scholarly journals Predicting detection performance with model observers: Fourier domain or spatial domain?

2016 ◽  
Author(s):  
Baiyu Chen ◽  
Lifeng Yu ◽  
Shuai Leng ◽  
James Kofler ◽  
Christopher Favazza ◽  
...  
Keyword(s):  
Spatial Domain ◽  
Detection Performance ◽  
Fourier Domain ◽  
Model Observers

Fast algorithms for Helmholtz Green's functions

2008 ◽  
Vol 464 (2100) ◽  
pp. 3301-3326 ◽  
Author(s):  
Gregory Beylkin ◽  
Christopher Kurcz ◽  
Lucas Monzón
Keyword(s):  
Helmholtz Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Fast Algorithm ◽  
Green's Functions ◽  
Green’S Functions ◽  
Mixed Boundary ◽  
Spatial Domain ◽  
Fourier Domain ◽  
Computational Domain

The formal representation of the quasi-periodic Helmholtz Green's function obtained by the method of images is only conditionally convergent and, thus, requires an appropriate summation convention for its evaluation. Instead of using this formal sum, we derive a candidate Green's function as a sum of two rapidly convergent series, one to be applied in the spatial domain and the other in the Fourier domain (as in Ewald's method). We prove that this representation of Green's function satisfies the Helmholtz equation with the quasi-periodic condition and, furthermore, leads to a fast algorithm for its application as an operator. We approximate the spatial series by a short sum of separable functions given by Gaussians in each variable. For the series in the Fourier domain, we exploit the exponential decay of its terms to truncate it. We use fast and accurate algorithms for convolving functions with this approximation of the quasi-periodic Green's function. The resulting method yields a fast solver for the Helmholtz equation with the quasi-periodic boundary condition. The algorithm is adaptive in the spatial domain and its performance does not significantly deteriorate when Green's function is applied to discontinuous functions or potentials with singularities. We also construct Helmholtz Green's functions with Dirichlet, Neumann or mixed boundary conditions on simple domains and use a modification of the fast algorithm for the quasi-periodic Green's function to apply them. The complexity, in dimension d ≥2, of these algorithms is ( κ d  log  κ + C (log  ϵ −1 ) d ), where ϵ is the desired accuracy, κ is proportional to the number of wavelengths contained in the computational domain and C is a constant. We illustrate our approach with examples.

EIGENVALUE ANALYSIS OF COMPASS GRADIENT EDGE OPERATORS IN FOURIER DOMAIN

1992 ◽  
Vol 02 (01) ◽  
pp. 67-74 ◽  
Author(s):  
RAE-HONG PARK ◽  
WOO YOUNG CHOI
Keyword(s):  
Spatial Domain ◽  
Simple Eigenvalue ◽  
Eigenvalue Analysis ◽  
Computation Method ◽  
Fourier Domain ◽  
Eigenvalue Computation ◽  
Transform Matrix

In this letter, we present a simple eigenvalue computation method of a transform matrix H introduced in a Fourier domain interpretation of the compass gradient edge operators.1 We derive the interrelation of three compass gradient edge operators in the spatial domain by explaining their eigenvalue structure in the transformed domain.

Correlation averaging of two-dimensional crystals: basic strategy and refinements

1986 ◽  
Vol 44 ◽  
pp. 136-139
Author(s):  
W. Baumeister ◽  
R. Rachel ◽  
R. Guckenberger ◽  
R. Hegerl
Keyword(s):  
Low Dose ◽  
Signal To Noise Ratio ◽  
Real Space ◽  
Fourier Domain ◽  
Two Dimensional ◽  
Signal To Noise ◽  
Unit Cells ◽  
Lateral Displacements ◽  
Established Technique ◽  
Crystalline Array

IntroductionCorrelation averaging (CAV) is meanwhile an established technique in image processing of two-dimensional crystals /1,2/. The basic idea is to detect the real positions of unit cells in a crystalline array by means of correlation functions and to average them by real space superposition of the aligned motifs. The signal-to-noise ratio improves in proportion to the number of motifs included in the average. Unlike filtering in the Fourier domain, CAV corrects for lateral displacements of the unit cells; thus it avoids the loss of resolution entailed by these distortions in the conventional approach. Here we report on some variants of the method, aimed at retrieving a maximum of information from images with very low signal-to-noise ratios (low dose microscopy of unstained or lightly stained specimens) while keeping the procedure economical.

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