98,711 research outputs found
Weighted Mean Curvature
In image processing tasks, spatial priors are essential for robust
computations, regularization, algorithmic design and Bayesian inference. In
this paper, we introduce weighted mean curvature (WMC) as a novel image prior
and present an efficient computation scheme for its discretization in practical
image processing applications. We first demonstrate the favorable properties of
WMC, such as sampling invariance, scale invariance, and contrast invariance
with Gaussian noise model; and we show the relation of WMC to area
regularization. We further propose an efficient computation scheme for
discretized WMC, which is demonstrated herein to process over 33.2
giga-pixels/second on GPU. This scheme yields itself to a convolutional neural
network representation. Finally, WMC is evaluated on synthetic and real images,
showing its superiority quantitatively to total-variation and mean curvature.Comment: 12 page
Mean curvature flow in a Ricci flow background
Following work of Ecker, we consider a weighted Gibbons-Hawking-York
functional on a Riemannian manifold-with-boundary. We compute its variational
properties and its time derivative under Perelman's modified Ricci flow. The
answer has a boundary term which involves an extension of Hamilton's Harnack
expression for the mean curvature flow in Euclidean space. We also derive the
evolution equations for the second fundamental form and the mean curvature,
under a mean curvature flow in a Ricci flow background. In the case of a
gradient Ricci soliton background, we discuss mean curvature solitons and
Huisken monotonicity.Comment: final versio
Remarks on the boundary curve of a constant mean curvature topological disc
We discuss some consequences of the existence of the holomorphic quadratic
Hopf differential on a conformally immersed constant mean curvature topological
disc with analytic boundary. In particular, we derive a formula for the mean
curvature as a weighted average of the normal curvature of the boundary curve,
and a condition for the surface to be totally umbilic in terms of the normal
curvature.Comment: 6 pages, 1 figure. Version 2: comments and references adde
The weighted Gaussian curvature derivative of a space-filling diagram
The morphometric approach [11, 14] writes the solvation free energy as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted Gaussian curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [4], and the weighted mean curvature in [1], this yields the derivative of the morphometric expression of solvation free energy
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