Fast Algorithm Research Articles

Numerical solution of the boundary value problems for the biharmonic equations via quasiseparable representations

AbstractThe paper incorporates new methods of numerical linear algebra for the approximation of the biharmonic equation with potential, namely, numerical solution of the Dirichlet problem for $$ \left( \frac{d}{dx}\right) ^4u(x)+c(x)u(x)=\phi (x),\quad 0<x<1. $$ d dx 4 u ( x ) + c ( x ) u ( x ) = ϕ ( x ) , 0 < x < 1 . High-order discrete finite difference operators are presented, constructed on the basis of discrete Hermitian derivatives, and the associated Discrete Biharmonic Operator (DBO). It is shown that the matrices associated with the discrete operator belong to a class of quasiseparable matrices of low rank matrices. The application of quasiseparable representation of rank structured matrices yields fast and stable algorithm for variable potentials c(x). Numerical examples corroborate the claim of high order accuracy of the algorithm, with optimal complexity O(N).

Деформирующие преобразования изображений и их применение при аугментации данных для обучения глубоких нейронных сетей

The paper focuses on the improvement of the quality of learning for deep neural networks for a small data set in a classification task. One of the possible approaches to improve the quality of learning is researched which is based on the use of data augmentation (artificial reproduction of the data set) by image warping. The presented mathematical model and fast algorithm for warping make it possible to transform the original image while preserving its structural basis. The proposed algorithm is used to augment image data sets containing a small number of training samples. The augmentation consists of two stages including horizontal mirroring and warping of each of the samples. The effectiveness of such augmentation is tested through the training of neural networks of various types: convolutional neural networks (CNN) of a standard architecture and deep residual networks (DRN). A specific feature of the implemented approach for the solution of the problem under consideration consists in the refusal to use pre-trained neural networks with a large number of layers as well as further transfer learning, since their application incurs costs in terms of the computational resources. The paper shows that the efficiency of image classification when implementing the proposed method of augmenting training data on small and medium-sized data sets increases to statistically significant values of the metric used.

DOT-type schemes for hybrid hyperbolic problems arising from free-surface, mobile-bed, shallow-flow models

Free-surface, mobile-bed, shallow-flow models may present Hybrid hyperbolic systems of partial differential equations characterised by conservative and non-conservative fluxes that can only be expressed in primitive variables. This paper presents the effort we made to derive DOT-type schemes (Osher-type schemes derived by Dumbser and Toro, 2011 [1]) for these kinds of systems formulated for the one-dimensional case. Firstly, for a Hybrid system, we managed to write a quasi-linear form characterised by the presence of a matrix, expressed as a function of the primitive variables, that multiplies the spatial derivative of the conserved variables. Next, we derived the first numerical flux by adapting the approach of Leibinger et al., 2016 [2] to this quasi-linear form. We called this result DOTHCP flux. To achieve a faster algorithm, instead of using an integration path in the space of conserved variables, as in the previous case, we employed a path in the space of primitive variables. We called this second formulation DOTHPP flux. Subsequently, we managed to account for certain physical constraints arising from the generalised Rankine-Hugoniot relations in the expression of one term of the previous flux formulation, thus obtaining the DOTHZR flux. Finally, we showed that these methods can also be applied to Combined systems characterised by conservative and non-conservative fluxes expressed in conserved variables. Several tests show the characteristics and good performances of the proposed methods when applied to Riemann problems of Hybrid and Combined systems deriving from free-surface models. Finally, thanks to the general formulation of the proposed DOT-type fluxes, these can also be applied to Hybrid and Combined hyperbolic systems deriving from different physical problems.

Mixed Matrix Completion in Complex Survey Sampling under Heterogeneous Missingness

Modern surveys with large sample sizes and growing mixed-type questionnaires require robust and scalable analysis methods. In this work, we consider recovering a mixed dataframe matrix, obtained by complex survey sampling, with entries following different canonical exponential distributions and subject to heterogeneous missingness. To tackle this challenging task, we propose a two-stage procedure: in the first stage, we model the entry-wise missing mechanism by logistic regression, and in the second stage, we complete the target parameter matrix by maximizing a weighted log-likelihood with a low-rank constraint. We propose a fast and scalable estimation algorithm that achieves sublinear convergence, and the upper bound for the estimation error of the proposed method is rigorously derived. Experimental results support our theoretical claims, and the proposed estimator shows its merits compared to other existing methods. The proposed method is applied to analyze the National Health and Nutrition Examination Survey data. Supplementary materials for this article are available online.

