Series Of Convolutions Research Articles

Fourier–Zernike Series of Convolutions on Disks

This paper presents a systematic study of the analytic aspects of Fourier–Zernike series of convolutions of functions supported on disks. We then investigate different aspects of the presented theory in the cases of zero-padded functions.

Deep Variational and Structural Hashing.

In this paper, we propose a deep variational and structural hashing (DVStH) method to learn compact binary codes for multimedia retrieval. Unlike most existing deep hashing methods which use a series of convolution and fully-connected layers to learn binary features, we develop a probabilistic framework to infer latent feature representation inside the network. Then, we design a struct layer rather than a bottleneck hash layer, to obtain binary codes through a simple encoding procedure. By doing these, we are able to obtain binary codes discriminatively and generatively. To make it applicable to cross-modal scalable multimedia retrieval, we extend our method to a cross-modal deep variational and structural hashing (CM-DVStH). We design a deep fusion network with a struct layer to maximize the correlation between image-text input pairs during the training stage so that a unified binary vector can be obtained. We then design modality-specific hashing networks to handle the out-of-sample extension scenario. Specifically, we train a network for each modality which outputs a latent representation that is as close as possible to the binary codes which are inferred from the fusion network. Experimental results on five benchmark datasets are presented to show the efficacy of the proposed approach.

PCNN: Deep Convolutional Networks for Short-Term Traffic Congestion Prediction

Traffic problems have seriously affected people's life quality and urban development, and forecasting the short-term traffic congestion is of great importance to both individuals and governments. However, understanding and modeling the traffic conditions can be extremely difficult, and our observations from real traffic data reveal that (1) similar traffic congestion patterns exist in the neighboring time slots and on consecutive workdays; (2) the levels of traffic congestion have clear multiscale properties. To capture these characteristics, we propose a novel method named PCNN based on deep Convolutional Neural Network, modeling Periodic traffic data for short-term traffic congestion prediction. PCNN has two pivotal procedures: time series folding and multi-grained learning. It first temporally folds the time series and constructs a two-dimensional matrix as the network input, such that both the real-time traffic conditions and past traffic patterns are well considered; then with a series of convolutions over the input matrix, it is able to model the local temporal dependency and multiscale traffic patterns. In particular, the global trend of congestion can be addressed at the macroscale; whereas more details and variations of the congestion can be captured at the microscale. Experimental results on a real-world urban traffic dataset confirm that folding time series data into a two-dimensional matrix is effective and PCNN outperforms the baselines significantly for the task of short-term congestion prediction.

Optimized staggered-grid finite-difference operators using window functions

The staggered-grid finite-difference (SGFD) method has been widely used in seismic forward modeling. The precision of the forward modeling results directly affects the results of the subsequent seismic inversion and migration. Numerical dispersion is one of the problems in this method. The window function method can reduce dispersion by replacing the finite-difference operators with window operators, obtained by truncating the spatial convolution series of the pseudospectral method. Although the window operators have high precision in the low-wavenumber domain, their precision decreases rapidly in the high-wavenumber domain. We develop a least squares optimization method to enhance the precision of operators obtained by the window function method. We transform the SGFD problem into a least squares problem and find the best solution iteratively. The window operator is chosen as the initial value and the optimized domain is set by the error threshold. The conjugate gradient method is also adopted to increase the stability of the solution. Approximation error analysis and numerical simulation results suggest that the proposed method increases the precision of the window function operators and decreases the numerical dispersion.

Power Spectrum Computation for an Arbitrary Phase Noise Using Middleton's Convolution Series: Implementation Guideline and Experimental Illustration.

In this paper, we revisit the convolution series initially introduced by Middleton several decades ago to determine the power spectrum (or spectral line shape) of a periodic signal from its phase noise power spectral density. This topic is of wide interest, as it has an important impact on many scientific areas that involve lasers and oscillators. We introduce a simple guideline that enables a fairly straightforward computation of the power spectrum corresponding to an arbitrary phase noise. We show the benefit of this approach on a computational point of view, and apply it to various types of experimental signals with different phase noise levels, showing a very good agreement with the experimental spectra. This approach also provides a qualitative and intuitive understanding of the power spectrum corresponding to different regimes of phase noise.

Fast bilateral filtering using exponential functions

It has been demonstrated that the bilateral filter can be computed using a series of fast convolutions by approximating its Gaussian range kernel using trigonometric functions. A novel approximation that can be applied to any range kernel is proposed. Specifically speaking, an exponential sum is exploited to approximate the range kernel of the bilateral filter, where the coefficients of the exponential basis are computed by solving a set of linear equations. The major advantage of the method is that the tradeoff between the run-time and the filtering accuracy can be controlled by the cardinality of the exponential basis. Experiments show that the method achieves state-of-the-art results in terms of accuracy and speed.

