Smooth Kernel Research Articles

EFFECTS OF SMOOTH, MEDIUM SMOOTH AND MEDIUM RECONSTRUCTION KERNELS ON IMAGE QUALITY IN THREE-PHASE CT OF LIVER

Reconstruction kernel is one of the parameters that affects the computed tomography (CT) image quality. This study aimed to evaluate the effects of applying three different reconstruction kernels on image quality in 3-phased CT of the liver. A total of 63 CT liver images including normal liver (n = 43) and liver lesion (n = 20) were retrospectively reviewed. Smooth (B20f), medium smooth (B30f) and medium (B40f) reconstruction kernels were employed in the image reconstruction process. Mean attenuation, image noise, and signal-to-noise ratio (SNR) values from each kernel reconstruction were quantified and compared among those kernels using One Way Analysis of Variance (ANOVA) statistical analysis. Significant changes in image noise and SNR were observed in the normal liver (p < 0.001, respectively) following the application of those reconstruction kernels. However, no significant changes in mean attenuation, image noise, and SNR were demonstrated in the liver lesion (p > 0.05). Application of smooth (B20f), medium smooth (B30f), and medium (B40f) kernel reconstructions would significantly affect the image noise and SNR in the normal liver of CT images instead of liver lesions. Hence, proper selection of reconstruction kernel is important in CT images reconstruction to improve precision in diagnostic CT interpretation.

An efficient wavelet-based numerical method to solve nonlinear Fredholm integral equation of second kind with smooth kernel

In this paper, a wavelet-based numerical algorithm is described to obtain approximate numerical solution of a class of nonlinear Fredholm integral equations of second kind having smooth kernels. The algorithm involves approximation of the unknown function in terms of Daubechies scale functions. The properties of Daubechies scale and wavelet functions together with one-point quadrature rule for the product of a smooth function and Daubechies scale functions are utilized to transform the integral equation to a system of nonlinear equations. The efficiency of the proposed method is demonstrated through three illustrative examples.

Influence of reconstruction kernels on the accuracy of CT-derived fractional flow reserve

ObjectivesWe evaluated the influence of image reconstruction kernels on the diagnostic accuracy of CT-derived fractional flow reserve (FFRCT) compared to invasive FFR in patients with coronary artery disease.MethodsSixty-nine patients, in whom coronary CT angiography was performed and who were further referred for invasive coronary angiography with FFR measurement via pressure wire, were retrospectively included. CT data sets were acquired using a third-generation dual-source CT system and rendered with medium smooth (Bv40) and sharp (Bv49) reconstruction kernels. FFRCT was calculated on-site using prototype software. Coronary stenoses with invasive FFR ≤ 0.80 were classified as significant. Agreement between FFRCT and invasive FFR was determined for both reconstruction kernels.ResultsOne hundred analyzed vessels in 69 patients were included. Twenty-five vessels were significantly stenosed according to invasive FFR. Using a sharp reconstruction kernel for FFRCT resulted in a significantly higher correlation with invasive FFR (r = 0.74, p < 0.01 vs. r = 0.58, p < 0.01; p = 0.04) and a higher AUC in ROC curve analysis to correctly identify/exclude significant stenosis (AUC = 0.92 vs. AUC = 0.82 for sharp vs. medium smooth kernel, respectively, p = 0.02). A FFRCT value of ≤ 0.8 using a sharp reconstruction kernel showed a sensitivity of 88% and a specificity of 92% for detecting ischemia-causing lesions, resulting in a diagnostic accuracy of 91%. The medium smooth reconstruction kernel performed worse (sensitivity 60%, specificity 89%, accuracy 82%).ConclusionCompared to invasively measured FFR, FFRCT using a sharp image reconstruction kernel shows higher diagnostic accuracy for detecting lesions causing ischemia, potentially altering decision-making in a clinical setting.Key Points• Image reconstruction parameters influence the diagnostic accuracy of simulated fractional flow reserve derived from coronary computed tomography angiography.• Using a sharp kernel image reconstruction algorithm delivers higher diagnostic accuracy compared to medium smooth kernel image reconstruction (gold standard invasive fractional flow reserve).

A numerical method for linear Volterra integral equations on infinite intervals and its application to the resolution of metastatic tumor growth models

A Nyström method for linear second kind Volterra integral equations on unbounded intervals, with sufficiently smooth kernels, is described. The procedure is based on the use of a truncated Lagrange interpolation process and of a truncated Gaussian quadrature formula. The stability and the convergence of the method in suitable weighted spaces of functions are studied and some numerical examples showing its reliability are presented. In particular, the proposed method has been tested for the numerical resolution of some Volterra integral equations arising from the reformulation of differential models describing metastatic tumor growth whose unknown solutions represent biological observables as the metastatic mass or the number of metastases.

