Two-dimensional Digital Filters Research Articles

A unified approach for the design of 2-D digital filters via semidefinite programming

This paper attempts to demonstrate that a modem optimization methodology known as semidefinite programming (SDP) can serve as the algorithmic core of a unified design tool for a variety of two-dimensional (2-D) digital filters. Representative SDP-based designs presented in the paper include minimax and weighted least-squares designs of FIR filters with continuous and discrete coefficients, and minimax design of stable separable-denominator IIR filters. Our studies are motivated by the fact that SDP as a subclass of convex programming can be solved efficiently using recently developed interior-point methods and, more importantly, constraints on amplitude/phase responses in certain frequency regions and on stability (for IIR filters), that are often encountered in many filter design problems, can be formulated in a natural way as linear matrix inequalities (LMI) which allow SDP to apply. Design examples for each class of filters are included to demonstrate that SDP-based methods can in many cases be useful in producing optimal or near-optimal 2-D filters with reduced computational complexity.

Spectral transformations for two-dimensional filters via FFT

In this paper, a new fast algorithm for spectral transformations for two-dimensional digital filters is presented. The algorithm is based on the use of the fast Fourier transform. The computational complexity of this algorithm is evaluated. The simplicity and efficiency of the algorithm is illustrated by a numerical example.

Error Spectrum Shaping in Two‐Dimensional Separable‐Denominator Digital Filters

AbstractIn the implementation of recursive digital filters, the characteristics might be degraded due to finite word length effects. The degradation generates noise at the filter output which is caused by roundoff errors in the product quantization. A technique called error spectrum shaping (ESS) can be applied to suppress the noise. ESS usually relies on the use of feedback of roundoff errors and sometimes their feedforward. In this paper, a technique is developed for ESS in two‐dimensional separable‐denominator digital filters by applying both feedback and feedforward of roundoff errors. A two‐dimensional separable denominator is implemented with the minimum number of delay elements and includes two quantizers. A genetic algorithm is also applied to obtain the suboptimal discrete solution of feedback–feedforward coefficients for the purpose of cost reduction. Finally, a numerical example is given to illustrate the utility of the proposed technique. © 2001 Scripta Technica, Electron Comm Jpn Pt 3, 85(4): 10–19, 2002

Design of Two-Dimensional Adaptive Digital Notch Filters

In this paper, a method for realizing a two-dimensional (2-D) adaptive notch filter is proposed. The obtained 2-D structure contains a pair of one-dimensional (1-D) second-order IIR notch filters and a pair of 1-D first-order allpass filters. The method has been successfully applied for the removal of a sinusoidal interference superimposed on an image.

Identification of 2-D Discrete Systems Using Neural Network

Abstract A Neural Network (NN) architecture is proposed for identification of two-dimensional (2-D) recursive digital filters in spatial domain using arbitrazy inputs and their corresponding output data. The method utilizes a recurrent neural network to obtain various parameters of a Local State-Space (LSS) 2-D model by solving the nonlinear optimization problem in l2 norm. The proposed identification technique could be applied to a general 2–D Roesser’s LSS model as well as a particular class of 2–D digital filters such as separable-in-denominator structure. Simulation results indeed demonstrate the effectiveness of the proposed methods.

H/sub ∞/ reduced-order approximation of 2-D digital filters

This paper considers the H/sub /spl infin// reduced-order approximation of two-dimensional (2-D) digital filters using the linear matrix inequality (LMI) approach, The 2-D digital filter is described by the 2-D Roesser model. We shall first establish LMI conditions for which a 2-D system is bounded real. This bounded realness property then allows us to derive solvability conditions for the 2-D H/sub /spl infin// reduced-order approximation which in general involves a nonconvex matrix rank minimization subject to LMI constraints. A numerical procedure is proposed to obtain a reduced-order H/sub /spl infin// approximation of the given 2-D filter using alternating projections. In particular, when a zeroth-order 2-D H/sub /spl infin// approximation is desired, it is shown that the approximation problem boils down to a convex LMI optimization problem, Numerical examples are given to demonstrate the proposed 2-D H/sub /spl infin//, reduced-order approximation approach.

Stability analysis of 2-D digital filters described by the Fornasini-Marchesini second model using overflow nonlinearities

This paper discusses new criteria for the global asymptotic stability of two-dimensional (2-D) digital filters described by the Fernasini-Marchesini second local state-space model subject to overflow nonlinearities. For saturation and triangular arithmetics, the presented approach will always lead to a larger overflow stability region in the parameter-space, as compared to a recent criterion due to Liu; for other overflow nonlinearities, new criteria may generally provide results as supplement to those obtainable from Liu's criterion. The approach leads to a more relaxed saturation overflow stability condition, as compared to a recent criterion due to Hinamoto. Finally, the approach is extended to the situations involving quantization nonlinearities.

