Projection Kernels Research Articles

A Cube-Based Dual-GPU List-Mode Reconstruction Algorithm for PET Imaging

Graphics processing units (GPUs) have been a popular choice for the acceleration of iterative tomographic reconstruction algorithms. In this work, two dual-GPU approaches to iterative list-mode reconstruction of PET data are studied. The developed techniques split the image into equal-size cubes and distribute them between the GPUs. Benefiting from caching a plane of the image into the fast shared memory, line-driven forward and back projection kernels are implemented incorporating a shift-invariant point spread function (PSF). In the first approach, lines were individually convolved with the PSF kernel whereas the second approach convolves the reconstructed image with the PSF before the forward projection and after the back projection. The algorithms were implemented using two CUDA-compatible devices with no CPU/GPU data transfer during the reconstruction process. List-mode data obtained from Monte Carlo simulations were used to evaluate the methods. One complete iteration of 107 events on an image of 220 × 220 × 110 voxels is done in less than 510 ms for a non-PSF reconstruction and in 520 to 5220 ms for a PSF reconstruction depending on the PSF and the reconstruction parameters. Dual-GPU memory layout, implementation details, sample images, and performance metrics of the developed algorithms are discussed.

A hierarchy of Palm measures for determinantal point processes with gamma kernels

The gamma kernels are a family of projection kernels $K^{(z,z’)}= K^{(z,z’)}(x,y)$ on a doubly infinite one-dimensional lattice. They are expressed through Euler’s gamma function and depend on two continuous parameters $z,z’$. The gamma kernels initia

Efficient projection kernels for discontinuous Galerkin simulations of disperse multiphase flows on arbitrary curved elements

In this work, we develop projection kernels for Euler-Lagrange point-particle simulations of disperse multiphase flows on arbitrary curved elements. These kernels are employed in a high-order discontinuous Galerkin framework for projecting the action of the particles to the Eulerian mesh. Instead of commonly used isotropic kernels, such as a Gaussian-type kernel, we construct an anisotropic polynomial-based smoothing function that preserves the compactness of the discontinuous Galerkin method on high-aspect-ratio elements and maintains an acceptable computational cost. At the same time, it mitigates inaccuracies associated with larger particles and numerical instabilities arising from the Dirac delta function and low-order kernels. Specifically, the geometric mapping of a physical element to the reference element is exploited to construct a kernel that is elliptical in 2D and ellipsoidal in 3D. We also develop a strategy to conserve interphase transfer near boundaries, particularly on curved elements. This strategy employs a high-order polynomial approximation to appropriately rescale the interphase source terms in an efficient manner. The compatibility of the proposed methodology with different types of meshes is investigated. We then apply it to a number of particle-laden flow configurations, including supersonic dusty flow over a flat plate, moving shocks interacting with clouds of particles, and hypersonic dusty flows over blunt bodies.

Aliasing mitigation in optical microscopy of dynamic biological samples by use of temporally modulated color illumination and a standard RGB camera.

.Significance: Despite recent developments in microscopy, temporal aliasing can arise when imaging dynamic samples. Modern sampling frameworks, such as generalized sampling, mitigate aliasing but require measurement of temporally overlapping and potentially negative-valued inner products. Conventional cameras cannot collect these directly as they operate sequentially and are only sensitive to light intensity.Aim: We aim to mitigate aliasing in microscopy of dynamic monochrome samples by implementing generalized sampling via the use of a color camera and modulated color illumination.Approach: We solve the overlap problem by spectrally multiplexing the acquisitions and using (positive) B-spline segments as projection kernels. Reconstruction involves spectral unmixing and inverse filtering. We implemented this method using a color LED illuminator. We evaluated its performance by imaging a rotating grid and its applicability by imaging the beating zebrafish embryo heart in transmission and light-sheet microscopes.Results: Compared to stroboscopic imaging, our method mitigates aliasing with performance improving as the projection order increases. The approach can be implemented in conventional microscopes but is limited by the number of available LED colors and camera channels.Conclusions: Generalized sampling can be implemented via color modulation in microscopy to mitigate temporal aliasing. The simple hardware requirements could make it applicable to other optical imaging modalities.

