SAMPLING SEQUENCES OF COMPACTLY SUPPORTED DISTRIBUTIONS IN Lp(R)

2005 ◽  
Vol 03 (03) ◽  
pp. 417-434 ◽  
Author(s):  
ISAAC PESENSON
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
Reconstruction Algorithm ◽  
Compactly Supported ◽  
Countable Sequence ◽  
Reconstruction Methods ◽  
Sampling Sequences

The aim of the paper is to obtain some generalizations of the so-called Plancherel–Polya inequalities which are also known as frame inequalities. By using these inequalities we show that a function f ∈ Lp(R), 1 ≤ p ≤ ∞, which is entire function of exponential type is uniquely determined by a set of numbers {Φj(f)}, j ∈ ℕ where {Φj}, j ∈ ℕ is a countable sequence of compactly supported distributions. In the case p = 2 we offer two reconstruction methods of a function f from a sequence of samples {Φj(f)}, j ∈ ℕ. The first reconstruction algorithm is given in terms of frames. To describe our second algorithm we introduce the so-called average variational splines.

Functions in the Schwartz algebra that are invertible in the sense of Ehrenpreis

2019 ◽  
Vol 484 (1) ◽  
pp. 7-11
Author(s):  
N. F. Abuzyarova
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
Zero Set ◽  
Schwartz Algebra

We consider the problem of obtaining the restrictions on the zero set of an entire function of exponential type under which this function belongs to the Schwartz algebra and invertible in the sense of Ehrenpreis.

scholarly journals Combining Acceleration Techniques for Low-Dose X-Ray Cone Beam Computed Tomography Image Reconstruction

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Hsuan-Ming Huang ◽  
Ing-Tsung Hsiao
Keyword(s):  
Computed Tomography ◽  
Image Reconstruction ◽  
Low Dose ◽  
Computational Cost ◽  
Reconstruction Algorithm ◽  
Reconstruction Algorithms ◽  
Acceleration Techniques ◽  
Reconstruction Methods ◽  
Speed Up ◽  
Phantom Studies

Background and Objective. Over the past decade, image quality in low-dose computed tomography has been greatly improved by various compressive sensing- (CS-) based reconstruction methods. However, these methods have some disadvantages including high computational cost and slow convergence rate. Many different speed-up techniques for CS-based reconstruction algorithms have been developed. The purpose of this paper is to propose a fast reconstruction framework that combines a CS-based reconstruction algorithm with several speed-up techniques.Methods. First, total difference minimization (TDM) was implemented using the soft-threshold filtering (STF). Second, we combined TDM-STF with the ordered subsets transmission (OSTR) algorithm for accelerating the convergence. To further speed up the convergence of the proposed method, we applied the power factor and the fast iterative shrinkage thresholding algorithm to OSTR and TDM-STF, respectively.Results. Results obtained from simulation and phantom studies showed that many speed-up techniques could be combined to greatly improve the convergence speed of a CS-based reconstruction algorithm. More importantly, the increased computation time (≤10%) was minor as compared to the acceleration provided by the proposed method.Conclusions. In this paper, we have presented a CS-based reconstruction framework that combines several acceleration techniques. Both simulation and phantom studies provide evidence that the proposed method has the potential to satisfy the requirement of fast image reconstruction in practical CT.

New Atom Probe Tomography Reconstruction Algorithm for Multilayered Samples: Beyond the Hemispherical Constraint

2017 ◽  
Vol 23 (2) ◽  
pp. 247-254 ◽  
Author(s):  
Nicolas Rolland ◽  
François Vurpillot ◽  
Sébastien Duguay ◽  
Baishakhi Mazumder ◽  
James S. Speck ◽  
...  
Keyword(s):  
Experimental Data ◽  
Atom Probe Tomography ◽  
State Of The Art ◽  
Reconstruction Algorithm ◽  
Atom Probe ◽  
Field Evaporation ◽  
Standard State ◽  
Reconstruction Methods ◽  
Tomography Reconstruction ◽  
Specimen Shape