Design of two-dimensional recursive filters by using neural networks

A new design method for two-dimensional (2-D) recursive digital filters is investigated. The design of the 2-D filter is reduced to a constrained minimization problem the solution of which is achieved by the convergence of an appropriate neural network. The method is tested on a numerical example and compared with previously published methods when applied to the same example. Advantages of the proposed method over the existing ones are discussed as well.

Synthesis of two-dimensional IIR digital filters with desired spatial domain specifications and no overflow oscillations

A design technique of two-dimensional (2D) recursive infinite-area impulse response (IIR) digital filters (DFs) which approximates desired spatial-domain specifications and guarantees no overflow oscillations is proposed in this paper. First, it is shown that the balanced approximation is not optimal for l2 model reduction. From this point of view, a new design technique of 2D separable-denominator (SD) digital filters is studied using reduced-dimensional decomposition and the l2 model reduction methods based on balanced gains. This result is used as a starting point of a design algorithm for 2D non-separable ND DFs. Second, a design problem of 2D ND DFs is formulated as a nonlinear minimization problem with equality and inequality constraints, and the Broyden-Fletcher-Goldfarb-Shanno algorithm is applied to solve it. Finally, two examples are given to show the effectiveness of the proposed design technique.

Stability analysis of 1-D and 2-D fixed-point state-space digital filters using any combination of overflow and quantization nonlinearities

A new criterion, together with its frequency-domain interpretation for the global asymptotic stability of zero-input one-dimensional (1-D) state-space digital filters under various combinations of overflow and quantization nonlinearities and for the situation where quantization occurs after summation only, is presented. A condition in closed form involving solely the parameters of the state transition matrix for the nonexistence of limit cycles in second-order digital filters is derived. Improved versions of some of the stability results due to Leclerc and Bauer (1994) are established. Finally, the approach is extended to two-dimensional (2-D) digital filters described by the Roesser and the Fornasini-Marchesini second local state-space models.

A constructive approach to stabilizability and stabilization of a class of n D D Systems

This paper presents a constructive approach to the problem of output feedback stabilizability and stabilization of a class of linear multidimensional ( nD, n>2) systems, whose varieties of the ideals generated by the reduced minors are infinite with respect to not more than two variables. The main idea of the proposed approach is to decompose the variety of an nD system in this class into a union of several varieties, each of which is defined by polynomials in just two variables. The new method can be considered as a combination of Grobner bases and existing results on two-dimensional (2D) digital filter stability tests and on stabilizability and stabilization of 2D systems. An example is illustrated.

Synthesis of two-dimensional IIR digital filters with desired spatial domain specifications and no overflow oscillations

A design technique of two-dimensional (2D) recursive infinite-area impulse response (IIR) digital filters (DFs) which approximates desired spatial-domain specifications and guarantees no overflow oscillations is proposed in this paper. First, it is shown that the balanced approximation is not optimal for l2 model reduction. From this point of view, a new design technique of 2D separable-denominator (SD) digital filters is studied using reduced-dimensional decomposition and the l2 model reduction methods based on balanced gains. This result is used as a starting point of a design algorithm for 2D non-separable ND DFs. Second, a design problem of 2D ND DFs is formulated as a nonlinear minimization problem with equality and inequality constraints, and the Broyden-Fletcher-Goldfarb-Shanno algorithm is applied to solve it. Finally, two examples are given to show the effectiveness of the proposed design technique.

Design of two-dimensional FIR digital filters by McClellan transform and quadratic programming

A quadratic programming (QP) approach for determining the coefficients of the McClellan transform is presented for the design of 2-D FIR digital filters. Three features of the proposed method are as follows. First, the transform parameters are determined by minimising the integration of the squared errors along the desired contour. Second, a set of linear constraints are incorporated into the QP formulation such that the conventional scaling problem of the transform can be avoided. Third, the optimal cutoff frequencies of a 1-D prototype filter are obtained directly from the QP solution. Several design examples, including fan filters, elliptic filters, diamond filters and bandpass filters, are illustrated to demonstrate the effectiveness of the QP method.

Design of separable-denominator variable 2-D digital filters with guaranteed stability

Stability guarantee is the most important issue in designing variable recursive digital filters since unstable variable filters cannot be applied in signal processing applications. This paper proposes a technique for designing separable-denominator variable two-dimensional (2-D) digital filters having variable magnitude characteristics, and the stability of the designed variable recursive 2-D filters is always guaranteed. The method assumes each coefficient of the variable 2-D recursive digital filter as a polynomial of the spectral parameters that specify different variable magnitude responses. Since the filter coefficients are represented by different polynomials, substitution of different values of the spectral parameters into the polynomials gives different filter coefficients and thus different magnitude responses. To guarantee the stability, we first perform coefficient transformations on the denominator coefficients of the recursive transfer function such that the stability conditions of the recursive filter are always satisfied for arbitrary values of the new coefficients. Then both the denominator and the numerator coefficients are determined as the polynomials of the spectral parameters. The proposed technique is effective in designing separable-denominator variable 2-D digital filters with guaranteed stability.