A General Method for Estimating Bulk 2D Projections of Ice Particle Shape: Theory and Applications

Abstract The orientation of falling ice particles directly influences estimates of microphysical and radiative bulk quantities as well as in situ retrievals of size, shape, and mass. However, retrieval efforts and bulk calculations often incorporate very basic orientations or ignore these effects altogether. To address this deficiency, this study develops a general method for projecting bulk distributions of particle shape for arbitrary orientations. The Amoroso distribution provides the most general bulk aspect ratio distribution for gamma-distributed particle axis lengths. The parameters that govern the behavior of this aspect ratio distribution depend on the assumed relationship between mass, maximum dimension, and aspect ratio. Individual spheroidal geometry allows for eccentricity quantities to linearly map onto ellipse analogs, whereas aspect ratio quantities map nonlinearly. For particles viewed from their side, this analytic distinction leads to substantially larger errors in projected aspect ratio than for projected eccentricity. Distribution transformations using these mapping equations and numerical integration of projection kernels show that both truncation of size distributions and changes in Gaussian dispersion can alter the modality and shape of projection distributions. As a result, the projection process can more than triple the relative entropy between the spheroidal and projection distributions for commonly assumed model and orientation parameters. This shape uncertainty is maximized for distributions of highly eccentric particles and for particles like aggregates that are thought to fall with large canting-angle deviations. As a result, the methods used to report projected aspect ratios and the corresponding values should be questioned.

Testing the cosmic shear spatially-flat universe approximation with generalized lensing and shear spectra

We introduce the Generalised Lensing and Shear Spectra GLaSS code which is available for download from https://github.com/astro-informatics/GLaSS It is a fast and flexible public code, written in Python, that computes generalized spherical cosmic shear spectra. The commonly used tomographic and spherical Bessel lensing spectra come as built-in run-mode options. GLaSS is integrated into the Cosmosis modular cosmological pipeline package. We outline several computational choices that accelerate the computation of cosmic shear power spectra. Using GLaSS, we test whether the assumption that using the lensing and projection kernels for a spatially-flat universe -- in a universe with a small amount of spatial curvature -- negligibly impacts the lensing spectrum. We refer to this assumption as The Spatially-Flat Universe Approximation, that has been implicitly assumed in all cosmic shear studies to date. We confirm that The Spatially-Flat Universe Approximation has a negligible impact on Stage IV cosmic shear experiments.

Free Fermions and the Classical Compact Groups

There is a close connection between the ground state of non-interacting fermions in a box with classical (absorbing, reflecting, and periodic) boundary conditions and the eigenvalue statistics of the classical compact groups. The associated determinantal point processes can be extended in two natural directions: (i) we consider the full family of admissible quantum boundary conditions (i.e., self-adjoint extensions) for the Laplacian on a bounded interval, and the corresponding projection correlation kernels; (ii) we construct the grand canonical extensions at finite temperature of the projection kernels, interpolating from Poisson to random matrix eigenvalue statistics. The scaling limits in the bulk and at the edges are studied in a unified framework, and the question of universality is addressed. Whether the finite temperature determinantal processes correspond to the eigenvalue statistics of some matrix models is, a priori, not obvious. We complete the picture by constructing a finite temperature extension of the Haar measure on the classical compact groups. The eigenvalue statistics of the resulting grand canonical matrix models (of random size) corresponds exactly to the grand canonical measure of free fermions with classical boundary conditions.

Polynomial Ensembles and Recurrence Coefficients

Polynomial ensembles are determinantal point processes associated with (non necessarily orthogonal) projections onto polynomial subspaces. The aim of this survey article is to put forward the use of recurrence coefficients to obtain the global asymptotic behavior of such ensembles in a rather simple way. We provide a unified approach to recover well-known convergence results for real OP ensembles. We study the mutual convergence of the polynomial ensemble and the zeros of its average characteristic polynomial; we discuss in particular the complex setting. We also control the variance of linear statistics of polynomial ensembles and derive comparison results, as well as asymptotic formulas for real OP ensembles. Finally, we reinterpret the classical algorithm to sample determinantal point processes so as to cover the setting of non-orthogonal projection kernels. A few open problems are also suggested.

Adaptive wavelet multivariate regression with errors in variables

In the multidimensional setting, we consider the errors-in- variables model. We aim at estimating the unknown nonparametric multivariate regression function with errors in the covariates. We devise an adaptive estimators based on projection kernels on wavelets and a deconvolution operator. We propose an automatic and fully data driven procedure to select the wavelet level resolution. We obtain an oracle inequality and optimal rates of convergence over anisotropic Hölder classes. Our theoretical results are illustrated by some simulations.

Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Caustic

We study the scaling asymptotics of the eigenspace projection kernels $\Pi_{\hbar, E}(x,y)$ of the isotropic Harmonic Oscillator $- \hbar ^2 \Delta + |x|^2$ of eigenvalue $E = \hbar(N + \frac{d}{2})$ in the semi-classical limit $\hbar \to 0$. The principal result is an explicit formula for the scaling asymptotics of $\Pi_{\hbar, E}(x,y)$ for $x,y$ in a $\hbar^{2/3}$ neighborhood of the caustic $\mathcal C_E$ as $\hbar \to 0.$ The scaling asymptotics are applied to the distribution of nodal sets of Gaussian random eigenfunctions around the caustic as $\hbar \to 0$. In previous work we proved that the density of zeros of Gaussian random eigenfunctions of $\hat{H}_{\hbar}$ have different orders in the Planck constant $\hbar$ in the allowed and forbidden regions: In the allowed region the density is of order $\hbar^{-1}$ while it is $\hbar^{-1/2}$ in the forbidden region. Our main result on nodal sets is that the density of zeros is of order $\hbar^{-\frac{2}{3}}$ in an $\hbar^{\frac{2}{3}}$-tube around the caustic. This tube radius is the `critical radius'. For annuli of larger inner and outer radii $\hbar^{\alpha}$ with $0< \alpha < \frac{2}{3}$ we obtain density results which interpolate between this critical radius result and our prior ones in the allowed and forbidden region. We also show that the Hausdorff $(d-2)$-dimensional measure of the intersection of the nodal set with the caustic is of order $\hbar^{- \frac{2}{3}}$.

On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators

By applying an idea of Borodin and Olshanski [J. Algebra 313 (2007), 40-60], we study various scaling limits of determinantal point processes with trace class projection kernels given by spectral projections of selfadjoint Sturm-Liouville operators. Instead of studying the convergence of the kernels as functions, the method directly addresses the strong convergence of the induced integral operators. We show that, for this notion of convergence, the Dyson, Airy, and Bessel kernels are universal in the bulk, soft-edge, and hard-edge scaling limits. This result allows us to give a short and unified derivation of the known formulae for the scaling limits of the classical random matrix ensembles with unitary invariance, that is, the Gaussian unitary ensemble (GUE), the Wishart or Laguerre unitary ensemble (LUE), and the MANOVA (multivariate analysis of variance) or Jacobi unitary ensemble (JUE).

On wavelet projection kernels and the integrated squared error in density estimation

Abstract It is shown that the integrated squared errors of wavelet projection estimators of a density f satisfy both the central limit theorem and the law of the iterated logarithm under the essentially minimal assumption f ∈ L p for some p > 2 and very mild conditions on the scaling function.

Abnormal Scene Change Detection from a Moving Camera Using Bags of Patches and Spider-Web Map

This paper proposes a novel surveillance system for detecting exceptional scene changes as abnormal events with a mobile camera mounted on a robot. In contrast to abnormal event analysis using fixed cameras, three key problems should be tackled in this system, i.e., scene construction, robot localization, and scene comparison. For the first problem, scene construction, a clustering scheme is proposed for extracting a set of key frames from the surveillance environment. Each key frame is further divided into a set of patches, which forms a sparse representation for representing scene contents. In addition to the compression effect, the scheme can tackle the effects of misalignment and lighting changes well. For the localization problem, a novel patch matching method is proposed to reduce not only the size of the search space but also the size of the feature dimensions in similarity matching. To prune the search space, a set of projection kernels is used to construct a ring structure. Then, one order of time complexity in the similarity calculation can be reduced from the structure. After scene searching, the robot location is not always guaranteed to be successfully registered to the scene map. Thus, a novel spider-web map is proposed to tackle the effect of misalignment and then detect different exceptional scene changes from the videos. The proposed method has been rigorously tested on a variety of videos to demonstrate its superiority in object detection and abnormal scene change detection.

Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections

Given an i.i.d. sample from a distribution $F$ on $ℝ$ with uniformly continuous density $p_0$, purely data-driven estimators are constructed that efficiently estimate $F$ in sup-norm loss and simultaneously estimate $p_0$ at the best possible rate of convergence over Hölder balls, also in sup-norm loss. The estimators are obtained by applying a model selection procedure close to Lepski’s method with random thresholds to projections of the empirical measure onto spaces spanned by wavelets or $B$-splines. The random thresholds are based on suprema of Rademacher processes indexed by wavelet or spline projection kernels. This requires Bernstein-type analogs of the inequalities in Koltchinskii [Ann. Statist. 34 (2006) 2593–2656] for the deviation of suprema of empirical processes from their Rademacher symmetrizations.

