On multi-degree splines

Beccari, Carolina Vittoria; Casciola, Giulio; Morigi, Serena

doi:10.1016/j.cagd.2017.10.003

Cited by 17 publications

(49 citation statements)

References 16 publications

(41 reference statements)

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“…We start with a pointwise recurrence. It is a direct extension of the integral relation presented in [2,42] for polynomial splines of non-uniform degree. For q = 0, .…”

Section: Integral Recurrence Relationsmentioning

confidence: 99%

“…1 2 R. R. HIEMSTRA, T. J. R. HUGHES, C. MANNI, H. SPELEERS, AND D. TOSHNIWAL spect to differentiation and integration makes them an appealing substitute for the rational NURBS model in the framework of both Galerkin and collocation isogeometric methods [1,23,25,26]. When the geometry is not an issue, Tchebycheffian splines can still provide an interesting problem-dependent alternative to classical polynomial B-splines/NURBS for solving differential problems: they allow for an efficient treatment of sharp gradients and thin layers [24,25] and are able to outperform classical polynomial B-splines in the spectral approximation of differential operators [25,26].The success of polynomial splines greatly relies on the famous B-spline basis which can also be defined in the multi-degree setting [2,33,38,39,41]. Most of the results known for polynomial splines extend in a natural way to Tchebycheffian splines.…”

mentioning

confidence: 99%

“…The success of polynomial splines greatly relies on the famous B-spline basis which can also be defined in the multi-degree setting [2,33,38,39,41]. Most of the results known for polynomial splines extend in a natural way to Tchebycheffian splines.…”

mentioning

confidence: 99%

“…The above contribution is complemented by additional results: we provide a knot insertion formula and a global integral recurrence relation for GTB-splines. While the former is in complete analogy with the one known for the multi-degree polynomial case [2,42], the latter is a new contribution also for the multi-degree polynomial case, where only local integral recurrence relations have been proposed so far in the literature [2,38]. The provided global integral recurrence relation completely mimics the one known for polynomial/Tchebycheffian splines of uniform degree/local dimension and is expressed in an elegant way by using an extension of the concept of weights.…”

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confidence: 99%

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A Tchebycheffian Extension of Multidegree B-Splines: Algorithmic Computation and Properties

Hiemstra¹,

Hughes²,

Manni³

et al. 2020

SIAM J. Numer. Anal.

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In this paper we present an efficient and robust approach to compute a normalized Bspline-like basis for spline spaces with pieces drawn from extended Tchebycheff spaces. The extended Tchebycheff spaces and their dimensions are allowed to change from interval to interval. The approach works by constructing a matrix that maps a generalized Bernstein-like basis to the B-spline-like basis of interest. The B-spline-like basis shares many characterizing properties with classical univariate B-splines and may easily be incorporated in existing spline codes. This may contribute to the full exploitation of Tchebycheffian splines in applications, freeing them from the restricted role of an elegant theoretical extension of polynomial splines. Numerical examples are provided that illustrate the procedure described. 1 2 R. R. HIEMSTRA, T. J. R. HUGHES, C. MANNI, H. SPELEERS, AND D. TOSHNIWAL spect to differentiation and integration makes them an appealing substitute for the rational NURBS model in the framework of both Galerkin and collocation isogeometric methods [1,23,25,26]. When the geometry is not an issue, Tchebycheffian splines can still provide an interesting problem-dependent alternative to classical polynomial B-splines/NURBS for solving differential problems: they allow for an efficient treatment of sharp gradients and thin layers [24,25] and are able to outperform classical polynomial B-splines in the spectral approximation of differential operators [25,26].The success of polynomial splines greatly relies on the famous B-spline basis which can also be defined in the multi-degree setting [2,33,38,39,41]. Most of the results known for polynomial splines extend in a natural way to Tchebycheffian splines. However, the possibility of representing the space in terms of a basis with similar properties to polynomial B-splines is not always guaranteed, even for pieces taken from ET-spaces of the same dimension. More precisely, there are two main categories of Tchebycheffian splines: the various pieces are drawn either from the same ET-space -see [33] for a proper meaning in case of different local dimensions -or from different ET-spaces. In the former case, Tchebycheffian splines always admit a representation in terms of basis functions with similar properties to polynomial Bsplines. The latter offers a much more general framework -sometimes referred to in the literature as piecewise Tchebycheffian splines [31] -and allows us to optimally benefit from the great diversity of ET-spaces, but the existence of a B-spline-like basis requires constraints on the various ET-spaces.Tchebycheffian splines can be easily incorporated in existing spline codes because the corresponding B-spline-like basis, whenever it exists, is compatible with classical B-splines as it enjoys the same structural properties. When all the ET-spaces have the same dimension, various approaches have been used in the Tchebycheffian setting to construct such a B-spline-like basis: generalized divided differences [32,34], Hermite interpolation [10,33], integral...

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“…We start with a pointwise recurrence. It is a direct extension of the integral relation presented in [2,42] for polynomial splines of non-uniform degree. For q = 0, .…”

Section: Integral Recurrence Relationsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

A Tchebycheffian Extension of Multidegree B-Splines: Algorithmic Computation and Properties

Hiemstra¹,

Hughes²,

Manni³

et al. 2020

SIAM J. Numer. Anal.

