Two-dimensional Discrete Fourier Research Articles

Deep Learning Soft-Decision GNSS Multipath Detection and Mitigation.

A technique is proposed to detect the presence of the multipath effect in Global Navigation Satellite Signal (GNSS) signals using a convolutional neural network (CNN) as the building block. The network is trained and validated, for a wide range of C/N0 values, with a realistic dataset constituted by the synthetic noisy outputs of a 2D grid of correlators associated with different Doppler frequencies and code delays (time-domain dataset). Multipath-disturbed signals are generated in agreement with the various scenarios encompassed by the adopted multipath model. It was found that pre-processing the outputs of the correlators grid with the two-dimensional Discrete Fourier Transform (frequency-domain dataset) enables the CNN to improve the accuracy relative to the time-domain dataset. Depending on the kind of CNN outputs, two strategies can then be devised to solve the equation of navigation: either remove the disturbed signal from the equation (hard decision) or process the pseudoranges with a weighted least-squares algorithm, where the entries of the weighting matrix are computed using the analog outputs of the neural network (soft decision).

Application of Wireless Network Remote Sensing Image and Video Processing Technology in Smart City Design and planning

With the rapid development of society, regional urban construction has become the main guarantee of economic development, and smart city construction has become the main trend of regional development. Smart cities consist of information images and urban development, which can meet the needs of the modern generation for material life, knowledge and scientific life. Therefore, wireless remote sensing (RS) and video image processing have become key tools for the development of smart cities, which can solve the problems encountered by many professionals before, and become an integral part of urban construction. On this basis, this paper summarized the video processing image and wireless network RS image, and analyzed their development and application in urban design and layout. Based on this, this paper expounded the application of wireless network RS image in smart city design and layout, and analyzed the application prospects of RS technology in building smart city space-time information framework, RS+urban management, RS+smart environmental protection, RS+smart city agriculture and video processing technology in smart city design and planning. It analyzed the application in intelligent security, intelligent transportation and management visualization, and then used the two-dimensional discrete Fourier transform (2D-DFT) definition to strengthen the design and layout of smart cities. According to experiments and surveys, introducing 2D-DFT into the design and layout of smart cities can build a more intelligent city and improve residents’ satisfaction by 30%.

Multiple sources localization with 2D-DFT under distributed massive antenna arrays

Traditional localization approaches, e.g., angle of arrival clustering (AOA-clustering), subspace data fusion, and minimum variance distortionless response, entail eigenvalue decomposition or global grid search within the target area, which could pose significant challenges when applied to massive antenna arrays as a result of their high computational complexity. In this paper, we present a low-complexity localization approach for multiple sources with two-dimensional discrete Fourier transform (2D-DFT). Firstly, we compute the cross-covariance, which could effectively suppress noise and obtain aligned AOAs via 2D-DFT. Then, we utilize phase offset method and total least square solution to obtain accurate position estimates. The proposed approach could achieve a good compromise between performance and computational complexity. The Cramér-Rao Bound (CRB) and simulation results are provided to corroborate the effectiveness and superiority of the proposed approach.

Striation-Based Beamforming with Two-Dimensional Filtering for Suppressing Tonal Interference

Based on the interference spectrogram in the element–frequency domain using the data measured by the horizontal linear array, the source range can be estimated through the striation-based beamforming (SBF) method and its variants. Estimation of the striation slope is the basis for these ranging methods. But in practical scenarios, the tonal interferences and other noise make it difficult to estimate the slope. In this paper, we proposed a two-dimensional low-pass filtering method after the two-dimensional discrete Fourier transform (2D-DFT) of the element–frequency domain spectrogram. The signals can be separated from the interference and noise using this filter. For the linear frequency-modulated signal without a known waveform, we also proposed an extraction and phase compensation method based on the time–frequency spectrogram, and the acoustic data obtained can be used for source ranging. The experimental results indicate that the methods proposed are feasible.

