This paper investigates two important analytical properties of hyperbolic-polynomial penalized sp... more This paper investigates two important analytical properties of hyperbolic-polynomial penalized splines, HP-splines for short. HPsplines, obtained by combining a special type of difference penalty with hyperbolic-polynomial B-splines (HB-splines), were recently introduced by the authors as a generalization of P-splines. HB-splines are bell-shaped basis functions consisting of segments made of real exponentials e αx , e −αx and linear functions multiplied by these exponentials, xe +αx and xe −αx. Here, we show that these type of penalized splines reproduce function in the space {e −αx , xe −αx }, that is they fit exponential data exactly. Moreover, we show that they conserve the first and second 'exponential' moments.
Univariate pseudo-splines are a generalization of uniform B-splines and interpolatory 2n-point su... more Univariate pseudo-splines are a generalization of uniform B-splines and interpolatory 2n-point subdivision schemes. Each pseudo-spline is characterized as the subdivision scheme with least possible support among all schemes with specific degrees of polynomial generation and reproduction. In this paper we consider the problem of constructing the symbols of the bivariate counterpart and provide a formula for the symbols of a family of symmetric four-directional bivariate pseudo-splines. All methods employed are of purely algebraic nature.
We present a new method for the stable reconstruction of a class of binary images from a small nu... more We present a new method for the stable reconstruction of a class of binary images from a small number of measurements. The images we consider are characteristic functions of algebraic domains, that is, domains defined as zero loci of bivariate polynomials, and we assume to know only a finite set of uniform samples for each image. The solution to such a problem can be set up in terms of linear equations associated to a set of image moments. However, the sensitivity of the moments to noise makes the numerical solution highly unstable. To derive a robust image recovery algorithm, we represent algebraic polynomials and the corresponding image moments in terms of bivariate Bernstein polynomials and apply polynomial-generating, refinable sampling kernels. This approach is robust to noise, computationally fast and simple to implement. We illustrate the performance of our reconstruction algorithm from noisy samples through extensive numerical experiments. Our code is released open source an...
This short note reviews how algebraic properties of subdivision symbols are linked to the analyti... more This short note reviews how algebraic properties of subdivision symbols are linked to the analytic/geometric properties of the corresponding schemes with particular emphasis on the reproduction capability. Also, with the aim at underlining the advantage of symbols manipulation, it shows, with an example in the Hermite case, how algebraic conditions for polynomial reproduction translate into a very simple algorithm.
We investigate properties of differential and difference operators annihilating certain finite-di... more We investigate properties of differential and difference operators annihilating certain finite-dimensional spaces of exponential functions in two variables that are connected to the representation of real-valued trigonometric and hyperbolic functions. Although exponential functions appear in a variety of contexts, the motivation behind this technical note comes from considering subdivision schemes where annihilation operators play an important role. Indeed, subdivision schemes with the capability of preserving exponential functions can be used to obtain an exact description of surfaces parametrized in terms of trigonometric and hyperbolic functions, and annihilation operators are useful to automatically detect the frequencies of such functions.
The main goal of this paper is to present some generalizations of polynomial B-splines, which inc... more The main goal of this paper is to present some generalizations of polynomial B-splines, which include exponential B-splines and the larger family of exponential pseudo-splines. We especially focus on their connections to subdivision schemes. In addition, we generalize a well-known result on the approximation order of exponential pseudo-splines, providing conditions to establish the approximation order of nonstationary subdivision schemes reproducing spaces of exponential polynomial functions. 2010 MSC: 65D17, 65D15, 41A25
In this paper we provide a complete and unifying characterization of compactly supported univaria... more In this paper we provide a complete and unifying characterization of compactly supported univariate scalar orthogonal wavelets and vector-valued or matrix-valued orthogonal multi-wavelets. This characterization is based on classical results from system theory and basic linear algebra. In particular, we show that the corresponding wavelet and multi-wavelet masks are identified with a transfer function F (z) = A+Bz(I −Dz) C, z ∈ D = {z ∈ C : |z| < 1}, of a conservative linear system. The complex matrices A, B, C, D define a block circulant unitary matrix. Our results show that there are no intrinsic differences between the elegant wavelet construction by Daubechies or any other construction of vector-valued or matrix-valued multiwavelets. The structure of the unitary matrix defined by A, B, C, D allows us to parametrize in a systematic way all classes of possible wavelet and multi-wavelet masks together with the masks of the corresponding refinable functions.
