937 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
Second-order Shape Optimization for Geometric Inverse Problems in Vision
We develop a method for optimization in shape spaces, i.e., sets of surfaces
modulo re-parametrization. Unlike previously proposed gradient flows, we
achieve superlinear convergence rates through a subtle approximation of the
shape Hessian, which is generally hard to compute and suffers from a series of
degeneracies. Our analysis highlights the role of mean curvature motion in
comparison with first-order schemes: instead of surface area, our approach
penalizes deformation, either by its Dirichlet energy or total variation.
Latter regularizer sparks the development of an alternating direction method of
multipliers on triangular meshes. Therein, a conjugate-gradients solver enables
us to bypass formation of the Gaussian normal equations appearing in the course
of the overall optimization. We combine all of the aforementioned ideas in a
versatile geometric variation-regularized Levenberg-Marquardt-type method
applicable to a variety of shape functionals, depending on intrinsic properties
of the surface such as normal field and curvature as well as its embedding into
space. Promising experimental results are reported
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