Abstract
We discuss a coordinate-free approach to the geometry of computer vision problems. The technique we use to analyse the three-dimensional transformations involved will be that of geometric algebra: a framework based on the algebras of Clifford and Grassmann. This is not a system designed specifically for the task in hand, but rather a framework for all mathematical physics. Central to the power of this approach is the way in which the formalism deals with rotations; for example, if we have two arbitrary sets of vectors, known to be related via a 3D rotation, the rotation is easily recoverable if the vectors are given. Extracting the rotation by conventional means is not as straightforward. The calculus associated with geometric algebra is particularly powerful, enabling one, in a very natural way, to take derivatives with respect to any multivector (general element of the algebra). What this means in practice is that we can minimize with respect to rotors representing rotations, vectors representing translations, or any other relevant geometric quantity. This has important implications for many of the least-squares problems in computer vision where one attempts to find optimal rotations, translations etc., given observed vector quantities. We will illustrate this by analysing the problem of estimating motion from a pair of images, looking particularly at the more difficult case in which we have available only 2D information and no information on range. While this problem has already been much discussed in the literature, we believe the present formulation to be the only one in which least-squares estimates of the motion and structure are derived simultaneously using analytic derivatives.
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Arun, K., Huang, T.S., and Blostein, S.D. 1987. Least squares fitting of two 3-D point sets. IEEE Trans. Pattern Anal. Mach. IntelligencePAMI-9, 698-700.
Bayro-Corrochano, E. and Lasenby, J. 1995. Object modelling and motion analysis using Clifford algebra. In Proc. of Europe-China Workshop on Geometric Modeling and Invariants for Computer Vision, Roger Mohr and Wu Chengke (Eds.), Xi'an, China.
Bayro-Corrochano, E., Lasenby, J., and Sommer, G. 1996. Geometric Algebra: a framework for computing point and line correspondences and projective structure using n-uncalibrated cameras. In Proc. of ICPR'96, Vienna.
Carlsson, S. 1994. The Double Algebra: and effective tool for computing invariants in computer vision. In Applications of Invariance in Computer Vision, Lecture Notes in Computer Science 825; In Proc. of 2nd-joint Europe-US workshop, Azores, October 1993, Mundy, Zisserman, and Forsyth (Eds.). Springer-Verlag.
Chevalier, D.P. 1991. Lie algebras, modules, dual quaternions and algebraic methods in kinematics. Mech. Mach. Theory26:350- 358.
Clifford, W.K. 1878. Applications of Grassmann's extensive algebra. Am. J. Math.26(6):613-627.
Csurka, G. and Faugeras, O. 1995. Computing three-dimensional projective invariants from a pair of images using the Grassmann-Cayley algebra. In Proc. of Europe-China Workshop on Geometric Modeling and Invariants for Computer Vision, Roger Mohr and Wu Chengke (Eds.), Xi'an, China.
Doran, C.J.L. 1994. Geometric algebra and its applications to mathematical physics. Ph.D. Thesis, University of Cambridge.
Doran, C.J.L., Hestenes, D., Sommen, F., and van Acker, N. 1993a. Lie groups as spin groups. J. Math. Phys., 34(8):3642.
Doran, C.J.L., Lasenby, A.N. and Gull, S.F. 1993b. Gravity as a gauge theory in the spacetime algebra. In Third International Conference on Clifford Algebras and their Applications in Mathematical Physics, F. Brackx and R. Delanghe. (Eds.), Kluwer: Dordrecht, p. 375.
Doran, C.J.L., Lasenby, A.N., Gull, S.F., Somaroo, S., and Challinor, A. 1996. Spacetime algebra and electron physics. Advances in Electronics and Electron Physics. 95:272-383.
Faugeras, O.D. and Hebert, M. 1983. A 3-D recognition and positioning algorithm using geometrical matching between primitive surfaces. In Proc. International Joint Conference on Artificial Intelligence, Karlsruhe, Germany, pp. 996-1002.
Faugeras, O.D., Lustman, F., and Toscani, G. 1987. Motion and structure from motion. In Proc. ICCV, pp. 25-34.
Faugeras, O. and Mourrain, B. 1995. On the geometry and algebra of the point and line correspondences between N images. In Proc. of Europe-China Workshop on Geometric Modeling and Invariants for Computer Vision, Roger Mohr and Wu Chengke (Eds.), Xi'an: China.
