Application of Majorization-Minimization Method to Chan --- Vese Algorithm in the Image Segmentation Problem

2019 ◽  
pp. 19-29
Author(s):  
I.S. Druzhitskiy ◽  
D.E. Bekasov
Keyword(s):  
Image Segmentation ◽  
Main Idea ◽  
Operation Time ◽  
Delta Function ◽  
Optimization Method ◽  
Step Function ◽  
Minimization Method ◽  
Heaviside Step Function ◽  
Dirac Delta ◽  
Wide Range

The purpose of the study was to modify Chan --- Vese algorithm in order to overcome its shortcomings, such as high computational complexity and the use of approximations. In the considered modification, optimization is carried out by the majorization-minimization method, the main idea of which is to reduce the complexity of the problem using the majority function. Due to the proposed optimization method, it is possible to use the Heaviside step function and Dirac delta function. This enabled the same or better saturation levels when optimization is done by the graph cut method in a smaller number of iterations, which reduced the operation time. The proposed algorithm was tested on a Caltech101 dataset. The algorithm is general, does not depend on the subject area and does not require prior training. This allows it to be used as the basis for a wide range of image segmentation algorithms.

A Universal Adjoint-Weighted Algorithm for Geometric Sensitivity Analysis of K-Eigenvalue Based on Continuous-Energy Monte Carlo Method

2018 ◽  
Author(s):  
Hao Li ◽  
Ganglin Yu ◽  
Shanfang Huang ◽  
Kan Wang
Keyword(s):  
Monte Carlo ◽  
Cross Section ◽  
Geometric Parameter ◽  
Neutron Transport ◽  
Delta Function ◽  
Step Function ◽  
Universal Method ◽  
Analytic Geometry ◽  
Heaviside Step Function ◽  
Dirac Delta

There exists a typical problem in Monte Carlo neutron transport: the effective multiplication factor sensitivity to geometric parameter. In several methods attempting to solve it, Monte Carlo adjoint-weighted theory has been proven to be quite effective. The major obstacle of adjoint-weighted theory is calculating derivative of cross section with respect to geometric parameter. In order to fix this problem, Heaviside step function and Dirac delta function are introduced to describe cross section and its derivative. This technique is crucial, and it establishes the foundation of further research. Based on above work, adjoint-weighted method is developed to solve geometric sensitivity. However, this method is limited to surfaces which are uniformly expanded or contracted with respect to its origin, such as vertical movement of plane or expansion of sphere. Rotation and translation are not allowed, while these two transformation types are more common and more important in engineering projects. In this paper, a more universal method, Cell Constraint Condition Perturbation (CCCP) method, is developed and validated. Different from traditional method, CCCP method for the first time explicitly articulates that the perturbed quantity is the parameter of spatial analytic geometry equations that used to describe surface. Thus, the CCCP can treat arbitrary one-parameter geometric perturbation of arbitrary surface as long as this surface can be described by spatial analytic geometry equation. Furthermore, CCCP can treat the perturbation of the whole cell, such as translation, rotation, expansion and constriction. Several examples are calculated to confirm the validity of CCCP method.

scholarly journals Motion blur invariant for estimating motion parameters of medical ultrasound images

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Barmak Honarvar Shakibaei Asli ◽  
Yifan Zhao ◽  
John Ahmet Erkoyuncu
Keyword(s):  
Image Quality Assessment ◽  
Medical Image Analysis ◽  
Main Idea ◽  
Delta Function ◽  
Motion Blur ◽  
Ultrasound Images ◽  
Medical Ultrasound ◽  
Fetal Ultrasound ◽  
Dirac Delta ◽  
Motion Parameters

