# zbMATH — the first resource for mathematics

Recovering an homogeneous polynomial from moments of its level set. (English) Zbl 1311.44009
Summary: Let $$K:=\{ x:g(x)\leq 1\}$$ be the compact (and not necessarily convex) sub-level set of some homogeneous polynomial $$g$$. Assume that the only knowledge about $$K$$ is the degree of $$g$$ as well as the moments of the Lebesgue measure on $$K$$ up to order $$2d$$. Then the vector of coefficients of is the solution of a simple linear system whose associated matrix is nonsingular. In other words, the moments up to order $$2d$$ of the Lebesgue measure on $$K$$ encode all information on the homogeneous polynomial $$g$$ that defines $$K$$ (in fact, only moments of order $$d$$ and $$2d$$ are needed).

##### MSC:
 44A60 Moment problems 26C10 Real polynomials: location of zeros 52A22 Random convex sets and integral geometry (aspects of convex geometry) 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
Full Text:
##### References:
 [1] Cuyt, A., Golub, G., Milanfar, P., Verdonk, B.: Multidimensional integral inversion, with applications in shape reconstruction. SIAM J. Sci. Comput. 27(3), 1058-1070 (2005) · Zbl 1099.65128 [2] Golub, G.H., Milanfar, P., Varah, J.: A stable numerical method for inverting shape from moments. SIAM J. Sci. Comput. 21(4), 1222-1243 (1999) · Zbl 0956.65030 [3] Gravin, N., Lasserre, J.B., Pasechnik, D.V., Robins, S.: The inverse moment problem for convex polytopes. Discrete Comput. Geom. 48(3), 596-621 (2012) · Zbl 1285.68198 [4] Morosov, A., Shakirov, S.: New and old results in resultant theory. Theor. Math. Phys. 163(2), 587 (2010). http://connection.ebscohost.com/c/articles/51280373/new-old-results-resultant-theory [5] Morosov, A., Shakirov, S.: Introduction to integral discriminants. J. High Energy Phys. 12, 002 (2009). http://iopscience.iop.org/1126-6708/2009/12/002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.