scholarly journals Stable recovery of planar regions with algebraic boundaries in Bernstein form

2021 ◽  
Vol 47 (2) ◽  
Author(s):  
Costanza Conti ◽  
Mariantonia Cotronei ◽  
Demetrio Labate ◽  
Wilfredo Molina
Keyword(s):  
Bernstein Polynomials ◽  
Linear Equations ◽  
Reconstruction Algorithm ◽  
Characteristic Functions ◽  
Bivariate Polynomials ◽  
Recovery Algorithm ◽  
Image Moments ◽  
Planar Regions ◽  
Set Up ◽  
Algebraic Domains

AbstractWe present a new method for the stable reconstruction of a class of binary images from a small number of measurements. The images we consider are characteristic functions of algebraic domains, that is, domains defined as zero loci of bivariate polynomials, and we assume to know only a finite set of uniform samples for each image. The solution to such a problem can be set up in terms of linear equations associated to a set of image moments. However, the sensitivity of the moments to noise makes the numerical solution highly unstable. To derive a robust image recovery algorithm, we represent algebraic polynomials and the corresponding image moments in terms of bivariate Bernstein polynomials and apply polynomial-generating, refinable sampling kernels. This approach is robust to noise, computationally fast and simple to implement. We illustrate the performance of our reconstruction algorithm from noisy samples through extensive numerical experiments. Our code is released open source and freely available.

scholarly journals Hybrid Bernstein Block-Pulse Functions Method for Second Kind Integral Equations with Convergence Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Mohsen Alipour ◽  
Dumitru Baleanu ◽  
Fereshteh Babaei
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Convergence Analysis ◽  
Bernstein Polynomials ◽  
Linear Equations ◽  
The Other ◽  
New Combination ◽  
System Of Linear Equations ◽  
Numerical Examples ◽  
Block Pulse Functions

We introduce a new combination of Bernstein polynomials (BPs) and Block-Pulse functions (BPFs) on the interval [0, 1]. These functions are suitable for finding an approximate solution of the second kind integral equation. We call this method Hybrid Bernstein Block-Pulse Functions Method (HBBPFM). This method is very simple such that an integral equation is reduced to a system of linear equations. On the other hand, convergence analysis for this method is discussed. The method is computationally very simple and attractive so that numerical examples illustrate the efficiency and accuracy of this method.

scholarly journals The non-modal onset of the tearing instability

2018 ◽  
Vol 84 (5) ◽  
Author(s):  
D. MacTaggart
Keyword(s):  
Initial Conditions ◽  
Linear Equations ◽  
Initial Perturbation ◽  
Transient Growth ◽  
Tearing Mode ◽  
Eigenvalue Spectrum ◽  
Tearing Instability ◽  
Set Up ◽  
Solution Behaviour ◽  
Physical Quantities

We investigate the onset of the classical magnetohydrodynamic (MHD) tearing instability (TI) and focus on non-modal (transient) growth rather than the tearing mode. With the help of pseudospectral theory, the operators of the linear equations are shown to be highly non-normal, resulting in the possibility of significant transient growth at the onset of the TI. This possibility increases as the Lundquist number$S$increases. In particular, we find evidence, numerically, that the maximum possible transient growth, measured in the$L_{2}$-norm, for the classical set-up of current sheets unstable to the TI, scales as$O(S^{1/4})$on time scales of$O(S^{1/4})$for$S\gg 1$. This behaviour is much faster than the time scale$O(S^{1/2})$when the solution behaviour is dominated by the tearing mode. The size of transient growth obtained is dependent on the form of the initial perturbation. Optimal initial conditions for the maximum possible transient growth are determined, which take the form of wave packets and can be thought of as noise concentrated at the current sheet. We also examine how the structure of the eigenvalue spectrum relates to physical quantities.

A new weighting scheme for arc based circle cone-beam CT reconstruction

2021 ◽  
pp. 1-19
Author(s):  
Wei Wang ◽  
Xiang-Gen Xia ◽  
Chuanjiang He ◽  
Zemin Ren ◽  
Jian Lu
Keyword(s):  
Characteristic Function ◽  
Weighting Function ◽  
Cone Beam Ct ◽  
Reconstruction Algorithm ◽  
Cone Beam ◽  
Characteristic Functions ◽  
Projection Data ◽  
Ct Reconstruction ◽  
Scanning Angle ◽  
Beam Geometry

In this paper, we present an arc based fan-beam computed tomography (CT) reconstruction algorithm by applying Katsevich’s helical CT image reconstruction formula to 2D fan-beam CT scanning data. Specifically, we propose a new weighting function to deal with the redundant data. Our weighting function ϖ ( x _ , λ ) is an average of two characteristic functions, where each characteristic function indicates whether the projection data of the scanning angle contributes to the intensity of the pixel x _ . In fact, for every pixel x _ , our method uses the projection data of two scanning angle intervals to reconstruct its intensity, where one interval contains the starting angle and another contains the end angle. Each interval corresponds to a characteristic function. By extending the fan-beam algorithm to the circle cone-beam geometry, we also obtain a new circle cone-beam CT reconstruction algorithm. To verify the effectiveness of our method, the simulated experiments are performed for 2D fan-beam geometry with straight line detectors and 3D circle cone-beam geometry with flat-plan detectors, where the simulated sinograms are generated by the open-source software “ASTRA toolbox.” We compare our method with the other existing algorithms. Our experimental results show that our new method yields the lowest root-mean-square-error (RMSE) and the highest structural-similarity (SSIM) for both reconstructed 2D and 3D fan-beam CT images.

