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Journal articles on the topic 'Non-Stationary subdivision scheme'

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Author: Grafiati
Published: 8 June 2024

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1

Siddiqi, S. S., and M. Younis. "A symmetric non-stationary subdivision scheme." LMS Journal of Computation and Mathematics 17, no. 1 (2014): 259–72. http://dx.doi.org/10.1112/s1461157013000375.

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AbstractThis paper proposes a new family of symmetric $4$-point ternary non-stationary subdivision schemes that can generate the limit curves of $C^3$ continuity. The continuity of this scheme is higher than the existing 4-point ternary approximating schemes. The proposed scheme has been developed using trigonometric B-spline basis functions and analyzed using the theory of asymptotic equivalence. It has the ability to reproduce or regenerate the conic sections, trigonometric polynomials and trigonometric splines as well. Some graphical and numerical examples are being considered, by choosing an appropriate tension parameter $0<\alpha <\pi /3 $, to show the usefulness of the proposed scheme. Moreover, the Hölder regularity and the reproduction property are also being calculated.
2

Daniel, Sunita, and P. Shunmugaraj. "An approximating non-stationary subdivision scheme." Computer Aided Geometric Design 26, no. 7 (October 2009): 810–21. http://dx.doi.org/10.1016/j.cagd.2009.02.007.

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3

Lamnii, Abdellah, Mohamed Yassir Nour, and Ahmed Zidna. "A Reverse Non-Stationary Generalized B-Splines Subdivision Scheme." Mathematics 9, no. 20 (October 18, 2021): 2628. http://dx.doi.org/10.3390/math9202628.

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In this paper, two new families of non-stationary subdivision schemes are introduced. The schemes are constructed from uniform generalized B-splines with multiple knots of orders 3 and 4, respectively. Then, we construct a third-order reverse subdivision framework. For that, we derive a generalized multi-resolution mask based on their third-order subdivision filters. For the reverse of the fourth-order scheme, two methods are used; the first one is based on least-squares formulation and the second one is based on solving a linear optimization problem. Numerical examples are given to show the performance of the new schemes in reproducing different shapes of initial control polygons.
4

Zhang, Baoxing, Yunkun Zhang, and Hongchan Zheng. "A Symmetric Non-Stationary Loop Subdivision with Applications in Initial Point Interpolation." Symmetry 16, no. 3 (March 21, 2024): 379. http://dx.doi.org/10.3390/sym16030379.

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Loop subdivision is a significant surface scheme with wide applications in fields like computer graphics and wavelet. As a type of stationary scheme, Loop subdivision cannot adjust the limit surface directly. In this paper, we present a new way to solve this problem by proposing a symmetric non-stationary Loop subdivision based on a suitable iteration. This new scheme can be used to adjust the limit surfaces freely and thus can generate surfaces with different shapes. For this new scheme, we show that it is C2 convergent in the regular part of mesh and is at least tangent plane continuous at the limit positions of the extraordinary points. Additionally, we present a non-uniform generalization of this new symmetric non-stationary subdivision so as to locally control the shape of the limit surfaces. More interestingly, we present the limit positions of the initial points, both for the symmetric non-stationary Loop subdivision and its non-uniform generalization. Such limit positions can be used to interpolate the initial points with different valences, generalizing the existing result. Several numerical examples are given to illustrate the performance of the new schemes.
5

Jena, M. K., P. Shunmugaraj, and P. C. Das. "A non-stationary subdivision scheme for curve interpolation." ANZIAM Journal 44 (January 13, 2008): 216. http://dx.doi.org/10.21914/anziamj.v44i0.494.

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6

Salam, Wardat us, Shahid S. Siddiqi, and Kashif Rehan. "Chaikin’s perturbation subdivision scheme in non-stationary forms." Alexandria Engineering Journal 55, no. 3 (September 2016): 2855–62. http://dx.doi.org/10.1016/j.aej.2016.07.002.