Theoretical and Empirical Analysis of a Fast Algorithm for Extracting Polygons from Signed Distance Bounds

Recently, there has been renewed interest in signed distance bound representations due to their unique properties for 3D shape modelling. This is especially the case for deep learning-based bounds. However, it is beneficial to work with polygons in most computer graphics applications. Thus, in this paper, we introduce and investigate an asymptotically fast method for transforming signed distance bounds into polygon meshes. This is achieved by combining the principles of sphere tracing (or ray marching) with traditional polygonization techniques, such as marching cubes. We provide theoretical and experimental evidence that this approach is of the O(N2logN) computational complexity for a polygonization grid with N3 cells. The algorithm is tested on both a set of primitive shapes and signed distance bounds generated from point clouds by machine learning (and represented as neural networks). Given its speed, implementation simplicity, and portability, we argue that it could prove useful during the modelling stage as well as in shape compression for storage.

Capi-score: a quantitative algorithm for identifying disease patterns in nailfold videocapillaroscopy.

The European Alliance of Associations for Rheumatology (EULAR) supports the use of nailfold videocapillaroscopy (NVC) to identify disease patterns (DPs) associated with systemic sclerosis (SSc) and Raynaud's phenomenon (RP). Recently, EULAR proposed an easy-to-manage procedure, a so-called Fast Track algorithm, to differentiate SSc from non-SSc patterns in NVC specimens. However, subjectivity among capillaroscopists remains a limitation. Our aim was to perform a software-based analysis of NVC peculiarities in a cohort of samples from SSc and RP patients and, subsequently, build a Fast Track-inspired algorithm to identify DPs without the constraint of interobserver variability. NVCs were examined by 9 capillaroscopists. Those NVCs whose DPs were consensually agreed (≥2 out of 3 interobservers) were subsequently analysed with an in-house developed software. Each variable's results were grouped according to the consensually agreed DPs in order to identify useful hallmarks to categorise them. Eight-hundred and fifty-one NVCs (21 957 images) whose DPs had been consensually agreed were software-analysed. Appropriate cut-offs set in capillary density and percentage of abnormal and giant capillaries, tortuosities and hemorrhages allowed DP categorization and the development of the CAPI-Score algorithm. This consisted of 4 rules: Rule 1, SSc vs non-SSc, accuracy 0.88; Rules 2 and 3, SSc-early vs SSc-active vs SSc-late, accuracy 0.82; Rule 4, non-SSc normal vs non-SSc non-specific, accuracy 0.73. Accuracy improved when the analysis was limited to NVCs whose DPs had achieved full consensus among interobservers. The CAPI-Score algorithm may become a useful tool to assign DPs by overcoming the limitations of subjectivity.

Improving model performance of shortest‐path‐based centrality measures in network models through scale space

SummaryThe quality of the solution in resolving a complex network depends on either the speed or accuracy of the results. While some health studies prioritize high performance, fast algorithms are favored in scenarios requiring rapid decision‐making. A comprehensive understanding of the problem necessitates a detailed analysis of the network and its individual components. Betweenness Centrality (BC) and Closeness Centrality (CC) are commonly employed measures in network studies. This study introduces a new strategy to compute BC and CC that assesses their sensitivity in the scale space while measuring the shortest path. The scale space is generated by incorporating a scale parameter that is shown to achieve up to 60% performance improvements for various datasets. The study provides in‐depth insights into the importance of the scale space analysis. Finally, a flexible measurement tool is provided that is suitable for various types of problems. To demonstrate the flexibility and applicability, we experimented with two methods for 10 different graphs using the proposed approach.

Exact Characterization of the Jointly Optimal Restocking and Auditing Policy in Inventory Systems with Record Inaccuracy

