183 research outputs found
Fractal image compression: A resolution independent representation for imagery
A deterministic fractal is an image which has low information content and no inherent scale. Because of their low information content, deterministic fractals can be described with small data sets. They can be displayed at high resolution since they are not bound by an inherent scale. A remarkable consequence follows. Fractal images can be encoded at very high compression ratios. This fern, for example is encoded in less than 50 bytes and yet can be displayed at resolutions with increasing levels of detail appearing. The Fractal Transform was discovered in 1988 by Michael F. Barnsley. It is the basis for a new image compression scheme which was initially developed by myself and Michael Barnsley at Iterated Systems. The Fractal Transform effectively solves the problem of finding a fractal which approximates a digital 'real world image'
Хаусдорфова фрактальна апроксимація функцій
Встановлено достатнi умови збiжностi iтерацiй оператора фрактального перетворення в просторi функцiй з хаусдорфовою метрикою. Наведено оцiнку похибки хаусдорфового фрактального наближення.Sufficient conditions for the convergence of fractal transform operator iterations in the space of functions with the Hausdorff metric are stated. An estimate for the error of the Hausdorff fractal approximation is given
Non-perturbative calculations for the effective potential of the symmetric and non-Hermitian field theoretic model
We investigate the effective potential of the symmetric
field theory, perturbatively as well as non-perturbatively. For the
perturbative calculations, we first use normal ordering to obtain the first
order effective potential from which the predicted vacuum condensate vanishes
exponentially as in agreement with previous calculations. For the
higher orders, we employed the invariance of the bare parameters under the
change of the mass scale to fix the transformed form totally equivalent to
the original theory. The form so obtained up to is new and shows that all
the 1PI amplitudes are perurbative for both and regions. For
the intermediate region, we modified the fractal self-similar resummation
method to have a unique resummation formula for all values. This unique
formula is necessary because the effective potential is the generating
functional for all the 1PI amplitudes which can be obtained via and thus we can obtain an analytic calculation for the 1PI
amplitudes. Again, the resummed from of the effective potential is new and
interpolates the effective potential between the perturbative regions.
Moreover, the resummed effective potential agrees in spirit of previous
calculation concerning bound states.Comment: 20 page
Self-Similar Factor Approximants
The problem of reconstructing functions from their asymptotic expansions in
powers of a small variable is addressed by deriving a novel type of
approximants. The derivation is based on the self-similar approximation theory,
which presents the passage from one approximant to another as the motion
realized by a dynamical system with the property of group self-similarity. The
derived approximants, because of their form, are named the self-similar factor
approximants. These complement the obtained earlier self-similar exponential
approximants and self-similar root approximants. The specific feature of the
self-similar factor approximants is that their control functions, providing
convergence of the computational algorithm, are completely defined from the
accuracy-through-order conditions. These approximants contain the Pade
approximants as a particular case, and in some limit they can be reduced to the
self-similar exponential approximants previously introduced by two of us. It is
proved that the self-similar factor approximants are able to reproduce exactly
a wide class of functions which include a variety of transcendental functions.
For other functions, not pertaining to this exactly reproducible class, the
factor approximants provide very accurate approximations, whose accuracy
surpasses significantly that of the most accurate Pade approximants. This is
illustrated by a number of examples showing the generality and accuracy of the
factor approximants even when conventional techniques meet serious
difficulties.Comment: 22 pages + 11 ps figure
COLLAGE-BASED INVERSE PROBLEMS FOR IFSM WITH ENTROPY MAXIMIZATION AND SPARSITY CONSTRAINTS
We consider the inverse problem associated with IFSM: Given a target function f, find an IFSM, such that its invariant fixed point f is sufficiently close to f in the Lp distance. In this paper, we extend the collage-based method developed by Forte and Vrscay (1995) along two different directions. We first search for a set of mappings that not only minimizes the collage error but also maximizes the entropy of the dynamical system. We then include an extra term in the minimization process which takes into account the sparsity of the set of mappings. In this new formulation, the minimization of collage error is treated as multi-criteria problem: we consider three different and conflicting criteria i.e., collage error, entropy and sparsity. To solve this multi-criteria program we proceed by scalarization and we reduce the model to a single-criterion program by combining all objective functions with different trade-off weights. The results of some numerical computations are presented. Numerical studies indicate that a maximum entropy principle exists for this approximation problem, i.e., that the suboptimal solutions produced by collage coding can be improved at least slightly by adding a maximum entropy criterion
Fractals and Self-Similarity in Economics: the Case of a Stochastic Two-Sector Growth Model
We study a stochastic, discrete-time, two-sector optimal growth model in which the production of the homogeneous consumption good uses a Cobb-Douglas technology, combining physical capital and an endogenously determined share of human capital. Education is intensive in human capital as in Lucas (1988), but the marginal returns of the share of human capital employed in education are decreasing, as suggested by Rebelo (1991). Assuming that the exogenous shocks are i.i.d. and affect both physical and human capital, we build specific configurations for the primitives of the model so that the optimal dynamics for the state variables can be converted, through an appropriate log-transformation, into an Iterated Function System converging to an invariant distribution supported on a generalized Sierpinski gasket.fractals, iterated function system, self-similarity, Sierpinski gasket, stochastic growth
Data comparison schemes for Pattern Recognition in Digital Images using Fractals
Pattern recognition in digital images is a common problem with application in
remote sensing, electron microscopy, medical imaging, seismic imaging and
astrophysics for example. Although this subject has been researched for over
twenty years there is still no general solution which can be compared with the
human cognitive system in which a pattern can be recognised subject to
arbitrary orientation and scale.
The application of Artificial Neural Networks can in principle provide a very
general solution providing suitable training schemes are implemented.
However, this approach raises some major issues in practice. First, the CPU
time required to train an ANN for a grey level or colour image can be very
large especially if the object has a complex structure with no clear geometrical
features such as those that arise in remote sensing applications. Secondly,
both the core and file space memory required to represent large images and
their associated data tasks leads to a number of problems in which the use of
virtual memory is paramount.
The primary goal of this research has been to assess methods of image data
compression for pattern recognition using a range of different compression
methods. In particular, this research has resulted in the design and
implementation of a new algorithm for general pattern recognition based on
the use of fractal image compression.
This approach has for the first time allowed the pattern recognition problem to
be solved in a way that is invariant of rotation and scale. It allows both ANNs
and correlation to be used subject to appropriate pre-and post-processing
techniques for digital image processing on aspect for which a dedicated
programmer's work bench has been developed using X-Designer
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