Magnitude and Delay Approximation of 1-D and 2-D Digital Filters

From the Publisher: The most outstanding feature of this book is that it treats the design of filters that approximate a constant group delay, and both, the prescribed magnitude and group delay response of one-dimensional as well as two-dimensional digital filters. It so fills a void in the literature, that almost solely deals with the magnitude response of the filter transfer function. The volume contains many of the important results that have appeared in professional journals only recently.

Parallel Image Processing with the Block Data Parallel Architecture

Many digital signal and image processing algorithms can be speeded up by executing them in parallel on multiple processors. The speed of parallel execution is limited by the need for communication and synchronization between processors. In this paper, we present a paradigm for parallel processing that we call the block data flow paradigm (BDFP). The goal of this paradigm is to reduce interprocessor communication, and relax the synchronization requirements for such applications. We present the block data parallel architecture which implements this paradigm, and we present methods for mapping algorithms onto this architecture. We illustrate this methodology for several applications including twodimensional (2-D) digital filters, the 2-D discrete cosine transform, QR decomposition of a matrix, and Cholesky factorization of a matrix. We analyze the resulting system performance for these applications with regard to speedup and efficiency as the number of processors increases. Our results demonstrate that the block data parallel architecture is a flexible, high-performance solution for numerous digital signal and image processing algorithms.

Minimax design of 2-D linear-phase FIR filters with continuous and powers-of-two coefficients

This paper deals with the minimax design problem of two-dimensional (2-D) linear-phase FIR digital filters with continuous and powers-of-two (POT) coefficients. First, the minimax continuous-coefficient design problem is formulated as a linear programming problem with inequality constraints. We present a method based on a variant of Karmarkar's algorithm to solve the resulting design problem. During each iteration, the main computing cost required for finding the filter coefficients is to solve a set of linear equations. Based on the obtained continuous coefficients, an efficient method is presented for designing a minimax 2-D FIR filter with POT coefficients in the spatial domain. Simulation results are provided for illustration and comparison.

Design of two-dimensional FIR digital filters for sampling structure conversion by genetic algorithm approach

By minimizing a quadratic measure of the error in the passband and stopband, a two-dimensional (2-D) diamond-shaped finite-duration impulse response (FIR) digital filter is designed for sampling structure conversion by genetic algorithm (GA) approach. This method is very easy to incorporate the frequency-domain constraints such that the design difficulty inherent to sampling conversion filters can be effectively solved. Several numerical design examples and simulation with test pictures are presented to demonstrate the effectiveness of this proposed approach.

An improved stability test algorithm for two-dimensional digital filters

In this paper, we show that the tabular algorithm given by Jury (1991) for obtaining inner determinants with numerical entries is also valid for the case of polynomial entries. Based on this algorithm, the new stability criterion recently proposed by Yang et al. (1998) for two-dimensional (2-D) digital filters can be efficiently implemented. As compared with the tabular algorithm given by Yang et al., the proposed algorithm leads to a great reduction in the order of polynomial entries of the array from n/sub 2/2/sup n1-1/ to n/sub 1/n/sub 2/.

A Numerical Calculation of the Contact Area and Pressure of Real Surfaces in Sliding Wear

A numerical simulation technique based on elastic asperity deformation is presented to analyze real surface contact and pressure distributions in sliding wear. The calculations have been applied to experimentally produced surfaces whose topography has been determined using an atomic force microscope. A variational approach is applied to minimize the stored energy in the contact, which determines contact area without additional iteration. Two-dimensional FIR digital filter techniques are used and an FFT procedure is used to improve the efficiency of the filter implementation. The model is used to obtain the pressure distribution, real contact area, and the distribution of real contact area under sliding conditions either parallel or perpendicular to the grinding lay. The calculated results of real contact stress at different stages of wear are given. The calculated and experimental results suggest that the stress distribution of real contact follows an exponential function. The contact stress distribution index β governs the stress distribution form and reflects the performance of the frictional components in sliding wear. The proportion of plastic contact deformation ψ is also related to β; therefore, reflecting the frictional property in sliding wear. The experimental and calculated results show that the smaller frictional coefficient and more homogeneous distribution of stress occur when the sliding direction is parallel rather than perpendicular to the grinding-lay direction.