The paper presents a new subdivision scheme, which constructs a surface approximating a given net... more The paper presents a new subdivision scheme, which constructs a surface approximating a given net of 3D-curves. Similar to the well known Chaikin algorithm for points, having a refinement step based on piecewise linear interpolation of the control points followed by evaluation at 1/4 and 3/4 of the local parameter value, the refinement step in the proposed subdivision scheme is based on piecewise Coons patch interpolation followed by evaluation at 1/4 and 3/4 of the local parameters values in both directions, which results in a refined net of curves. We prove the convergence of the scheme to a continuous surface. The proof is based on the ”proximity” of the scheme to a new, convergent subdivision scheme for points. Some examples, illustrating the performance of our scheme, are given. §
In this paper we construct a family of ternary interpolatory Hermite subdivision schemes of order... more In this paper we construct a family of ternary interpolatory Hermite subdivision schemes of order 1 with small support and ${\mathscr{H}}\mathcal {C}^{2}$ H C 2 -smoothness. Indeed, leaving the binary domain, it is possible to derive interpolatory Hermite subdivision schemes with higher regularity than the existing binary examples. The family of schemes we construct is a two-parameter family whose ${\mathscr{H}}\mathcal {C}^{2}$ H C 2 -smoothness is guaranteed whenever the parameters are chosen from a certain polygonal region. The construction of this new family is inspired by the geometric insight into the ternary interpolatory scalar three-point subdivision scheme by Hassan and Dodgson. The smoothness of our new family of Hermite schemes is proven by means of joint spectral radius techniques.
The terms and conditions for the reuse of this version of the manuscript are specified in the pub... more The terms and conditions for the reuse of this version of the manuscript are specified in the publishing policy. For all terms of use and more information see the publisher's website.
The regularity of refinable functions has been investigated deeply in the past 25 years using Fou... more The regularity of refinable functions has been investigated deeply in the past 25 years using Fourier analysis, wavelet analysis, restricted and joint spectral radii techniques. However the shift-invariance of the underlying regular setting is crucial for these approaches. We propose an efficient method based on wavelet tight frame decomposition techniques for estimating Hölder-Zygmund regularity of univariate semi-regular refinable functions generated, e.g., by subdivision schemes defined on semi-regular meshes t = −h ℓ N ∪ {0} ∪ h r N, h ℓ , h r ∈ (0, ∞). To ensure the optimality of this method, we provide a new characterization of Hölder-Zygmund spaces based on suitable irregular wavelet tight frames. Furthermore, we present proper tools for computing the corresponding frame coefficients in the semi-regular setting. We also propose a new numerical approach for estimating the optimal Hölder-Zygmund exponent of refinable functions which is more efficient than the linear regression method. We illustrate our results with several examples of known and new semi-regular subdivision schemes with a potential use in blending curve design.
In this paper we provide a complete and unifying characterization of compactly supported univaria... more In this paper we provide a complete and unifying characterization of compactly supported univariate scalar orthogonal wavelets and vector-valued or matrix-valued orthogonal multi-wavelets. This characterization is based on classical results from system theory and basic linear algebra. In particular, we show that the corresponding wavelet and multi-wavelet masks are identified with a transfer function F (z) = A + Bz(I − Dz) −1 C, z ∈ D = {z ∈ C : |z| < 1}, of a conservative linear system. The complex matrices A, B, C, D define a block circulant unitary matrix. Our results show that there are no intrinsic differences between the elegant wavelet construction by Daubechies or any other construction of vector-valued or matrix-valued multiwavelets. The structure of the unitary matrix defined by A, B, C, D allows us to parametrize in a systematic way all classes of possible wavelet and multi-wavelet masks together with the masks of the corresponding refinable functions.
