scholarly journals Accurate and Efficient Evaluation of Chebyshev Tensor Product Surface

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Keshan He ◽  
Peibing Du ◽  
Hao Jiang ◽  
Chongwen Duan ◽  
Hongxia Wang ◽  
...  
Keyword(s):  
Image Analysis ◽  
Error Analysis ◽  
Tensor Product ◽  
Condition Number ◽  
Numerical Experiments ◽  
Numerical Approximation ◽  
Classical Approach ◽  
Accurate Evaluation ◽  
Product Surface ◽  
Tensor Product Surface

A Chebyshev tensor product surface is widely used in image analysis and numerical approximation. This article illustrates an accurate evaluation for the surface in form of Chebyshev tensor product. This algorithm is based on the application of error-free transformations to improve the traditional Clenshaw Chebyshev tensor product algorithm. Our error analysis shows that the error bound is u+Ou2×condP,x,y in contrast to classic scheme u×cond(P,x,y), where u is working precision and condP,x,y is a condition number of bivariate polynomial P(x,y), which means that the accuracy of the computed result is similar to that produced by classical approach with twice working precision. Numerical experiments verify that the proposed algorithm is stable and efficient.

scholarly journals Tensor-product surface patches with Pythagorean-hodograph isoparametric curves

2015 ◽  
Vol 36 (3) ◽  
pp. 1389-1409 ◽  
Author(s):  
Rida T. Farouki ◽  
Francesca Pelosi ◽  
Maria Lucia Sampoli ◽  
Alessandra Sestini
Keyword(s):  
Tensor Product ◽  
Surface Patches ◽  
Pythagorean Hodograph ◽  
Product Surface ◽  
Tensor Product Surface

scholarly journals A Family of Binary Approximating Subdivision Schemes based on Binomial Distribution

2019 ◽  
Vol 38 (4) ◽  
pp. 1087-1100 ◽  
Author(s):  
Muhammad Asghar ◽  
Ghulam Mustafa
Keyword(s):  
Tensor Product ◽  
Binomial Distribution ◽  
Probability Generating Function ◽  
Visual Performance ◽  
Complete Analysis ◽  
Subdivision Schemes ◽  
B Splines ◽  
Surface Subdivision ◽  
Product Surface ◽  
Tensor Product Surface

A simplest way is introduced to generate a generalized algorithm of univariate and bivariate subdivision schemes. This generalized algorithm is based on the symbol of uniform B-splines subdivision schemes and probability generating function of Binomial distribution. We present a family of binary approximating subdivision schemes which has maximum continuity and less support size. Our proposed family members P4, P5, P6, and P7, have C7, C9, C11 and C13 continuities respectively. In fact, we use Binomial probability distribution to increase the continuity of uniform B-splines subdivision schemes up to more than double. We present the complete analysis of one family member of proposed schemes and give a visual performance to check smoothness graphically. In our analysis, we present continuity, Holder regularity, degree of generation, degree of reproduction and limit stencils analysis of proposed family of subdivision schemes. We also present a survey of high continuity subdivision schemes. Comparison shows that our proposed family of subdivision schemes gives high continuity of subdivision schemes comparative to existing subdivision schemes. We have found that as complexity increases the continuity also increases. In the last, we give the general formula for tensor product surface subdivision schemes and also present the visual performance of proposed tensor product surface subdivision schemes.

scholarly journals Accurate Evaluation of Polynomials in Legendre Basis

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Peibing Du ◽  
Hao Jiang ◽  
Lizhi Cheng
Keyword(s):  
Error Analysis ◽  
Numerical Experiments ◽  
Floating Point ◽  
Polynomial Evaluation ◽  
Accurate Evaluation ◽  
Forward Error ◽  
Floating Point Numbers

This paper presents a compensated algorithm for accurate evaluation of a polynomial in Legendre basis. Since the coefficients of the evaluated polynomial are fractions, we propose to store these coefficients in two floating point numbers, such as double-double format, to reduce the effect of the coefficients’ perturbation. The proposed algorithm is obtained by applying error-free transformation to improve the Clenshaw algorithm. It can yield a full working precision accuracy for the ill-conditioned polynomial evaluation. Forward error analysis and numerical experiments illustrate the accuracy and efficiency of the algorithm.

scholarly journals An Overview of the Topological Gradient Approach in Image Processing: Advantages and Inconveniences

2010 ◽  
Vol 2010 ◽  
pp. 1-19 ◽  
Author(s):  
Lamia Jaafar Belaid
Keyword(s):  
Image Processing ◽  
Image Analysis ◽  
Numerical Experiments ◽  
Grey Level ◽  
Major Drawback ◽  
Topological Gradient ◽  
Boundary Measurements ◽  
Topological Asymptotic Expansion ◽  
Gradient Approach ◽  
Segmentation Problem

Image analysis by topological gradient approach is a technique based upon the historic application of the topological asymptotic expansion to crack localization problem from boundary measurements. This paper aims at reviewing this methodology through various applications in image processing; in particular image restoration with edge detection, classification and segmentation problems for both grey level and color images is presented in this work. The numerical experiments show the efficiency of the topological gradient approach for modelling and solving different image analysis problems. However, the topological gradient approach presents a major drawback: the identified edges are not connected and then the results obtained particularly for the segmentation problem can be degraded. To overcome this inconvenience, we propose an alternative solution by combining the topological gradient approach with the watershed technique. The numerical results obtained using the coupled method are very interesting.

Shape parameter selection for multi-quadrics function method in solving electromagnetic boundary value problems

2016 ◽  
Vol 35 (1) ◽  
pp. 64-79 ◽  
Author(s):  
Huaiqing Zhang ◽  
Chunxian Guo ◽  
Xiangfeng Su ◽  
Lin Chen
Keyword(s):  
Boundary Value Problems ◽  
Condition Number ◽  
Numerical Experiments ◽  
Shape Parameter ◽  
Early Stage ◽  
Coefficient Matrix ◽  
Parameter Selection ◽  
Selection Strategy ◽  
Node Number ◽  
Content Type

Purpose – The multi-quadrics (MQ) function is a kind of radial basis function. And the MQ method has been successfully adopted as a type of meshless method in solving electromagnetic boundary value problems. However, the accuracy of MQ interpolation or solving equations is severely influenced by shape parameter. Thus the purpose of this paper is to propose a case-independent shape parameter selection strategy from the aspect of coefficient matrix condition number analysis. Design/methodology/approach – The condition number of coefficient matrix is investigated. It is shown that the condition number is only a function of shape parameter and MQ node number, and is irrelevant to the interpolated function which means case-independent. The effective condition number which takes into account the interpolated function is introduced. Then, the relation between the relative root mean square error and condition number is analyzed. Three numerical experiments as transmission line, cable channel and grounding metal box model were carried out. Findings – In the numerical experiments, there is an approximate linear relationship between the logarithm of the condition number and shape parameter, an approximate quadratic relationship with node number. And the optimal shape parameter is corresponding to the early stage of condition number oscillation. Originality/value – This paper proposed a case-independent shape parameter selection strategy. For a finite precision computation, the upper limit of the condition number is predetermined. Therefore, the shape parameter can be chosen where condition number oscillates in early stage.

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