Frequency-domain Volterra kernel-based adaptation: Formulations and algorithms

For the correlated input, the Volterra kernel-based least mean-square (LMS) algorithm in the time-domain exhibits a slow learning rate caused by the large eigenvalue spread of the input covariance matrix. To tackle such an issue, this paper develops a novel frequency-domain Volterra kernel-based filter, resulting in the periodic update constrained frequency-domain second-order Volterra normalized LMS (named as P-CFDSOV-NLMS1) algorithm. Subsequently, by using one- and two-dimensional discrete Fourier transforms (DFTs) simultaneously, another frequency-domain implementation and corresponding P-CFDSOV-NLMS2 algorithm are constructed. In contrast, the P-CFDSOV-NLMS1 scheme only requires one-dimensional DFT operations and takes advantage of the joint information between the block input vectors. Then, the mean and mean-square convergence behaviors of the P-CFDSOV-NLMS1 algorithm are investigated. Furthermore, the designed frequency-domain method is extended to three different widely complex-valued Volterra kernel-based models. Finally, computer simulations reveal that the suggested algorithms outperform the previously reported frequency-domain techniques in terms of convergence speed and tracking ability.

A Robust Video Watermarking Algorithm Based on Two-Dimensional Discrete Fourier Transform

Due to the continuous development and popularity of digital video technology, the copyright protection of digital video content has become an increasingly prominent issue. Digital video watermarking, as an effective means of digital copyright protection, has attracted widespread attention from academia and industry. The two-dimensional discrete Fourier transform (2D-DFT) template-based method has the advantages of good real-time performance and robustness, but the embedding capacity is small and cannot resist frame-dropping attack. To address this problem, a new template construction method is proposed in this paper, which can extend the embedding capacity from 1 bit each group of pictures (GOP) to theoretically n bits each GOP. In addition, by changing the GOP pattern, the method gains the ability to resist frame-dropping attacks. Experimental results show that the proposed method can achieve larger watermark capacity while maintaining strong robustness against image-processing attacks, geometric attacks, video-processing attacks and compression attacks.

An Improved Automation System for Destructive and Visual Measurements of Cross-Sectional Geometric Parameters of Microdrills

Microdrills are specific cutting tools widely used to drill microholes and microvias. For certain microdrill manufacturers, a conventional sampling inspection procedure is still manually operated for carrying out the destructive and visual measurements of two essential cross-sectional geometric parameters (CSGPs), called the cross-sectional web thickness (CSWT) and the cross-sectional outer diameter (CSOD), of their straight (ST) and undercut (UC) type microdrill products. In order to comprehensively automate the conventional sampling inspection procedure, a destructive and visual measuring system improved from an existing vision-aided automation system, for both the hardware and the automated measuring process (AMP), is presented in this paper. The major improvement of the hardware is characterized by a machine vision module consisting of several conventional machine vision components in combination with an innovative and lower cost optical subset formed by a set of plano-concave achromatic (PCA) lenses and a reflection mirror, so that the essential functions of visually positioning the drilltip and visually measuring the CSGPs can both be achieved via the use of merely one machine vision module. The major improvement of the AMP is characterized by the establishment of specific image processing operations for an auto-focusing (AF) sub-process based on two-dimensional discrete Fourier transform (2D-DFT), for a web thickness measuring (WTM) sub-process based on an iterative least-square (LS) circle-fitting approach, and for an outer diameter measuring (ODM) sub-process based on integrated applications of an iterative LS circle-fitting approach and an LS line-fitting-based group-dividing approach, respectively. Experiments for measuring the CSGPs of microdrill samples were conducted to evaluate the actual effectiveness of the developed system. It showed that the developed system could achieve good repeatability and accuracy for the measurements of the CSWTs and CSODs of both ST and UC type microdrills. Therefore, the developed system could effectively and comprehensively automate the conventional sampling inspection procedure.

