Project-Homepage

Project: Geometric Algebras and the Perception-Action Cycle

Researchers: Banarer V. , Buchholz S. , Bülow T. , Felsberg M. , Perwass C. , Rosenhahn B. , Sommer G.

One research project deals with applications of geometric algebras in computer science. The perception-action cycle (PAC) as the frame for autonomous behavior relates perception and action in a purposive manner. The implementation of artificial PACs demands on the fusion of signal theory, computer vision, robotics and neural computing. In this research work we use an interpretation of the Clifford algebra called geometric algebra.
Clifford (or geometric) algebras are well known to pure mathematicans. The elements in geometric algebras are called multivectors which can be multiplied together using a geometric product. Euclidean, projective and conformal geometry find in geometric algebra the frame where they can reconcile and express their potential. This opens a new alternative for the mathematical treatment of the stratification of the 3D visual space.
Since 1995 the Kiel GA applications group set up theoretic bases for dealing with tasks of signal processing, projective geometry, robot kinematics and geometric neural computing. Since 1999 the work is extended to practical applications and numerical experiences with respect to different research topics. Please check out the other project pages for futher information

2D-3D Pose Estimation Neural Computing Signal processing Teaching GAs
Tree 2Spirals Signal 3spheres

We use different geometric algebras to model geometric scenarios. The most expressive one we use so-far is the conformal geometric algebra. It provides a homogeneous model for stereographically projected points on a hypersphere and therefore couples kinematics with projective geometry. The geometric idea behing this algebra are stereographic projections:
Simply speaking, a stereographic projection is one way to generate a flat map of the earth. The rule for a stereographic projection has a nice geometric description: Think of the earth as transparent sphere, intersected on the equator by an equatorial plane. Now imagine a light bulb at the north pole n, which shines through the sphere. Each point on the sphere casts a shadow on the paper and that is where it is drawn on the map.
Using a homogeneous model for stereographic projected points leads to a cone in space. Embedding this model in a Clifford algebra leads to the conformal geometric algebra (CGA). It is suited to describe conformal geometry, it contains spheres as entities and the conformal transformations as geometric manipulations.
Several other research groups deal with Clifford algebras (mainly mathematicans and physicists). Please check out the following links to the GA-community:

Publications:

