Fourier Transforms of B-Splines and Fundamental Splines for Cardinal Hermite Interpolations

1976 ◽  
Vol 57 (2) ◽  
pp. 291 ◽  
Author(s):  
S. L. Lee
Keyword(s):  
Fourier Transforms ◽  
B Splines ◽  
Hermite Interpolations

Approximation based on elliptic splines

2014 ◽  
Vol 12 (05) ◽  
pp. 1461014 ◽  
Author(s):  
Zhuyuan Yang ◽  
Zongwen Yang
Keyword(s):  
Sobolev Spaces ◽  
Besov Spaces ◽  
Fourier Transforms ◽  
Projection Operators ◽  
Order Of Approximation ◽  
B Splines ◽  
Interpolation Operators ◽  
Polynomial Reproducing

In this paper, we study the elliptic splines which include the well-known polyharmonic B-splines. We analyze their Fourier transforms, decay behaviors and polynomial reproducing properties. We also study the order of approximation in Sobolev spaces and consider their characterizations of Besov spaces by the scale projection operators, quasi-interpolation operators and wavelet operators.

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.

C Hermite interpolations with spatial Pythagorean hodograph B-splines

2021 ◽  
pp. 101992
Author(s):  
Michal Bizzarri ◽  
Miroslav Lávička ◽  
Jan Vršek
Keyword(s):  
B Splines ◽  
Pythagorean Hodograph ◽  
Hermite Interpolations

Computer processing of high-resolution images of periodic specimens

1986 ◽  
Vol 44 ◽  
pp. 2-5
Author(s):  
W. Chiu ◽  
M.F. Schmid ◽  
T.-W. Jeng
Keyword(s):  
High Resolution ◽  
Fourier Transforms ◽  
Complex Structure ◽  
Distribution Functions ◽  
Multiple Image ◽  
Cryo Electron Microscopy ◽  
Structure Factors ◽  
Unit Cells ◽  
High Resolution Images ◽  
Phase Probability Distribution

Cryo-electron microscopy has been developed to the point where one can image thin protein crystals to 3.5 Å resolution. In our study of the crotoxin complex crystal, we can confirm this structural resolution from optical diffractograms of the low dose images. To retrieve high resolution phases from images, we have to include as many unit cells as possible in order to detect the weak signals in the Fourier transforms of the image. Hayward and Stroud proposed to superimpose multiple image areas by combining phase probability distribution functions for each reflection. The reliability of their phase determination was evaluated in terms of a crystallographic “figure of merit”. Grant and co-workers used a different procedure to enhance the signals from multiple image areas by vector summation of the complex structure factors in reciprocal space.

The extended Fourier algorithm: Application in discrete optics and electron holography

1994 ◽  
Vol 52 ◽  
pp. 918-919
Author(s):  
E. Voelkl ◽  
L. F. Allard
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Sampling Rate ◽  
Electron Holography ◽  
Optical Bench ◽  
Fourier Space ◽  
Ccd Cameras ◽  
Simulated Image ◽  
Initial Sampling ◽  
Fourier Algorithm

The conventional discrete Fourier transform can be extended to a discrete Extended Fourier transform (EFT). The EFT allows to work with discrete data in close analogy to the optical bench, where continuous data are processed. The EFT includes a capability to increase or decrease the resolution in Fourier space (thus the argument that CCD cameras with a higher number of pixels to increase the resolution in Fourier space is no longer valid). Fourier transforms may also be shifted with arbitrary increments, which is important in electron holography. Still, the analogy between the optical bench and discrete optics on a computer is limited by the Nyquist limit. In this abstract we discuss the capability with the EFT to change the initial sampling rate si of a recorded or simulated image to any other(final) sampling rate sf.

Observations of Frozen-Hydrated Microtubules

1990 ◽  
Vol 48 (1) ◽  
pp. 504-505
Author(s):  
D. Chrétien ◽  
D. Job ◽  
R.H. Wade
Keyword(s):  
Fourier Transforms ◽  
Structural Complexity ◽  
Image Contrast ◽  
Phase Behaviour ◽  
Experimental Conditions ◽  
Thin Sections ◽  
Surface Lattice ◽  
Cilia And Flagella ◽  
Outstanding Question

Microtubules are filamentary structures found in the cytoplasm of eukaryotic cells, where, together with actin and intermediate filaments, they form the components of the cytoskeleton. They have many functions and show various levels of structural complexity as witnessed by the singlet, doublet and triplet structures involved in the architecture of centrioles, basal bodies, cilia and flagella. The accepted microtubule model consists of a 25 nm diameter hollow tube with a wall made up of 13 paraxial protofilaments (pf). Each pf is a string of aligned tubulin dimers. Some results have suggested that the pfs follow a superhelix. To understand how microtubules function in the cell an accurate model of the surface lattice is one of the requirements. For example the 9x2 architecture of the axoneme will depend on the organisation of its component microtubules. We should also note that microtubules with different numbers of pfs have been observed in thin sections of cellular and of in-vitro material. An outstanding question is how does the surface lattice adjust to these different pf numbers?We have been using cryo-electron microscopy of frozen-hydrated samples to study in-vitro assembled microtubules. The experimental conditions are described in detail in this reference. The results obtained in conjunction with thin sections of similar specimens and with axoneme outer doublet fragments have already allowed us to characterise the image contrast of 13, 14 and 15 pf microtubules on the basis of the measured image widths, of the the image contrast symmetry and of the amplitude and phase behaviour along the equator in the computed Fourier transforms. The contrast variations along individual microtubule images can be interpreted in terms of the geometry of the microtubule surface lattice. We can extend these results and make some reasonable predictions about the probable surface lattices in the case of other pf numbers, see Table 1. Figure 1 shows observed images with which these predictions can be compared.

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