Finite Resolutions Research Articles

Cone-Traced Supersampling With Subpixel Edge Reconstruction.

While signed distance fields (SDFs) in theory offer infinite level of detail, they are typically rendered using the sphere tracing algorithm at finite resolutions, which causes the common rasterized image synthesis problem of aliasing. Most existing optimized antialiasing solutions rely on polygon mesh representations; SDF-based geometry can only be directly antialiased with the computationally expensive supersampling or with post-processing filters that may produce undesirable blurriness and ghosting. In this work, we present cone-traced supersampling (CTSS), an efficient and robust spatial antialiasing solution that naturally complements the sphere tracing algorithm, does not require casting additional rays per pixel or offline pre-filtering, and can be easily implemented in existing real-time SDF renderers. CTSS performs supersampling along the traced ray near surfaces with partial visibility - object contours - identified by evaluating cone intersections within a pixel's view frustum. We further introduce subpixel edge reconstruction (SER), a technique that extends CTSS to locate and resolve complex pixels with geometric edges in relatively flat regions, which are otherwise undetected by cone intersections. Our combined solution relies on a specialized sampling strategy to minimize the number of shading computations and correlates sample visibility to aggregate the samples. With comparable antialiasing quality at significantly lower computational cost, CTSS is a reliable practical alternative to conventional supersampling.

Effect of finite spatial and temporal resolutions on super-resolution particle tracking velocimetry for pressure-driven flow in a nanochannel

Effect of finite spatial and temporal resolutions on super-resolution particle tracking velocimetry for pressure-driven flow in a nanochannel

RIS-Assisted Self-Interference Mitigation for In-Band Full-Duplex Transceivers

The wireless in-band full-duplex (IBFD) technology can in theory double the system capacity over the conventional frequency division duplex (FDD) or time-division duplex (TDD) alternatives. But the strong self-interference of the IBFD can cause excessive quantization noise at the output of the analog-to-digital converters (ADCs), which hampers its real implementation. Fortunately, the reconfigurable intelligent surface (RIS) can reconfigure the wireless channels by dynamically adjusting the reflection coefficients, which makes it a promising solution to self-interference mitigation (SIM) for the IBFD system. This paper considers a RIS-assisted IBFD (RAIBFD) system where a full-duplex base station (BS), assisted by a co-located RIS, receives and transmits signals on the same frequency band simultaneously. We propose to jointly design the DL precoding matrix and the RIS coefficients to mitigate the strong self-interference before the ADCs of finite bit resolutions. To gauge the performance of the proposed RAIBFD system, we also consider an idealistic RIS-assisted IBFD (I-RAIBFD) system with ADCs of infinite bit resolutions and RIS-assisted FDD (RAFDD) system, and take them as performance benchmarks. The effectiveness of the proposed RAIBFD system is verified by simulation studies, even if the phases of the RIS have only finite resolutions.

THE SMALL FINITISTIC DIMENSIONS OF COMMUTATIVE RINGS

Let R be a commutative ring with identity. The small finitistic dimension fPD(R) of R is defined to be the supremum of projective dimensions of R-modules with finite projective resolutions. In this paper, we characterize a ring R with fPD(R)≤n using finitely generated semiregular ideals, tilting modules, cotilting modules of cofinite type and vaguely associated prime ideals. As an application, we obtain that if R is a Noetherian ring, then fPD(R)=sup{grade(𝔪,R)|𝔪∈Max(R)} where grade(𝔪,R) is the grade of 𝔪 on R. We also show that a ring R satisfies fPD(R)≤1 if and only if R is a DW ring. As applications, we show that the small finitistic dimensions of strong Prüfer rings and LPVDs are at most one. Moreover, for any given n∈ℕ, we obtain a total ring of quotients R satisfying fPD(R)=n.

Tracking decay positrons in a magnetic field for muon microscope applications

We present a theoretical calculation for a feasibility study of the Muon Microscope, which is intended to add positional resolution within the sample by tracking down the positron trajectories to its source positions. In the presence of a magnetic field, any positrons whose trajectories have components which are perpendicular to the magnetic field will start to move in a helical path due to the Lorentz force. Taking special relativity into account, we have analytically determined the trajectories of the positrons in a uniform magnetic field. We also evaluate more realistic cases, such as finite detector spatial resolutions, as well as the effect of positron scattering from the materials in its trajectory.