FDTD modeling of induced polarization phenomena in transient electromagnetics

The finite-difference time-domain scheme is augmented in order to treat the modelling of transient electromagnetic signals containing induced polarization effects from 3-D distributions of polarizable media. Compared to the non-dispersive problem, the discrete dispersive Maxwell system contains costly convolution operators. Key components to our solution for highly digitized model meshes are Debye decomposition and composite memory variables. We revert to the popular Cole–Cole model of dispersion to describe the frequency-dependent behaviour of electrical conductivity. Its inversely Laplace-transformed Debye decomposition results in a series of time convolutions between electric field and exponential decay functions, with the latter reflecting each Debye constituents’ individual relaxation time. These function types in the discrete-time convolution allow for their substitution by memory variables, annihilating the otherwise prohibitive computing demands. Numerical examples demonstrate the efficiency and practicality of our algorithm.

Optimization of finite difference forward modeling for elastic waves based on optimum combined window functions

Full waveform inversion and reverse time migration are active research areas for seismic exploration. Forward modeling in the time domain determines the precision of the results, and numerical solutions of finite difference have been widely adopted as an important mathematical tool for forward modeling. In this article, the optimum combined of window functions was designed based on the finite difference operator using a truncated approximation of the spatial convolution series in pseudo-spectrum space, to normalize the outcomes of existing window functions for different orders. The proposed combined window functions not only inherit the characteristics of the various window functions, to provide better truncation results, but also control the truncation error of the finite difference operator manually and visually by adjusting the combinations and analyzing the characteristics of the main and side lobes of the amplitude response. Error level and elastic forward modeling under the proposed combined system were compared with outcomes from conventional window functions and modified binomial windows. Numerical dispersion is significantly suppressed, which is compared with modified binomial window function finite-difference and conventional finite-difference. Numerical simulation verifies the reliability of the proposed method.

Asymptotic bounds for special values of shifted convolution Dirichlet series

Hoffstein and Hulse defined the shifted convolution series of two cusp forms by shifting the usual Rankin-Selberg convolution L-series by a parameter h. We use the theory of harmonic Maass forms to study the behavior in h-aspect of certain values of these series and prove a polynomial bound as h approaches infinity. Our method relies on a result of Mertens and Ono, who showed that these values are Fourier coefficients of mixed mock modular forms.

Improving the convergence order of the regularization method for Fredholm integral equations of the second kind

We build a numerical approximation method, for Fredholm integral equation solution of the second type. This method is based on the regularization by convolution and Fourier series expansion. It provides a better convergence order.

On Fast Bilateral Filtering Using Fourier Kernels

It was demonstrated in earlier work that, by approximating its range kernel using shiftable functions, the non-linear bilateral filter can be computed using a series of fast convolutions. Previous approaches based on shiftable approximation have, however, been restricted to Gaussian range kernels. In this work, we propose a novel approximation that can be applied to any range kernel, provided it has a pointwise-convergent Fourier series. More specifically, we propose to approximate the Gaussian range kernel of the bilateral filter using a Fourier basis, where the coefficients of the basis are obtained by solving a series of least-squares problems. The coefficients can be efficiently computed using a recursive form of the QR decomposition. By controlling the cardinality of the Fourier basis, we can obtain a good tradeoff between the run-time and the filtering accuracy. In particular, we are able to guarantee sub-pixel accuracy for the overall filtering, which is not provided by most existing methods for fast bilateral filtering. We present simulation results to demonstrate the speed and accuracy of the proposed algorithm.

New convolution structure for the linear canonical transform and its application in filter design

The convolution and product theorem for the Fourier transform (FT) plays an important role in signal processing theory and application. The linear canonical transform (LCT), which is a generalization of the FT and the fractional Fourier transform (FRFT), has found many applications in optics and non-stationary signal processing. Recently, some scholars have formulated a series of convolution and product theorems for the LCT, however, both of them do not maintain the convolution theorem for the FT. The purpose of this paper is to present a new convolution structure for the LCT having the elegance and simplicity in both time and LCT domains comparable to that of the FT. We also show that with the new convolution theorem it is convenient to implement in the designing of multiplicative filters through both the new convolution in the time domain and the product in the LCT domain.

Gerber–Shiu functionals for classical risk processes perturbed by an [formula omitted]-stable motion

We study the Gerber–Shiu functional of the classical risk process perturbed by a spectrally negative α-stable motion. We provide representations of the scale functions of the process as an infinite series of convolutions of given functions. This, together with a result from Biffis and Kyprianou (2010), allows us to obtain a representation of the Gerber–Shiu functional as an infinite series of convolutions. Moreover, we calculate the Laplace transform and derive a defective renewal equation for the Gerber–Shiu functional, thus extending previous work of Furrer (1998) and of Tsai and Willmot (2002). We also obtain asymptotic expressions for the joint tail distribution of the severity of ruin and the surplus before ruin.

