Sensor networks play a crucial role in smart cities, with applications such as remote environmental monitoring and target tracking. A major challenge in data collection is the corruption of measurement data by unwanted noise, necessitating the development of effective data denoising methods. In this paper, the sensor network is modeled as a graph consisting of nodes and edges, allowing the application of graph filtering techniques from graph signal processing to address the denoising problem. The proposed approach involves several key steps: First, the transfer matrix of the graph filter is expressed as a linear combination of Gegenbauer polynomials, which generalize Legendre and Chebyshev polynomials. Second, the filter coefficients are determined by minimizing the integral weighted least squares error between actual and ideal spectral responses. Third, design examples of frequency-selective graph filters are presented. Fourth, an implementation structure for the designed filter is derived using the recurrence relation of Gegenbauer polynomials. Finally, the effectiveness of the proposed method is demonstrated through its application to temperature data denoising in the sensor networks at USA and Taiwan.

Heat kernel smoothing (HKS) is a prominent method in graph signal processing (GSP) tailored for handling irregular data from complex networks. This paper introduces a generalized HKS filter designed for enhanced signal denoising over graphs. Initially, we generalize the graph Laplacian matrix (GLM) in HKS method to the p-power GLM. This allows modification of the spectral response of the HKS filter, enabling a flatter passband and a steeper transient band through adjustments to the value of positive integer p. Subsequently, two methods are presented to implement the generalized HKS filter. One is the centralized implementation method that is based on eigen-decomposition of GLM; the other is the distributed implementation method that is based on Bernstein polynomial approximation. As a case study, we apply the HKS filter to temperature data from sensor networks, demonstrating its efficacy. Experimental results show that the proposed HKS filter can provides higher signal to noise ratio (SNR) than original HKS filter.

In this paper, the digraph filter design based on directed Laplacian matrix (DLM) and least squares method is presented. First, the eigen-decomposition of DLM is used to define the digraph Fourier transform (DGFT). Then, the spectral properties of DGFT are studied and applied to specify the ideal spectral response of the digraph filter. Next, the coefficients of polynomial and rational digraph filters are determined by the least squares method which minimizes the integral absolute squared errors between ideal response and actual spectral response of filter. The matrix inversion can be used to compute the optimal solution. Finally, the proposed method is compared with conventional design methods to evaluate performance and the signal denoising application examples are demonstrated to show the effectiveness of the designed digraph filters.

In this paper, a study of total variation (TV) regularization in digraph signal denoising is presented. First, four definitions of the TV of digraph signals are described to measure the variation or smoothness of digraph signals. Then, an optimization method is presented to solve the digraph signal denoising problem by minimizing the signal distortion with TV regularization. The optimal solution can be obtained by solving the matrix inversion. Two iterative methods are also derived to implement the denoising operator by using Neumann series method and steepest descent method such that matrix inversion can be avoided. Finally, the sensor network data is used to make experiments to study which definition of TV provides the highest signal to noise ratio (SNR) in digraph signal denoising application.

In this paper, node classification using graph convolutional network (GCN) is studied. First, problem formulation of node classification is described. Then, the graph convolutional operator (GCO) is constructed and it is combined with nonlinear activation function to obtain the two-layer GCN for tackling the node classification problem. Next, the dropout regularization is incorporated into the GCN for solving the overfitting problem. Because input feature data is very sparse, sparse dropout is used in the first layer and general dropout is employed in the second layer. Finally, citation network datasets, t-SNE data visualization, ablation study, and classification accuracy are used to evaluate the performance of the GCN with dropout regularization.

In this paper, rational graph filters are designed by using conventional IIR digital filters and spectral transformation. First, the specification of the rational graph filter design is converted to that of IIR digital filter by using a spectral transformation. Then, conventional IIR digital filter design methods are directly employed to design converted digital filter without needing to develop new design algorithms. Next, the coefficients of rational graph filter are obtained from those of the designed digital filter by using Chebyshev polynomials and spectral transformation. Finally, one design example is demonstrated to show the effectiveness of the proposed method, and the application to temperature data denoising illustrates the usefulness of graph filter to practical problem.

In the conventional designs of orthogonal graph filter bank, both eigenvalues and eigenvectors of Laplacian matrix are assumed to be unknown such that graph filter bank can be efficiently implemented by using polynomial graph filters; that is, the output signal at each vertex is computed only from the signal at that vertex and its K ‐hop neighbourhoods. However, the disadvantage of these methods is that the designed orthogonal filter bank does not have the exact perfect reconstruction (PR). In this study, the authors first assume that the eigenvalues of Laplacian matrix are known but eigenvectors are still unknown. Then, the orthogonal graph filter bank is designed by solving the simultaneous equation or Lagrange multiplier method such that the PR condition at all eigenvalues can be achieved for a given graph structure. Although the proposed filter bank is a graph‐structure‐dependent design, it can be efficiently implemented by using polynomial graph filters. Finally, experimental results are shown to demonstrate the effectiveness of the proposed design method.