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7

Zhang, Zeze, Hongchan Zheng, and Lulu Pan. "Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials." Open Mathematics 19, no. 1 (January 1, 2021): 909–26. http://dx.doi.org/10.1515/math-2021-0058.

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Abstract In this paper, we propose a family of non-stationary combined ternary ( 2 m + 3 ) \left(2m+3) -point subdivision schemes, which possesses the property of generating/reproducing high-order exponential polynomials. This scheme is obtained by adding variable parameters on the generalized ternary subdivision scheme of order 4. For such a scheme, we investigate its support and exponential polynomial generation/reproduction and get that it can generate/reproduce certain exponential polynomials with suitable choices of the parameters and reach 2 m + 3 2m+3 approximation order. Moreover, we discuss its smoothness and show that it can produce C 2 m + 2 {C}^{2m+2} limit curves. Several numerical examples are given to show the performance of the schemes.
8

Daniel, Sunita, and P. Shunmugaraj. "An interpolating 6-point C2 non-stationary subdivision scheme." Journal of Computational and Applied Mathematics 230, no. 1 (August 2009): 164–72. http://dx.doi.org/10.1016/j.cam.2008.11.006.

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9

Tan, Jieqing, Jiaze Sun, and Guangyue Tong. "A non-stationary binary three-point approximating subdivision scheme." Applied Mathematics and Computation 276 (March 2016): 37–43. http://dx.doi.org/10.1016/j.amc.2015.12.002.

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10

Zheng, Hongchan, and Baoxing Zhang. "A non-stationary combined subdivision scheme generating exponential polynomials." Applied Mathematics and Computation 313 (November 2017): 209–21. http://dx.doi.org/10.1016/j.amc.2017.05.066.

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11

Siddiqi, Shahid S., Wardat us Salam, and Kashif Rehan. "A new non-stationary binary 6-point subdivision scheme." Applied Mathematics and Computation 268 (October 2015): 1227–39. http://dx.doi.org/10.1016/j.amc.2015.07.031.

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12

Akram, Ghazala, Khalida Bibi, Kashif Rehan, and Shahid S. Siddiqi. "Shape preservation of 4-point interpolating non-stationary subdivision scheme." Journal of Computational and Applied Mathematics 319 (August 2017): 480–92. http://dx.doi.org/10.1016/j.cam.2017.01.026.

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13

Zhang, Baoxing, Hongchan Zheng, and Weijie Song. "A non-stationary Catmull–Clark subdivision scheme with shape control." Graphical Models 106 (November 2019): 101046. http://dx.doi.org/10.1016/j.gmod.2019.101046.

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14

Bari, Mehwish. "A Family of 2n-Point Ternary Non-Stationary Interpolating Subdivision Scheme." Mehran University Research Journal of Engineering and Technology 36, no. 4 (October 1, 2017): 921–32. http://dx.doi.org/10.22581/muet1982.1704.17.

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15

Mukhtar, Uzma, and Kashif Rehan. "A UNIQUE COMBINATION OF MASK IN BINARY FOUR-POINT SUBDIVISION SCHEME." Journal of Mountain Area Research 8 (July 8, 2023): 82. http://dx.doi.org/10.53874/jmar.v8i0.168.

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A unique binary four-point approximating subdivision scheme has been developed in which one part of binary formula have stationary mask and other part have the non-stationary mask. The resulting curves have the smoothness of C3 continuous for the wider range of shape control parameter. The role of the parameter has been depicted using the square form of discrete control points.
16

Beccari, C., G. Casciola, and L. Romani. "An interpolating 4-point ternary non-stationary subdivision scheme with tension control." Computer Aided Geometric Design 24, no. 4 (May 2007): 210–19. http://dx.doi.org/10.1016/j.cagd.2007.02.001.