Improvement of low contrast images in the frequency domain using fuzzy intensification

Due to the variety of types of images, there are currently no universal methods that provide a guaranteed result of solving the problem of improving the quality of low-contrast digital images. Medical systems often produce images of insufficient quality for reliable visual analysis. In particular, X-ray images, characterized by low intensity, un-even background, high level of noise, poor contrast and weakly defined boundaries of structures, are particularly difficult to analyze and choose an effective processing meth-od. The paper presents the information possibilities of the method of processing half-tone medical images aimed at improving the contrast and increasing the detail of ob-jects of interest in order to increase the reliability of diagnosis based on them. The pro-posed algorithm is based on a multi-stage processing process, which includes the use of two-dimensional frequency Fourier transformation and the method of fuzzy intensifica-tion in the spatial domain. The use of two-dimensional discrete Fourier transformation changes not the im-age, but the form of its representation, converting the output signal into its components of different frequencies and amplitudes. In this form, it is much easier to carry out filter-ing or amplification of individual components of the signal. Fuzzy set theory has the ability to quantitatively and qualitatively model problems related to uncertainty and imprecision, which are always present in digital images. Their presence is determined both by the features of the physical processes of image forming systems and by the stage of creating a digital image. The application of the proposed method provides improved contrast and increased detailing of objects of interest and affects the reliability of visual analysis. Only the ar-guments of complex Fourier transform coefficients can be corrected. The method of fuzzy intensification is used as a refinement for the second stage of frequency conversion. The effect of frequency conversion parameters on the detail of the resulting image was stud-ied. The results of the algorithm are presented on the example of real X-ray images.

Elucidation of the mechanism of chatter mark formation during end-milling and inverse analysis of chatter vibration by two-dimensional discrete Fourier transform of chatter mark

Regenerative chatter vibration is a typical machining problem during cutting. The frequency and phase shift of regenerative chatter can be used as indicators to suppress regenerative chatter. However, direct measurement of chatter vibration during cutting is difficult for end milling because the tool blade rotates. Therefore, chatter vibration is generally measured indirectly, for example, from cutting noise or vibration transmitted to the spindle. The type and position of the installed sensors must be appropriately selected to accurately measure chatter vibration with these methods. This selection must account for the machine tool structure, cutting conditions, and the position where the chatter vibration occurs, making the measurement methods not universally applicable. When chatter vibration occurs during cutting, a periodic pattern called a chatter mark is formed on the machined surface. The chatter mark is formed by vibration during cutting, i.e., vibration information during cutting is directly recorded. Therefore, this paper proposes a method to analyze the frequency and phase shift of chatter vibration using a two-dimensional discrete Fourier analysis of the chatter mark based on the modeled formation mechanism of chatter marks, and further validates the proposed method by elucidating the mechanism of chatter mark formed by chatter vibration. First, a cutting simulation was performed based on the vibration displacement of the tool during chatter cutting obtained experimentally. This simulation elucidates the mechanism by which the vibration information of chatter vibration can be analyzed from the chatter mark. Subsequently, the chatter vibration parameters were analyzed from the chatter mark using the proposed method. The parameters were consistent with those analyzed from the vibration displacement of the tool, confirming the effectiveness of the proposed method.

Algorithm for Solving Discrete Contact Problems for an Elastic Layer

The problems of discrete contact between an elastic layer and a rigid punch with unknown areas of actual contact are considered. A variational formulation of the problems is obtained in the form of a boundary variational inequality using the Poincaré-Steklov operator, which maps normal stresses into normal displacements on a part of the elastic layer boundary. To approximate this operator, a two-dimensional discrete Fourier transform is used, for the numerical implementation of which algorithms of the fast Fourier transform are used. A minimization problem equivalent to the variational inequality is presented. As a result of its approximation, a quadratic programming problem with constraints in the form of equalities and inequalities is obtained. To numerically solve this problem, we used an algorithm based on the conjugate gradient method, which takes into account the specifics of the set of constraints. Two-parameter families of punches rectangular in plan with surface relief are constructed. As a result of computational experiments, the existence of a single envelope of contact pressure, a single envelope of normalized contact forces, and a single envelope of the relative values of the actual contact areas of microprotrusions has been established for each family of punches. The shape and position of these envelopes for a family of punches depend on the parameters of the external load and the ratio of the dimensions of the nominal contact area to the layer thickness.

