14,311 research outputs found
Locally Adaptive Frames in the Roto-Translation Group and their Applications in Medical Imaging
Locally adaptive differential frames (gauge frames) are a well-known
effective tool in image analysis, used in differential invariants and
PDE-flows. However, at complex structures such as crossings or junctions, these
frames are not well-defined. Therefore, we generalize the notion of gauge
frames on images to gauge frames on data representations defined on the extended space of positions and
orientations, which we relate to data on the roto-translation group ,
. This allows to define multiple frames per position, one per
orientation. We compute these frames via exponential curve fits in the extended
data representations in . These curve fits minimize first or second
order variational problems which are solved by spectral decomposition of,
respectively, a structure tensor or Hessian of data on . We include
these gauge frames in differential invariants and crossing preserving PDE-flows
acting on extended data representation and we show their advantage compared
to the standard left-invariant frame on . Applications include
crossing-preserving filtering and improved segmentations of the vascular tree
in retinal images, and new 3D extensions of coherence-enhancing diffusion via
invertible orientation scores
New Exact and Numerical Solutions of the (Convection-)Diffusion Kernels on SE(3)
We consider hypo-elliptic diffusion and convection-diffusion on , the quotient of the Lie group of rigid body motions SE(3) in
which group elements are equivalent if they are equal up to a rotation around
the reference axis. We show that we can derive expressions for the convolution
kernels in terms of eigenfunctions of the PDE, by extending the approach for
the SE(2) case. This goes via application of the Fourier transform of the PDE
in the spatial variables, yielding a second order differential operator. We
show that the eigenfunctions of this operator can be expressed as (generalized)
spheroidal wave functions. The same exact formulas are derived via the Fourier
transform on SE(3). We solve both the evolution itself, as well as the
time-integrated process that corresponds to the resolvent operator.
Furthermore, we have extended a standard numerical procedure from SE(2) to
SE(3) for the computation of the solution kernels that is directly related to
the exact solutions. Finally, we provide a novel analytic approximation of the
kernels that we briefly compare to the exact kernels.Comment: Revised and restructure
Quotients of
We consider closed topological 4-manifolds with universal cover
and Euler characteristic . All such manifolds
with are homotopy equivalent. In this case,
we show that there are four homeomorphism types, and propose a candidate for a
smooth example which is not homeomorphic to the geometric quotient. If
, we show that there are three
homotopy types (and between 6 and 24 homeomorphism types).Comment: 18 page
Abelian Fourfolds of Weil type and certain K3 Double Planes
Double planes branched in 6 lines give a famous example of K3 surfaces. Their
moduli are well understood and related to abelian fourfolds of Weil type. We
compare these two moduli interpretations and in particular divisors on the
moduli spaces. On the K3 side, this is achieved with the help of elliptic
fibrations. We also study the Kuga-Satake correspondence on these special
divisors.Comment: 48 pages, 4 figures; v3: final version with several additions
suggested by the refere
Non-simply-laced Lie algebras via F theory strings
In order to describe the appearance in F theory of the non--simply--laced Lie
algebras, we use the representation of symmetry enhancements by means of string
junctions. After an introduction to the techniques used to describe symmetry
enhancement, that is algebraic geometry, BPS states analysis and string
junctions, we concentrate on the latter. We give an explicit description of the
folding of D_{2n} to B_n of the folding of E_6 to F_4 and that of D_4 to G_2 in
terms of junctions and Jordan strings. We also discuss the case of C_n, but we
are unable in this case to provide a string interpretation.Comment: 24 pages, 3 figure
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