Journal of Mathematical Analysis and Applications, 2016
Pseudo-splines are a rich family of functions that allows the user to meet various demands for ba... more Pseudo-splines are a rich family of functions that allows the user to meet various demands for balancing polynomial reproduction (i.e., approximation power), regularity and support size. Such a family includes, as special members, B-spline functions, universally known for their usefulness in different fields of application. When replacing polynomial reproduction by exponential polynomial reproduction, a new family of functions is obtained. This new family is here constructed and called the family of exponential pseudo-splines. It is the nonstationary counterpart of (polynomial) pseudo-splines and includes exponential B-splines as a special subclass. In this work we provide a computational strategy for deriving the explicit expression of the Laurent polynomial sequence that identifies the family of exponential pseudo-spline nonstationary subdivision schemes. For this family we study its symmetry properties and perform its convergence and regularity analysis. Finally, we also show that the family of primal exponential pseudo-splines fills in the gap between exponential B-splines and interpolatory cardinal functions. This extends the analogous property of primal pseudo-spline stationary subdivision schemes.
The terms and conditions for the reuse of this version of the manuscript are specified in the pub... more The terms and conditions for the reuse of this version of the manuscript are specified in the publishing policy. For all terms of use and more information see the publisher's website.
In this paper we focus on Hermite subdivision operators that act on vector valued data interpreti... more In this paper we focus on Hermite subdivision operators that act on vector valued data interpreting their components as function values and associated consecutive derivatives. We are mainly interested in studying the exponential and polynomial preservation capability of such kind of operators, which can be expressed in terms of a generalization of the spectral condition property in the spaces generated by polynomials and exponential functions. The main tool for our investigation are convolution operators that annihilate the aforementioned spaces, which apparently is a general concept in the study of various types of subdivision operators. Based on these annihilators, we characterize the spectral condition in terms of factorization of the subdivision operator. Keywords Hermite subdivision • factorization • annihilators • Taylor operator • exponentials Mathematics Subject Classification (2000) 65D15 • 65D10 • 41A05 1 Introduction Subdivision schemes are iterative procedures based on the repeated application of subdivision operators, which can even di↵er at di↵erent levels of iteration, on discrete data. More specifically, subdivision operators act on bi-infinite sequences
Subdivision schemes are iterative methods for the design of smooth curves and surfaces. Any linea... more Subdivision schemes are iterative methods for the design of smooth curves and surfaces. Any linear subdivision scheme can be identified by a sequence of Laurent polynomials, also called subdivision symbols, which describe the linear rules determining successive refinements of coarse initial meshes. One important property of subdivision schemes is their capability of exactly reproducing in the limit specific types of functions from which the data is sampled. Indeed, this property is linked to the approximation order of the scheme and to its regularity. When the capability of reproducing polynomials is required, it is possible to define a family of subdivision schemes that allows to meet various demands for balancing approximation order, regularity and support size. The members of this family are known in the literature with the name of pseudo-splines. In case reproduction of exponential polynomials instead of polynomials is requested, the resulting family turns out to be the non-stationary counterpart of the one of pseudo-splines, that we here call the family of exponential pseudo-splines. The goal of this work is to derive the explicit expressions of the subdivision symbols of exponential pseudo-splines and to study their symmetry properties as well as their convergence and regularity.
An algorithm for the problem of transforming a regular array of tabular points into a monotonous ... more An algorithm for the problem of transforming a regular array of tabular points into a monotonous array of tabular points is presented.
Convergence and regularity analysis of a bivariate non-stationary (level-dependent) subdivision s... more Convergence and regularity analysis of a bivariate non-stationary (level-dependent) subdivision scheme for 2-manifold meshes with arbitrary topology is still an open issue. Exploiting ideas from the theory of asymptotically equivalent subdivision schemes, in this paper we derive new sufficient conditions for establishing convergence and $C^1$-regularity of any symmetric, non-stationary, subdivision scheme near an extraordinary vertex/face.
This paper describes an algebraic construction of bivariate interpolatory subdivision masks induc... more This paper describes an algebraic construction of bivariate interpolatory subdivision masks induced by three-directional box spline subdivision schemes. Specifically, given a three-directional box spline, we address the problem of defining a corresponding interpolatory subdivision scheme by constructing an appropriate correction mask to convolve with the three-directional box spline mask. The proposed approach is based on the analysis of certain polynomial identities in two variables and leads to interesting new interpolatory bivariate subdivision schemes.
