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 & 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).   Google Scholar

[2]

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V. V. Volchkov, "Integral Geometry and Convolution Equations,", Kluwer Academic Publishers, (2003).   Google Scholar

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show all references

References:
[1]

T. Fidler, M. Grasmair, H. Pottmann and O. Scherzer, Inverse problems of integral invariants and shape signatures,, Preprint, (2007).   Google Scholar

[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.  doi: 10.1007/s002110050158.  Google Scholar

[3]

H. Krim and A. Yezzi, Jr., "Statistics and Analysis of Shapes,", Birkhäuser, (2006).  doi: 10.1007/0-8176-4481-4.  Google Scholar

[4]

S. Manay, D. Cremers, B. W. Hong, A. Yezzi, Jr. and S. Soatto, Integral invariants and shape matching,, In, (): 137.   Google Scholar

[5]

H. Pottmann, J. Wallner, Q.-X. Huang and Y.-L. Yang, Integral invariants for robust geometry processing,, Submitted for publication, (2007).   Google Scholar

[6]

W. Rudin, "Functional Analysis,", McGraw-Hill Book Co., (1973).   Google Scholar

[7]

W. Rudin, "Real and Complex Analysis,", McGraw-Hill Book Co., (1987).   Google Scholar

[8]

R. Schneider, Über eine Integralgleichung in der Theorie der konvexen Körper,, Math. Nachr., 44 (1970), 55.  doi: 10.1002/mana.19700440105.  Google Scholar

[9]

V. V. Volchkov, "Integral Geometry and Convolution Equations,", Kluwer Academic Publishers, (2003).   Google Scholar

[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.  doi: 10.1109/TVCG.2007.19.  Google Scholar

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