AbstractHigh-quality medical ultrasound imaging is definitely concerning motion blur, while medical image analysis requires motionless and accurate data acquired by sonographers. The main idea of this paper is to establish some motion blur invariant in both frequency and moment domain to estimate the motion parameters of ultrasound images. We propose a discrete model of point spread function of motion blur convolution based on the Dirac delta function to simplify the analysis of motion invariant in frequency and moment domain. This model paves the way for estimating the motion angle and length in terms of the proposed invariant features. In this research, the performance of the proposed schemes is compared with other state-of-the-art existing methods of image deblurring. The experimental study performs using fetal phantom images and clinical fetal ultrasound images as well as breast scans. Moreover, to validate the accuracy of the proposed experimental framework, we apply two image quality assessment methods as no-reference and full-reference to show the robustness of the proposed algorithms compared to the well-known approaches.

scholarly journals Unified approach to the impulse response and green function in the circuit and field theory part II : multi–dimensional case

2012 ◽  
Vol 63 (6) ◽  
pp. 341-348
Author(s):  
L’ubomír Šumichrast
Keyword(s):  
Field Theory ◽  
Impulse Response ◽  
Transmission Lines ◽  
Delta Function ◽  
Step Function ◽  
Unified Approach ◽  
Dirac Delta ◽  
Time Space ◽  
Previous Part ◽  
Heaviside Unit Step Function

In the circuit theory the concept of the impulse response of a linear system due to its excitation by the Dirac delta function δ(t) together with the convolution principle is widely used and accepted. The rigorous theory of symbolic functions, sometimes called distributions, where also the delta function belongs, is rather abstract and requires subtle mathematical tools [1-4]. Nevertheless, the most people intuitively well understand the delta function as a derivative of the (Heaviside) unit step function 1(t) without too much mathematical rigor. In the previous part [5] the concept of the impulse response of linear systems was approached in a unified manner and generalized to the time-space phenomena in one dimension (transmission lines). Here the phenomena in more dimensions (static and dynamic electromagnetic fields) are treated. It is shown that many formulas in the field theory, which are often postulated in an inductive way as results of the experiments, and therefore appear as “deux ex machina” effects, can be mathematically deduced from a few starting equations.

The Exact Solution of the Translational Acceleration of a Low Modulus Elastic Medium in Rigid Spherical Shells—Implications for Head Injury Models

1975 ◽  
Vol 42 (4) ◽  
pp. 759-762 ◽  
Author(s):  
K. B. Chandran ◽  
Y. King Liu ◽  
D. U. von Rosenberg
Keyword(s):  
Exact Solution ◽  
Head Injury ◽  
Elastic Medium ◽  
Closed Head Injury ◽  
Delta Function ◽  
Step Function ◽  
Spherical Shells ◽  
Dirac Delta ◽  
Low Modulus ◽  
Translational Acceleration

The exact solution in the form of a finite series has been obtained for the problem of low modulus elastic medium contained in rigid spherical shells subjected to translational acceleration about its diametrical axis. Laplace transformation technique and the shifting theorem were used to obtain the Green’s functions for the potentials when the external acceleration is a Dirac delta function. The solutions are formally extended to external accelerations which are general functions of time by the convolution integral. The shear stress distribution for a unit step function acceleration is illustrated. The results obtained are used to judge the adequacy of this and other similar models for the study of closed head injury mechanism.

Unified approach to the impulse response and green function in the circuit and field theory, part I: one–dimensional case

2012 ◽  
Vol 63 (5) ◽  
pp. 273-280 ◽  
Author(s):  
L’Ubomír Šumichrast
Keyword(s):  
Impulse Response ◽  
Transmission Lines ◽  
Delta Function ◽  
Step Function ◽  
Subsequent Paper ◽  
Unified Approach ◽  
One Dimensional ◽  
Dirac Delta ◽  
Time Space ◽  
Heaviside Unit Step Function

