An Example of Two-Dimensional Interpolation Using a Linear Combination of Bicubic B-Splines

2011 ◽  
Vol 57 (3) ◽  
pp. 293-299
Author(s):  
Stanisław Rosłoniec
Keyword(s):  
Linear Combination ◽  
Short Description ◽  
Two Dimensional ◽  
Interpolation Algorithm ◽  
Rectangular Grid ◽  
B Spline ◽  
One Dimensional ◽  
The Real ◽  
B Splines

An Example of Two-Dimensional Interpolation Using a Linear Combination of Bicubic B-Splines The paper describes how a linear combination of bicubic B-splines can be effectively used in a two-dimensional interpolation. It is assumed that values of a function to be interpolated are evaluated at the uniformly located nodes of a corresponding rectangular grid. All formulae of importance have been derived step by step and are presented in a form convenient for computer implementations. To ensure clarity of considerations a short description of one-dimensional B-spline is also given in Appendix 1. The usefulness of the presented interpolation algorithm has been confirmed by the real engineering example of applications.

The Results of the Sub-Pixel Efficacy Region Based B-Spline Interpolation Functions

2009 ◽  
pp. 239-337
Author(s):  
Carlo Ciulla
Keyword(s):  
Spectral Power ◽  
Spline Interpolation ◽  
Image Resolution ◽  
B Spline ◽  
One Dimensional ◽  
B Splines ◽  
Sampling Locations ◽  
Characteristic Features ◽  
Analysis Of Error ◽  
Interpolation Functions

The results obtained processing the MRI database with classic and SRE-based one dimensional quadratic and cubic B-Splines are presented in this chapter. The chapter opens up with information relevant to the image resolution of the MRI database employed for validation. The assessment of the performance of the two classes of interpolators (classic and SRE-based) is conducted both quantitatively and qualitatively. The RSME Ratio is plotted to ascertain which ones of the classic or the SRE-based models deliver the smaller interpolation error. Also, the analysis of error images obtained after processing with either of the two model interpolators and the display of the maps of novel re-sampling locations along with spectral power evolutions corroborates the presentation of the characteristic features of the performances of the interpolation functions treated in this chapter.

scholarly journals Separation Of Non-Periodic And Periodic 2D Profile Features Using B-Spline Functions

2015 ◽  
Vol 22 (2) ◽  
pp. 289-302 ◽  
Author(s):  
Dariusz Janecki ◽  
Leszek Cedro ◽  
Jarosław Zwierzchowski
Keyword(s):  
Spline Function ◽  
Fourier Transforms ◽  
Spline Functions ◽  
Transition Band ◽  
B Spline ◽  
One Dimensional ◽  
B Splines ◽  
Measured Profile ◽  
Spline Filter ◽  
Filter Frequency

Abstract The form, waviness and roughness components of a measured profile are separated by means of digital filters. The aim of analysis was to develop an algorithm for one-dimensional filtering of profiles using approximation by means of B-splines. The theory of B-spline functions introduced by Schoenberg and extended by Unser et al. was used. Unlike the spline filter proposed by Krystek, which is described in ISO standards, the algorithm does not take into account the bending energy of a filtered profile in the functional whose minimization is the principle of the filter. Appropriate smoothness of a filtered profile is achieved by selecting an appropriate distance between nodes of the spline function. In this paper, we determine the Fourier transforms of the filter impulse response at different impulse positions, with respect to the nodes. We show that the filter cutoff length is equal to half of the node-to-node distance. The inclination of the filter frequency characteristic in the transition band can be adjusted by selecting an appropriate degree of the B-spline function. The paper includes examples of separation of 2D roughness, as well as separation of form and waviness of roundness profiles.

The Extension of Theory and Methodology to B-Splines

2009 ◽  
pp. 223-238
Author(s):  
Carlo Ciulla
Keyword(s):  
Methodological Approach ◽  
The Novel ◽  
B Spline ◽  
One Dimensional ◽  
B Splines ◽  
Sampling Locations ◽  
Starting Point ◽  
Before And After ◽  
The One

The organization of the chapter is similar to that of Chapters VII and X. The methodological approach to extend the unifying theory to the one dimensional quadratic and cubic B-Splines is herein reported along with the most relevant mathematical details. This chapter should be read along with Appendix VI where proofs are given to the assertions herein presented. In either of the two cases: quadratic and cubic B-Spline the math process starts from the calculation of the Intensity-Curvature Functional and continues with the calculation of the Sub-pixel Efficacy Region. Finally, the math process arrives to the calculation of the novel re-sampling locations through the formulas of the unifying theory seen in equations (23) and (33) for the quadratic and the cubic models respectively. The chapter concludes with a section that addresses specifically the theoretical proposition of resilient interpolation for the two classes of B-Splines. This is conducted consistently with Chapters VII and XII of the book choosing to equate the two intensity-curvature terms (before and after interpolation) as the starting point of the math deduction.