AbstractAccuracy of atom probe tomography measurements is strongly degraded by the presence of phases that have different evaporation fields. In particular, when there are perpendicular interfaces to the tip axis in the specimen, layers thicknesses are systematically biased and the resolution is degraded near the interfaces. Based on an analytical model of field evaporated emitter end-form, a new algorithm dedicated to the 3D reconstruction of multilayered samples was developed. Simulations of field evaporation of bilayer were performed to evaluate the effectiveness of the new algorithm. Compared to the standard state-of-the-art reconstruction methods, the present approach provides much more accurate analyzed volume, and the resolution is clearly improved near the interface. The ability of the algorithm to handle experimental data was also demonstrated. It is shown that the standard algorithm applied to the same data can commit an error on the layers thicknesses up to a factor 2. This new method is not constrained by the classical hemispherical specimen shape assumption.

scholarly journals Weighted Markov-Bernstein inequalities for entire functions of exponential type

2014 ◽  
Vol 96 (110) ◽  
pp. 181-192 ◽  
Author(s):  
Doron Lubinsky
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
Entire Functions ◽  
Functions Of Exponential Type ◽  
Bernstein Inequalities

We prove weighted Markov-Bernstein inequalities of the form ???? |f?(x)|pw(x) dx ? C(? + 1)p ???? |f(x)|pw(x) dx Here w satisfies certain doubling type properties, f is an entire function of exponential type ? ?, p > 0, and C is independent of f and ?. For example, w(x) = (1 + x2)? satisfies the conditions for any ? ? R. Classical doubling inequalities of Mastroianni and Totik inspired this result.

scholarly journals A Practical Statistical Approach to the Reconstruction Problem Using a Single Slice Rebinning Method

2020 ◽  
Vol 10 (2) ◽  
pp. 137-149 ◽  
Author(s):  
Robert Cierniak ◽  
Piotr Pluta ◽  
Andrzej Kaźmierczak
Keyword(s):  
Data Model ◽  
Statistical Approach ◽  
Reconstruction Algorithm ◽  
Reconstruction Method ◽  
Ct Scanner ◽  
Reconstruction Problem ◽  
Single Slice ◽  
Reconstruction Methods ◽  
3D Geometry

AbstractThe paper presented here describes a new practical approach to the reconstruction problem applied to 3D spiral x-ray tomography. The concept we propose is based on a continuous-to-continuous data model, and the reconstruction problem is formulated as a shift invariant system. This original reconstruction method is formulated taking into consideration the statistical properties of signals obtained by the 3D geometry of a CT scanner. It belongs to the class of nutating reconstruction methods and is based on the advanced single slice rebinning (ASSR) methodology. The concept shown here significantly improves the quality of the images obtained after reconstruction and decreases the complexity of the reconstruction problem in comparison with other approaches. Computer simulations have been performed, which prove that the reconstruction algorithm described here does indeed significantly outperforms conventional analytical methods in the quality of the images obtained.

scholarly journals Reducing the Uncertainty of Lidar Measurements in Complex Terrain Using a Linear Model Approach

2018 ◽  
Vol 10 (9) ◽  
pp. 1465 ◽  
Author(s):  
Martin Hofsäß ◽  
Andrew Clifton ◽  
Po Cheng
Keyword(s):  
Linear Model ◽  
Complex Terrain ◽  
Linear Method ◽  
Reconstruction Algorithm ◽  
Opening Angle ◽  
Reconstruction Methods ◽  
Lidar Measurements ◽  
In Situ Sensors ◽  
Ground Based Lidar ◽  
The Individual