Simulation of image formation in x-ray coded aperture microscopy with polycapillary optics.

In x-ray coded aperture microscopy with polycapillary optics (XCAMPO), the microstructure of focusing polycapillary optics is used as a coded aperture and enables depth-resolved x-ray imaging at a resolution better than the focal spot dimensions. Improvements in the resolution and development of 3D encoding procedures require a simulation model that can predict the outcome of XCAMPO experiments. In this work we introduce a model of image formation in XCAMPO which enables calculation of XCAMPO datasets for arbitrary positions of the object relative to the focal plane as well as to incorporate optics imperfections. In the model, the exit surface of the optics is treated as a micro-structured x-ray source that illuminates a periodic object. This makes it possible to express the intensity of XCAMPO images as a convolution series and to perform simulations by means of fast Fourier transforms. For non-periodic objects, the model can be applied by enforcing artificial periodicity and setting the spatial period larger then the field-of-view. Simulations are verified by comparison with experimental data.

Caroline Gordon’s Ghosts: The Women on the Porch as Southern Gothic Literature

Soon after its publication in 1944, Caroline Gordon's The Women on Porch--her sixth novel--generated two reviews in The New York Times. The first, by Orville Prescott, noted that novel's dust jacket depicted the figure of young woman fleeing from nameless terrors of dark forest, and that encircling gloom that menaces her ... billows and eddies pages of [the] cryptic and peculiar (17). Prescott also conveyed that Gordon's latest work wandered through series of spirals and convolutions of time and place and thought, slipping from stream of consciousness of one character to that of another, from Tennessee to New York, from present to past (17). The result was an elusive [and] haunting, taut and twisted, work that like novels of a number of her fellow Southern[ers] shared a preoccupation with death and decay and destruction ... [While] she does not engage in ghoulish melodramatics of Faulkner ... her sense of doom and frustration is [just] as great (Prescott 17). The second review, written by Lorine Pruette, similarly underscored novel's modernist and gothic elements, specifically its thick shadows of past, its depiction of struggle between Old and New South, and its intricate narrative form--a modified stream-of-consciousness technique ... [which] admirably evokes mood (BR6). Much of other criticism that followed initial publication of The Women of Porch focused on mythological framework of novel, which is based on saga of Orpheus's and Eurydice's descent into, and escape from, Hades; heroic quests of protagonists; and its innovation in structure. Agrarians Andrew Lytle and Brainard Cheney were among first to recognize literary merits of Gordon's novel. Cheney was particularly laudatory, describing it as profoundly informed by tradition and history. The Women on Porch, he predicted, would be novel that would last because it not only gives us realism of [its] time but [also] characters, places, and situations which tend toward [the] symbolic representation of our experience (Cheney 149-150). Despite this initial excitement, over years The Women on Porch has been marginalized in favor of earlier Gordon works such as Penhally (1931), Aleck Maury, Sportsman (1934), and None Shall Look Back (1937), which have come to dominate Gordon criticism. The Women on Porch has been categorized either, at worst, as succession of frozen moments, each with its own prearranged meaning ... [with] no progression, no plot ... [just] elegiac pattern of decline and fall (Gray 157) or, at best, as transitionary in that it signals Gordon's movement away from grand historical novels about Old South (Brown 367), and toward critiquing Agrarian agenda espoused in earlier works. What has been lost in these analyses, however, is Prescott's and Pruette's original emphasis on novel's gothic elements, which not only position it among many of greatest contributions to Southern Renaissance but also serve as framework to examine some of Gordon's concerns with Agrarian manifesto. In particular, gothic elements in The Women on Porch address unresolved antagonism between Old and New South, feasibility of their agenda in an increasing urban and industrial mid-twentieth-century South, and role of women within this shifting social dynamic. This essay seeks to recuperate and expand upon these initial interpretations of The Women on Porch by considering novel as an example of Southern Renaissance literature which uses gothic elements to question Agrarian cause--quite daring project for Gordon given that she was married to one of movement's founders and an unofficial member herself (Tunc, Recuperating 182-183). Only by placing novel in dialogue with other prominent works of Renaissance can its literary lineage be restored and its deep criticism of southern society be revealed. …

On a convolution series attached to a Siegel Hecke cusp form of degree 2

We prove that the “naive” convolution Dirichlet series $D_{2}(s)$ attached to a degree 2 Siegel Hecke cusp form $F$ , has a pole at $s=1$ . As an application, we write down the asymptotic formula for the partial sums of the squares of the eigenvalues of $F$ with an explicit error term. Further, as a corollary, we are able to show that the abscissa of absolute convergence of the (normalized) spinor-zeta function attached to $F$ is $s = 1$ .

