Supremum/Infimum and Nonlinear Averaging of Positive Definite Symmetric Matrices

Jesus Angulo (1)

1 CMM - Centre de Morphologie Mathématique

Mathematical morphology is a nonlinear image processing methodology based on the computation of supremum (dilation operator) and infimum (erosion operator) in local neighborhoods called structuring elements. This chapter deals with definition of supremum and infimum operators for positive definite symmetric (PDS) matrices, which are the basic ingredients for the extension mathematical morphology to PDS matrices-valued images. The problem is tackled under three different paradigms. Firstly, total orderings using lexicographic cascades of eigenvalues as well as kernelized distances to matrix references are studied. Secondly, by decoupling the shape and the orientation of the ellipsoid associated to each PDS matrix, the supremum and infimum can be obtained by using a marginal supremum/infimum for the eigenvalues and a geometric matrix mean for the orthogonal basis. Thirdly, an estimate of the supremum and infimum associated to the Löwner ellipsoids are computed as the asymptotic cases of nonlinear averaging using the original notion of counter-harmonic mean for PDS matrices. Properties of the three introduced approaches are explored in detail, including also some numerical examples.
Chapter contents:
- Total Orderings for Sup-Inf Input-Preserving Sets of PDS Matrices
- Partial Spectral Ordering for PDS Matrices and Inverse Eigenvalue Problem
- Asymptotic Nonlinear Averaging Using Counter-Harmonic
- Mean for PDS Matrices
- Application to Nonlinear Filtering of Matrix-Valued Images

Domaines: Traitement des images [eess.IV]

Jesus Angulo. Supremum/Infimum and Nonlinear Averaging of Positive Definite Symmetric Matrices. Frank Nielsen and Rajendra Bhatia. Matrix Information Geometry, Springer, pp.3-33, 2013, 978-3-642-30231-2 (Print) / 978-3-642-30232-9 (Online). ⟨10.1007/978-3-642-30232-9_1⟩. ⟨hal-00835158⟩