Grassmann, H. 1877. Der ort der Hamilton'schen quaternionen in der ausdehnungslehre. Math. Ann., 12:375.
Gull, S.F., Lasenby, A.N., and Doran, C.J.L. 1993. Imaginary numbers are not real-the geometric algebra of spacetime. Found. Phys., 23(9):1175.
Hestenes, D. 1966. Space-Time Algebra.Gordon and Breach.
Hestenes, D. 1986a. New Foundations for Classical Mechanics, D. Reidel: Dordrecht.
Hestenes, D. 1986b. Aunified language for mathematics and physics. In Clifford Algebras and Their Applications in Mathematical Physics, J.S.R. Chisolm and A.K. Common(Eds.), D. Reidel: Dordrecht, p. 1.
Hestenes, D. 1994. Invariant body kinematics: II. reaching and neurogeometry. Neural Networks, 7(1):79-88.
Hestenes, D. and Sobczyk, G. 1984. Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, D. Reidel: Dordrecht.
Hestenes, D. and Ziegler, R. 1991. Projective geometry with clifford algebra. Acta Applicandae Mathematicae, 23:25-63.
Horn, B.K.P. 1987. Closed-form solution of absolute orientation using unit quaternions. J. Opt. Soc. Am., 4:629-642.
Horn, B.K.P., Hilden, H.M., and Negahdaripour, S. 1988. Closed-form solution of absolute orientation using orthonormal matrices. J. Opt. Soc. Am., 5:1127-1135.
Huang, T.S. 1986. Determining three-dimensional motion and structure from two perspective views. Handbook of Pattern Recognition and Image Processing, Academic Press, Chapter 14.
Huang, T.S. and Netravali, A.N. 1994. Motion and structure from feature correspondences: A review. In Proc. of the IEEE, Vol. 82, No.2, pp. 252-268.
Lasenby, J. 1996. Geometric algebra: Applications in engineering. In Geometric (Clifford) Algebras in Physics, W.E. Baylis (Ed.), Birkhauser: Boston.
Lasenby, A.N., Doran, C.J.L., and Gull, S.F. 1993. Grassmann calculus, pseudoclassical mechanics and geometric algebra. J. Math. Phys., 34(8):3683.
Lasenby, A.N., Doran, C.J.L., and Gull, S.F. 1994. Astrophysical and cosmological consequences of a gauge theory of gravity. Advances in Astrofundamental Physics. In Proc. of 1994 International School of Astrophysics, Erice, N. Sanchez and A. Zichichi (Eds.), World Scientific: Singapore, p. 359.
Lasenby, J., Bayro-Corrochano, E., Lasenby, A.N., and Sommer, G. 1996. A new methodology for computing invariants in computer vision. In Proc. of ICPR'96, Vienna.
Lasenby, A.N., Doran, C.J.L., and Gull, S.F. 1997. Gravity, gauge theories and geometric algebra. Phil. Trans. Roy. Soc. A. In press.
Lin, Z., Huang, T.S., Blostein, S.D., Lee, H., and Margerum, E.A. 1986. Motion estimation from 3-D point sets with and without correspondences. In Proc. IEEE Conf. on Computer Vision and Pattern Recognition, Miami Beach, Florida, pp. 194-201.
Longuet-Higgins, H.C. 1981. A computer algorithm for reconstructing a scene from two projections. Nature, 293:133-138.
Mitchie, A. and Aggarwal, J.K. 1986. A computational analysis of time-varying images. Handbook of Pattern Recognition and Image Processing, T.Y. Young and K.S. Fu (Eds.), Academic Press: New York.
Sabata, B. and Aggarwal, J.K. 1991. Estimation of motion from a pair of range images: A review. CVGIP: Image Understanding, 54(3):309-324.
Walker, M.W., Shao, L. and Volz, R.A. 1991. Estimating 3-D location parameters using dual number quaternions. CVGIP: Image Understanding, 54(3):358-367.
Weng, J., Huang, T.S., and Ahuja, N. 1989. Motion and structure from two perspective views: Algorithms, error analysis and error estimation. IEEE Trans. Pattern Anal. Mach. Intelligence, 11(5):451-476.
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Lasenby, J., Fitzgerald, W., Lasenby, A. et al. New Geometric Methods for Computer Vision: An Application to Structure and Motion Estimation. International Journal of Computer Vision 26, 191–213 (1998). https://doi.org/10.1023/A:1007901028047
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DOI: https://doi.org/10.1023/A:1007901028047