A fast and oblivious matrix compression algorithm for Volterra integral operators

The numerical solution of dynamical systems with memory requires the efficient evaluation of Volterra integral operators in an evolutionary manner. After appropriate discretization, the basic problem can be represented as a matrix-vector product with a lower diagonal but densely populated matrix. For typical applications, like fractional diffusion or large-scale dynamical systems with delay, the memory cost for storing the matrix approximations and complete history of the data then becomes prohibitive for an accurate numerical approximation. For Volterra integral operators of convolution type, the fast and oblivious convolution quadrature method of Schädle, Lopez-Fernandez, and Lubich resolves this issue and allows to compute the discretized evaluation with N time steps in O(N log N) complexity and only requires O(log N) active memory to store a compressed version of the complete history of the data. We will show that this algorithm can be interpreted as an {{mathscr{H}}}-matrix approximation of the underlying integral operator. A further improvement can thus be achieved, in principle, by resorting to {{mathscr{H}}}^{2}-matrix compression techniques. Following this idea, we formulate a variant of the {{mathscr{H}}}^{2}-matrix-vector product for discretized Volterra integral operators that can be performed in an evolutionary and oblivious manner and requires only O(N) operations and O(log N) active memory. In addition to the acceleration, more general asymptotically smooth kernels can be treated and the algorithm does not require a priori knowledge of the number of time steps. The efficiency of the proposed method is demonstrated by application to some typical test problems.

The Lp-to-Lq boundedness of commutators with applications to the Jacobian operator

Supplying the missing necessary conditions, we complete the characterisation of the Lp→Lq boundedness of commutators [b,T] of pointwise multiplication and Calderón–Zygmund operators, for arbitrary pairs of 1<p,q<∞ and under minimal non-degeneracy hypotheses on T.For p≤q (and especially p=q), this extends a long line of results under more restrictive assumptions on T. In particular, we answer a recent question of Lerner, Ombrosi, and Rivera-Ríos by showing that b∈BMO is necessary for the Lp-boundedness of [b,T] for any non-zero homogeneous singular integral T. We also deal with iterated commutators and weighted spaces.For p>q, our results are new even for special classical operators with smooth kernels. As an application, we show that every f∈Lp(Rd) can be represented as a convergent series of normalised Jacobians Ju=det⁡∇u of u∈W˙1,dp(Rd)d. This extends, from p=1 to p>1, a result of Coifman, Lions, Meyer and Semmes about J:W˙1,d(Rd)d→H1(Rd), and supports a conjecture of Iwaniec about the solvability of the equation Ju=f∈Lp(Rd).

Cones, rectifiability, and singular integral operators

Let \mu be a Radon measure on \mathbb{R}^d . We define and study conical energies \mathcal{E}_{\mu,p}(x,V,\alpha) , which quantify the portion of \mu lying in the cone with vertex x\in\mathbb{R}^d , direction V\in G(d,d-n) , and aperture \alpha\in (0,1) . We use these energies to characterize rectifiability and the big pieces of Lipschitz graphs property. Furthermore, if we assume that \mu has polynomial growth, we give a sufficient condition for L^2(\mu) -boundedness of singular integral operators with smooth odd kernels of convolution type.

Propriety of the reference posterior distribution in Gaussian process modeling

In a seminal article, Berger, De Oliveira and Sansó [J. Amer. Statist. Assoc. 96 (2001) 1361–1374] compare several objective prior distributions for the parameters of Gaussian process models with isotropic correlation kernel. The reference prior distribution stands out among them insofar as it always leads to a proper posterior. They prove this result for rough correlation kernels: Spherical, Exponential with power ρ<2, Matérn with smoothness ν<1. This paper provides a proof for smooth correlation kernels: Exponential with power ρ=2, Matérn with smoothness ν≥1, Rational Quadratic, along with tail rates of the reference prior for these kernels.

Unraveling the interplay of image formation, data representation and learning in CT-based COPD phenotyping automation: The need for a meta-strategy.