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17

Zhang, Zeze, Hongchan Zheng, Weijie Song, and Baoxing Zhang. "A Non-stationary Combined Ternary 5-point Subdivision Scheme with $C^{4}$ Continuity." Taiwanese Journal of Mathematics 24, no. 5 (October 2020): 1259–81. http://dx.doi.org/10.11650/tjm/200303.

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18

Jena, M. K., P. Shunmugaraj, and P. C. Das. "A non-stationary subdivision scheme for generalizing trigonometric spline surfaces to arbitrary meshes." Computer Aided Geometric Design 20, no. 2 (May 2003): 61–77. http://dx.doi.org/10.1016/s0167-8396(03)00008-6.

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19

Badoual, Anaïs, Paola Novara, Lucia Romani, Daniel Schmitter, and Michael Unser. "A non-stationary subdivision scheme for the construction of deformable models with sphere-like topology." Graphical Models 94 (November 2017): 38–51. http://dx.doi.org/10.1016/j.gmod.2017.10.001.

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20

Tan, Jieqing, Bingyao Huang, and Jun Shi. "Non-Stationary Four-Point Binary Blending Subdivision Schemes." Journal of Computer-Aided Design & Computer Graphics 31, no. 4 (2019): 629. http://dx.doi.org/10.3724/sp.j.1089.2019.17366.

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21

Charina, Maria, and Costanza Conti. "Convergence of multivariate non-stationary vector subdivision schemes." Applied Numerical Mathematics 49, no. 3-4 (June 2004): 343–54. http://dx.doi.org/10.1016/j.apnum.2003.12.012.

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22

Abdul Karim, Samsul Ariffin, Faheem Khan, Ghulam Mustafa, Aamir Shahzad, and Muhammad Asghar. "An Efficient Computational Approach for Computing Subdivision Depth of Non-Stationary Binary Subdivision Schemes." Mathematics 11, no. 11 (May 25, 2023): 2449. http://dx.doi.org/10.3390/math11112449.

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Subdivision schemes are equipped with some rules that take a polygon as an input and produce smooth curves or surfaces as an output. This presents the issue of how accurately the polygon approximates the limit curve and surface. What number of iterations/levels would be necessary to achieve the required shape at a user-specified error tolerance? In fact, several methods have been introduced in the case of stationary schemes to address the issue in terms of the error bounds (distance between polygon/polyhedron and limiting shape) and subdivision depth (the number of iterations required to obtain the result at a user-specified error tolerance). However, in the case of non-stationary schemes, this topic needs to be further studied to meet the requirements of new practical applications. This paper highlights a new approach based on a convolution technique to estimate error bounds and subdivision depth for non-stationary schemes. The given technique is independent of any condition on the coefficient of the non-stationary subdivision schemes, and it also produces the best results with the least amount of computational effort. In this paper, we first associated constants with the vectors generated by the given non-stationary schemes, then formulated an expression for the convolution product. This expression gives real values, which monotonically decrease with the increase in the order of the convolution in both the curve and surface cases. This convolution feature plays an important role in obtaining the user-defined error tolerance with fewer iterations. It achieves a trade-off between the number of iterations and user-specified errors. In practice, more iterations are needed to achieve a lower error rate, but we achieved this goal by using fewer iterations.
23

Rehan, Kashif, and Waqas Ali Tanveer. "Curve Variations in Non-Stationary Three-Point Subdivision Schemes." Research Journal of Applied Sciences, Engineering and Technology 15, no. 6 (June 15, 2018): 212–18. http://dx.doi.org/10.19026/rjaset.15.5860.

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24

Conti, C., N. Dyn, C. Manni, and M. L. Mazure. "Convergence of univariate non-stationary subdivision schemes via asymptotic similarity." Computer Aided Geometric Design 37 (August 2015): 1–8. http://dx.doi.org/10.1016/j.cagd.2015.06.004.