We present a continuous-time stochastic model of an inventory system with record inaccuracy. In this formulation, demand is modeled by a point process and is observable only when it leads to sales. In addition to demand that can reduce the stock, an unobservable stochastic loss process can also reduce the stock. The retailer’s goal is to identify the restocking and auditing policy that minimizes the expected discounted cost of carrying a product over an infinite horizon. We analytically characterize the optimal restocking and jointly optimal auditing policy. We prove that the optimal restocking policy is a threshold policy. Our proof of this result is based on a coupling argument that is valid for any demand and loss model. Unlike the optimal restocking policy, the jointly optimal auditing policy is not of threshold type. We show that a complete proof of this statement cannot be obtained by solely resorting to the first-order stochastic dominance property of the Bayesian shelf stock distribution induced by the demand and loss process. Instead, our characterization of the jointly optimal auditing policy is based on proving that the dynamics of the shelf stock distribution constitute a (strictly) sign-regular kernel. To our knowledge, this is the first paper that characterizes the optimal policy of a complex control problem by establishing sign regularity of its underlying Markovian dynamics. Our theoretical results lead to a fast algorithm for computing the exact jointly optimal auditing/restocking policy over the problem’s entire state space. This enables comparative statics analysis, which allows us to determine how inventory record inaccuracy affects the economic significance of various cost drivers. This, in turn, allows us to determine when or, better, under what conditions auditing can be an effective tool for reducing the total cost.

Back to basics: Fast denoising iterative algorithm

We introduce Back to Basics (BTB), a fast iterative algorithm for noise reduction. Our method is computationally efficient, does not require training or ground truth data, and can be applied in the presence of independent noise, as well as correlated (coherent) noise, where the noise level is unknown. We examine three study cases: natural image denoising in the presence of additive white Gaussian noise, Poisson-distributed image denoising, and speckle suppression in optical coherence tomography (OCT). Experimental results demonstrate that the proposed approach can effectively improve image quality, in challenging noise settings. Theoretical guarantees are provided for convergence stability.

A modified atmospheric scattering model and degradation image clarification algorithm for haze environments

Addressing the limitation of existing image dehazing algorithms based on atmospheric scattering models, which often overlook the underlying physical causes of color distortion in hazy images, resulting in suboptimal algorithm robustness and image clarity when faced with varying levels of haze pollution, this study introduces a refined atmospheric scattering model. It also presents a haze image clarity algorithm based on this modified atmospheric scattering model by analyzing the fundamental factors contributing to image degradation and distortion caused by haze. Initially, the hazy image undergoes conversion to a new HSI color space. The hazy image is aligned more closely with human visual perception characteristics by implementing an adaptive saturation enhancement technique and adjusting contribution coefficients for the three RGB channels to the luminance component. Subsequently, the hazy image is partitioned into dense haze and non-dense haze regions based on haze distribution characteristics within the image. A global light intensity computation method is devised for the hazy image utilizing a quad-tree segmentation algorithm. Following this, calculations are performed to determine the atmospheric light value and transmittance for dense and non-dense haze regions separately. The global atmospheric light and transmittance values are then derived based on the probability density function of the dense haze region using Alpha Fusion. Finally, corrections are applied to the global atmospheric light values using an atmospheric light correction vector, while denoising the global transmittance is achieved through a fast non-local mean filtering algorithm. Experimental findings demonstrate that the proposed haze image clarity algorithm, grounded in the modified atmospheric scattering model, exhibits adaptability across various degrees and types of haze images. Comparative analysis against other prominent image dehazing algorithms reveals its efficacy in addressing severe color distortion in dense haze scenarios, achieving a more natural clarity effect, and showcasing superior performance across all numerical evaluation metrics.

RESEARCH OF ALGORITHMS FOR SEARCHING PRIMITIVE ELEMENTS OF A FINITE FIELD OF HIGH ORDER

One of the most important unsolved and notoriously difficult problems in computational finite field theory is the development of a fast algorithm for constructing primitive roots in a finite field. It is known that for many applications, instead of a primitive root, just an element of high multiplicative order is sufficient. Such applications include, but are not limited to, cryptography, coding theory, pseudorandom number generation, and combinatorial schemes. Explicit constructions of high-order elements usually rely on combinatory methods that can provide a provable lower bound on the order, but this does not compute the exact order. Its execution usually implies knowledge of the factorization of the order. Ideally, we should be able to get a primitive element for any finite field in a reasonable amount of time. However, if the simple factorization of the group order is unknown, it is difficult to achieve the goal. Thus, we set the task of constructing an element, probably of a high order. This article discusses various algorithms that find a high-order element for general or special finite fields. This work also represents another contribution to the theory of Gauss periods over finite fields and their generalizations and analogues, which have already proven their usefulness for a number of different applications.