Hotspot Detection with Machine Learning Based on Pixel-Based Feature Extraction

The complexity of physical verification increases rapidly with fast shrinking technology nodes. Considering only design rule checking (DRC) constraints or lithography models cannot capture the side physical effects in the fabrication process well. Thus, it is desirable to consider a more general physical verification problem with various types of hotspots. In this paper, we apply machine learning which is based on pixel-based feature extraction to deal with the generalized hotspot detection problem. First, a two-dimensional discrete Fourier transformation-based pixel extraction method is proposed to alleviate the shifting effect and produce stable hotspot features. Then, a pattern-based layout scanning approach is developed to enhance the program efficiency while preserving good detection accuracy. Finally, we design two false alarm reduction strategies to effectively reduce the number of detected nonhotspots and further improve the accuracy of hotspot position. Experimental results based on the industrial benchmarks show that our algorithm outperforms three competitive works in terms of accuracy, false alarm rate, efficiency, and time.

Inverse Analysis of Chatter Vibration Information of Circular End Milling Based on Two Dimensional Discrete Fourier Transform of Machined Surface Pattern Reconstructed from Multiple Images

It is known that chatter marks are formed on the machined surface when chatter vibration occurs during cutting. This paper proposes a method for inverse analysis of the frequency and phase difference of chatter vibration during cutting based on the periodicity of the chatter mark on circular machined surface analyzed by the two-dimensional discrete Fourier transform. The chatter mark on the circular machined surface was reconstructed on the image by projecting multiple images taken by a digital camera of the machined surface onto the circular machined surface on the computer, based on the camera calibration using the accurate table feed of the machine tool. The chatter vibration information, which was obtained by the proposed method based on the two-dimensional discrete Fourier transform of this reconstructed chatter mark image, was compared with the information analyzed from the tool displacement during cutting. As a result, the proposed process is found to be effective as an inverse analysis method to estimate the chatter vibration when end milling the curved surface.

Two‐dimensional sparse fractional Fourier transform and its applications

The discrete fractional Fourier transform is an excellent tool in non-stationary signal processing. And an efficient and accurate computation is important for the two-dimensional discrete fractional Fourier transform (2D DFRFT). Inspired by the sparse Fourier transform algorithm, we propose a two-dimensional sparse fractional Fourier transform (2D SFRFT) algorithm to estimate the fractional Fourier spectrum efficiently. Compared with existing methods, we have achieved the lowest runtime and sample complexity. Moreover, by analyzing the errors due to noises, the 2D SFRFT algorithm is improved to be robust. The applications in image fusion, parameter estimation of multicomponent 2D chirp signal and complex maneuvering targets in SAR radar demonstrate the effectiveness of the proposed algorithms.

Machine Learning-Based Optical Performance Monitoring for Super-Channel Optical Networks

In this paper, and for the first time in literature, optical performance monitoring (OPM) of super-channel optical networks is considered. In particular, we propose a novel machine learning OPM technique based on the use of transformed in-phase quadrature histogram (IQH) features and support vector regressor (SVR) to estimate different optical parameters such as optical signal-to-noise ratio (OSNR) and chromatic dispersion (CD). Two transformation methods, the two-dimensional (2D) discrete Fourier transform (DFT) and 2D discrete cosine transform (DCT), are applied to the IQH to extract features with a considerably reduced dimensionality. For the purpose of simulation, the OPM of a 7 × 20 Gbaud dual-polarization–quadrature phase shift keying (DP-QPSK) is considered. Simulations reveal that it can accurately estimate the various optical parameters (i.e., OSNR and CD) with a coefficient of determination value greater than 0.98. In addition, the effectiveness of proposed OPM scheme is examined under different values of polarization mode dispersion and frequency offset, as well as the utilization of different higher order modulation formats. Moreover, proof-of-concept experiments are performed for validation. The results show an excellent matching between the simulation and experimental findings.