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25

Jeong, Byeongseon, Yeon Ju Lee, and Jungho Yoon. "A family of non-stationary subdivision schemes reproducing exponential polynomials." Journal of Mathematical Analysis and Applications 402, no. 1 (June 2013): 207–19. http://dx.doi.org/10.1016/j.jmaa.2013.01.026.

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26

Jeong, Byeongseon, and Jungho Yoon. "Analysis of non-stationary Hermite subdivision schemes reproducing exponential polynomials." Journal of Computational and Applied Mathematics 349 (March 2019): 452–69. http://dx.doi.org/10.1016/j.cam.2018.07.050.

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27

Mustafa, Ghulam, and Pakeeza Ashraf. "A family of 4-point odd-ary non-stationary subdivision schemes." SeMA Journal 67, no. 1 (January 2015): 77–91. http://dx.doi.org/10.1007/s40324-014-0029-2.

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28

Lee, Yeon Ju, and Jungho Yoon. "Non-stationary subdivision schemes for surface interpolation based on exponential polynomials." Applied Numerical Mathematics 60, no. 1-2 (January 2010): 130–41. http://dx.doi.org/10.1016/j.apnum.2009.10.005.

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29

Zhang, Baoxing, Hongchan Zheng, and Yingwei Chen. "Multiple-Function Systems Based on Regular Subdivision." Fractal and Fractional 6, no. 11 (November 16, 2022): 677. http://dx.doi.org/10.3390/fractalfract6110677.

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Self-similar fractals can be generated using subdivision and the subdivision curves/surfaces are actually attractors. Such a connection has been studied between fractals and an extended family of subdivision including stationary and non-stationary schemes. This paper aims to move one step further on such a connection and introduce multiple-function systems, which has a set of function systems and choose one for each step of iteration. These multiple-function systems can be obtained by deriving the iterated function systems based on the subdivision operators and applying some modifications, including deleting some transformations, to them. Such multiple-function systems can be arranged in a tree structure and can generate different attractors along different paths in the tree. Several examples are presented to illustrate the performance of these multiple-function systems.
30

Jeong, Byeongseon, and Jungho Yoon. "A new family of non-stationary hermite subdivision schemes reproducing exponential polynomials." Applied Mathematics and Computation 366 (February 2020): 124763. http://dx.doi.org/10.1016/j.amc.2019.124763.

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31

Siddiqi, Shahid S., Wardat us Salam, and Kashif Rehan. "Hyperbolic forms of ternary non-stationary subdivision schemes originated from hyperbolic B-splines." Journal of Computational and Applied Mathematics 301 (August 2016): 16–27. http://dx.doi.org/10.1016/j.cam.2016.01.001.

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32

Conti, Costanza, Luca Gemignani, and Lucia Romani. "From approximating to interpolatory non-stationary subdivision schemes with the same generation properties." Advances in Computational Mathematics 35, no. 2-4 (July 21, 2011): 217–41. http://dx.doi.org/10.1007/s10444-011-9175-6.

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33

Siddiqi, Shahid S., Wardat us Salam, and Kashif Rehan. "Binary 3-point and 4-point non-stationary subdivision schemes using hyperbolic function." Applied Mathematics and Computation 258 (May 2015): 120–29. http://dx.doi.org/10.1016/j.amc.2015.01.091.

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34

Jeong, Byeongseon, Hong Oh Kim, Yeon Ju Lee, and Jungho Yoon. "Exponential polynomial reproducing property of non-stationary symmetric subdivision schemes and normalized exponential B-splines." Advances in Computational Mathematics 38, no. 3 (December 13, 2011): 647–66. http://dx.doi.org/10.1007/s10444-011-9253-9.

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35

Siddiqi, Shahid S., Wardat us Salam, and Kashif Rehan. "Construction of binary four and five point non-stationary subdivision schemes from hyperbolic B-splines." Applied Mathematics and Computation 280 (April 2016): 30–38. http://dx.doi.org/10.1016/j.amc.2016.01.020.

