SCIENTIA SINICA Mathematica, Volume 46 , Issue 5 : 747-768(2016) https://doi.org/10.1360/N012015-00339

Compensated convex transforms and geometric singularity\\ extraction from semiconvex functions

More info
  • ReceivedNov 4, 2015
  • AcceptedMar 24, 2016
  • PublishedApr 25, 2016



[1] Zhang K. Compensated convexity and its applications. Ann Inst H Poincaré Anal Non Linéaire, 2008, 25: 743-771. Google Scholar

[2] ZhangK. Convex analysis based smooth approximations of maximum functions and squared-distance functions. J Nonlinear Convex Anal, 2008, 9: 379-406. Google Scholar

[3] ZhangK, Orlando A, Crooks E C M. Compensated convexity and Hausdorff stable geometric singularity extraction. Math Models Methods Appl Sci, 2015, 25: 747-801. Google Scholar

[4] ZhangK, Orlando A, Crooks E C M. Compensated convexity and Hausdorff stable extraction of intersections for smooth manifolds. Math Models Methods Appl Sci, 2015, 25: 839-873. Google Scholar

[5] Zhang K, Crooks E C M, Orlando A. Compensated convexity, multiscale medial axis maps and sharp regularity of the squared distance function. SIAM J Math Anal, 2015, 47: 4289-4331. Google Scholar

[6] Zhang K, Orlando A, Crooks E C M. Image processing. UK Patent, GB2488294, 2015. Google Scholar

[7] Albano P. Some properties of semiconcave functions with general modulus. J Math Anal Appl, 2002, 271: 217-231. Google Scholar

[8] AlbertiG, Ambrosio L, Cannarsa P. On the singularities of convex functions. Manuscripta Math, 1992, 76: 421-435. Google Scholar

[9] AlbanoP, Cannarsa P. Structural properties of singularities of semiconcave functions. Ann Sc Norm Super Pisa Cl Sci (5), 1999, 28: 719-740. Google Scholar

[10] CannarsaP, Sinestrari C. Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control. Boston: Birkhäuser, 2004. Google Scholar

[11] HartmanP. On functions representable as a difference of convex functions. Pacific J Math, 1959, 9: 707-713. Google Scholar

[12] Hiriart-UrrutyJ B. Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. In: Convexity and Duality in Optimization (Groningen, 1984). Lecture Notes in Economics and Mathematical Systems, vol. 256. Berlin: Springer, 1985, 37-70. Google Scholar

[13] BlumH. A transformation for extracting new descriptors of shape. In: Proceedings of the Symposium on Models for the Perception of Speech and Visual Form. Cambridge: MIT Press, 1967, 362-380. Google Scholar

[14] SiddiqiK, Pizer S M. Medial Representations. New York: Springer, 2008. Google Scholar

[15] OkabeA, Boots B, Sugihara K, et al. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. 2nd ed. New York: Wiley, 2000. Google Scholar

[16] Hiriart-Urruty J B, Lemaréchal C. Fundamentals of Convex Analysis. New York: Springer, 2001. Google Scholar

[17] RockafellarR T. Convex Analysis. Princeton: Princeton University Press, 1966. Google Scholar

[18] Attouch H, Aze D. Approximations and regularizations of arbitrary functions in Hilbert spaces by the Lasry-Lions methods. Ann Inst H Poincaré Anal Non Linéaire, 1993, 10: 289-312. Google Scholar

[19] LasryJ M, Lions P L. A remark on regularization in Hilbert spaces. Israel Math J, 1986, 55: 257-266. Google Scholar

[20] MoreauJ J. Proximaté dualité dans un espace Hilbertien. Bull Soc Math France, 1965, 93: 273-299. Google Scholar

[21] MoreauJ J. Fonctionnelles convexes. Http://eudml.org/doc/112529, 1966. Google Scholar

[22] Jackway P T. Morphological scale-space. In: Proceedings of the 11th IAPR International Conference on Pattern Recognition. Los Alamitos: IEEE Computer Society Press, 1992, 252-255. Google Scholar

[23] SerraJ. Image Analysis and Mathematical Morphology, vol. 1. London: Academic Press, 1982. Google Scholar

[24] EvansL C, Gariepy R F. Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. Boca Raton: CRC Press, 1992. Google Scholar

[25] SylvesterJ J. A question in the geometry of situation. Quart J Pure Appl Math, 1857, 1: 79-79. Google Scholar

[26] SylvesterJ J. On Poncelet's approximate valuation of surd forms. Philos Mag, 1860, 20: 203-222. Google Scholar

[27] Jung H W E. Über die kleinste Kugel, die eine räumliche Figur einschliesst. J Reine Angew Math, 1901, 123: 241-257. Google Scholar

[28] BlumenthalL M, Wahlin G E. On the spherical surface of smallest radius enclosing a bounded subset of $n$-dimensional euclidean space. Bull Amer Math Soc (NS), 1941, 47: 771-777. Google Scholar

[29] DanzerL, Grünbaum B, Klee V. Helly's theorem and its relatives. In: Proceedings of Symposia in Pure Mathematics, vol. VII. Providence: Amer Math Soc, 1963, 101-180. Google Scholar

[30] VerblunskyS. On the circumradius of a bounded set. J Lond Math Soc (2), 1952, 27: 505-507. Google Scholar

[31] HellyE. Über Mengen konvexer Körper mit gemeinschaftichen Punkten. Jahresber Deutsch Math Verein, 1923, 32: 175-176. Google Scholar

[32] FischerK, Gärtner B. The smallest enclosing ball of balls: Combinatorial structure and algorithms. Internat J Comput Geom Appl, 2004, 14: 341-378. Google Scholar

[33] KirchheimB, Kristensen J. Differentiability of convex envelopes. C R Acad Sci Paris Sér I Math, 2001, 333: 725-728. Google Scholar

[34] EvansL C. Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. Providence:Amer Math Soc, 2010. Google Scholar

[35] BallJ M, Kirchheim B, Kristensen J. Regularity of quasiconvex envelopes. Calc Var Partial Differential Equations, 2000, 11: 333-359. Google Scholar


Contact and support