Восстановление радиоизображений по многочастотной мультистатической радиоголограмме методом неэквидистантного БПФ

The article considers the transformation of the back-projection method (focusing) for recovering geometry from the knowledge of the scattered field received with a multi-static radar system (multi-static radio hologram). It is shown that it can be expressed as a two-dimensional non-equidistant Discrete Fourier Transform (NDFT). An example of focusing a multi-frequency multi-static radio hologram of a flat multipoint object is given, and a reconstructed image is obtained based on the back-projection algorithm and the NDFT algorithms. A modification of NDFT based on the fast Gaussian gridding of non-uniform nodes (NuFFT), implemented in MATLAB, is used to speed up image reconstruction. A numerical experiment indicates similar reconstructed images obtained by the backpropagation method and with the two-dimensional NuFFT method. However, the NuFFT method was 740 times faster. The results show a significant reduction.

Ownership Protection on Digital Elevation Model (DEM) Using Transform-Based Watermarking

This research aims to protect Digital Elevation Model (DEM) data from piracy or counterfeiting. An invisible watermark inserted into the data, which will not considerably change the data value, is necessary. The proposed method involves the use of the two-dimensional discrete cosine transform (2D DCT), a combination of 2D DCT and discrete wavelet transform (DWT), and two-dimensional discrete Fourier transform (2D DFT) in the frequency domain. The data used include a National DEM file downloaded from the geoportal of the Geospatial Information Agency (Badan Informasi Geospasial—BIG). Three files represent mountainous, lowland/urban, and coastal areas. An “attack” is also conducted on the watermarked DEM by cropping. The results indicate that the watermarked DEM is well recognized. The watermark can be read 100% for 2D DCT, while that for 2D DFT can be read 90.50%. The distortion value of the elevation data under the DCT technique demonstrates the smallest maximum value of 0.1 m compared with 4.5 and 1.1 m for 2D DFT and 2D DCT–DWT. Meanwhile, the height difference (Max Delta), the peak signal-to-noise ratio, and the root mean squared error (RMSE) are highest in mountainous, lowland, and coastal areas, respectively. Overall, the 2D DCT is also superior to the 2D DFT and the2D DCT–DWT. Although only one can recognize the nine watermarks inserted on each sheet, DEMs attacked by the cropping process can still be identified. However, this finding can sufficiently confirm that DEMs belong to BIG.

Application of two-dimensional warped discrete Fourier transform to nonlinear two-dimensional amplitude demodulation

This study introduces a novel nonlinear two-dimensional (2D) warped discrete Fourier transform (WDFT) amplitude demodulation method that allows higher-resolution measurements on specific two-dimensional regions of interest. The specific application in this study addresses the need for the nonlinear 2D amplitude demodulation of fringe-wave fields acquired by a deformed Mach–Zehnder interferometer. The proposed method exploits the properties based on which the WDFT can obtain frequency spectra sampled at non-equidistant intervals and the symmetry of the Fourier transform. As a result, this 2D-WDFT-based method can be used to nonlinearly demodulate the amplitude of the 2D interference fringe wave field to obtain a high-resolution localized fringe envelope. This study describes the proposed analysis procedure and its application using this 2D WDFT-based demodulation scheme. Experimental verification of the proposed approach was conducted with the use of deformed Mach–Zehnder interferometry. The successful experimental validation proves the efficiency of the proposed algorithm for amplitude demodulation with dense sampling points located in the central part of the envelope. To the best of the authors’ knowledge, this is the first investigational report on a 2D nonlinear envelope demodulator.