Asymptotic analysis of domain decomposition for optimal transport

Large optimal transport problems can be approached via domain decomposition, i.e. by iteratively solving small partial problems independently and in parallel. Convergence to the global minimizers under suitable assumptions has been shown in the unregularized and entropy regularized setting and its computational efficiency has been demonstrated experimentally. An accurate theoretical understanding of its convergence speed in geometric settings is still lacking. In this article we work towards such an understanding by deriving, via varGamma -convergence, an asymptotic description of the algorithm in the limit of infinitely fine partition cells. The limit trajectory of couplings is described by a continuity equation on the product space where the momentum is purely horizontal and driven by the gradient of the cost function. Global optimality of the limit trajectories remains an interesting open problem, even when global optimality is established at finite scales. Our result provides insights about the efficiency of the domain decomposition algorithm at finite resolutions and in combination with coarse-to-fine schemes.

Finite permutation resolutions

We prove that every finite-dimensional representation of a finite group over a field of characteristic p admits a finite resolution by p-permutation modules. The proof involves a reformulation in terms of derived categories.

Experimental Evaluation of a 3-D CZT Imaging Spectrometer for Potential Use in Compton-Enhanced PET Imaging.

We constructed a prototype positron emission tomography (PET) system and experimentally evaluated large-volume 3-D cadmium zinc telluride (CZT) detectors for potential use in Compton-enhanced PET imaging. The CZT spectrometer offers sub-0.5-mm spatial resolution, an ultrahigh energy resolution (~1% @ 511 keV), and the capability of detecting multiple gamma-ray interactions that simultaneously occurred. The system consists of four CZT detector panels with a detection area of around 4.4 cm × 4.4 cm. The distance between the front surfaces of the two opposite CZT detector panels is ~80 mm. This system allows us to detect coincident annihilation photons and Compton interactions inside the detectors and then, exploit Compton kinematics to predict the first Compton interaction site and reject chance coincidences. We have developed a numerical integration technique to model the near-field Compton response that incorporates Doppler broadening, detector's finite resolutions, and the distance between the first and second interactions. This method was used to effectively reject random and scattered coincidence events. In the preliminary imaging studies, we have used point sources, line sources, a custom-designed resolution phantom, and a commercial image quality (IQ) phantom to demonstrate an imaging resolution of approximately 0.75 mm in PET images, and Compton-based enhancement.

Minuscule Schubert varieties of exceptional type

We study the finite free resolutions of intersections of exceptional minuscule Schubert varieties with the opposite big open cell. We investigate the patterns which arise and relate the resolutions we obtain to certain basic cases which have been already extensively studied by commutative algebraists.

A universal hybrid precoding scheme for massive MIMO communications

Based on an analog radio frequency (RF) network, hybrid precoding (HPC) for massive MIMO can achieve very high spectral efficiencies with moderate hardware cost and power consumption. Despite the extensive research efforts in recent years, the practioners are still looking for HPCs that are efficient and easy-to-implement. In this paper, we present a new method termed as the universal hybrid precoding (UHP), which is nearly optimal, computationally efficient, and applicable to various types of RF network (thus, the name universal): the components of the network can be phase shifters (with finite or infinite resolutions), switches, or their combinations; the topology of the network can be fully-connected or partially-connected. Besides the standard UHP, we also propose a simplified version termed as sUHP to trade a negligible performance loss for much reduced computational complexity. The analysis shows that the computational complexity of the proposed UHP/sUHP is one to two orders of magnitude lower than the state-of-the-art methods. Simulation results verify the (near-) optimality of the proposed UHP scheme for various forms of the analog networks.

Recurrent Neural Networks (RNNs) with dimensionality reduction and break down in computational mechanics; application to multi-scale localization step

Artificial Neural Networks (NNWs) are appealing functions to substitute high dimensional and non-linear history-dependent problems in computational mechanics since they offer the possibility to drastically reduce the computational time. This feature has recently been exploited in the context of multi-scale simulations, in which the NNWs serve as surrogate model of micro-scale finite element resolutions. Nevertheless, in the literature, mainly the macro-stress–macro-strain response of the meso-scale boundary value problem was considered and the micro-structure information could not be recovered in a so-called localization step. In this work, we develop Recurrent Neural Networks (RNNs) as surrogates of the RVE response while being able to recover the evolution of the local micro-structure state variables for complex loading scenarios. The main difficulty is the high dimensionality of the RNNs output which consists in the internal state variable distribution in the micro-structure. We thus propose and compare several surrogate models based on a dimensionality reduction: (i) direct RNN modeling with implicit NNW dimensionality reduction, (ii) RNN with PCA dimensionality reduction, and (iii) RNN with PCA dimensionality reduction and dimensionality break down, i.e. the use of several RNNs instead of a single one. Besides, we optimize the sequential training strategy of the latter surrogate for GPU usage in order to speed up the process. Finally, through RNN modeling of the principal components coefficients, the connection between the physical state variables and the hidden variables of the RNN is revealed, and exploited in order to select the hyper-parameters of the RNN-based surrogate models in their design stage.

