14,264 research outputs found
Unified Heat Kernel Regression for Diffusion, Kernel Smoothing and Wavelets on Manifolds and Its Application to Mandible Growth Modeling in CT Images
We present a novel kernel regression framework for smoothing scalar surface
data using the Laplace-Beltrami eigenfunctions. Starting with the heat kernel
constructed from the eigenfunctions, we formulate a new bivariate kernel
regression framework as a weighted eigenfunction expansion with the heat kernel
as the weights. The new kernel regression is mathematically equivalent to
isotropic heat diffusion, kernel smoothing and recently popular diffusion
wavelets. Unlike many previous partial differential equation based approaches
involving diffusion, our approach represents the solution of diffusion
analytically, reducing numerical inaccuracy and slow convergence. The numerical
implementation is validated on a unit sphere using spherical harmonics. As an
illustration, we have applied the method in characterizing the localized growth
pattern of mandible surfaces obtained in CT images from subjects between ages 0
and 20 years by regressing the length of displacement vectors with respect to
the template surface.Comment: Accepted in Medical Image Analysi
Coherence Filtering to Enhance the Mandibular Canal in Cone-Beam CT data
Segmenting the mandibular canal from cone beam CT data, is difficult due to low edge contrast and high image noise. We introduce 3D coherence filtering as a method to close the interrupted edges and denoise the structure of the mandibular canal. Coherence Filtering is an anisotropic non-linear tensor based diffusion algorithm for edge enhancing image filtering. We test different numerical schemes of the tensor diffusion equation, non-negative, standard discretization and also a rotation invariant scheme of Weickert [1]. Only the\ud
scheme of Weickert did not blur the high spherical images frequencies on the image diagonals of our test volume. Thus this scheme is chosen to enhance the small curved mandibular canal structure. The best choice of the diffusion equation parameters c1 and c2, depends on the image noise. Coherence filtering on the CBCT-scan works well, the noise in the mandibular canal is gone and the edges are connected. Because the algorithm is tensor based it cannot deal with edge joints or splits, thus is less fit for more complex image structures
Discrete spherical means of directional derivatives and Veronese maps
We describe and study geometric properties of discrete circular and spherical
means of directional derivatives of functions, as well as discrete
approximations of higher order differential operators. For an arbitrary
dimension we present a general construction for obtaining discrete spherical
means of directional derivatives. The construction is based on using the
Minkowski's existence theorem and Veronese maps. Approximating the directional
derivatives by appropriate finite differences allows one to obtain finite
difference operators with good rotation invariance properties. In particular,
we use discrete circular and spherical means to derive discrete approximations
of various linear and nonlinear first- and second-order differential operators,
including discrete Laplacians. A practical potential of our approach is
demonstrated by considering applications to nonlinear filtering of digital
images and surface curvature estimation
Planet migration in three-dimensional radiative discs
The migration of growing protoplanets depends on the thermodynamics of the
ambient disc. Standard modelling, using locally isothermal discs, indicate in
the low planet mass regime an inward (type-I) migration. Taking into account
non-isothermal effects, recent studies have shown that the direction of the
type-I migration can change from inward to outward. In this paper we extend
previous two-dimensional studies, and investigate the planet-disc interaction
in viscous, radiative discs using fully three-dimensional radiation
hydrodynamical simulations of protoplanetary accretion discs with embedded
planets, for a range of planetary masses.
We use an explicit three-dimensional (3D) hydrodynamical code NIRVANA that
includes full tensor viscosity. We have added implicit radiation transport in
the flux-limited diffusion approximation, and to speed up the simulations
significantly we have newly adapted and implemented the FARGO-algorithm in a 3D
context.
First, we present results of test simulations that demonstrate the accuracy
of the newly implemented FARGO-method in 3D. For a planet mass of 20 M_earth we
then show that the inclusion of radiative effects yields a torque reversal also
in full 3D. For the same opacity law used the effect is even stronger in 3D
than in the corresponding 2D simulations, due to a slightly thinner disc.