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show abstract

“…This further simplifies the construction, and might open the door to the practical use of multi-degree (B-)splines in a wide range of application fields, from CAGD and IGA to data fitting and compression. We remark that some computational aspects of multi-degree splines were also addressed in [Beccari et al 2017].The remainder of the paper is organized as follows. We start by defining notation for standard B-splines in Section 2.…”

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confidence: 99%

Algorithm 999

Speleers

2019

ACM Trans. Math. Softw.

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Multi-degree splines are smooth piecewise-polynomial functions where the pieces can have different degrees. We describe a simple algorithmic construction of a set of basis functions for the space of multi-degree splines, with similar properties to standard B-splines. These basis functions are called multi-degree B-splines (or MDB-splines). The construction relies on an extraction operator that represents all MDB-splines as linear combinations of local B-splines of different degrees. This enables the use of existing efficient algorithms for B-spline evaluations and refinements in the context of multi-degree splines. A Matlab implementation is provided to illustrate the computation and use of MDB-splines.An extension of the standard concept of (B-)splines is to consider polynomial pieces of different degrees glued together with certain smoothness. This idea of multi-degree (B-)splines dates back to Nürnberger et al. [1984] but has been rejuvenated several times under different forms (and names). A special case was investigated by Sederberg et al. [2003] for degrees (1, 2, 3) and (1, n) in the context of CAGD. Later, Shen and Wang [2010a,b] constructed multi-degree B-splines with general degrees by means of an integral recurrence relation. Li et al. [2012] observed the latter as being difficult to compute, and instead presented a geometric-oriented recursive procedure. Recently, multi-degree B-splines were revisited by Beccari et al. [2017] illustrating their capabilities in CAGD, but still based on an integral recurrence relation and so-called transition functions identified through Hermite interpolation. Even though the theoretical framework is very elegant, multi-degree B-splines are a burden to implement when dealing with recursive integration or when solving several Hermite interpolation problems. This might explain why they have not been popular in a practical context, despite their (theoretical) potential. Besides CAGD, multi-degree splines were also applied in IGA, in the frame of T-splines [Liu et al. 2016] and polar splines [Toshniwal et al. 2017]. Non-polynomial extensions can be found in [Buchwald and Mühlbach 2003;Nürnberger et al. 1984;Sommer and Strauss 1988] considering local spaces generated by ECT-systems and Descartes systems.The purpose of this paper is to show that the implementation of multi-degree B-splines (in short, MDB-splines) can be painless, and we provide a small Matlab toolbox as illustration. We closely follow the algorithmic construction proposed by Toshniwal et al. [2017] in the context of rational multi-degree polar splines, but reformulate the algorithms in a computationally more efficient way. This further simplifies the construction, and might open the door to the practical use of multi-degree (B-)splines in a wide range of application fields, from CAGD and IGA to data fitting and compression. We remark that some computational aspects of multi-degree splines were also addressed in [Beccari et al. 2017].The remainder of the paper is organized as follows. We start by defining no...

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A fuzzy and spline based dynamic histogram equalization for contrast enhancement of brain images

Saravanan

Karthigaivel

2020

Int J Imaging Syst Tech

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In general, medical images acquired at insufficient lighting conditions are suffering from low contrast issues which are inadequate for image analysis steps. One standard solution to improve the image contrast is to change its intensity distribution with the help of an image histogram. The histogram‐based contrast enhancement methods treat the images as regions rather than objects which would be more useful for applications like brain image enhancement. In this paper, a Fuzzy and Spline based Histogram Equalization (FSDHE) is proposed to perform contrast enhancement with medical images. The proposed FSDHE method partitions the image into connected components, and the type of components are identified with a fuzzy membership function. The dynamic histogram equalization is applied to each component individually. The equalized sub‐histograms are combined to drive the global histogram, which is inconsistent as dynamic histogram equalization treated the intensity range for each connected component differently. Hence, a spline‐based histogram smoothing is proposed here in this research work. The equalized intensity mapping is received as control points for the polynomial curve, and a smooth intensity transformation is interpolated as a spline curve. The proposed FSDHE model is analyzed with MRI‐brain image dataset of 3064 images, which consists of both benign and malignant cases. The contrast enhancement performance of the proposed FSDHE method is quantified by various measures like Absolute Mean Brightness Error (AMBE), Peak Signal to Noise Ratio (PSNR), Contrast (C), Weber Contrast (WC), Entropy, Hausdorff Distance (HD) and Texture Preservation (TP) measures. The performance of the FSDHE method is compared against with other histogram equalization methods, and the results indicate that the FSDHE method achieves better quality measures of 3.1401, 30.5499, 21.5486, 0.7779, 4.0252, 0.2777, and 0.7836 for the seven‐performance metrics.

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On multi-degree splines

Cited by 17 publications

References 16 publications

A Tchebycheffian Extension of Multidegree B-Splines: Algorithmic Computation and Properties

A Tchebycheffian Extension of Multidegree B-Splines: Algorithmic Computation and Properties

Algorithm 999

A fuzzy and spline based dynamic histogram equalization for contrast enhancement of brain images

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