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36

Charina, Maria, Costanza Conti, and Lucia Romani. "Reproduction of exponential polynomials by multivariate non-stationary subdivision schemes with a general dilation matrix." Numerische Mathematik 127, no. 2 (October 24, 2013): 223–54. http://dx.doi.org/10.1007/s00211-013-0587-8.

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37

Jena, Hrushikesh, and Mahendra Kumar Jena. "A Hybrid Non-Stationary Subdivision Scheme Based on Triangulation." International Journal of Applied and Computational Mathematics 7, no. 4 (August 2021). http://dx.doi.org/10.1007/s40819-021-01114-2.

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38

Barrera, Domingo, Abdellah Lamnii, Mohamed‐Yassir Nour, and Ahmed Zidna. "A mixed hyperbolic/trigonometric non‐stationary subdivision scheme for arbitrary topology meshes." Mathematical Methods in the Applied Sciences, May 13, 2022. http://dx.doi.org/10.1002/mma.8350.

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39

Ashraf, Pakeeza, Mehak Sabir, Abdul Ghaffar, Kottakkaran Sooppy Nisar, and Ilyas Khan. "Shape-Preservation of the Four-Point Ternary Interpolating Non-stationary Subdivision Scheme." Frontiers in Physics 7 (January 31, 2020). http://dx.doi.org/10.3389/fphy.2019.00241.

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40

Bibi, Khalida, Ghazala Akram, and Kashif Rehan. "Level Set Shape Analysis of Binary 4-Point Non-stationary Interpolating Subdivision Scheme." International Journal of Applied and Computational Mathematics 5, no. 6 (October 23, 2019). http://dx.doi.org/10.1007/s40819-019-0732-x.

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41

Siddiqi, Shahid, and Muhammad Younis. "Ternary approximating non-stationary subdivision schemes for curve design." Open Engineering 4, no. 4 (January 1, 2014). http://dx.doi.org/10.2478/s13531-013-0149-y.

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AbstractIn this paper, an algorithm has been introduced to produce ternary 2m-point (for any integer m ≥ 1) approximating non-stationary subdivision schemes which can generate the linear spaces spanned by {1; cos(α.); sin(α.)}. The theory of asymptotic equivalence is being used to analyze the convergence and smoothness of the schemes. The proposed algorithm can be consider as the non-stationary counter part of the 2-point and 4-point existing ternary stationary approximating schemes, for different values of m. Moreover, the proposed algorithm has the ability to reproduce or regenerate the conic sections, trigonometric polynomials and trigonometric splines.
42

Ghaffar, Abdul, Zafar Ullah, Mehwish Bari, Kottakkaran Sooppy Nisar, and Dumitru Baleanu. "Family of odd point non-stationary subdivision schemes and their applications." Advances in Difference Equations 2019, no. 1 (May 6, 2019). http://dx.doi.org/10.1186/s13662-019-2105-5.

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43

Ghaffar, Abdul, Zafar Ullah, Mehwish Bari, Kottakkaran Sooppy Nisar, Maysaa M. Al-Qurashi, and Dumitru Baleanu. "A new class of 2m-point binary non-stationary subdivision schemes." Advances in Difference Equations 2019, no. 1 (August 7, 2019). http://dx.doi.org/10.1186/s13662-019-2264-4.

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44

Jena, Hrushikesh, and Mahendra Kumar Jena. "Construction of Trigonometric Box Splines and the Associated Non-Stationary Subdivision Schemes." International Journal of Applied and Computational Mathematics 7, no. 4 (June 23, 2021). http://dx.doi.org/10.1007/s40819-021-01069-4.

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45

Barrera, D., A. Lamnii, M. Y. Nour, and A. Zidna. "α-B-splines non-stationary subdivision schemes for grids of arbitrary topology design." Computers & Graphics, September 2022. http://dx.doi.org/10.1016/j.cag.2022.09.004.

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