1,721,036 research outputs found
Data approximation using shape preserving parametric surfaces
No full textIn this paper we present a new, composite, method for the construction of shapepreserving
surfaces approximating a set of spatial data, obtained by combining together: a scheme
for detecting the shape of the data, a new class of tensor-product splines, and a suitable linear least-squares strategy. This method is mainly conceived for the reconstruction of objects in reverse engineering
Shape-preserving approximation by space curves
No full textThis paper describes a new method for the construction of C-2 shape-preserving curves which approximate an ordered set of data in R-3. The curves are obtained using the variable degree polynomial spline spaces recently described in [5]
Data approximation using shape-preserving parametric surfaces
No full textIn this paper we present a new, composite, method for the
construction of shape-preserving surfaces approximating a set of
spatial data, obtained by combining together: a scheme for
detecting the shape of the data, a new class of tensor-product
splines, and a suitable linear least-squares strategy. This method
is mainly conceived for the reconstruction of objects in reverse
engineering
Shape-preserving data approximation using new spline spaces
No full textAfter a brief description of a new class of splines
generated by five dimensional polynomial spaces which, for limit
values of the shape parameters, tend to quadratic splines,
we discuss their application in the construction of curves
approximating spatial data and subject to shape constraints
Construction of G2 planar Hermite interpolants with prescribed arc lengths
Full text linkIn this paper we address the problem of constructing G^2 planar Pythagorean–hodograph (PH) spline curves, that interpolate points, tangent directions and curvatures, and have prescribed arc-length. The interpolation scheme is completely local. Each spline segment is defined as a PH biarc curve of degree 7, which results in having a closed form solution of the G^2 interpolation equations depending on four free parameters. By fixing two of them to zero, it is proven that the length constraint can be satisfied for any data and any chosen ratio between the two boundary tangents. Length interpolation equation reduces to one algebraic equation with four solutions in general. To select the best one, the value of the bending energy is observed. Several numerical examples are provided to illustrate the obtained theoretical results and to numerically confirm that the approximation order is 5
Fractal properties of 4-point interpolatory subdivision schemes and wavelet scattering transform for signal classification
Full text linkWavelet scattering is a recent time-frequency transform that shares the convolutional architecture with convolutional neural networks, but it allows for a faster training and it often requires smaller training sets. It consists of a multistage non-linear transform that allows us to compute the deep spectrum of a signal by cascading convolution, non-linear operator and pooling at each stage, resulting a powerful tool for signal classification when embedded in machine learning architectures. One of the most delicate parameters in convolutional architectures is the temporal sampling that strongly affects the computational load as well as the classification rate. In this paper the role of sampling in the wavelet scattering transform is studied for signal classification purposes. In particular, the role of subdivision schemes in properly compensating the information lost when using sampling at each stage of the transform is investigated. Preliminary experimental results show that, starting from coarse grids, interpolatory subdivision schemes reproduce copies of the original scattering coefficients at a fixed full grid that still represent distinctive features for signal classes. In fact, thanks to the ability of the scheme in reproducing similar fractal properties of the transform through an efficient iterative refinement procedure, the reproduced coefficients enable to obtain classification rates similar to those provided by the native wavelet scattering transform. The relationships between the tension parameter of the scheme and the fractal dimension of its limit curve are also investigated
Spline surfaces with C1 quintic PH isoparametric curves
No full textGiven two spatial PH spline curves, aim of this paper is to study the construction of a tensor–product spline surface which has the two curves as assigned boundaries and which in addition incorporates a single family of isoparametric PH spline curves. Such a construction is carried over in two steps. In the first step a bi–patch is determined in a ‘Coons–like’ way having as boundaries two quintic PH curves forming a single section of given spline curves, and two polynomial quartic curves. In the second step the bi–patches are put together to form a globally continuous surface. In order to determine the final shape of the resulting surface, some free parameters are set by minimizing suitable shape functionals. The method can be extended to general boundary curves by preliminary approximating them with quintic PH splines
Isogeometric analysis in advection-diffusion problems: Tension splines approximation
No full textWe present a novel approach, within the new paradigm of isogeometric analysis introduced by Hughes et al. (2005) [6], to deal with advection dominated advection-diffusion problems. The key ingredient is the use of Galerkin approximating spaces of functions with high smoothness, as in IgA based on classical B-splines, but particularly well suited to describe sharp layers involving very strong gradient
Integrating subdivision schemes into SVM for improved signal classification
Full text linkThe integration of advanced signal processing techniques into machine learning models has gained increasing attention due to its potential to improve model performance, particularly for classification tasks. Support Vector Machine (SVM) is widely recognized as a powerful tool for signal classification due to its robust mathematical foundation and effectiveness in handling high-dimensional data. Subdivision schemes, originally developed in computer graphics for geometric modeling, offer a novel and parametric approach to feature preprocessing by iteratively refining input data through an efficient computational procedure. This paper studies the impact of subdivision schemes on SVM performance in terms of class separability and provides insights into the relationship between feature transformation and SVM response. Specifically, it investigates the theoretical and empirical implications of applying subdivision schemes to input features in SVM-based classification. The conditions under which these schemes preserve or enhance class separability are analyzed, focusing on the tension parameter which governs both the smoothness properties of the limit curve and the subdivision rule at each iteration. An estimation method for the tension parameter from the training data is also provided. Experimental results, performed in the context of signal classification based on the wavelet scattering transform, demonstrate that the appropriate selection of the tension parameter of the scheme can significantly enhance class separability, highlighting that subdivision schemes are a promising tool for improving classification accuracy in machine learning workflows
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