This paper investigates two important analytical properties of hyperbolic-polynomial penalized sp... more This paper investigates two important analytical properties of hyperbolic-polynomial penalized splines, HP-splines for short. HPsplines, obtained by combining a special type of difference penalty with hyperbolic-polynomial B-splines (HB-splines), were recently introduced by the authors as a generalization of P-splines. HB-splines are bell-shaped basis functions consisting of segments made of real exponentials e αx , e −αx and linear functions multiplied by these exponentials, xe +αx and xe −αx. Here, we show that these type of penalized splines reproduce function in the space {e −αx , xe −αx }, that is they fit exponential data exactly. Moreover, we show that they conserve the first and second 'exponential' moments.
Univariate pseudo-splines are a generalization of uniform B-splines and interpolatory 2n-point su... more Univariate pseudo-splines are a generalization of uniform B-splines and interpolatory 2n-point subdivision schemes. Each pseudo-spline is characterized as the subdivision scheme with least possible support among all schemes with specific degrees of polynomial generation and reproduction. In this paper we consider the problem of constructing the symbols of the bivariate counterpart and provide a formula for the symbols of a family of symmetric four-directional bivariate pseudo-splines. All methods employed are of purely algebraic nature.
We present a new method for the stable reconstruction of a class of binary images from a small nu... more We present a new method for the stable reconstruction of a class of binary images from a small number of measurements. The images we consider are characteristic functions of algebraic domains, that is, domains defined as zero loci of bivariate polynomials, and we assume to know only a finite set of uniform samples for each image. The solution to such a problem can be set up in terms of linear equations associated to a set of image moments. However, the sensitivity of the moments to noise makes the numerical solution highly unstable. To derive a robust image recovery algorithm, we represent algebraic polynomials and the corresponding image moments in terms of bivariate Bernstein polynomials and apply polynomial-generating, refinable sampling kernels. This approach is robust to noise, computationally fast and simple to implement. We illustrate the performance of our reconstruction algorithm from noisy samples through extensive numerical experiments. Our code is released open source an...
This short note reviews how algebraic properties of subdivision symbols are linked to the analyti... more This short note reviews how algebraic properties of subdivision symbols are linked to the analytic/geometric properties of the corresponding schemes with particular emphasis on the reproduction capability. Also, with the aim at underlining the advantage of symbols manipulation, it shows, with an example in the Hermite case, how algebraic conditions for polynomial reproduction translate into a very simple algorithm.
We investigate properties of differential and difference operators annihilating certain finite-di... more We investigate properties of differential and difference operators annihilating certain finite-dimensional spaces of exponential functions in two variables that are connected to the representation of real-valued trigonometric and hyperbolic functions. Although exponential functions appear in a variety of contexts, the motivation behind this technical note comes from considering subdivision schemes where annihilation operators play an important role. Indeed, subdivision schemes with the capability of preserving exponential functions can be used to obtain an exact description of surfaces parametrized in terms of trigonometric and hyperbolic functions, and annihilation operators are useful to automatically detect the frequencies of such functions.
The main goal of this paper is to present some generalizations of polynomial B-splines, which inc... more The main goal of this paper is to present some generalizations of polynomial B-splines, which include exponential B-splines and the larger family of exponential pseudo-splines. We especially focus on their connections to subdivision schemes. In addition, we generalize a well-known result on the approximation order of exponential pseudo-splines, providing conditions to establish the approximation order of nonstationary subdivision schemes reproducing spaces of exponential polynomial functions. 2010 MSC: 65D17, 65D15, 41A25
In this paper we provide a complete and unifying characterization of compactly supported univaria... more In this paper we provide a complete and unifying characterization of compactly supported univariate scalar orthogonal wavelets and vector-valued or matrix-valued orthogonal multi-wavelets. This characterization is based on classical results from system theory and basic linear algebra. In particular, we show that the corresponding wavelet and multi-wavelet masks are identified with a transfer function F (z) = A+Bz(I −Dz) C, z ∈ D = {z ∈ C : |z| < 1}, of a conservative linear system. The complex matrices A, B, C, D define a block circulant unitary matrix. Our results show that there are no intrinsic differences between the elegant wavelet construction by Daubechies or any other construction of vector-valued or matrix-valued multiwavelets. The structure of the unitary matrix defined by A, B, C, D allows us to parametrize in a systematic way all classes of possible wavelet and multi-wavelet masks together with the masks of the corresponding refinable functions.
