Tomáš Oberhuber

Michal Benes, Karol Mikula, Tomas Oberhuber et al.

The main goal of this paper is to present results of comparison study for the level set and direct Lagrangian methods for computing evolution of the Willmore flow of embedded planar curves. To perform such a study we construct new numerical approximation schemes for both Lagrangian as well as level set methods based on semi-implicit in time and finite/complementary volume in space discretizations. The Lagrangian scheme is stabilized in tangential direction by the asymptotically uniform grid point redistribution. Both methods are experimentally second order accurate. Moreover, we show precise coincidence of both approaches in case of various elastic curve evolutions provided that solving the linear systems in semi-implicit level set method is done in a precise way, redistancing is performed occasionally and the influence of boundary conditions on the level set function is eliminated.

Tomas Oberhuber

The Willmore flow is well known problem from the differential geometry. It minimizes the Willmore functional defined as integral of the mean-curvature square over given manifold. For the graph formulation, we derive modification of the Willmore flow with anisotropic mean curvature. We define the weak solution and we prove an energy equality. We approximate the solution numerically by the complementary finite volume method. To show the stability, we re-formulate the resulting scheme in terms of the finite difference method. By using simple framework of FDM we show discrete version of the energy equality. The time discretization is done by the method of lines and the resulting system of ODEs is solved by the Runge-Kutta-Merson solver with adaptive integration step. We also show experimental order of convergence as well as results of the numerical experiments, both for several different anisotropies.

Vladimír Klement, Tomáš Oberhuber, Daniel Ševčovič

We propose and analyze a constrained level-set method for semi-automatic image segmentation. Our level-set model with constraints on the level-set function enables us to specify which parts of the image lie inside respectively outside the segmented objects. Such a-priori information can be expressed in terms of upper and lower constraints prescribed for the level-set function. Constraints have the same conceptual meaning as initial seeds of the popular graph-cuts based methods for image segmentation. A numerical approximation scheme is based on the complementary-finite volumes method combined with the Projected successive over-relaxation method adopted for solving constrained linear complementarity problems. The advantage of the constrained level-set method is demonstrated on several artificial images as well as on cardiac MRI data.

Tomas Pevny, Vasek Smidl, Martin Trapp et al.

In this work, we propose Sum-Product-Transform Networks (SPTN), an extension of sum-product networks that uses invertible transformations as additional internal nodes. The type and placement of transformations determine properties of the resulting SPTN with many interesting special cases. Importantly, SPTN with Gaussian leaves and affine transformations pose the same inference task tractable that can be computed efficiently in SPNs. We propose to store affine transformations in their SVD decompositions using an efficient parametrization of unitary matrices by a set of Givens rotations. Last but not least, we demonstrate that G-SPTNs achieve state-of-the-art results on the density estimation task and are competitive with state-of-the-art methods for anomaly detection.

Vladimír Klement, Tomáš Oberhuber, Daniel Ševčovič

We propose and analyze a constrained level-set method for semi-automatic image segmentation. Our level-set model with constraints on the level-set function enables us to specify which parts of the image lie inside respectively outside the segmented objects. Such a-priori information can be expressed in terms of upper and lower constraints prescribed for the level-set function. Constraints have the same conceptual meaning as initial seeds of the popular graph-cuts based methods for image segmentation. A numerical approximation scheme is based on the complementary-finite volumes method combined with the Projected successive over-relaxation method adopted for solving constrained linear complementarity problems. The advantage of the constrained level-set method is demonstrated on several artificial images as well as on cardiac MRI data.