e}}_{-1} - {\mathbf {e}}_{+1})\), are defined, as in [5, 16]. But in general any factor \(\lambda \in {\mathbb {R}}\setminus \{0\}\), could be fixed and define \({\mathbf {e}}_{\infty } = \frac{1}{\lambda \sqrt{2}} ({\mathbf {e}}_{+1} + {\mathbf {e}}_{-1})\), \({\mathbf {e}}_{0} = \frac{\lambda }{\sqrt{2}} ({\mathbf {e}}_{-1} - {\mathbf {e}}_{+1})\), while preserving the scalar products of Table 1. This freedom to operate with a continuously parametrized basis (equivalent to a continuously parametrized set of horospheres) has e.g., been used advantageously by El Mir et al for elegant algebraic view point change representation in [6]. On the other hand [19] showed that for the modelling of quadrics \(\lambda =1\) is of advantage.

  3. Note that the left- and right contraction \(\rfloor \) and \(\lfloor \), respectively, are needed essentially.

  4. We use the notation \({\mathbf {e}}_{31}\) for the bivector of the \({\mathbf {e}}_1\), \({\mathbf {e}}_3\)-plane, since then we can conveniently compute \({\mathbf {e}}_{31}={\mathbf {e}}_2 {\mathbf {e}}_{123}\), preserving the cyclic order of the indexes 31|2 on the left and right. And cyclic index interchange gives the other two bivector and normal vector relationships without need for sign considerations. Software implementations of GA, like GAALOP [13], may rather use a lexicographical order, that is \({\mathbf {e}}_{13}=-{\mathbf {e}}_{31}\), which may need to be taken care of when implementing an algorithm.

  5. In the readjustment of the boresight misalignment in CGA one can simply directly optimize with respect to the Euclidean bivector \(\varphi {\mathbf {e}}_{si}\).

  6. Product of two parallel planes (29), perpendicular to \({\mathbf {a}}^i\) and at distance \(\tfrac{1}{2}|{\mathbf {a}}^i|\).

  7. Note that the same rotation operators (rotors), e.g., of (37), are used in the geometric algebra Cl(3, 0) of three-dimensional space \({\mathbb {R}}^3\) and in conformal geometric algebra Cl(4, 1).

  8. The notation longitude \(\lambda \) and latitude \(\varphi \) simply follows equ. (5) of [10].

  9. Taking the conformal vector representations of \(Plane^*_i\) from one ALS strip and \(Plane^*_j\) from a second overlapping ALS strip allows to compute their angle as \(\alpha _{ij} = \cos ^{-1}(Plane^*_i\cdot Plane^*_j)\) and thus to decide on correspondence rejection beyond a certain angular threshold, e.g., \(5^{\circ }\) (see p. 77 of [10]).

  10. This also allows to implement based on CGA the roughness limit criterion for the rejection of correspondences (p. 77 of [10]).

  11. Equation (51) also allows to directly compute the point-to-plane standard deviation estimator (16) of [10].

  12. In order to make it easier to identify a variable as a vector, we now switch for the rest of this work to the conventional notation with an arrow on top.


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In the beginning, God created the heavens and the earth [9]. E.H. thanks his colleagues at ICU (Tokyo, Japan) for the opportunity of a sabbatical with AHM GmbH (Innsbruck, Austria), his colleagues at AHM GmbH for stimulating discussions and collaboration on Lidar data processing, and H. Salchner (Gschnitz, Austria). Note that this paper is an extended version of the conference paper [20].

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Correspondence to Eckhard Hitzer.

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“This article is part of the Topical Collection T.C. : SCIS&ISIS2020 - Modern Applications of Clifford Algebra edited by Eckhard Hitzer, Kanta Tachibana and Kazunori Uruma”

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Hitzer, E., Benger, W., Niederwieser, M. et al. Foundations for Strip Adjustment of Airborne Laserscanning Data with Conformal Geometric Algebra. Adv. Appl. Clifford Algebras 32, 1 (2022).

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  • Conformal geometric algebra
  • Georeferencing
  • Iterative closest point algorithm
  • Orientation
  • Calibration

Mathematics Subject Classification

  • Primary 15A66
  • Secondary 15A23
  • 15A16