The paper presents a new subdivision scheme, which constructs a surface approximating a given net... more The paper presents a new subdivision scheme, which constructs a surface approximating a given net of 3D-curves. Similar to the well known Chaikin algorithm for points, having a refinement step based on piecewise linear interpolation of the control points followed by evaluation at 1/4 and 3/4 of the local parameter value, the refinement step in the proposed subdivision scheme is based on piecewise Coons patch interpolation followed by evaluation at 1/4 and 3/4 of the local parameters values in both directions, which results in a refined net of curves. We prove the convergence of the scheme to a continuous surface. The proof is based on the ”proximity” of the scheme to a new, convergent subdivision scheme for points. Some examples, illustrating the performance of our scheme, are given. §
In this paper we construct a family of ternary interpolatory Hermite subdivision schemes of order... more In this paper we construct a family of ternary interpolatory Hermite subdivision schemes of order 1 with small support and ${\mathscr{H}}\mathcal {C}^{2}$ H C 2 -smoothness. Indeed, leaving the binary domain, it is possible to derive interpolatory Hermite subdivision schemes with higher regularity than the existing binary examples. The family of schemes we construct is a two-parameter family whose ${\mathscr{H}}\mathcal {C}^{2}$ H C 2 -smoothness is guaranteed whenever the parameters are chosen from a certain polygonal region. The construction of this new family is inspired by the geometric insight into the ternary interpolatory scalar three-point subdivision scheme by Hassan and Dodgson. The smoothness of our new family of Hermite schemes is proven by means of joint spectral radius techniques.
The terms and conditions for the reuse of this version of the manuscript are specified in the pub... more The terms and conditions for the reuse of this version of the manuscript are specified in the publishing policy. For all terms of use and more information see the publisher's website.
The regularity of refinable functions has been investigated deeply in the past 25 years using Fou... more The regularity of refinable functions has been investigated deeply in the past 25 years using Fourier analysis, wavelet analysis, restricted and joint spectral radii techniques. However the shift-invariance of the underlying regular setting is crucial for these approaches. We propose an efficient method based on wavelet tight frame decomposition techniques for estimating Hölder-Zygmund regularity of univariate semi-regular refinable functions generated, e.g., by subdivision schemes defined on semi-regular meshes t = −h ℓ N ∪ {0} ∪ h r N, h ℓ , h r ∈ (0, ∞). To ensure the optimality of this method, we provide a new characterization of Hölder-Zygmund spaces based on suitable irregular wavelet tight frames. Furthermore, we present proper tools for computing the corresponding frame coefficients in the semi-regular setting. We also propose a new numerical approach for estimating the optimal Hölder-Zygmund exponent of refinable functions which is more efficient than the linear regression method. We illustrate our results with several examples of known and new semi-regular subdivision schemes with a potential use in blending curve design.
In this paper we provide a complete and unifying characterization of compactly supported univaria... more In this paper we provide a complete and unifying characterization of compactly supported univariate scalar orthogonal wavelets and vector-valued or matrix-valued orthogonal multi-wavelets. This characterization is based on classical results from system theory and basic linear algebra. In particular, we show that the corresponding wavelet and multi-wavelet masks are identified with a transfer function F (z) = A + Bz(I − Dz) −1 C, z ∈ D = {z ∈ C : |z| < 1}, of a conservative linear system. The complex matrices A, B, C, D define a block circulant unitary matrix. Our results show that there are no intrinsic differences between the elegant wavelet construction by Daubechies or any other construction of vector-valued or matrix-valued multiwavelets. The structure of the unitary matrix defined by A, B, C, D allows us to parametrize in a systematic way all classes of possible wavelet and multi-wavelet masks together with the masks of the corresponding refinable functions.