In complex terrain, ground-based lidar wind speed measurements sometimes show noticeable differences compared to measurements made with in-situ sensors mounted on meteorological masts. These differences are mostly caused by the inhomogeneities of the flow field and the applied reconstruction methods. This study investigates three different methods to optimize the reconstruction algorithm in order to improve the agreement between lidar measurements and data from sensors on meteorological masts. The methods include a typical velocity azimuth display (VAD) method, a leave-one-out cross-validation method, and a linear model which takes into account the gradients of the wind velocity components. In addition, further aspects such as the influence of the half opening angle of the scanning cone and the scan duration are considered. The measurements were carried out with two different lidar systems, that measured simultaneously. The reference was a 100 m high meteorological mast. The measurements took place in complex terrain characterized by a 150 m high escarpment. The results from the individual methods are quantitatively compared with the measurements of the cup anemometer mounted on the meteorological mast by means of the three parameters of a linear regression (slope, offset, R 2 ) and the width of the 5th–95th quantile. The results show that expanding the half angle of the scanning cone from 20 ∘ to 55 ∘ reduces the offset by a factor of 14.9, but reducing the scan duration does not have an observable benefit. The linear method has the lowest uncertainty and the best agreement with the reference data (i.e., lowest offset and scatter) of all of the methods that were investigated.

Remarks on Entire Functions of Exponential Type

1986 ◽  
Vol 29 (3) ◽  
pp. 365-371
Author(s):  
Clément Frappier
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
Classical Result ◽  
Entire Functions ◽  
Functions Of Exponential Type

AbstractA classical result of Laguerre says that if P is a polynomial of degree n such that P(z) ≠ 0 for | z | < 1 then (ξ - z)P' (z) + nP(z) ≠ 0 for | z | < 1 and | ξ | < 1. Rahman and Schmeisser have obtained an extension of that result to entire functions of exponential type: if f is an entire function of exponential type τ, bounded on ℝ, such that hf(π/2) = 0 then (ξ- l)f'(z) + iτ(z) ≠ 0 for Im(z) > 0 and | ξ | < 1, whenever f(z) ≠ 0 if Im(z) > 0. We obtain a new proof of that result. We also obtain a generalization, to entire functions of exponential type, of a result of Szegö according to which the inequality | P(Rz) — P(z) | < Rn - 1, | z | ≤ 1, R ≥ 1, holds for all polynomials P, of degree ≤ n, such that | P(z) | ≤ 1 for | z | ≤ 1.

Functions of Exponential Type are Differences of Functions of Bounded Index

1977 ◽  
Vol 20 (4) ◽  
pp. 479-483 ◽  
Author(s):  
Shantilal N. Shah
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
Functions Of Exponential Type ◽  
Image Position

The notion of entire function of Bounded Index is by now well established. It may be stated as follows.An entire function f(z) is said to be of Bounded Index if for some fixed sfor all n and all z. (See [1], [2].)

Reconstruction of EIT Images Using Fish School Search and Non-Blind Search

2021 ◽  
Vol 10 (1) ◽  
pp. 89-103
Author(s):  
Valter Augusto de Freitas Barbosa ◽  
David Edson Ribeiro ◽  
Clarisse Lins de Lima ◽  
Maíra Araújo de Santana ◽  
Ricardo Emmanuel de Souza ◽  
...  
Keyword(s):  
Breast Cancer ◽  
Early Detection ◽  
Ionizing Radiation ◽  
Reconstruction Algorithm ◽  
Breast Cancer Diagnosis ◽  
Element Mesh ◽  
Impedance Tomography ◽  
Fish School ◽  
Reconstruction Methods ◽  
Blind Search

Breast cancer is the most common type of cancer among women, affecting 2.1 million women per year worldwide. The best strategy for decreasing disease morbidity and mortality is early detection. Mammography is the most used exam for the diagnosis of breast cancer. However, this technique uses ionizing radiation and causes discomfort to the patient. One promising technique that can be used for early detection of breast cancer is electrical impedance tomography (EIT), which is an imaging technique free of ionizing radiation. Yet, its images still have low resolution, making it difficult to use in breast cancer diagnosis. Thus, the development of new reconstruction methods aiming better resolution is necessary. This work evaluates the performance of the reconstruction algorithm based on fish school search with non-blind search in a 3,190 finite element mesh.

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