In the literature on automated phenotyping of chronic obstructive pulmonary disease (COPD), there is a multitude of isolated classical machine learning and deep learning techniques, mostly investigating individual phenotypes, with small study cohorts and heterogeneous meta-parameters, e.g., different scan protocols or segmented regions. The objective is to compare the impact of different experimental setups, i.e., varying meta-parameters related to image formation and data representation, with the impact of the learning technique for subtyping automation for a variety of phenotypes. The identified associations of these parameters with automation performance and their interactions might be a first step towards a determination of optimal meta-parameters, i.e., a meta-strategy. A clinical cohort of 981 patients (53.8±15.1years, 554male) was examined. The inspiratory CT images were analyzed to automate the diagnosis of 13 COPD phenotypes given by two radiologists. A benchmark feature set that integrates many quantitative criteria was extracted from the lung and trained a variety of learning algorithms on the first 654 patients (two thirds) and the respective algorithm retrospectively assessed the remaining 327 patients (one third). The automation performance was evaluated by the area under the receiver operating characteristic curve (AUC). 1717 experiments were conducted with varying meta-parameters such as reconstruction kernel, segmented regions and input dimensionality, i.e., number of extracted features. The association of the meta-parameters with the automation performance was analyzed by multivariable general linear model decomposition of the automation performance in the contributions of meta-parameters and the learning technique. The automation performance varied strongly for varying meta-parameters. For emphysema-predominant phenotypes, an AUC of 93%-95% could be achieved for the best meta-configuration. The airways-predominant phenotypes led to a lower performance of 65%-85%, while smooth kernel configurations on average were unexpectedly superior to those with sharp kernels. The performance impact of meta-parameters, even that of often neglected ones like the missing-data imputation, was in general larger than that of the learning technique. Advanced learning techniques like 3D deep learning or automated machine learning yielded inferior automation performance for non-optimal meta-configurations in comparison to simple techniques with suitable meta-configurations. The best automation performance was achieved by a combination of modern learning techniques and a suitable meta-configuration. Our results indicate that for COPD phenotype automation, study design parameters such as reconstruction kernel and the model input dimensionality should be adapted to the learning technique and may be more important than the technique itself. To achieve optimal automation and prediction results, the interaction between input those meta-parameters and the learning technique should be considered. This might be particularly relevant for the development of specific scan protocols for novel learning algorithms, and towards an understanding of good study design for automated phenotyping.

Evaluating a Convolutional Neural Network Noise Reduction Method When Applied to CT Images Reconstructed Differently Than Training Data.

The aim of this study was to evaluate a narrowly trained convolutional neural network (CNN) denoising algorithm when applied to images reconstructed differently than training data set. A residual CNN was trained using 10 noise inserted examinations. Training images were reconstructed with 275 mm of field of view (FOV), medium smooth kernel (D30), and 3 mm of thickness. Six examinations were reserved for testing; these were reconstructed with 100 to 450 mm of FOV, smooth to sharp kernels, and 1 to 5 mm of thickness. When test and training reconstruction settings were not matched, there was either reduced denoising efficiency or resolution degradation. Denoising efficiency was reduced when FOV was decreased or a smoother kernel was used. Resolution loss occurred when the network was applied to an increased FOV, sharper kernel, or decreased image thickness. The CNN denoising performance was degraded with variations in FOV, kernel, or decreased thickness. Denoising performance was not affected by increased thickness.

Low Rank Education of Cascade Loss Sensitivity to Unsteady Parameters by Proper Orthogonal Decomposition

AbstractIn this study, proper orthogonal decomposition (POD) has been applied to a large dataset describing the profile losses of low-pressure turbine (LPT) cascades, thus allowing (i) the identification of the most influencing parameters that affect the loss generation; (ii) the identification of the minimum number of requested conditions useful to educate a model with a reduced number of data. The dataset is constituted by the total pressure loss coefficient distributions in the pitchwise direction. The experiments have been conducted varying the flow Reynolds number, the reduced frequency, and the flow coefficient. Two cascades are considered: the first for tuning the procedure and identifying the number of really requested tests, and the second for the verification of the proposed model. They are characterized by the same axial chord but different pitch-to-chord ratio and different flow angles, hence two Zweifel numbers. The POD mode distributions indicate the spatial region where losses occur, the POD eigenvectors provide how such losses vary for different design conditions and the POD eigenvalues provide the rank of the approximation. Since the POD space shows an optimal basis describing the overall process with a low-rank representation (LRR), a smooth kernel is educated by means of least-squares method (LSM) on the POD eigenvectors. Particularly, only a subset of data (equal to the rank of the problem) has been used to generate the POD modes and related coefficients. Thanks to the LRR of the problem in the POD space, predictors are low-order polynomials of the independent variables (Re, f+, and ϕ). It will be shown that the smooth kernel adequately estimates the loss distribution in points that do not participate to the education. In addition, keeping the same steps for the education of the kernel on another cascade, loss distribution and magnitude are still well captured. Thus, the analysis show that the rank of the problem is much lower than the tested conditions, and consequently, a reduced number of tests are really necessary. This could be useful to reduce the number of hi-fidelity simulations or detailed experiments in the future, thus further contributing to optimize LPT blades.