In the circuit theory the concept of the impulse response of a linear system due to its excitation by the Dirac delta function ƍ(t) together with the convolution principle is widely used and accepted. The rigorous theory of symbolic functions, sometimes called distributions, where also the delta function belongs, is rather abstract and requires subtle mathematical tools [1], [2], [3], [4]. Nevertheless, the most people intuitively well understand the delta function as a derivative of the (Heaviside) unit step function 1(t) without too much mathematical rigor. The concept of the impulse response of linear systems is here approached in a unified manner and generalized to the time-space phenomena in one dimension (transmission lines), as well as in a subsequent paper [5] to the phenomena in more dimensions (static and dynamic electromagnetic fields).

scholarly journals Плотность энергетического спектра электрона в поле изображения и запирающем электрическом поле

2021 ◽  
Vol 129 (2) ◽  
pp. 161
Author(s):  
П.А. Головинский ◽  
М.А. Преображенский ◽  
А.А. Дробышев
Keyword(s):  
Energy Spectrum ◽  
Metal Surface ◽  
Semiclassical Approximation ◽  
Energy Parameter ◽  
Step Function ◽  
Logarithmic Correction ◽  
Elementary Functions ◽  
Heaviside Step Function ◽  
Wide Range ◽  
Spectrum Density

In the semiclassical approximation, the density of the electron energy spectrum near the metal surface is described, when electron is bound by the image field and the blocking electrostatic field. In the system under consideration, the confinement mechanism is realized, and the energy spectrum for the motion of an electron in the direction perpendicular to the metal surface is completely discrete. The density of states of the energy spectrum is expressed in terms of elliptic integrals, the argument of which is a sigmoidal function. When the field is turned off, it becomes the Heaviside step function. A dimensionless energy parameter is introduced, which determines the intervals with qualitatively different changes in the width of the classically accessible region of motion. For large positive values of the energy parameter, the spectrum density asymptotically tends to the density in the triangular potential with the addition of the Coulomb logarithmic correction, and for negative values of the energy parameter, the spectrum density tends to dependence for a one-dimensional Coulomb potential. Approximate expressions are obtained for the spectrum density in terms of elementary functions in a wide range of electron energies and electric field strength.

scholarly journals An Analytic Exact form of Heaviside Function

2019 ◽  
Author(s):  
John Venetis
Keyword(s):  
Special Functions ◽  
Error Function ◽  
Operational Calculus ◽  
Delta Function ◽  
Step Function ◽  
Heaviside Function ◽  
Exact Form ◽  
Dirac Delta ◽  
Computational Procedures ◽  
Engineering Practices

In this paper, the author obtains an analytic exact form of Heaviside function, which is also known as Unit Step function and constitutes a fundamental concept of the Operational Calculus.In particulat, this function is explicitly expressed in a very simple manner by the aid of purely algebraic representations. The novelty of this work is that the proposed explicit formula is not performed in terms of non – elementary special functions, e.g. Dirac delta function or Error function and also is neither the limit of a function, nor the limit of a sequence of functions with point wise or uniform convergence. Hence, it may be much more appropriate and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

scholarly journals Proposed standardized definitions for vertical resolution and uncertainty in the NDACC lidar ozone and temperature algorithms – Part 1: Vertical resolution

2016 ◽  
Vol 9 (8) ◽  
pp. 4029-4049 ◽  
Author(s):  
Thierry Leblanc ◽  
Robert J. Sica ◽  
Joanna A. E. van Gijsel ◽  
Sophie Godin-Beekmann ◽  
Alexander Haefele ◽  
...  
Keyword(s):  
Data Processing ◽  
Programming Languages ◽  
Delta Function ◽  
Step Function ◽  
Atmospheric Composition ◽  
Vertical Resolution ◽  
Heaviside Step Function ◽  
Impulse Function ◽  
Definition Of ◽  
The Impact

