2,530 research outputs found
Ellipse-preserving Hermite interpolation and subdivision
We introduce a family of piecewise-exponential functions that have the
Hermite interpolation property. Our design is motivated by the search for an
effective scheme for the joint interpolation of points and associated tangents
on a curve with the ability to perfectly reproduce ellipses. We prove that the
proposed Hermite functions form a Riesz basis and that they reproduce
prescribed exponential polynomials. We present a method based on Green's
functions to unravel their multi-resolution and approximation-theoretic
properties. Finally, we derive the corresponding vector and scalar subdivision
schemes, which lend themselves to a fast implementation. The proposed vector
scheme is interpolatory and level-dependent, but its asymptotic behaviour is
the same as the classical cubic Hermite spline algorithm. The same convergence
properties---i.e., fourth order of approximation---are hence ensured
Neural-network-based curve fitting using totally positive rational bases
This paper proposes a method for learning the process of curve fitting through a general class of totally positive rational bases. The approximation is achieved by finding suitable weights and control points to fit the given set of data points using a neural network and a training algorithm, called AdaMax algorithm, which is a first-order gradient-based stochastic optimization. The neural network presented in this paper is novel and based on a recent generalization of rational curves which inherit geometric properties and algorithms of the traditional rational BĂ©zier curves. The neural network has been applied to different kinds of datasets and it has been compared with the traditional least-squares method to test its performance. The obtained results show that our method can generate a satisfactory approximation
On The Continuous Steering of the Scale of Tight Wavelet Frames
In analogy with steerable wavelets, we present a general construction of
adaptable tight wavelet frames, with an emphasis on scaling operations. In
particular, the derived wavelets can be "dilated" by a procedure comparable to
the operation of steering steerable wavelets. The fundamental aspects of the
construction are the same: an admissible collection of Fourier multipliers is
used to extend a tight wavelet frame, and the "scale" of the wavelets is
adapted by scaling the multipliers. As an application, the proposed wavelets
can be used to improve the frequency localization. Importantly, the localized
frequency bands specified by this construction can be scaled efficiently using
matrix multiplication
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