Mixed Fourier Legendre spectral Galerkin methods for two-dimensional Fredholm integral equations of the second kind

In this article, the mixed Fourier Legendre spectral Galerkin (MFLSG) methods are considered to solve the two-dimensional Fredholm integral equations (fies) on the Banach spaces with smooth kernel. The same methods are also considered to find the eigenvalues of the eigenvalue problems (evps) associated with the two-dimensional fies. Making use of these methods, we establish the error between the approximated solution as well as iterated approximate solution versus exact solution for two-dimensional fies in both L2 and L∞ norms. We also establish the error between approximated eigen-values, eigen-vectors and iterated eigen-vectors and exact eigen-elements by MFLSG methods in L2 and L∞ norms. The numerical illustrations are introduced for the error of these methods.

Convergence analysis of Galerkin and multi-Galerkin methods for nonlinear-Hammerstein integral equations on the half-line using Laguerre polynomials

In this paper, we consider Galerkin and multi-Galerkin methods and their iterated versions for solving the nonlinear Hammerstein-type integral equation on the half-line with sufficiently smooth kernels, using Laguerre polynomials as basis functions. We obtain optimal convergence results in iterated-Galerkin method in both infinity and weighted -norms. We also obtain the superconvergence results in both multi-Galerkin and iterated multi-Galerkin methods, respectively, in weighted -norm. Numerical results are presented to validate the theoretical results.

Gradient bounds for radial maximal functions

In this paper we study the regularity properties of certain maximal operators of convolution type at the endpoint \(p=1\), when acting on radial data. In particular, for the heat flow maximal operator and the Poisson maximal operator, when the initial datum \(u_0 \in W^{1,1}(\mathbf{R}^d)\) is a radial function, we show that the associated maximal function \(u^*\) is weakly differentiable and \[\|\nabla u^*\|_{L^1(\mathbf{R}^d)} \lesssim_d \|\nabla u_0\|_{L^1(\mathbf{R}^d)}.\] This establishes the analogue of a recent result of Luiro for the uncentered Hardy-Littlewood maximal operator, now in a centered setting with smooth kernels. In a second part of the paper, we establish similar gradient bounds for maximal operators on the sphere \(\mathbf{S}^d\), when acting on polar functions. Our study includes the uncentered Hardy-Littlewood maximal operator, the heat flow maximal operator and the Poisson maximal operator on \(\mathbf{S}^d\).

Detection of Microcalcifications in Spiral Breast Computed Tomography with Photon-Counting Detector Is Feasible: A Specimen Study.

The primary objective of the study was to compare a spiral breast computed tomography system (SBCT) to digital breast tomosynthesis (DBT) for the detection of microcalcifications (MCs) in breast specimens. The secondary objective was to compare various reconstruction modes in SBCT. In total, 54 breast biopsy specimens were examined with mammography as a standard reference, with DBT, and with a dedicated SBCT containing a photon-counting detector. Three different reconstruction modes were applied for SBCT datasets (Recon1 = voxel size (0.15 mm)3, smooth kernel; Recon2 = voxel size (0.05 mm)3, smooth kernel; Recon3 = voxel size (0.05 mm)3, sharp kernel). Sensitivity and specificity of DBT and SBCT for the detection of suspicious MCs were analyzed, and the McNemar test was used for comparisons. Diagnostic confidence of the two readers (Likert Scale 1 = not confident; 5 = completely confident) was analyzed with ANOVA. Regarding detection of MCs, reader 1 had a higher sensitivity for DBT (94.3%) and Recon2 (94.9%) compared to Recon1 (88.5%; p < 0.05), while sensitivity for Recon3 was 92.4%. Respectively, reader 2 had a higher sensitivity for DBT (93.0%), Recon2 (92.4%), and Recon3 (93.0%) compared to Recon1 (86.0%; p < 0.05). Specificities ranged from 84.7–94.9% for both readers (p > 0.05). The diagnostic confidence of reader 1 was better with SBCT than with DBT (DBT 4.48 ± 0.88, Recon1 4.77 ± 0.66, Recon2 4.89 ± 0.44, and Recon3 4.75 ± 0.72; DBT vs. Recon1/2/3: p < 0.05), while reader 2 found no differences. Sensitivity and specificity for the detection of MCs in breast specimens is equal for DBT and SBCT when a small voxel size of (0.05 mm)3 is used with an equal or better diagnostic confidence for SBCT compared to DBT.