Faraday depolarization and induced circular polarization by multipath propagation with application to FRBs

ABSTRACT We describe how the observed polarization properties of an astronomical object are related to its intrinsic polarization properties and the finite temporal and spectral resolutions of the observing device. Moreover, we discuss the effect that a scattering screen, with non-zero magnetic field, between the source and observer has on the observed polarization properties. We show that the polarization properties are determined by the ratio of observing bandwidth and coherence bandwidth of the scattering screen and the ratio of temporal resolution of the instrument and the variability time of screen, as long as the length over which the Faraday rotation induced by the screen changes by ∼π is smaller than the size of the screen visible to the observer. We describe the conditions under which a source that is 100 per cent linearly polarized intrinsically might be observed as partially depolarized, and how the source’s temporal variability can be distinguished from the temporal variability induced by the scattering screen. In general, linearly polarized waves passing through a magnetized scattering screen can develop a significant circular polarization. We apply the work to the observed polarization properties of a few fast radio bursts (FRBs), and outline potential applications to pulsars.

Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry

In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A fairly complete picture of the representation theory for CW left regular bands is obtained.

Development of skeletal systems for ICRP pediatric mesh-type reference computational phantoms

In 2016, the International Commission on Radiological Protection (ICRP) launched Task Group 103 (TG 103) for the explicit purpose of developing a new generation of adult and pediatric reference computational phantoms, named ‘mesh-type reference computational phantoms (MRCPs)’, that can overcome the limitations of voxel-type reference computational phantoms (VRCPs) of ICRP Publications 110 and 143 due to their finite voxel resolutions and the nature of voxel geometry. After completing the development of the adult MRCPs, TG 103 has started the development of pediatric MRCPs comprising 10 phantoms (male and female versions of the reference newborn, 1-year-old, 5-year-old, 10-year-old, and 15-year-old). As part of the TG 103 project, within the present study, the skeletal systems, one of the most important and complex organ systems of the body, were developed for each phantom age and sex. The developed skeletal systems, while closely preserving the original bone topology of the pediatric VRCPs, present substantial improvements in the anatomy of complex and/or small bones. In order to investigate the dosimetric impact of the developed skeletons, the average absorbed doses and the specific absorbed fractions for radiosensitive skeletal tissues (i.e. active marrow and bone endosteum) were computed for some selected external and internal exposure cases, which were then compared with those calculated with the skeletons of pediatric VRCPs. The comparison result showed that the dose values of the pediatric MRCPs were generally similar to those of the pediatric VRCPs for highly penetrating radiations (e.g. photons >200 keV); however, for weakly penetrating radiations (e.g. photons ⩽200 keV and electrons), significant differences up to a factor of 140 were observed.

Schubert varieties and finite free resolutions of length three

In this paper we describe the relationship between the finite free resolutions of perfect ideals in split format (for Dynkin formats) and certain intersections of opposite Schubert varieties with the big cell for homogeneous spaces G / P G/P , where P P is a maximal parabolic subgroup.

Detailed Voxel-Based Implicit Modeling With Local Boolean Composition of Discrete Level Sets

Compared with the functionally-based implicit models, the level set models are usually defined by discretely sampling the implicit function values over the volume grid to represent the shapes, which are suitable for scientific computing and engineering. However, the voxel-based level set methods usually lose some features and details to represent the solids due to the finite grid resolutions. In this paper, we propose a more accurate voxel-based implicit modeling method based on discrete level sets. Compared with the traditional level set methods, more than one level set function can be defined in an implicit model with our approach by classifying the cells of the volume grid. The local Boolean compositions conducted over the compound cells are used to depict the interactions of different implicit surfaces represented by corresponding level set functions, and the sharp and thin details caused by Boolean operations are preserved owing to the definition of the compound cells. We show in our experiments that our method is robust to provide the more detailed implicit description of the shapes with less loss of the features than the conventional voxel-based level set methods.