Headings of UCAV Based on Nash Equilibrium

2018 ◽  
Vol 6 (3) ◽  
pp. 269-276
Author(s):  
Li Dai ◽  
Zheng Xie
Keyword(s):  
Nash Equilibrium ◽  
Linear Equations ◽  
Theoretic Model ◽  
Bounded Curvature ◽  
Smooth Path ◽  
Game Theoretic ◽  
Strategy Game ◽  
Simulation Results ◽  
Game Theoretic Model ◽  
Set Up

Abstract Given n vertices in a plane and UCAV going through each vertex once and only once and then coming back, the objective is to find the direction (heading) of motion in each vertex to minimize the smooth path of bounded curvature. This paper studies the headings of UCAV. First, the optimal headings for two vertices were given. On this basis, an n-player two-strategy game theoretic model was established. In addition, in order to obtain the mixed Nash equilibrium efficiently, n linear equations were set up. The simulation results demonstrated that the headings given in this paper are effective.

scholarly journals A New Regularized Reconstruction Algorithm Based on Compressed Sensing for the Sparse Underdetermined Problem and Applications of One-Dimensional and Two-Dimensional Signal Recovery

Algorithms ◽  
2019 ◽  
Vol 12 (7) ◽  
pp. 126 ◽  
Author(s):  
Bin Wang ◽  
Li Wang ◽  
Hao Yu ◽  
Fengming Xin
Keyword(s):  
Compressed Sensing ◽  
Reconstruction Algorithm ◽  
Steepest Descent Method ◽  
Signal Recovery ◽  
Two Dimensional ◽  
Reconstruction Algorithms ◽  
Experimental Conditions ◽  
One Dimensional ◽  
Recovery Algorithm ◽  
Reconstruction Performance

The compressed sensing theory has been widely used in solving undetermined equations in various fields and has made remarkable achievements. The regularized smooth L0 (ReSL0) reconstruction algorithm adds an error regularization term to the smooth L0(SL0) algorithm, achieving the reconstruction of the signal well in the presence of noise. However, the ReSL0 reconstruction algorithm still has some flaws. It still chooses the original optimization method of SL0 and the Gauss approximation function, but this method has the problem of a sawtooth effect in the later optimization stage, and the convergence effect is not ideal. Therefore, we make two adjustments to the basis of the ReSL0 reconstruction algorithm: firstly, we introduce another CIPF function which has a better approximation effect than Gauss function; secondly, we combine the steepest descent method and Newton method in terms of the algorithm optimization. Then, a novel regularized recovery algorithm named combined regularized smooth L0 (CReSL0) is proposed. Under the same experimental conditions, the CReSL0 algorithm is compared with other popular reconstruction algorithms. Overall, the CReSL0 algorithm achieves excellent reconstruction performance in terms of the peak signal-to-noise ratio (PSNR) and run-time for both a one-dimensional Gauss signal and two-dimensional image reconstruction tasks.

Analytic solution of a finite dam governed by a general input

1974 ◽  
Vol 11 (01) ◽  
pp. 134-144 ◽  
Author(s):  
S. K. Srinivasan
Keyword(s):  
Total Duration ◽  
Random Variables ◽  
Rational Functions ◽  
Characteristic Functions ◽  
Dry Period ◽  
Laplace Transform Technique ◽  
Independent Families ◽  
Set Up ◽  
Input Form ◽  
Finite Dam

A stochastic model of a finite dam in which the epochs of input form a renewal process is considered. It is assumed that the quantities of input at different epochs and the inter-input times are two independent families of random variables whose characteristic functions are rational functions. The release rate is equal to unity. An imbedding equation is set up for the probability frequency governing the water level in the first wet period and the resulting equation is solved by Laplace transform technique. Explicit expressions relating to the moments of the random variables representing the number of occasions in which the dam becomes empty as well as the total duration of the dry period in any arbitrary interval of time are indicated for negative exponentially distributed inter-input times.

Near Real-Time X-Ray Cone-Beam Microtomography

1999 ◽  
Vol 5 (S2) ◽  
pp. 940-941
Author(s):  
Shih Ang ◽  
Wang Ge ◽  
Cheng Ping-Chin
Keyword(s):  
Reconstruction Algorithm ◽  
Cone Beam ◽  
Reconstruction Process ◽  
X Ray ◽  
Beam Geometry ◽  
Cone Beam Reconstruction ◽  
Cylindrical Coordinate ◽  
Contrast Mechanism ◽  
Penetration Ability ◽  
Set Up

Due to the penetration ability and absorption contrast mechanism, cone-beam X-ray microtomography is a powerful tool in studying 3D microstructures in opaque specimens. In contrast to the conventional parallel and fan-beam geometry, the cone-beam tomography set up is highly desirable for faster data acquisition, build-in magnification, better radiation utilization and easier hardware implementation. However, the major draw back of the cone-beam reconstruction is its computational complexity. In an effort to maximize the reconstruction speed, we have developed a generalized Feldkamp cone-beam reconstruction algorithm to optimize the reconstruction process. We report here the use of curved voxels in a cylindrical coordinate system and mapping tables to further improve the reconstruction efficiency.The generalized Feldkamp cone-beam image reconstruction algorithm is reformulated utilizing mapping table in the discrete domain as: , where .