Multilevel augmentation methods for eigen-problems of compact integral operators

In this paper, we combine the ideas of multilevel augmentation methods for solving integral equations and the shifted-inverse power method to develop a new multilevel augmentation method for solving eigen-problem of compact integral operators with smooth kernels. We first solve an eigen-problem in a suitable initial coarse level and then seek a more accurate approximation from solving a linear system on a finer mesh. Moreover, we need only to deal with a small linear system corresponding to the initial coarse level when we solve the linear system on a finer mesh. The method exhibits to be convenient for implementing adaptivity, and reduces the computational cost greatly and leads to the method faster. Numerical examples are presented to illustrate the theoretical estimates for the error of these methods.

Spatial Shrinkage Via the Product Independent Gaussian Process Prior

We study the problem of sparse signal detection on a spatial domain. We propose a novel approach to model continuous signals that are sparse and piecewise-smooth as the product of independent Gaussian (PING) processes with a smooth covariance kernel. The smoothness of the PING process is ensured by the smoothness of the covariance kernels of the Gaussian components in the product, and sparsity is controlled by the number of components. The bivariate kurtosis of the PING process implies that more components in the product results in the thicker tail and sharper peak at zero. We develop an efficient computation algorithm based on spectral methods. The simulation results demonstrate superior estimation using the PING prior over Gaussian process prior for different image regressions. We apply our method to a longitudinal magnetic resonance imaging dataset to detect the regions that are affected by multiple sclerosis computation in this domain. Supplementary materials for this article are available online.

Chebyshev Spectral Projection Methods for Two-Dimensional Fredholm Integral Equations of Second Kind

In this paper, we approximate the two-dimensional linear Fredholm integral equations (fies) with smooth kernels using Chebyshev spectral Galerkin and collocation methods. The existence and convergence analysis for the problem have been discussed. The errors between approximated solutions with exact solution have been evaluated in $$L^2_\omega $$ norm applying both these methods. We also solve the associated eigenvalue problems (evps) by using above methods and obtain the errors between approximated eigenelements with the exact eigenelements in $$L^2_\omega $$ and $$L^\infty _\omega $$ norms. Numerical examples are presented to illustrate the theoretical results.

An integral model based on slender body theory, with applications to curved rigid fibers

We propose a novel integral model describing the motion of both flexible and rigid slender fibers in viscous flow and develop a numerical method for simulating dynamics of curved rigid fibers. The model is derived from nonlocal slender body theory (SBT), which approximates flow near the fiber using singular solutions of the Stokes equations integrated along the fiber centerline. In contrast to other models based on (singular) SBT, our model yields a smooth integral kernel which incorporates the (possibly varying) fiber radius naturally. The integral operator is provably negative definite in a nonphysical idealized geometry, as expected from the partial differential equation theory. This is numerically verified in physically relevant geometries. We discuss the convergence and stability of a numerical method for solving the integral equation. The accuracy of the model and method is verified against known models for ellipsoids. Finally, we develop an algorithm for computing dynamics of rigid fibers with complex geometries in the case where the fiber density is much greater than that of the fluid, for example, in turbulent gas-fiber suspensions.

Lagrangian–Eulerian multidensity topology optimization with the material point method

AbstractIn this paper, a hybrid Lagrangian–Eulerian topology optimization (LETO) method is proposed to solve the elastic force equilibrium with the Material Point Method (MPM). LETO transfers density information from freely movable Lagrangian carrier particles to a fixed set of Eulerian quadrature points. This transfer is based on a smooth radial kernel involved in the compliance objective to avoid the artificial checkerboard pattern. The quadrature points act as MPM particles embedded in a lower‐resolution grid and enable a subcell multidensity resolution of intricate structures with a reduced computational cost. A quadrature‐level connectivity graph‐based method is adopted to avoid the artificial checkerboard issues commonly existing in multiresolution topology optimization methods. Numerical experiments are provided to demonstrate the efficacy of the proposed approach.