Seminar Details
2019-02-19 (14:00) : Minkowski Set Operations and Morphological Openings of Ellipsoids - Application for Processing Positive Definite Matrix-Valued Images
At Euler building (room A.002)
Organized by Mathematical Engineering
Speaker :
Jesús Angulo (CMM-Centre de Morphologie Mathématique, MINES ParisTech, PSL-Research University, France)
Abstract :
Morphological dilation and erosion for sets are based on (geometric) Minkowski sum and difference of two sets. In this talk we deal with the particular case of ellipsoids. Indeed ellipsoidal sets appear nowadays in different image and data representations, e.g., in diffusion tensor imaging each voxel is valued with a 3D ellipsoid; the dispersion of a scatter set of points can be described by a multivariate Gaussian distribution where the covariance matrix may be seen as ellipsoidal shape centered at the mean position.
The Minkowski sum and difference of two ellipsoidal sets are in general not ellipsoidal. However, in many applications, we are interested in computing the ellipsoidal set which approximates in a certain sense the Minkowski operations.
A closed-form characterization of Minkowski operations of two ellipsoids has been recently introduced (Yan and Chirikjian, 2015), which provide implicit surface expression and parametric formulas for the boundary of the Minkowski sum and difference of two arbitrary oriented ellipsoids. We adopt in this study a different approach based on the so-called ellipsoidal calculus (Kurzhanski and Valyi, 1996). Ellipsoidal calculus is a method for solving problems in control and estimation theory having unknown but bounded errors in terms of sets of approximating ellipsoidal-value functions.
Using the theory and algorithms from ellipsoidal calculus, we consider parameterized families of external and internal ellipsoids that tightly approximate the Minkowski sum and difference of ellipsoids. Approximations are tight along a direction l in the sense that the support functions on l of the ellipsoids are equal to the support function on l of the sum and difference. External (resp. internal) support function-based approximation can be then selected according to minimal (resp. maximal) measures of volume or trace of the corresponding ellipsoid. We focus in particular on the ellipsoidal approximations to the morphological opening between two ellipsoids and its interest in practical examples.
We discuss also the connection between the approximations to the Minkowski sum and difference of two positive definite matrices and their mean using their Euclidean or Riemannian geometries, which is also related to their Bures-Wasserstein mean (Bhatia et al., 2018).
References
Bhatia R., Jain T., Lim Y. (2018) On the Bures-Wasserstein Distance Between Positive Definite Matrices. Expositiones Mathematicae, to appear.
Kurzhanski A.B, Valyi I (1996). Ellipsoidal Calculus for Estimation and Control. Birkhäuser, Boston, MA, 1996.
Mazure M.-L (1991) Equations de convolution et formes quadratiques. Annali di Maematica pura et applicata (IV), Vol. CLVIII, 75-97.
Yan Y, Chirikjian G.S (2015) Closed characterization of the Minkowski sum and difference of two ellipsoids. Geom. Dedicata 177:103-128.