FedNS: A Fast Sketching Newton-Type Algorithm for Federated Learning

Recent Newton-type federated learning algorithms have demonstrated linear convergence with respect to the communication rounds. However, communicating Hessian matrices is often unfeasible due to their quadratic communication complexity. In this paper, we introduce a novel approach to tackle this issue while still achieving fast convergence rates. Our proposed method, named as Federated Newton Sketch methods (FedNS), approximates the centralized Newton's method by communicating the sketched square-root Hessian instead of the exact Hessian. To enhance communication efficiency, we reduce the sketch size to match the effective dimension of the Hessian matrix. We provide convergence analysis based on statistical learning for the federated Newton sketch approaches. Specifically, our approaches reach super-linear convergence rates w.r.t. the communication rounds for the first time. We validate the effectiveness of our algorithms through various experiments, which coincide with our theoretical findings.

Optimal Transport with Tempered Exponential Measures

In the field of optimal transport, two prominent subfields face each other: (i) unregularized optimal transport, ``a-la-Kantorovich'', which leads to extremely sparse plans but with algorithms that scale poorly, and (ii) entropic-regularized optimal transport, ``a-la-Sinkhorn-Cuturi'', which gets near-linear approximation algorithms but leads to maximally un-sparse plans. In this paper, we show that an extension of the latter to tempered exponential measures, a generalization of exponential families with indirect measure normalization, gets to a very convenient middle ground, with both very fast approximation algorithms and sparsity, which is under control up to sparsity patterns. In addition, our formulation fits naturally in the unbalanced optimal transport problem setting.

Optimal Transport with Cyclic Symmetry

We propose novel fast algorithms for optimal transport (OT) utilizing a cyclic symmetry structure of input data. Such OT with cyclic symmetry appears universally in various real-world examples: image processing, urban planning, and graph processing. Our main idea is to reduce OT to a small optimization problem that has significantly fewer variables by utilizing cyclic symmetry and various optimization techniques. On the basis of this reduction, our algorithms solve the small optimization problem instead of the original OT. As a result, our algorithms obtain the optimal solution and the objective function value of the original OT faster than solving the original OT directly. In this paper, our focus is on two crucial OT formulations: the linear programming OT (LOT) and the strongly convex-regularized OT, which includes the well-known entropy-regularized OT (EROT). Experiments show the effectiveness of our algorithms for LOT and EROT in synthetic/real-world data that has a strict/approximate cyclic symmetry structure. Through theoretical and experimental results, this paper successfully introduces the concept of symmetry into the OT research field for the first time.

Denoising image-based experimental data without clean targets based on deep autoencoders

Linear data-driven methods have demonstrated being effective denoising filters for image-based techniques. On the downside, they are still challenged by conditions of extremely low signal-to-noise ratio. A procedure based on denoising autoencoder (AE) is proposed here. The main challenge is the absence of a “clean” target, thus the training is referred to as “blind”. The method is validated on two synthetic datasets, based on the flow around a circular cylinder, and a fabricated pattern. The reconstruction accuracy is found to be mostly sensitive to the size of the latent space of the AE. A criterion to determine the optimal latent vector size based solely on the information available during the training process is proposed. The reconstruction error is shown to be up to one order of magnitude smaller than the case of a denoising Proper Orthogonal Decomposition. Furthermore, a fast approach using an attention mechanism is proposed. This fast alternative algorithm saves up to 98% training time with only a limited reconstruction accuracy loss if compared to the original approach. Finally, the superiority of the proposed AE blind denoising over linear data-driven filtering has been proved on Pressure Sensitive Paint data of an inclined impinging jet, showing a better performance than the POD denoising.

Hardware motion estimation methods survey and example real-time architecture design

Motion estimation is essential in video processing, computer vision, and other related fields due to its applications in improving data compression, enhancing image/video quality, increasing computer vision task accuracy, and conserving computational resources. Typical motion estimation methods include FS (Full Search) and DS (Diamond Search). Traditional motion estimation methods will face the challenge of high complexity in practical real-time applications. Therefore, this paper focuses on fast motion estimation algorithms and hardware architecture design for real-time applications survey and research. Aiming at the need for 1920x1080@60fps processing speed, this paper explores the key ideas of FS and DS hardware structure design based on the Vivado HLS high-level synthesis tool and provides a feasible hardware scheme to meet the real-time requirements.