Finite amplitude cellular convection

1958 ◽  
Vol 4 (3) ◽  
pp. 225-260 ◽  
Author(s):  
W. V. R. Malkus ◽  
G. Veronis
Keyword(s):  
Rayleigh Number ◽  
Prandtl Number ◽  
Heat Transport ◽  
Stability Problem ◽  
Dominant Role ◽  
Linear Equations ◽  
Finite Amplitude ◽  
Set Up ◽  
Rayleigh Numbers ◽  
Linear Stability Problem

When a layer of fluid is heated uniformly from below and cooled from above, a cellular regime of steady convection is set up at values of the Rayleigh number exceeding a critical value. A method is presented here to determine the form and amplitude of this convection. The non-linear equations describing the fields of motion and temperature are expanded in a sequence of inhomogeneous linear equations dependent upon the solutions of the linear stability problem. We find that there are an infinite number of steady-state finite amplitude solutions (having different horizontal plan-forms) which formally satisfy these equations. A criterion for ‘relative stability’ is deduced which selects as the realized solution that one which has the maximum mean-square temperature gradient. Particular conclusions are that for a large Prandtl number the amplitude of the convection is determined primarily by the distortion of the distribution of mean temperature and only secondarily by the self-distortion of the disturbance, and that when the Prandtl number is less than unity self-distortion plays the dominant role in amplitude determination. The initial heat transport due to convection depends linearly on the Rayleigh number; the heat transport at higher Rayleigh numbers departs only slightly from this linear dependence. Square horizontal plan-forms are preferred to hexagonal plan-forms in ordinary fluids with symmetric boundary conditions. The proposed finite amplitude method is applicable to any model of shear flow or convection with a soluble stability problem.

X-ray Nano- and Micro-tomography in an SEM

2013 ◽  
Vol 21 (2) ◽  
pp. 24-28 ◽  
Author(s):  
Bart Pauwels ◽  
Alexander Sasov
Keyword(s):  
Computer Tomography ◽  
Small Angle ◽  
Reconstruction Algorithm ◽  
X Ray ◽  
Projection Images ◽  
Micro Tomography ◽  
Ray Projection ◽  
Set Up ◽  
Rotation Stage ◽  
Non Destructive

X-ray microfocus computer tomography (μ-CT) is a non-destructive experimental technique that reveals the 3D internal microstructure of the sample under study. The experimental set-up consists of an X-ray source, an X-ray detector, and set in between is a sample that is placed on a rotation stage. With this set-up multiple X-ray projection images can be obtained from the sample at different angles. In between the acquisition of two successive images, the sample is rotated over a small angle, typically between 0.2° and 1°. This set of projection images is then used as input for the reconstruction algorithm, which calculates a reconstruction of the internal microstructure of the sample with (sub-) micrometer sensitivity.

scholarly journals Application of a Modified Generative Adversarial Network in the Superresolution Reconstruction of Ancient Murals

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Jianfang Cao ◽  
Zibang Zhang ◽  
Aidi Zhao
Keyword(s):  
High Resolution ◽  
Signal To Noise Ratio ◽  
Reconstruction Algorithm ◽  
Structural Similarity ◽  
Image Feature ◽  
Generative Adversarial Network ◽  
Resolution Image ◽  
High Resolution Image ◽  
Adversarial Network ◽  
Set Up

Considering the problems of low resolution and rough details in existing mural images, this paper proposes a superresolution reconstruction algorithm for enhancing artistic mural images, thereby optimizing mural images. The algorithm takes a generative adversarial network (GAN) as the framework. First, a convolutional neural network (CNN) is used to extract image feature information, and then, the features are mapped to the high-resolution image space of the same size as the original image. Finally, the reconstructed high-resolution image is output to complete the design of the generative network. Then, a CNN with deep and residual modules is used for image feature extraction to determine whether the output of the generative network is an authentic, high-resolution mural image. In detail, the depth of the network increases, the residual module is introduced, the batch standardization of the network convolution layer is deleted, and the subpixel convolution is used to realize upsampling. Additionally, a combination of multiple loss functions and staged construction of the network model is adopted to further optimize the mural image. A mural dataset is set up by the current team. Compared with several existing image superresolution algorithms, the peak signal-to-noise ratio (PSNR) of the proposed algorithm increases by an average of 1.2–3.3 dB and the structural similarity (SSIM) increases by 0.04 = 0.13; it is also superior to other algorithms in terms of subjective scoring. The proposed method in this study is effective in the superresolution reconstruction of mural images, which contributes to the further optimization of ancient mural images.

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