410 research outputs found
Incremental and Transitive Discrete Rotations
A discrete rotation algorithm can be apprehended as a parametric application
from \ZZ[i] to \ZZ[i], whose resulting permutation ``looks
like'' the map induced by an Euclidean rotation. For this kind of algorithm, to
be incremental means to compute successively all the intermediate rotate d
copies of an image for angles in-between 0 and a destination angle. The di
scretized rotation consists in the composition of an Euclidean rotation with a
discretization; the aim of this article is to describe an algorithm whic h
computes incrementally a discretized rotation. The suggested method uses o nly
integer arithmetic and does not compute any sine nor any cosine. More pr
ecisely, its design relies on the analysis of the discretized rotation as a
step function: the precise description of the discontinuities turns to be th e
key ingredient that will make the resulting procedure optimally fast and e
xact. A complete description of the incremental rotation process is provided,
also this result may be useful in the specification of a consistent set of
defin itions for discrete geometry
Bijective rigid motions of the 2D Cartesian grid
International audienceRigid motions are fundamental operations in image processing. While they are bijective and isometric in R^2, they lose these properties when digitized in Z^2. To investigate these defects, we first extend a combinatorial model of the local behavior of rigid motions on Z^2, initially proposed by Nouvel and Rémila for rotations on Z^2. This allows us to study bijective rigid motions on Z^2, and to propose two algorithms for verifying whether a given rigid motion restricted to a given finite subset of Z^2 is bijective
Criteria of measure-preserving for -adic dynamical systems in terms of the van der Put basis
This paper is devoted to (discrete) -adic dynamical systems, an important
domain of algebraic and arithmetic dynamics. We consider the following open
problem from theory of -adic dynamical systems. Given continuous function
Let us represent it via special convergent series, namely van
der Put series. How can one specify whether this function is measure-preserving
or not for an arbitrary ? In this paper, for any prime we present a
complete description of all compatible measure-preserving functions in the
additive form representation. In addition we prove the criterion in terms of
coefficients with respect to the van der Put basis determining whether a
compatible function preserves the Haar measure
Shift Radix Systems - A Survey
Let be an integer and . The {\em shift radix system} is defined by has the {\em finiteness
property} if each is eventually mapped to
under iterations of . In the present survey we summarize
results on these nearly linear mappings. We discuss how these mappings are
related to well-known numeration systems, to rotations with round-offs, and to
a conjecture on periodic expansions w.r.t.\ Salem numbers. Moreover, we review
the behavior of the orbits of points under iterations of with
special emphasis on ultimately periodic orbits and on the finiteness property.
We also describe a geometric theory related to shift radix systems.Comment: 45 pages, 16 figure
Ergodicity criteria for non-expanding transformations of 2-adic spheres
In the paper, we obtain necessary and sufficient conditions for ergodicity
(with respect to the normalized Haar measure) of discrete dynamical systems
on 2-adic spheres of radius
, , centered at some point from the ultrametric space of
2-adic integers . The map is
assumed to be non-expanding and measure-preserving; that is, satisfies a
Lipschitz condition with a constant 1 with respect to the 2-adic metric, and
preserves a natural probability measure on , the Haar measure
on which is normalized so that
Rotation on the digital plane
Let Aϕ denote the matrix of rotation with angle ϕ of the Euclidean plane, FLOOR the function, which rounds a real point to the nearest lattice point down on the left and ROUND the function for rounding off a vector to the nearest node of the lattice. We prove under the natural assumption ϕ 6= k π 2 that the functions F LOOR ◦ Aϕ and ROUND ◦ Aϕ are neither surjective nor injective. More precisely we prove lower and upper estimates for the size of the sets of lattice points, which are the image of two lattice points as well as of lattice points, which have no preimages. It turns out that the density of that sets are positive except when sin ϕ 6= ± cos ϕ + r, r ∈ Q
Lie Group Algebra Convolutional Filters
In this paper we propose a framework to leverage Lie group symmetries on
arbitrary spaces exploiting \textit{algebraic signal processing} (ASP). We show
that traditional group convolutions are one particular instantiation of a more
general Lie group algebra homomorphism associated to an algebraic signal model
rooted in the Lie group algebra for given Lie group . Exploiting
this fact, we decouple the discretization of the Lie group convolution
elucidating two separate sampling instances: the filter and the signal. To
discretize the filters, we exploit the exponential map that links a Lie group
with its associated Lie algebra. We show that the discrete Lie group filter
learned from the data determines a unique filter in , and we show how
this uniqueness of representation is defined by the bandwidth of the filter
given a spectral representation. We also derive error bounds for the
approximations of the filters in with respect to its learned
discrete representations. The proposed framework allows the processing of
signals on spaces of arbitrary dimension and where the actions of some elements
of the group are not necessarily well defined. Finally, we show that multigraph
convolutional signal models come as the natural discrete realization of Lie
group signal processing models, and we use this connection to establish
stability results for Lie group algebra filters. To evaluate numerically our
results, we build neural networks with these filters and we apply them in
multiple datasets, including a knot classification problem
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