CN ADI fast algorithm on non-uniform meshes for the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamics

This paper proposes a Crank-Nicolson alternating direction implicit (CN-ADI) finite difference scheme for solving the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamic for the first time. Due to the weakly singular behavior of the exact solution near the initial time t=0, we use the non-uniform meshes to capture the rapid change of the solution at t=0. The Crank-Nicolson method and product-integration (PI) rule are proposed to approximate temporal derivative and the Riemann-Liouville (R-L) fractional integral term, respectively. The fully discrete scheme is obtained by the standard central finite difference method (FDM) in space. The stability in L2-norm and convergence of the CN-ADI difference scheme are strictly proved, where the convergence reached O(τ2+hx2+hy2+hz2). The ADI algorithm greatly reduces the computational cost of the three-dimensional problems in viscoelastic dynamics. At last, the results of numerical examples verify the correctness of the theoretical analysis and prove the effectiveness of the proposed method.

An Integrated Thresholding and Morphological Process with Histogram-based Method for Brain Tumor Analysis and MRI Tumor Detection

INTRODUCTION: Over the past several years analysis of image has moved from larger system to pervasive portable devices. For example, in pervasive biomedical systems like PACS-Picture achieving and Communication system, computing is the main element. Image processing application for biomedical diagnosis needs efficient and fast algorithms and architecture for their functionality. Future pervasive systems designed for biomedical application should provide computational efficiency and portability. The discrete wavelet transform (DWT) designed in on-chip been used in several applications like data, audio signal processing and machine learning.
 OBJECTIVES: The conventional convolution based scheme is easy to implement but occupies more memory , power and delay. The conventional lifting based architecture has multiplier blocks which increase the critical delay. Designing the wavelet transform without multiplier is a effective task especially for the 2-D image analysis. Without multiplier Daubechies wavelet implementation in forward and inverse transforms may find efficient. The objective of the work is on obtaining low power and less delay architecture.
 METHODS: The proposed lifting scheme for two dimensional architecture reduces critical path through multiplier less and provides low power, area and high throughput. The proposed multiplier is delay efficient.
 RESULTS: The architecture is Multiplier less in the predict and update stage and the implementation carried out in FPGA by the use of Quartus II 9.1 and it is found that there is reduction in consumption of power at approximately 56%. There is reduction in delay due to multiplier less architecture.
 CONCLUSION: multiplier less architecture provides less delay and low power. The power observed is in milliwatts and suitable for high speed application due to low critical path delay.

Secure and Fast Image Encryption Algorithm Based on Modified Logistic Map

In the past few decades, the transmission of data over an unsecure channel has resulted in an increased rate of hacking. The requirement to make multimedia data more secure is increasing day by day. Numerous algorithms have been developed to improve efficiency and robustness in the encryption process. In this article, a novel and secure image encryption algorithm is presented. It is based on a modified chaotic logistic map (CLM) that provides the advantage of taking less computational time to encrypt an input image. The encryption algorithm is based on Shannon’s idea of using a substitution–permutation and one-time pad network to achieve ideal secrecy. The CLM is used for substitution and permutation to improve randomness and increase dependency on the encryption key. Various statistical tests are conducted, such as keyspace analysis, complexity analysis, sensitivity analysis, strict avalanche criteria (SAC), histogram analysis, entropy analysis, mean of absolute deviation (MAD) analysis, correlation analysis, contrast analysis and homogeneity, to give a comparative analysis of the proposed algorithm and verify its security. As a result of various statistical tests, it is evident that the proposed algorithm is more efficient and robust as compared to previous ones.

A fast gangue detection algorithm based on multi-head self-attention mechanism and anchor frame optimization strategy

ABSTRACT Because of the problem of difficult identification, model acceleration, and localization for coal gangue detection, a fast coal gangue detection algorithm based on a multi-head self-attention mechanism and anchor frame optimization strategy is designed. Firstly, the multi-head self-attention mechanism MHSA is combined with the CNN backbone structure of YOLOv5s, which is used to enhance the algorithm’s ability to perceive global information to strengthen the model’s recognition of coal and gangue; Secondly, the lightweight Ghost module will be used to reconfigure the backbone of YOLOv5s and the feature fusion layer to reduce the complexity of the model and increase the speed of the coal gangue image detection; And lastly, the selection of an appropriate strategy of input image scale and anchor frame size to improve the performance of the detection algorithm to solve the problem of localization of coal and gangue. Experiments show that the average accuracy of the improved MYL model mAP_0.5 reaches 91.8%, the detection accuracy of mAP_0.5 is improved by 3.6%, and the detection speed is 80 FPS. The improved MYL model can show stronger robustness and superior performance in the detection of coal gangue and can meet the requirements of effective industrial detection.