In this paper, the design of two-dimensional (2D) notch filter using bandpass filter and fractional delay filter is presented. First, the design of 2D notch filter is decomposed into the designs of 2D parallel-line filter and straight-line filter. Then, parallel-line filter is designed by bandpass filter and straight-line filter is designed by fractional delay filter. Next, the FIR fractional delay filters are used to implement the designed notch filter. Finally, several numerical examples are demonstrated to show the effectiveness of the proposed design method.

In this paper, the closed-form design of fractional order differentiator (FOD) using discrete cosine transform (DCT) is presented. First, the definition of fractional derivative is reviewed briefly. Then, the DCT-based interpolation method is applied to compute the fractional derivative of a given digital signal. Next, the transfer function of FOD is obtained from the DCT computational results by using suitable index mapping. Finally, several numerical examples are demonstrated to show the effectiveness of the proposed design method.

In this paper, the computation of partial fractional derivatives of a digital image using discrete cosine transform (DCT) is presented. First, the definition of partial fractional derivatives of two dimensional (2D) signal is defined. Then, the DCT-based method is used to compute the partial fractional derivatives of a given digital image. Next, the proposed computation method is used to develop the image sharpening and edge detection algorithms. Finally, several numerical examples are demonstrated to show the effectiveness of the proposed computation method.

In this paper, a new digital image sharpening method is presented by using fractional derivative and Mach band effect. First, fractional order digital differentiator is designed by using the Grünwald-Letnikov derivative. Next, the Mach band effect and fractional order digital differentiator are used to construct an image sharpening algorithm. Finally, the numerical examples are shown to demonstrate the effectiveness of the proposed method.

In this paper, a modular cascade structure is presented to design the variable fractional order differentiator (VFOD). First, a product expansion of exponential function is used to obtain a modular cascade implementation structure of VFOD. This structure is easily implemented by using only one basic modular cell. Then, the digital first order logarithm differentiator designed by the least squares method is applied to realize basic cell in the proposed structure. Finally, several numerical examples are demonstrated to show the effectiveness of the proposed VFOD design method.

In this paper, the radial basis function (RBF) interpolation method is used to design fractional delay FIR filter. As a result, the design error can be reduced by suitably choosing the shape parameter of radial basis function. Design examples are demonstrated to show that the proposed RBF-based method has a smaller design error than those of the conventional Lagrange and window fractional delay FIR filters when using the same design parameters.

In this paper, the design of digital integrator is investigated. First, the interpolation formula of discrete-time sequence is derived by using discrete Hartley transform (DHT). Then, the numerical integration rules and DHT interpolation are applied to obtain the transfer function of digital integrator. The filter coefficients are easily computed because closed-form design is gotten. Finally, some numerical comparisons with conventional digital integrators are made to demonstrate the effectiveness of this new design approach.

This paper presents a hierarchical lane detection system with the ability to deal with both structured and unstructured roads. The proposed system classifies the environment first before applying suitable algorithms for different types of roads. For environment classification, pixels with lane- marking colors are extracted as feature points. Eigenvalue decomposition regularized discriminant analysis is utilized in model selection and maximum likelihood estimation of Gaussian parameters in high dimensional feature space. For structured roads, the extracted feature points are reused for lane detection. For unstructured roads, mean-shift segmentation is applied to divide the scene into regions. Possible road boundary candidates are selected, and Bayes rule is used to choose the most probable boundary pairs. The experimental results show that the system is able to robustly find the boundaries of the lanes on different types of roads and various weather conditions.

In this paper, the design of second order differentiator is investigated. First, the backward difference formula in numerical differentiation is applied to derive the transfer function of second order differentiator. Then, the Richardson extrapolation is used to generate high-accuracy results while using low-order formulas. Next, conventional Lagrange FIR fractional delay filter is directly applied to implement the designed second order differentiator. Finally, several numerical examples are illustrated to demonstrate the effectiveness of this new design approach.

In this paper, the design of fractional delay FIR filter is investigated. First, the interpolation formula of a discrete-time sequence is derived by using discrete Fourier transform (DFT). Then, the DFT-based interpolation formula is applied to design fractional delay FIR filter by using suitable index mapping. The filter coefficients are easily computed because a closed form design is obtained. Finally, design examples are demonstrated to show that the proposed DFT method has smaller design error than those of the conventional window and Lagrange fractional delay FIR filters when using the same design parameters.

In this paper, the design of half sample delay IIR filter is presented. First, the continued fraction expansion (CFE) and its recursive computation are reviewed briefly. Then, the CFE of square root function is applied to design half sample delay IIR filters with various orders. Next, the comparison with conventional maximally flat IIR allpass filter is made and implementation issue is also addressed. Finally, several design examples are illustrated to demonstrate the effectiveness of the proposed method.

