Search Results (4)

The linear Weingarten condition with ellipticity for the mean curvature and the extrinsic Gaussian curvature on a surface in the three-sphere can define a Riemannian metric which is called the elliptic linear Weingarten metric. We established some local characterizations of the round spheres and the tori immersed in the 3-dimensional unit sphere, along with the Laplace operator, the spherical Gauss map and the Gauss map associated with the elliptic linear Weingarten metric. Full article

Different arguments led to supposing that the deep origin of phase transitions has to be identified with suitable topological changes of potential related submanifolds of configuration space of a physical system. An important step forward for this approach was achieved with two theorems stating that, for a wide class of physical systems, phase transitions should necessarily stem from topological changes of energy level submanifolds of the phase space. However, the sufficiency conditions are still a wide open question. In this study, a first important step forward was performed in this direction; in fact, a differential equation was worked out which describes how entropy varies as a function of total energy, and this variation is driven by the total energy dependence of a topology-related quantity of the relevant submanifolds of the phase space. Hence, general conditions can be in principle defined for topology-driven loss of differentiability of the entropy. Full article

We characterize spheres and the tori, the product of the two plane circles immersed in the three-dimensional unit sphere, which are associated with the Laplace operator and the Gauss map defined by the elliptic linear Weingarten metric defined on closed surfaces in the three-dimensional sphere. Full article

Image segmentation is an important process that separates objects from the background and also from each other. Applied to cells, the results can be used for cell counting which is very important in medical diagnosis and treatment, and biological research that is often used by scientists and medical practitioners. Segmenting 3D confocal microscopy images containing cells of different shapes and sizes is still challenging as the nuclei are closely packed. The watershed transform provides an efficient tool in segmenting such nuclei provided a reasonable set of markers can be found in the image. In the presence of low-contrast variation or excessive noise in the given image, the watershed transform leads to over-segmentation (a single object is overly split into multiple objects). The traditional watershed uses the local minima of the input image and will characteristically find multiple minima in one object unless they are specified (marker-controlled watershed). An alternative to using the local minima is by a supervised technique called seeded watershed, which supplies single seeds to replace the minima for the objects. Consequently, the accuracy of a seeded watershed algorithm relies on the accuracy of the predefined seeds. In this paper, we present a segmentation approach based on the geometric morphological properties of the ‘landscape’ using curvatures. The curvatures are computed as the eigenvalues of the Shape matrix, producing accurate seeds that also inherit the original shape of their respective cells. We compare with some popular approaches and show the advantage of the proposed method. Full article