The problem of reconstructing a piecewise constant function from a finite number of its Fourier coefficients perturbed by noise is considered. A reconstruction method, based on the computation of the Padè approximants to the Z-transform of the sequence of the noisy Fourier coefficients is proposed. The method is based on the remark that the distribution of the poles of the Padè approximants shows, asymptotically, clusters in the complex plane which allow the identification of the discontinuities of the function. It turns out that the Z-transform is a multiple-valued function and the location of the clusters corresponds to the branch points of such a function. By using this property of the Padè poles, a very effective reconstruction method can be developed. Some numerical experiments are presented to show the feasibility of the method.
Pad ?e approximants
signal processing
singular integral equations
Riemann sur- faces
We consider a geometric motion associated with the minimization
of a curvature dependent functional, which is related to the Willmore
functional. Such a functional arises in connection with the
image segmentation problem in computer vision theory.
We show by using formal asymptotics that the geometric motion
can be approximated by the evolution of the zero
level set of the solution of a nonlinear fourth-order
equation related to the Cahn-Hilliard and Allen-Cahn equations.
Geometric evolution equations; level sets; image segmentation; motion by curvature
An integro-differential equation of Prandtl's type and a collocation method as well as a collocationquadrature
method for its approximate solution is studied in weighted spaces of continuous functions.
AVHRR is an imager flying on polar meteorological satellites that, due to its spatial resolution (about 1 km at Nadir view),
produces a huge amount of data. A method based on the wavelet packet transform is devised to compress AVHRR images. The
method is driven by the compression error (application dependent), so that different types of images have the same quality. The
best basis wavelet packet is chosen by L1
norm criterion that was found to be the most suited for the problem at hand. Ability of
the compression method to preserve most fine structures of the images even at the highest resolution is demonstrated based on
some examples. Comparison with wavelet algorithms is performed.
Let $q>1$. Initiated by P. Erd\H os et al. in \cite{ErdJooKom1}, several authors
studied the numbers $l^m(q)=\inf \{y\ :\ y\in\Lambda_m,\ y\ne 0\}$,
$m=1,2,\dots$, where $\Lambda_m$ denotes the set of all finite sums of the form
$y=\eps_0 + \eps_1 q + \eps_2 q^2 + \dots + \eps_n q^n$
with integer coefficients $-m\le \eps_i \le m$. It is
known (\cite{Bug}, \cite{ErdJooKom1}, \cite{ErdKom}) that $q$ is a Pisot number
if and only if $l^m(q)>0$ for all $m$. The value of $l^1(q)$ was determined for
many particular Pisot numbers, but the general case remains widely open. In this
paper we determine the value of $l^m(q)$ in other cases.
