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 & Imaging, 2008, 2 (3) : 341-354. doi: 10.3934/ipi.2008.2.341
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References:
 [1] Plamen Stefanov, Yang Yang. Multiwave tomography with reflectors: Landweber's iteration. Inverse Problems & Imaging, 2017, 11 (2) : 373-401. doi: 10.3934/ipi.2017018 [2] Konovenko Nadiia, Lychagin Valentin. Möbius invariants in image recognition. Journal of Geometric Mechanics, 2017, 9 (2) : 191-206. doi: 10.3934/jgm.2017008 [3] David Maxwell. Kozlov-Maz'ya iteration as a form of Landweber iteration. Inverse Problems & Imaging, 2014, 8 (2) : 537-560. doi: 10.3934/ipi.2014.8.537 [4] Barbara Kaltenbacher, Ivan Tomba. Enhanced choice of the parameters in an iteratively regularized Newton-Landweber iteration in Banach space. Conference Publications, 2015, 2015 (special) : 686-695. doi: 10.3934/proc.2015.0686 [5] Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133 [6] Inês Cruz, M. Esmeralda Sousa-Dias. Reduction of cluster iteration maps. Journal of Geometric Mechanics, 2014, 6 (3) : 297-318. doi: 10.3934/jgm.2014.6.297 [7] André de Carvalho, Toby Hall. Decoration invariants for horseshoe braids. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 863-906. doi: 10.3934/dcds.2010.27.863 [8] Sobhan Seyfaddini. Spectral killers and Poisson bracket invariants. Journal of Modern Dynamics, 2015, 9: 51-66. doi: 10.3934/jmd.2015.9.51 [9] Rémi Leclercq. Spectral invariants in Lagrangian Floer theory. Journal of Modern Dynamics, 2008, 2 (2) : 249-286. doi: 10.3934/jmd.2008.2.249 [10] BronisŁaw Jakubczyk, Wojciech Kryński. Vector fields with distributions and invariants of ODEs. Journal of Geometric Mechanics, 2013, 5 (1) : 85-129. doi: 10.3934/jgm.2013.5.85 [11] Yong Fang. Thermodynamic invariants of Anosov flows and rigidity. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1185-1204. doi: 10.3934/dcds.2009.24.1185 [12] Joep H.M. Evers, Sander C. Hille, Adrian Muntean. Modelling with measures: Approximation of a mass-emitting object by a point source. Mathematical Biosciences & Engineering, 2015, 12 (2) : 357-373. doi: 10.3934/mbe.2015.12.357 [13] Roman Chapko. On a Hybrid method for shape reconstruction of a buried object in an elastostatic half plane. Inverse Problems & Imaging, 2009, 3 (2) : 199-210. doi: 10.3934/ipi.2009.3.199 [14] Xianchao Xiu, Ying Yang, Wanquan Liu, Lingchen Kong, Meijuan Shang. An improved total variation regularized RPCA for moving object detection with dynamic background. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-14. doi: 10.3934/jimo.2019024 [15] Mark Comerford, Todd Woodard. Orbit portraits in non-autonomous iteration. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 2253-2277. doi: 10.3934/dcdss.2019144 [16] George Papadopoulos, Holger R. Dullin. Semi-global symplectic invariants of the Euler top. Journal of Geometric Mechanics, 2013, 5 (2) : 215-232. doi: 10.3934/jgm.2013.5.215 [17] Walter D. Neumann and Jun Yang. Invariants from triangulations of hyperbolic 3-manifolds. Electronic Research Announcements, 1995, 1: 72-79. [18] Michael C. Sullivan. Invariants of twist-wise flow equivalence. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 475-484. doi: 10.3934/dcds.1998.4.475 [19] Kingshook Biswas. Complete conjugacy invariants of nonlinearizable holomorphic dynamics. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 847-856. doi: 10.3934/dcds.2010.26.847 [20] Paul Loya and Jinsung Park. On gluing formulas for the spectral invariants of Dirac type operators. Electronic Research Announcements, 2005, 11: 1-11.

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