2004 Pose Estimation of Free-form Objects
Rosenhahn B., Sommer G., Klette R.
Technical Report 0401, Christian-Albrechts-Universität zu Kiel, Institut für Informatik und Praktische Mathemati k, März 2004
PDF, Bibtex
2003 The hypersphere neuron
Banarer V., Perwass C., Sommer G.
In Proc. 11th European Symposium on Artificial Neural Networks, ESANN 2003, Bruges, pp. 469-474. d-side pu blications, Evere, Belgium, 2003
PDF, Bibtex
2003 Pose estimation of free-form surface models
Rosenhahn B., Perwass C., Sommer G.
In 25. Symposium für Mustererkennung, DAGM 2003, Magdeburg, Vol. 2781 of LNCS, pp. 574-581, Springer-Verla g, Berlin, 2003.
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2003 Implementation of a Clifford algebra co-processor design on a field programmable gate array
Perwass C., Gebken C., Sommer G. < br> In R. Ablamowicz, Editor, 6th International Conference on Clifford Algebras, and Applications, , Cookevil le, TN, CLIFFORD ALGEBRAS: Application to Mathematics, Physics, and Engineering, pp. 561-575, Birkhäuser, Boston, Progress in Mathe matical Physics, 2003
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2003 Pose Estimation Revisited
Rosenhahn B.
Dissertation, Institut für Informatik und Praktische Mathematik, Christian-Albrechts-Universität zu Kiel, 2003.
PDF, Bibtex, Abtract
2002 Low-Level Image Processing with the Structure Multivector
Felsberg M.
Dissertation, Institut für Informatik und Praktische Mathematik, Christian-Albrechts-Universität zu Kiel, 2002. -
PDF, PS, Bibtex
2002 Pose Estimation of 3D Free-form Contours
Rosenhahn B., Perwass C., Sommer G.
Technical Report 0207, Christian-Albrechts-Universität zu Kiel, Institut für Informatik und Praktische Mathematik, August 2002
PDF, Bibtex
2002 Pose estimation of 3D free-form contours in conformal geometry
Rosenhahn B., Perwass C., Sommer G.
In D. Kenwright, editor, Proceedings of Image and Vision Computing, IVCNZ, Auckland, NZ, pp. 29-34. 2002 -
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2002 Adaptive pose estimation for different corresponding entities
Rosenhahn B., Sommer G.
In L. Van Gool, editor, Pattern Recognition, 24. Symposium für Mustererkennung, Zürich, September 2002, Vo l. 2449 of LNCS, pp. 265-273. Springer-Verlag, Berlin Heidelberg, 2002
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2002 Monocular pose estimation of kinematic chains
Rosenhahn B., Granert O., Sommer G.
In L. Dorst, C. Doran and J. Lasenby, editors, Applications of Geometric Algebra in Computer Science and Engin eering, pp. 373-375. Proc. AGACSE 2001, Cambridge, UK, Birkhäuser Boston, 2002
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2002 A geometric approach for the analysis and computation of the intrinsic camera parameters
Bayro-Corrochano E., Rosenhahn B.
Pattern Recognition, 35:169-186, 2002
PDF, Bibtex
2001 Geometric Computing with Clifford Algebras
Sommer G.
Springer-Verlag, Heidelberg, 2001
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2001 Pose estimation using geometric constraints
Sommer G., Rosenhahn B., Zhang Y.
In R. Klette, T. Huang and G. GimelŽfarb, editors, Multi-image analysis, Vol. 2032 of LNCS, pp. 153-170. P roc. Dagstuhl Workshop on Theoretical Foundations of Computer Vision, Springer-Verlag, Berlin, 2001
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2001 Introduction to neural computation in Clifford Algebra
Buchholz S., Sommer G.
In G. Sommer, editor, Geometric Computing with Clifford Algebra, pp. 291-314. Springer-Verlag, Heidelberg, 2001
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2001 Local hypercomplex signal representation and applications
Bülow T., Sommer G.
In G. Sommer, editor, Geometric Computing with Clifford Algebra, pp. 255-289. Springer-Verlag, Heidelberg, 2001
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2001 Commutative hypercomplex Fourier transforms of multidimensional signals
Felsberg M., Bülow T., Sommer G. < br> In G. Sommer, editor, Geometric Computing with Clifford Algebra, pp. 209-229. Springer-Verlag, Heidelberg, 2001
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2001 The monogenic signal
Felsberg M., Sommer G.
IEEE Transactions on Signal Processing, 49(12):3136-3144, December 2001
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2001 Coordinate-free projective geometry for computer vision
Li H., Sommer G.
In G. Sommer, editor, Geometric Computing with Clifford Algebra, pp. 415-454. Springer-Verlag, Heidelberg, 2001
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2001 3D-reconstruction from vanishing points
Perwass C., Lasenby J.
In G. Sommer, editor, Geometric Computing with Clifford Algebra, pp. 371-392. Springer-Verlag, Heidelberg, 2001
PDF, PS, Bibtex
2001 The motor extended Kalman filter for dynamic rigid motion estimation from line observations
Zhang Y., Sommer G., Bayro-Corrochano E.< /span>
In G. Sommer, editor, Geometric Computing with Clifford Algebra, pp. 501-530. Springer-Verlag, Heidelberg, 2001
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2001 Analysis and computation of the intrinsic camera parameters
Bayro-Corrochano E., Rosenhahn B.
In G. Sommer, editor, Geometric Computing with Clifford Algebra, pp. 393-414. Springer-Verlag, Heidelberg, 2001
PDF, PS, Bibtex
2000 Applications of Geometric Algebra in Computer Vision
Perwass C.
Dissertation, Cambridge University, 2000.
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2000 Extended Kalman filter design for motion estimation by point and line observations
Zhang Y., Rosenhahn B., Sommer G.
In G. Sommer and Y. Zeevi, editors, 2nd International Workshop on Algebraic Frames for the Perception-Action C ycle, AFPAC 2000, Kiel, Vol. 1888 of LNCS, pp. 339-348. Springer-Verlag, 2000
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2000 Motor algebra for 3D kinematics: The case of hand-eye calibration
Bayro-Corrochano E., Daniilidis K., Somme r G.
Journal of Mathematical Imaging and Vision, 13:79-100, 2000
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1999 The global algebraic frame of the perception-action cycle
Sommer G.
In B. Jähne, H. Haussecker and P. Geissler, editors, Handbook of Computer Vision and Applications, pp. 221 -264. Academic Press, San Diego, 1999
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1999 Hypercomplex Spectral Signal Representations for Image Processing and Analysis
Bülow T.
Dissertation, Institut für Informatik und Praktische Mathematik, Christian-Albrechts-Universität zu Kiel, 1999. -
PS, Bibtex
1999 Hand-eye calibration using dual quaternions
Daniilidis K.
Int. Journal Robotics Research, 18:286-298, 1999
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1997 A unified language for computer vision and robotics
Bayro-Corrochano E., Lasenby J.
In G. Sommer and J.J. Koenderink, editors, Algebraic Frames for the Perception-Action Cycle, Vol. 1315 of Lecture Notes in Computer Science, pp. 219-234. Int. Workshop AFPACŽ97, Kiel, Springer-Verlag, Heidelberg, 1997
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1997 What can Grassmann, Hamilton and Clifford tell us about computer vision and robotics
Bayro-Corrochano E., Lasenby J., Sommer G .
In E. Paulus and F.M. Wahl, editors, Mustererkennung 1997, Tagungsband 19. DAGM-Symposium, pp. 164-171. DA GMŽ97, Springer-Verlag, 1997
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1995 Object modelling and motion analysis using Clifford algebra
Bayro-Corrochano E., Lasenby J.
In R. Mohr and W. Chengke, editors, Proc. Europe-China Workshop on Geometrical Modelling and Invariants for Computer Vision, pp. 143-149. 1995
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