Journal of Mathematical Analysis and Applications, 2016
Pseudo-splines are a rich family of functions that allows the user to meet various demands for ba... more Pseudo-splines are a rich family of functions that allows the user to meet various demands for balancing polynomial reproduction (i.e., approximation power), regularity and support size. Such a family includes, as special members, B-spline functions, universally known for their usefulness in different fields of application. When replacing polynomial reproduction by exponential polynomial reproduction, a new family of functions is obtained. This new family is here constructed and called the family of exponential pseudo-splines. It is the nonstationary counterpart of (polynomial) pseudo-splines and includes exponential B-splines as a special subclass. In this work we provide a computational strategy for deriving the explicit expression of the Laurent polynomial sequence that identifies the family of exponential pseudo-spline nonstationary subdivision schemes. For this family we study its symmetry properties and perform its convergence and regularity analysis. Finally, we also show that the family of primal exponential pseudo-splines fills in the gap between exponential B-splines and interpolatory cardinal functions. This extends the analogous property of primal pseudo-spline stationary subdivision schemes.
The terms and conditions for the reuse of this version of the manuscript are specified in the pub... more The terms and conditions for the reuse of this version of the manuscript are specified in the publishing policy. For all terms of use and more information see the publisher's website.
In this paper we focus on Hermite subdivision operators that act on vector valued data interpreti... more In this paper we focus on Hermite subdivision operators that act on vector valued data interpreting their components as function values and associated consecutive derivatives. We are mainly interested in studying the exponential and polynomial preservation capability of such kind of operators, which can be expressed in terms of a generalization of the spectral condition property in the spaces generated by polynomials and exponential functions. The main tool for our investigation are convolution operators that annihilate the aforementioned spaces, which apparently is a general concept in the study of various types of subdivision operators. Based on these annihilators, we characterize the spectral condition in terms of factorization of the subdivision operator. Keywords Hermite subdivision • factorization • annihilators • Taylor operator • exponentials Mathematics Subject Classification (2000) 65D15 • 65D10 • 41A05 1 Introduction Subdivision schemes are iterative procedures based on the repeated application of subdivision operators, which can even di↵er at di↵erent levels of iteration, on discrete data. More specifically, subdivision operators act on bi-infinite sequences
Subdivision schemes are iterative methods for the design of smooth curves and surfaces. Any linea... more Subdivision schemes are iterative methods for the design of smooth curves and surfaces. Any linear subdivision scheme can be identified by a sequence of Laurent polynomials, also called subdivision symbols, which describe the linear rules determining successive refinements of coarse initial meshes. One important property of subdivision schemes is their capability of exactly reproducing in the limit specific types of functions from which the data is sampled. Indeed, this property is linked to the approximation order of the scheme and to its regularity. When the capability of reproducing polynomials is required, it is possible to define a family of subdivision schemes that allows to meet various demands for balancing approximation order, regularity and support size. The members of this family are known in the literature with the name of pseudo-splines. In case reproduction of exponential polynomials instead of polynomials is requested, the resulting family turns out to be the non-stationary counterpart of the one of pseudo-splines, that we here call the family of exponential pseudo-splines. The goal of this work is to derive the explicit expressions of the subdivision symbols of exponential pseudo-splines and to study their symmetry properties as well as their convergence and regularity.
An algorithm for the problem of transforming a regular array of tabular points into a monotonous ... more An algorithm for the problem of transforming a regular array of tabular points into a monotonous array of tabular points is presented.
Convergence and regularity analysis of a bivariate non-stationary (level-dependent) subdivision s... more Convergence and regularity analysis of a bivariate non-stationary (level-dependent) subdivision scheme for 2-manifold meshes with arbitrary topology is still an open issue. Exploiting ideas from the theory of asymptotically equivalent subdivision schemes, in this paper we derive new sufficient conditions for establishing convergence and $C^1$-regularity of any symmetric, non-stationary, subdivision scheme near an extraordinary vertex/face.
This paper describes an algebraic construction of bivariate interpolatory subdivision masks induc... more This paper describes an algebraic construction of bivariate interpolatory subdivision masks induced by three-directional box spline subdivision schemes. Specifically, given a three-directional box spline, we address the problem of defining a corresponding interpolatory subdivision scheme by constructing an appropriate correction mask to convolve with the three-directional box spline mask. The proposed approach is based on the analysis of certain polynomial identities in two variables and leads to interesting new interpolatory bivariate subdivision schemes.
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