Diffuse Lung Disease: CT of the Chest with Adaptive Statistical Iterative Reconstruction Technique

To compare visualization of subtle normal and abnormal findings at computed tomography (CT) of the chest for diffuse lung disease with images reconstructed with filtered back projection and adaptive statistical iterative reconstruction (ASIR) techniques. In this HIPAA-compliant, institutional review board-approved study, 24 patients underwent 64-section multi-detector row CT of the chest for evaluation of diffuse lung disease. Scanning parameters included a pitch of 0.984:1 and 120 kVp in thin-section mode, with 2496 views per rotation compared with 984 views acquired for normal mode. The 0.625-mm-thick images were reconstructed with filtered back projection, ASIR, and ASIR high-definition (ASIR-HD) kernels. Two thoracic radiologists independently assessed the filtered back projection, ASIR, and ASIR-HD images for small anatomic details (interlobular septa, centrilobular region, and small bronchi and bronchioles), abnormal findings (reticulation, tiny nodules, altered attenuation, bronchiectasis), image quality (graded by using a six-point scale, where 1 = excellent image quality, and 5 = interpretation impossible), image noise, and artifacts. Data were tabulated for statistical testing. For visualization of normal and pathologic structures, CT image series reconstructed with ASIR-HD were rated substantially better than those reconstructed with filtered back projection and ASIR (P < .001). ASIR-HD images were superior to filtered back projection images in 15 of 24 (62%) patients for visualization of normal structures and in 24 of 24 (100%) patients for pathologic findings. ASIR-HD was superior to ASIR in three of 24 (12%) images for normal anatomic findings and in seven of 24 (29%) images for pathologic evaluation. None of the images in the three groups were rated as unacceptable for noise (P < .001). ASIR-HD reconstruction results in superior visualization of subtle and tiny anatomic structures and lesions in diffuse lung disease compared with ASIR and filtered back projection reconstructions.

Supervised learning of local projection kernels

We formulate a supervised, localized dimensionality reduction method using a gating model that divides up the input space into regions and selects the dimensionality reduction projection separately in each region. The gating model, the locally linear projections, and the kernel-based supervised learning algorithm which uses them in its kernels are coupled and their training is performed with an alternating optimization procedure. Our proposed local projection kernel projects a data instance into different feature spaces by using the local projection matrices, combines them with the gating model, and performs the dot product in the combined feature space. Empirical results on benchmark data sets for visualization and classification tasks validate the idea. The method is generalizable to regression estimation and novelty detection.

Bergman kernels and equilibrium measures for line bundles over projective manifolds

Let $L$ be a holomorphic line bundle over a compact complex projective Hermitian manifold $X.$ Any fixed smooth hermitian metric $\phi$ on $L$ induces a Hilbert space structure on the space of global holomorphic sections with values in the $k$th tensor power of $L.$ In this paper various convergence results are obtained for the corresponding Bergman kernels (i.e., orthogonal projection kernels). The convergence is studied in the large $k$ limit and is expressed in terms of the equilibrium metric $\phi_{e}$ associated to the fixed metric $\phi,$ as well as in terms of the Monge-Ampere measure of the metric $\phi$ itself on a certain support set. It is also shown that the equilibrium metric is ${\cal C}^{1,1}$ on the complement of the augmented base locus of $L.$ For $L$ ample these results give generalizations of well-known results concerning the case when the curvature of $\phi$ is globally positive (then $\phi_{e}=\phi$). In general, the results can be seen as local metrized versions of Fujita's approximation theorem for the volume of $L$.

Polynomial functions on subdirect products

A congruence preserving function on a subdirect product of two finite Mal’cev algebras is polynomial if it induces polynomial functions on the subdirect factors and there are no skew congruences between the projection kernels. As a special case, if the direct product A × B of finite algebras A and B in a congruence permutable variety has no skew congruences, then the polynomial functions on A × B are exactly direct products of polynomials on A and on B. These descriptions apply in particular to classical polynomial functions on nonassociative rings. Also, for finite algebras A, B in a variety with majority term, the polynomial functions on A × B are exactly the direct products of polynomials on A and on B. However in arbitrary congruence distributive varieties the corresponding result is not true.

Moment analysis for localization in random Schrödinger operators

We study localization effects of disorder on the spectral and dynamical properties of Schrodinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weak-L 1 estimate concerning the boundary-value distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.

SUPPORT VECTOR MACHINES BASED ON THE ORTHOGONAL PROJECTION KERNEL OF FATHER WAVELET

Recently the study on the theory of wavelets shows that the wavelets have not only the multi-resolution property both in frequency and time domain, but also the good approximation ability. SVMs based on the statistical learning theory are a kind of general and effective learning machines, and have described for us the nice application blueprint in machine learning domain. There exists a bottleneck problem, or the pre-selection of kernel parameter for SVMs. In this paper, the orthogonal projection kernels of father wavelet (OPFW kernels) are introduced into SVMs. In doing so SVMs based on the OPFW kernels can have good performance in both approximation and generalisation. Simultaneously the parameter pre-selection of wavelet kernels can be implemented by discrete wavelet transform. Experiments on regression estimation illustrate the approximation and generalisation ability of our method.