Effective aperture of X-ray compound refractive lenses

2017 ◽  
Vol 24 (3) ◽  
pp. 609-614 ◽  
Author(s):  
V. G. Kohn
Keyword(s):  
Integral Intensity ◽  
Two Dimensional ◽  
One Dimensional ◽  
X Ray ◽  
The Real ◽  
A Value ◽  
Effective Aperture ◽  
The One ◽  
Definition Of ◽  
Focusing Properties

A new definition of the effective aperture of the X-ray compound refractive lens (CRL) is proposed. Both linear (one-dimensional) and circular (two-dimensional) CRLs are considered. It is shown that for a strongly absorbing CRL the real aperture does not influence the focusing properties and the effective aperture is determined by absorption. However, there are three ways to determine the effective aperture in terms of transparent CRLs. In the papers by Kohn [(2002). JETP Lett. 76, 600–603; (2003). J. Exp. Theor. Phys. 97, 204–215; (2009). J. Surface Investig. 3, 358–364; (2012). J. Synchrotron Rad. 19, 84–92; Kohn et al. (2003). Opt. Commun. 216, 247–260; (2003). J. Phys. IV Fr, 104, 217–220], the FWHM of the X-ray beam intensity just behind the CRL was used. In the papers by Lengeler et al. [(1999). J. Synchrotron Rad. 6, 1153–1167; (1998). J. Appl. Phys. 84, 5855–5861], the maximum intensity value at the focus was used. Numerically, these two definitions differ by 50%. The new definition is based on the integral intensity of the beam behind the CRL over the real aperture. The integral intensity is the most physical value and is independent of distance. The new definition gives a value that is greater than that of the Kohn definition by 6% and less than that of the Lengeler definition by 41%. A new approximation for the aperture function of a two-dimensional CRL is proposed which allows one to calculate the two-dimensional CRL through the one-dimensional CRL and to obtain an analytical solution for a complex system of many CRLs.

Covalent linkage of [Mn2(μ-O2PPh2)2] units by trans-1,4-bis(4-pyridyl)ethene ligands into one-dimensional and two-dimensional polymers

2018 ◽  
Vol 73 (12) ◽  
pp. 1023-1028
Author(s):  
Ying Zhang ◽  
Ai-Quan Jia ◽  
Jing-Jing Zhang ◽  
Zhifeng Xin ◽  
Qian-Feng Zhang
Keyword(s):  
Single Crystal ◽  
Coordination Polymers ◽  
Two Dimensional ◽  
Rectangular Grid ◽  
X Ray Diffraction ◽  
Covalent Linkage ◽  
One Dimensional ◽  
X Ray ◽  
Distorted Octahedral ◽  
Metal Organic

AbstractTwo coordination polymers, [Mn2(μ-O2PPh2)2(η1-O2PPh2)2(η1-HOCH3)2(μ-bpe)2·CH3OH]n (1) and [Mn2(μ-O2PPh2)4(μ-bpe)2]n (2), were assembled in single-pot reactions from [Mn(CH3COO)2·4H2O], Ph2P(O)OK and trans-1,4-bis(4-pyridyl)ethene (bpe). The products were characterized by single-crystal X-ray diffraction, which revealed a one-dimensional metal-organic ladder type in 1 and a two-dimensional rectangular grid type in 2. Both 1 and 2 are constructed from six-coordinate Mn(II) nodes that adopt distorted octahedral (MnN4O2) environments; the adjacent nodes are connected by the μ-bpe linkers in 1 or μ-bpe and μ-O2PPh2 linkers in 2 to form different metal-organic networks. The catalytic property of complex 1 for selective thioether oxidation was also investigated in this work.

scholarly journals Key-Point Interpolation: A Sparse Data Interpolation Algorithm based on B-splines

2021 ◽  
Vol 2068 (1) ◽  
pp. 012010
Author(s):  
Bolun Wang ◽  
Xin Jiang ◽  
Guanying Huo ◽  
Cheng Su ◽  
Dongming Yan ◽  
...  
Keyword(s):  
Sparse Data ◽  
Shanxi Province ◽  
Interpolation Algorithm ◽  
Data Interpolation ◽  
Dynamic Surface ◽  
B Spline ◽  
B Splines ◽  
Surface Interpolation ◽  
Spline Surface ◽  
Point Interpolation

Abstract B-splines are widely used in the fields of reverse engineering and computer-aided design, due to their superior properties. Traditional B-spline surface interpolation algorithms usually assume regularity of the data distribution. In this paper, we introduce a novel B-spline surface interpolation algorithm: KPI, which can interpolate sparsely and non-uniformly distributed data points. As a two-stage algorithm, our method generates the dataset out of the sparse data using Kriging, and uses the proposed KPI (Key-Point Interpolation) method to generate the control points. Our algorithm can be extended to higher dimensional data interpolation, such as reconstructing dynamic surfaces. We apply the method to interpolating the temperature of Shanxi Province. The generated dynamic surface accurately interpolates the temperature data provided by the weather stations, and the preserved dynamic characteristics can be useful for meteorology studies.

scholarly journals The Research of the NURBS Curve Interpolation Algorithm

2011 ◽  
Vol 2-3 ◽  
pp. 614-618
Author(s):  
Wen Jie Zhang
Keyword(s):  
Adaptive Control ◽  
Real Time ◽  
Feed Rate ◽  
Interpolation Algorithm ◽  
Nurbs Curve ◽  
B Spline ◽  
The Real ◽  
Spline Curves ◽  
Curve Interpolation ◽  
Rate Adaptive

In this article, the definition, the nature and the parameter of the NURBS (Non-Uniform Rational B-spline) curves are defined, the more thorough analysis and research of the real-time NURBS curve interpolation algorithm and the feed rate adaptive control are carried out.

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