Impact of the vibration measurement points geometric coordinates uncertainties on two-dimensional k-space identification: Application to a sandwich plate with honeycomb core

The purpose of the present paper is to investigate the wave propagation features in a sandwich plate with honeycomb core over a large frequency domain. Robust experiment-based identification processes are proposed to estimate propagation direction-dependent and frequency-dependent wavenumber-space (k-space) characteristics in presence of parametric uncertainties. These processes combine structural identification and uncertainty propagation methods. A special emphasis is put on wave correlation methods compared to two-dimensional Discrete Fourier Transform. The Variant of the Inhomogeneous Wave Correlation method is extended here to two-dimensional identification problems. The vibration field is experimentally measured at points which geometric coordinates are supposed to vary randomly. Statistical investigations are then carried out to quantify the impact of the measurement points geometric coordinates’ variability on the identified parameters and evaluate the robustness of the proposed identification processes against uncertainties. Valuable insights into k-space profiles, damping loss factor and equivalent mechanical parameters, regarding the structural orthotropic behavior, are highlighted. The obtained results show the large variability of the identified parameters and reveal a significant identification sensitivity to the measurement points geometric coordinates’ uncertainties. The use of the generalized Polynomial Chaos method allows robust identification with an interesting computing time reduction regarding the Latin Hypercube Sampling.

An Enhanced Optimal Technique For Accurate Detection Of Color Face Images With Different Illuminations

Secured authentication system is one of the most challenging tasks focused now-a-days by many researchers and greatly achieved by means of face detection technique. Face image recognition is currently realized by Adaptive singular value decomposition in two-dimensional discrete Fourier domain (ASVDF). The recognition systems ability enhancement is attained for face images recognition by side light influence reduction on a color face image for inadequate light. The prevailing researches does not focuses the following points: No correct output during face recognition process, Face spoofing is not concentrated thereby face recognition may effect in imprecise result, Optimal feature extraction. Optimized Face Recognition System with Illumination and Rotation Consideration (OFRS-IRC) is one of the promising solutions for mitigating all those issues. Various methods are presented for ensuring accurate face recognition. Additive White Gaussian Noise removal technique is utilized for eliminating noise when the image is captured through sensor devices. Illuminate invariant features and locality preserving projection approach is exploited for segmented image recognition. As a final step, Fuzzy neural network is deployed for precise prediction on the basis of locality preserving projection approach results. MATLAB simulation tool is exploited for evaluating this research, where improved performance are attained by proposed method than prevailing methods. The proposed method shows 7.42% better detection rate than the existing work.

Two‐dimensional DFT with sliding and hopping windows for edge map generation of road images

An edge map generation technique based on two-dimensional discrete Fourier transform with sliding and hopping windows is proposed for road images to detect the lanes and road markings. A 2 × $\times$ 2 sliding/hopping window DFT with bin indices ( k 1 , k 2 = 0 , 1 $k_1, k_2 = 0,1$ ) for horizontal edge detection and ( k 1 , k 2 = 1 , 0 $k_1, k_2 = 1,0$ ) for vertical edge detection has been proposed. The 2-D SDFT/HDFT-based edge detector has been proved to be more efficient for lane and road marking images in comparison with conventional edge detectors and cellular neural network based edge detectors. In the presence of noise and various signal to noise ratio conditions, the horizontal and vertical edges have been efficiently recovered with good Pratt figure of merit without applying any pre-processing, noise removal process, and post processing techniques. The PFOM was found to be quite stable with wide threshold range for various noise levels. The consistent performance of the proposed edge detector is proved with MSE and PSNR determination of detected edge images. Moreover, the proposed 2-D SDFT/HDFT-based edge detector performs well in developing edge maps for real-time road videos. The system-on-chip implementation of the 2-D SDFT/HDFT edge detector on Cyclone IV FPGA chip is also carried out for detecting the lane and road markings.