Binary Compressed Imaging

Compressed sensing can substantially reduce the number of samples required for conventional signal acquisition at the expense of an additional reconstruction procedure. It also provides robust reconstruction when using quantized measurements, including in the one-bit setting. In this paper, our goal is to design a framework for binary compressed sensing that is adapted to images. Accordingly, we propose an acquisition and reconstruction approach that complies with the high dimensionality of image data and that provides reconstructions of satisfactory visual quality. Our forward model describes data acquisition and follows physical principles. It entails a series of random convolutions performed optically followed by sampling and binary thresholding. The binary samples that are obtained can be either measured or ignored according to predefined functions. Based on these measurements, we then express our reconstruction problem as the minimization of a compound convex cost that enforces the consistency of the solution with the available binary data under total-variation regularization. Finally, we derive an efficient reconstruction algorithm relying on convex-optimization principles. We conduct several experiments on standard images and demonstrate the practical interest of our approach.

Determination of finite-difference weights using scaled binomial windows

The finite-difference method evaluates a derivative through a weighted summation of function values from neighboring grid nodes. Conventional finite-difference weights can be calculated either from Taylor series expansions or by Lagrange interpolation polynomials. The finite-difference method can be interpreted as a truncated convolutional counterpart of the pseudospectral method in the space domain. For this reason, we also can derive finite-difference operators by truncating the convolution series of the pseudospectral method. Various truncation windows can be employed for this purpose and they result in finite-difference operators with different dispersion properties. We found that there exists two families of scaled binomial windows that can be used to derive conventional finite-difference operators analytically. With a minor change, these scaled binomial windows can also be used to derive optimized finite-difference operators with enhanced dispersion properties.

Asynchronous CDMA Under Single-Carrier Block Transmission—Architectures and Signal Models

Single-carrier (SC) block transmission with frequency-domain equalization (FDE) is an efficient scheme that avoids complex time-domain equalization in multipath channels. There are also several systems that are proposed that apply SC-FDE in code-division multiple access (CDMA). However, the present CDMA systems with FDE may not be suitable for asynchronous users for lack of a signal model that can suitably express the despread signal in the frequency domain. In this paper, we first propose circularly sliding (CS) despreading that can attain preliminary separation of interference among asynchronous users. Then, we derive a signal model that represents spreading, channel effect, and despreading as a series of circular convolutions. This signal model helps to derive a new FD minimum mean-square error (MMSE) equalization algorithm for SC block transmission asynchronous CDMA. We further extend the signal model by including equalization as another circular convolution. The extended signal model indicates several equivalent architectures of the receiver. For these architectures, despreading and equalization can be implemented in either the time or frequency domain, and the order is exchangeable. Based on the proposed signal model, we also derive a RAKE combiner that can be implemented in the frequency domain. Our architectures and signal model can be applied for further development of multiuser-detection algorithms.

Multi-Level Ewald: A Hybrid Multigrid/Fast Fourier Transform Approach to the Electrostatic Particle-Mesh Problem

We present a new method for decomposing the one convolution required by standard Particle-Particle Particle-Mesh (P(3)M) electrostatic methods into a series of convolutions over slab-shaped subregions of the original simulation cell. Most of the convolutions derive data from separate regions of the cell and can thus be computed independently via FFTs, in some cases with a small amount of zero padding so that the results of these sub-problems may be reunited with minimal error. A single convolution over the entire cell is also performed, but using a much coarser mesh than the original problem would have required. This "Multi-Level Ewald" (MLE) method therefore requires moderately more FFT work plus the tasks of interpolating between different sizes of mesh and accumulating the results from neighboring sub-problems, but we show that the added expense can be less than 10% of the total simulation cost. We implement MLE as an approximation to the Smooth Particle Mesh Ewald (SPME) style of P(3)M, and identify a number of tunable parameters in MLE. With reasonable settings pertaining to the degree of overlap between the various sub-problems and the accuracy of interpolation between meshes, the errors obtained by MLE can be smaller than those obtained in molecular simulations with typical SPME settings. We compare simulations of a box of water molecules performed with MLE and SPME, and show that the energy conservation, structural, and dynamical properties of the system are more affected by the accuracy of the SPME calculation itself than by the additional MLE approximation. We anticipate that the MLE method's ability to break a single convolution into many independent sub-problems will be useful for extending the parallel scaling of molecular simulations.