In this paper, the Newton-Cotes quadrature rule and fractional sample delay is used to obtain the closed-form design of IIR digital integrators. Although the proposed IIR digital integrators involves the implementation of fractional sample delay, this problem is easily solved by applying well-documented design techniques of the FIR Lagrange and IIR allpass fractional delay filters. Several design examples are illustrated to demonstrate the effectiveness of the proposed method.

In this paper, the Taylor series expansion is used to transform the design problem of a fractional delay filter into the one of a first-order differentiator such that the conventional finite-impulse response and infinite-impulse response differentiators can be applied to design a fractional delay filter directly. The proposed structure is more efficient than the well-known Farrow structure in terms of filter coefficient storage because only one first-order differentiator needs to be designed and implemented. Moreover, one design example is demonstrated to illustrate the effectiveness of this new design approach.

In this paper, the design of a fractional order FIR differentiator is investigated. First, the fractional derivative of the power function is defined. Then, the impulse response of the fractional order differentiator is obtained by solving linear equations of the Vandermonde form. Finally, one example is used to demonstrate that the fractional derivatives of digital signals are easily computed by using the proposed filtering technique.

In this paper, the design problems of the fractional Hilbert transformer (FHT) are investigated. First, the conventional Hilbert transformer is generalized to FHT and its corresponding analytic signal is also defined to construct a single side-band (SSB) signal for saving communication bandwidth. Then, several methods are presented to design FIR and IIR fractional Hilbert transformers including a Hilbert transformer based method and an allpass filter based method. Finally, several design examples are presented to demonstrate the effectiveness of the proposed methods.

This paper is concerned with the definition of the continuous fractional Hartley transform. First, a general theory of the linear fractional transform is presented to provide a systematic procedure to define the fractional version of any well-known linear transforms. Then, the results of general theory are used to derive the definitions of the fractional Fourier transform (FRFT) and fractional Hartley transform (FRHT) which satisfy the boundary conditions and additive property simultaneously. Next, an important relationship between FRFT and FRHT is described. Finally, a numerical example is illustrated to demonstrate the transform results of the delta function of FRHT.

This paper is concerned with the definition of the discrete fractional Fourier transform (DFRFT). First, an eigendecomposition of the discrete Fourier transform (DFT) matrix is derived by sampling the Hermite Gauss functions which are eigenfunctions of the continuous Fourier transform and by performing a novel error removal procedure. Then, the result of the eigendecomposition of the DFT matrix is used to define a new DFRFT. Finally, a numerical example is illustrated to demonstrate the proposed DFRFT is a better approximation to the continuous fractional Fourier transform than the conventional defined DFRFT.

Iterated function systems (IFSs) have received great attention in encoding and decoding fractal images. Barnsley (1988) has shown that IFSs for image compression can achieve a very high compression ratio for a single image. However, the major drawback of such a technique is the large computation load required to both encode and decode a fractal image. We provide a novel algorithm to decode IFS codes. The main features of this algorithm are that it is very suitable for parallel implementation and has no transient behavior. Also, from the decoding process of this method we can understand the encoding procedure explicitly. One example is illustrated to demonstrate the quality of its performance.

In this correspondence, we consider wavelet transform and scale space filtering as special cases of general scale shift mapping (SSM). Then, the SSM is used as a tool for characterizing the geometrical complexity of uniform fractals. This tool can reveal the construction rule of fractals and estimate fractal dimension. Finally, we use the results obtained from scale shift mapping to estimate iterated function system (IFS) codes of a class of fractal images. Some examples have been illustrated to demonstrate its good performance.

A technique for suppressing the transient states of IIR notch filters is presented. This technique mainly uses the vector projection to find the best initialization of a notch filter. When a notch filter is used to cancel 60 Hz interference in the recording of electrocardiograms (ECG), the performance of the notch filter with transient suppression is better than that of the notch filter with arbitrary initialization. The improvements realized by this technique is at the cost of little additional computational load at the beginning of filtering.>

A new two dimensional (2D) adaptive filter structure is developed. It is based on modeling the impulse response of the 2D IIR systems as a finite linear combination of a set of 2D Laguerre orthonormal functions. When it is required to identify a system with a long impulse response, computer simulation shows that the 2D Laguerre filter has a better performance than the conventional 2D FIR filter at the same computation cost.>

An adaptive FIR filter based on the least mean p-power error (MPE) criterion is investigated. First, some useful properties of MPE function are studied. Three main results are as follows: 1) MPE function is a convex function of filter coefficients; so it has no local minima. 2) When input process and desired process are both Gaussian processes, then MPE function has the same optimum solution as the conventional Wiener solution for any p. 3) When input process and desired process are non-Gaussian processes, then MPE function may have better optimum solution than Wiener solution. Next, a least mean p-power (LMP) error adaptive algorithm is derived and some application examples are presented. Consequently, when the signal is corrupted by an impulsive noise, the adaptive algorithm with p=1 is preferred. Furthermore, when the signal is corrupted by noise or interference, the adaptive algorithm with proper choice of p may be preferred.>