Abstract. A standardized approach for the definition and reporting of vertical resolution of the ozone and temperature lidar profiles contributing to the Network for the Detection for Atmospheric Composition Change (NDACC) database is proposed. Two standardized definitions homogeneously and unequivocally describing the impact of vertical filtering are recommended. The first proposed definition is based on the width of the response to a finite-impulse-type perturbation. The response is computed by convolving the filter coefficients with an impulse function, namely, a Kronecker delta function for smoothing filters, and a Heaviside step function for derivative filters. Once the response has been computed, the proposed standardized definition of vertical resolution is given by Δz = δz  ×  HFWHM, where δz is the lidar's sampling resolution and HFWHM is the full width at half maximum (FWHM) of the response, measured in sampling intervals. The second proposed definition relates to digital filtering theory. After applying a Laplace transform to a set of filter coefficients, the filter's gain characterizing the effect of the filter on the signal in the frequency domain is computed, from which the cut-off frequency fC, defined as the frequency at which the gain equals 0.5, is computed. Vertical resolution is then defined by Δz = δz∕(2fC). Unlike common practice in the field of spectral analysis, a factor 2fC instead of fC is used here to yield vertical resolution values nearly equal to the values obtained with the impulse response definition using the same filter coefficients. When using either of the proposed definitions, unsmoothed signals yield the best possible vertical resolution Δz = δz (one sampling bin). Numerical tools were developed to support the implementation of these definitions across all NDACC lidar groups. The tools consist of ready-to-use “plug-in” routines written in several programming languages that can be inserted into any lidar data processing software and called each time a filtering operation occurs in the data processing chain. When data processing implies multiple smoothing operations, the filtering information is analytically propagated through the multiple calls to the routines in order for the standardized values of vertical resolution to remain theoretically and numerically exact at the very end of data processing.

scholarly journals Proposed standardized definitions for vertical resolution and uncertainty in the NDACC lidar ozone and temperature algorithms – Part 1: Vertical resolution

2016 ◽  
Author(s):  
Thierry Leblanc ◽  
Robert J. Sica ◽  
J. Anne E. van Gijsel ◽  
Sophie Godin-Beekman ◽  
Alexander Haefele ◽  
...  
Keyword(s):  
Data Processing ◽  
Programming Languages ◽  
Delta Function ◽  
Step Function ◽  
Atmospheric Composition ◽  
Vertical Resolution ◽  
Heaviside Step Function ◽  
Impulse Function ◽  
Definition Of ◽  
The Impact

Abstract. A standardized approach for the definition and reporting of vertical resolution of the ozone and temperature lidar profiles contributing to the Network for the Detection for Atmospheric Composition Change (NDACC) database is proposed. Two standardized definitions describing homogeneously and unequivocally the impact of vertical filtering are recommended. The first proposed definition is based on the width of the response to a Finite Impulse-type perturbation. The response is computed by convolving the filter coefficients with an impulse function, namely, a Kronecker Delta function for smoothing filters, and a Heaviside Step function for derivative filters. Once the response has been computed, the proposed standardized definition of vertical resolution is given by Δz = δz * HFWHM, where δz is the lidar’s sampling resolution and HFWHM is the full-width at half-maximum (FWHM) of the response, measured in sampling intervals. The second proposed definition relates to digital filtering theory. After applying a Laplace Transform to a set of filter coefficients, the filter’s gain characterizing the effect of the filter on the signal in the frequency-domain is computed, from which the cut-off frequency fC, defined as the frequency at which the gain equals 0.5, is computed. Vertical resolution is then defined by Δz = δz ⁄ (2fC). Unlike common practice in the field of spectral analysis, a factor 2fC instead of fC is constrained here to yield vertical resolution values nearly equal to the values obtained with the impulse response definition using the same filter coefficients. When using either of the proposed definitions, unsmoothed signals yield the best possible vertical resolution Δz = δz (one sampling bin). Numerical tools were developed to support the implementation of these definitions across all NDACC lidar groups. The tools consist of ready-to-use “plug-in” routines written in several programming languages that can be inserted into any lidar data processing software and called each time a filtering operation occurs in the data processing chain.

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