# American Institute of Mathematical Sciences

August  2008, 2(3): 341-354. doi: 10.3934/ipi.2008.2.341

## Identifiability and reconstruction of shapes from integral invariants

 1 Department of Mathematics, University of Innsbruck, Technikerstr.21a, A-6020 Innsbruck, Austria, Austria 2 Department of Computer Science, University of Innsbruck, Technikerstrasse 21a, A-6020 Innsbruck

Received  October 2007 Revised  January 2008 Published  July 2008

Integral invariants have been proven to be useful for shape matching and recognition, but fundamental mathematical questions have not been addressed in the computer vision literature. In this article we are concerned with the identifiability and numerical algorithms for the reconstruction of a star-shaped object from its integral invariants. In particular we analyse two integral invariants and prove injectivity for one of them. Additionally, numerical experiments are performed.
Citation: Thomas Fidler, Markus Grasmair, Otmar Scherzer. Identifiability and reconstruction of shapes from integral invariants. Inverse Problems and Imaging, 2008, 2 (3) : 341-354. doi: 10.3934/ipi.2008.2.341
##### References:
 [1] T. Fidler, M. Grasmair, H. Pottmann and O. Scherzer, Inverse problems of integral invariants and shape signatures, Preprint, 2007. [2] M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37. doi: 10.1007/s002110050158. [3] H. Krim and A. Yezzi, Jr., "Statistics and Analysis of Shapes," Birkhäuser, Boston, 2006. doi: 10.1007/0-8176-4481-4. [4] S. Manay, D. Cremers, B. W. Hong, A. Yezzi, Jr. and S. Soatto, Integral invariants and shape matching, In "[3]," 137-166. [5] H. Pottmann, J. Wallner, Q.-X. Huang and Y.-L. Yang, Integral invariants for robust geometry processing, Submitted for publication, 2007. [6] W. Rudin, "Functional Analysis," McGraw-Hill Book Co., New York, 1973.McGraw-Hill Series in Higher Mathematics. [7] W. Rudin, "Real and Complex Analysis," McGraw-Hill Book Co., New York, third edition, 1987. [8] R. Schneider, Über eine Integralgleichung in der Theorie der konvexen Körper, Math. Nachr., 44 (1970), 55-75. doi: 10.1002/mana.19700440105. [9] V. V. Volchkov, "Integral Geometry and Convolution Equations," Kluwer Academic Publishers, Dordrecht/Boston/London, 2003. [10] L. Yu-Kun, Z. Qian-Yi, H. Shi-Min, J. Wallner and H. Pottmann, Robust feature classification and editing, IEEE Trans. Vis. Comp. Graphics, 13 (2007), 34-45. doi: 10.1109/TVCG.2007.19.

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##### References:
 [1] T. Fidler, M. Grasmair, H. Pottmann and O. Scherzer, Inverse problems of integral invariants and shape signatures, Preprint, 2007. [2] M. Hanke, A. Neubauer and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37. doi: 10.1007/s002110050158. [3] H. Krim and A. Yezzi, Jr., "Statistics and Analysis of Shapes," Birkhäuser, Boston, 2006. doi: 10.1007/0-8176-4481-4. [4] S. Manay, D. Cremers, B. W. Hong, A. Yezzi, Jr. and S. Soatto, Integral invariants and shape matching, In "[3]," 137-166. [5] H. Pottmann, J. Wallner, Q.-X. Huang and Y.-L. Yang, Integral invariants for robust geometry processing, Submitted for publication, 2007. [6] W. Rudin, "Functional Analysis," McGraw-Hill Book Co., New York, 1973.McGraw-Hill Series in Higher Mathematics. [7] W. Rudin, "Real and Complex Analysis," McGraw-Hill Book Co., New York, third edition, 1987. [8] R. Schneider, Über eine Integralgleichung in der Theorie der konvexen Körper, Math. Nachr., 44 (1970), 55-75. doi: 10.1002/mana.19700440105. [9] V. V. Volchkov, "Integral Geometry and Convolution Equations," Kluwer Academic Publishers, Dordrecht/Boston/London, 2003. [10] L. Yu-Kun, Z. Qian-Yi, H. Shi-Min, J. Wallner and H. Pottmann, Robust feature classification and editing, IEEE Trans. Vis. Comp. Graphics, 13 (2007), 34-45. doi: 10.1109/TVCG.2007.19.
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