Finally, we demonstrate the extent of the torque reversal by calculating a
sequence of planet masses. Through full 3D simulations of embedded planets in
viscous, radiative discs we confirm that the migration can be directed outwards
up to planet masses of about 33 M_earth. Hence, the effect may help to resolve
the problem of too rapid inward migration of planets during their type-I phase.Comment: 16 pages, Astronomy&Astrophysics, in pres
A new code for orbit analysis and Schwarzschild modelling of triaxial stellar systems
We review the methods used to study the orbital structure and chaotic
properties of various galactic models and to construct self-consistent
equilibrium solutions by Schwarzschild's orbit superposition technique. These
methods are implemented in a new publicly available software tool, SMILE, which
is intended to be a convenient and interactive instrument for studying a
variety of 2D and 3D models, including arbitrary potentials represented by a
basis-set expansion, a spherical-harmonic expansion with coefficients being
smooth functions of radius (splines), or a set of fixed point masses. We also
propose two new variants of Schwarzschild modelling, in which the density of
each orbit is represented by the coefficients of the basis-set or spline
spherical-harmonic expansion, and the orbit weights are assigned in such a way
as to reproduce the coefficients of the underlying density model. We explore
the accuracy of these general-purpose potential expansions and show that they
may be efficiently used to approximate a wide range of analytic density models
and serve as smooth representations of discrete particle sets (e.g. snapshots
from an N-body simulation), for instance, for the purpose of orbit analysis of
the snapshot. For the variants of Schwarzschild modelling, we use two test
cases - a triaxial Dehnen model containing a central black hole, and a model
re-created from an N-body snapshot obtained by a cold collapse. These tests
demonstrate that all modelling approaches are capable of creating equilibrium
models.Comment: MNRAS, 24 pages, 18 figures. Software is available at
http://td.lpi.ru/~eugvas/smile
Hydrodynamics of embedded planets' first atmospheres - III. The role of radiation transport for super-Earth planets
The population of close-in super-Earths, with gas mass fractions of up to 10%
represents a challenge for planet formation theory: how did they avoid runaway
gas accretion and collapsing to hot Jupiters despite their core masses being in
the critical range of ? Previous
three-dimensional (3D) hydrodynamical simulations indicate that atmospheres of
low-mass planets cannot be considered isolated from the protoplanetary disc,
contrary to what is assumed in 1D-evolutionary calculations. This finding is
referred to as the recycling hypothesis. In this Paper we investigate the
recycling hypothesis for super-Earth planets, accounting for realistic 3D
radiation hydrodynamics. Also, we conduct a direct comparison in terms of the
evolution of the entropy between 1D and 3D geometries. We clearly see that 3D
atmospheres maintain higher entropy: although gas in the atmosphere loses
entropy through radiative cooling, the advection of high entropy gas from the
disc into the Bondi/Hill sphere slows down Kelvin-Helmholtz contraction,
potentially arresting envelope growth at a sub-critical gas mass fraction.
Recycling, therefore, operates vigorously, in line with results by previous
studies. However, we also identify an "inner core" -- in size 25% of
the Bondi radius -- where streamlines are more circular and entropies are much
lower than in the outer atmosphere. Future studies at higher resolutions are
needed to assess whether this region can become hydrodynamically-isolated on
long time-scales.Comment: 16 pages, 12 figures, accepted for publication at MNRA
3D time series analysis of cell shape using Laplacian approaches
Background:
Fundamental cellular processes such as cell movement, division or food uptake critically depend on cells being able to change shape. Fast acquisition of three-dimensional image time series has now become possible, but we lack efficient tools for analysing shape deformations in order to understand the real three-dimensional nature of shape changes.
Results:
We present a framework for 3D+time cell shape analysis. The main contribution is three-fold: First, we develop a fast, automatic random walker method for cell segmentation. Second, a novel topology fixing method is proposed to fix segmented binary volumes without spherical topology. Third, we show that algorithms used for each individual step of the analysis pipeline (cell segmentation, topology fixing, spherical parameterization, and shape representation) are closely related to the Laplacian operator. The framework is applied to the shape analysis of neutrophil cells.
Conclusions:
The method we propose for cell segmentation is faster than the traditional random walker method or the level set method, and performs better on 3D time-series of neutrophil cells, which are comparatively noisy as stacks have to be acquired fast enough to account for cell motion. Our method for topology fixing outperforms the tools provided by SPHARM-MAT and SPHARM-PDM in terms of their successful fixing rates. The different tasks in the presented pipeline for 3D+time shape analysis of cells can be solved using Laplacian approaches, opening the possibility of eventually combining individual steps in order to speed up computations
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