Table‐based PWM for all‐digital RF transmitters

SummaryIn this paper, a novel pulse‐width modulator (PWM) for an all‐digital PWM transmitter based on switching amplification is presented. In this approach, each symbol is represented by a sequence of pre‐computed PWM pulse widths stored in a table that results in reduced error vector magnitude (EVM). A high‐resolution pulse former is then used to build the bipolar radio frequency (RF) signal. Simulations and comparisons to other approaches reported in the literature under different oversampling ratios and finite time resolutions are presented. The impact of nonlinearities in the variable duty‐cycle generator is also analyzed to demonstrate the robustness of the proposed technique.

On Finite Resolutions in Triangulated Categories

Let $$\mathcal {X}$$ be an extension-closed subcategory of a triangulated category such that $$\mathcal {X}$$ admits a weak-cogenerator $$\mathcal {W}$$. We investigate finite resolutions by objects in $$\mathcal {X}$$ and $$\mathcal {X}$$-resolution dimension of objects. It is shown that if $$\mathcal {W}$$ is also a weak-generator in $$\mathcal {X}$$, then any finite resolution by objects in $$\mathcal {X}$$ of an object gives rise to another finite resolution such that one of the objects in the resolution belongs to $$\mathcal {X}$$, while all the rest objects are in $$\mathcal {W}$$. We establish two additive functors between additive quotient categories related to $$\mathcal {X}$$ and $$\mathcal {W}$$ when $$\mathcal {W}$$ is further $$\mathcal {X}$$-injective. In particular, we show that both two functors possess certain adjointness and that one can characterize the objects with a finite resolution by objects in $$\mathcal {X}$$ having a finite resolution by objects in $$\mathcal {W}$$ (resp., when an object with a finite resolution by objects in $$\mathcal {X}$$ belongs to $$\mathcal {X}$$) by vanishing of the counit (resp., unit) of the adjunction with respect to one of these functors (resp., another one). For any object with a finite resolution by objects in $$\mathcal {X}$$, we construct a cellular tower with respect to $$\mathcal {X}$$ and show that such a cellular tower can be used to detect the value of $$\mathcal {X}$$-resolution dimension provided that $$\mathcal {X}$$ is also closed under cocones.

PyUnfold: A Python package for iterative unfolding

PyUnfold is a Python package for incorporating imperfections of the measurement process into a data analysis pipeline. In an ideal world, we would have access to the perfect detector: an apparatus that makes no error in measuring a desired quantity. However, in real life, detectors have finite resolutions, characteristic biases that cannot be eliminated, less than full detection efficiencies, and statistical and systematic uncertainties. By building a matrix that encodes a detector's smearing of the desired true quantity into the measured observable(s), a deconvolution can be performed that provides an estimate of the true variable. This deconvolution process is known as unfolding. The unfolding method implemented in PyUnfold accomplishes this deconvolution via an iterative procedure, providing results based on physical expectations of the desired quantity. Furthermore, tedious book-keeping for both statistical and systematic errors produces precise final uncertainty estimates.

Dissipation, intermittency, and singularities in incompressible turbulent flows.

We examine the connection between the singularities or quasisingularities in the solutions of the incompressible Navier-Stokes equation (INSE) and the local energy transfer and dissipation, in order to explore in detail how the former contributes to the phenomenon of intermittency. We do so by analyzing the velocity fields (a) measured in the experiments on the turbulent von Kármán swirling flow at high Reynolds numbers and (b) obtained from the direct numerical simulations of the INSE at a moderate resolution. To compute the local interscale energy transfer and viscous dissipation in experimental and supporting numerical data, we use the weak solution formulation generalization of the Kármán-Howarth-Monin equation. In the presence of a singularity in the velocity field, this formulation yields a nonzero dissipation (inertial dissipation) in the limit of an infinite resolution. Moreover, at finite resolutions, it provides an expression for local interscale energy transfers down to the scale where the energy is dissipated by viscosity. In the presence of a quasisingularity that is regularized by viscosity, the formulation provides the contribution to the viscous dissipation due to the presence of the quasisingularity. Therefore, our formulation provides a concrete support to the general multifractal description of the intermittency. We present the maps and statistics of the interscale energy transfer and show that the extreme events of this transfer govern the intermittency corrections and are compatible with a refined similarity hypothesis based on this transfer. We characterize the probability distribution functions of these extreme events via generalized Pareto distribution analysis and find that the widths of the tails are compatible with a similarity of the second kind. Finally, we make a connection between the topological and the statistical properties of the extreme events of the interscale energy transfer field and its multifractal properties.