A speed preserving Hilbert gradient flow for generalized integral Menger curvature

We establish long-time existence for a projected Sobolev gradient flow of generalized integral Menger curvature in the Hilbert case and provide C 1 , 1 C^{1,1} -bounds in time for the solution that only depend on the initial curve. The self-avoidance property of integral Menger curvature guarantees that the knot class of the initial curve is preserved under the flow, and the projection ensures that each curve along the flow is parametrized with the same speed as the initial configuration. Finally, we describe how to simulate this flow numerically with substantially higher efficiency than in the corresponding numerical L 2 L^{2} gradient descent or other optimization methods.

Extraction of Cerebral Hemorrhage on CT Images Using Level Set Algorithm and Otsu Threshold

Extraction of cerebral hemorrhage on CT images has always been the focus of several research hotspots and is still challenging as it does not show clear boundary. In this paper, a novel segmentation framework is presented for extracting the cerebral hemorrhage in brain CT images with weak boundary. Firstly, we utilize the Otsu threshold algorithm to get the coarse outline approximate to the target boundary as the initial curve of level set algorithm. Then, the active contour model is employed using both edge information and global Gaussian distribution fitting energy of images to modify energy function of level set. The proposed approach is applied on real images which from Quzhou People’s Hospital. Compared to manual delineation, the proposed technique shows a higher JS value than the existing methods and requires less interaction which is listed in the literature.

Scoliosis and Prognosis—a systematic review regarding patient-specific and radiological predictive factors for curve progression

Introduction Idiopathic scoliosis, defined as a > 10° curvature of the spine in the frontal plane, is one of the most common spinal deformities. Age, initial curve magnitude and other parameters define whether a scoliotic deformity will progress or not. Still, their interactions and amounts of individual contribution are not fully elaborated and were the aim of this systematic review. Methods A systematic literature search was conducted in the common databases using MESH terms, searching for predictive factors of curve progression in adolescent idiopathic scoliosis (“adolescent idiopathic scoliosis” OR “ais” OR “idiopathic scoliosis”) AND (“predictive factors” OR “progression” OR “curve progression” OR “prediction” OR “prognosis”). The identified and analysed factors of each study were rated to design a top five scale of the most relevant factors. Results Twenty-eight investigations with 8255 patients were identified by literature search. Patient-specific risk factors for curve progression from initial curve were age (at diagnosis < 13 years), family history, bone mineral status (< 110 mg/cm3 in quantitative CT) and height velocity (7–8 cm/year, peak 11.6 ± 1.4 years). Relevant radiological criteria indicating curve progression included skeletal maturity, marked by Risser stages (Risser < 1) or Sanders Maturity Scale (SMS < 5), the initial extent of the Cobb angle (> 25° progression) and curve location (thoracic single or double curve). Discussion This systematic review summarised the current state of knowledge as the basis for creation of patient-specific algorithms regarding a risk calculation for a progressive scoliotic deformity. Curve magnitude is the most relevant predictive factor, followed by status of skeletal maturity and curve location.

High order curvature flows of plane curves with generalised Neumann boundary conditions

We consider the parabolic polyharmonic diffusion and the L 2 {L^{2}} -gradient flow for the square integral of the m-th arclength derivative of curvature for regular closed curves evolving with generalised Neumann boundary conditions. In the polyharmonic case, we prove that if the curvature of the initial curve is small in L 2 {L^{2}} , then the evolving curve converges exponentially in the C ∞ {C^{\infty}} topology to a straight horizontal line segment. The same behaviour is shown for the L 2 {L^{2}} -gradient flow provided the energy of the initial curve is sufficiently small. In each case the smallness conditions depend only on m.

A non-local expanding flow of convex closed curves in the plane

This paper presents a new non-local expanding flow for convex closed curves in the Euclidean plane which increases both the perimeter of the evolving curves and the enclosed area. But the flow expands the evolving curves to a finite circle smoothly if they do not develop singularity during the evolving process. In addition, it is shown that an additional assumption about the initial curve will ensure that the flow exists on the time interval [Formula: see text]. Meanwhile, a numerical experiment reveals that this flow may blow up for some initial convex curves.

On the Isoperimetric Inequality and Surface Diffusion Flow for Multiply Winding Curves

In this paper we establish a general form of the isoperimetric inequality for immersed closed curves (possibly non-convex) in the plane under rotational symmetry. As an application, we obtain a global existence result for the surface diffusion flow, providing that an initial curve is $$H^2$$ H 2 -close to a multiply covered circle and is sufficiently rotationally symmetric.

A Harm Reduction Approach to the Ethical Management of the COVID-19 Pandemic

The post-confinement phase of the COVID-19 pandemic will require that governments navigate more complex ethical questions than had occurred in the initial, ‘curve-flattening’ phase, and that will occur when the pandemic is in the past. By looking at the unavoidable harms involved in the confinement and quarantine methods employed during the initial phase of the pandemic, we can develop a harm reduction approach to managing the phase during which society will be gradually reopened in a context of managed risk. The principles that are at the heart of such an approach include a reckoning with all of the harms involved in policy choice, including harms that might be given rise to by policy implementation itself; a focus on the harms to which already vulnerable populations are susceptible; and a strong preference for policies that economize on the use of prohibitions and of coercive state enforcement, and that instead emphasize the agency of citizens in realizing health-promoting behavior change. This framework is applied to a policy proposal that has been discussed in policy circles in a number of countries, that of immunity ‘passports’, and to policies that emphasize the creative use of space and time to achieve physical distancing goals.

The Curve Shortening Flow in the Metric-Affine Plane

We investigated, for the first time, the curve shortening flow in the metric-affine plane and prove that under simple geometric condition (when the curvature of initial curve dominates the torsion term) it shrinks a closed convex curve to a “round point” in finite time. This generalizes the classical result by M. Gage and R.S. Hamilton about convex curves in a Euclidean plane.