On filtering by means of generalized integral images: a review and applications

In this paper the method of multidimensional (n-D) filtering based on prior signal integration is analyzed. This method has the advantage that the computational complexity for filtering is independent of the filter kernel size. An overview of recent 2-D image processing systems is presented where these types of filters are applied. Based on this overview a framework that covers this class of filters is derived using repeated integration. These filters include for example rect and triangle-filters which can be used to approximate Gaussian derivative filters. Furthermore the normalization of the filters, computational complexity, and storage cost are discussed. Finally, two image processing systems which benefit from the application of the filters are presented. They belong to the topic of advanced driver assistance systems.     

A general state-space representation of n-variable bilinear transformation

This paper proposes a general formulation of the relationship between the state-space representations of a multidimensional (n-D) continuous system and an n-D discrete system which are related by the n-variable bilinear transformation such that the state-space representations of these two related systems can be directly calculated from each other. The new formulation is derived based on the theory of linear fractional transformation (LFT) and the resultant form is simple and concise. Moreover, the relations among the proposed formulation and the existing 1-D and 2-D results are investigated and it turns out that the new formulation includes these existing results as special cases. A numerical example is presented to illustrate the effectiveness of the proposed formulation.     

Fan filters, the 3-D Radon transform, and image sequence analysis

This paper develops a theory for the application of fan filters to moving objects. In contrast to previous treatments of the subject based on the 3-D Fourier transform, simplicity and insight are achieved by using the 3-D Radon transform. With this point of view, the Radon transform decomposes the image sequence into a set of plane waves that are parameterized by a two-component slowness vector. Fan filtering is equivalent to a multiplication in the Radon transform domain by a slowness response function, followed by an inverse Radon transform. The plane wave representation of a moving object involves only a restricted set of slownesses such that the inner product of the plane wave slowness vector and the moving object velocity vector is equal to one. All of the complexity in the application of fan filters to image sequences results from the velocity-slowness mapping not being one-to-one; therefore, the filter response cannot be independently specified at all velocities. A key contribution of this paper is to elucidate both the power and the limitations of fan filtering in this new application. A potential application of 3-D fan filters is in the detection of moving targets in clutter and noise. For example, an appropriately designed fan filter can reject perfectly all moving objects whose speed, irrespective of heading, is less than a specified cut-off speed, with only minor attenuation of significantly faster objects. A simple geometric construction determines the response of the filter for speeds greater than the cut-off speed.     

Coefficient-dependent direct-construction approach to realization of multidimensional systems in Roesser model

The purpose of this paper is to present a significant generalization to the direct-construction realization approach proposed recently by the authors, with an emphasis on the coefficient-dependent nature of the realization problem of multidimensional (n-D) systems in Roesser model. The central idea in the approach is to reduce the n-D realization problem to the construction of an admissible n-D polynomial matrix where careful treating of the structure-dependent and coefficient-dependent properties is required. It is shown that though the direct construction of an admissible n-D polynomial matrix is very difficult, it is possible to construct first an admissible n-D monomial matrix, say ??, which can be constructively obtained based only on the structure-dependent property, and then to build the polynomial one by combining some monomial entries into polynomial ones when the corresponding coefficients satisfy certain proportional conditions. Specifically, the relationship among the monomial entries of ?? is first investigated thoroughly and some new concepts and notations necessary for the technique development are introduced. Then, some fundamental facts on combinability conditions and combination techniques are clarified, and algorithms for the generation of an admissible n-D polynomial matrix and the corresponding realization are established. Furthermore, a preprocessing strategy on entry re-assignment is explored, which can lead to possible further significant reduction in the realization order. Several symbolic and numerical examples are presented to illustrate the basic ideas as well as the effectiveness of the proposed approach.     

Insufficiencies of practical BIBO stable n-D systems

This paper is focused on analyzing the practical bounded input bounded output (BIBO) stability concept originally introduced by Agathoklis and Bruton with respect to its applicability in n-D signal processing applications. By comparing the frequency response expected on basis of the transfer function to the Fourier transform of the actually measured impulse response of exemplary systems it is shown that the concept of practical stability cannot be a general n-D stability criterion for systems, in particular for signal processing applications, if roots of the given transfer function are not guaranteed to be close to the respective n-D BIBO stability region.     

A set of necessary stability conditions form-D nonlinear digital filters

This paper addresses the problem of global asymptotic stability in recursive first hyperquadrant causalm-D digital filters. A set of simple-to-check 1-D conditions necessary for stability of them-D system is given. Generalizations tov-dimensional (v<m) subsystems are also provided. These derived conditions are useful in determining asymptotic convergence ofm-D digital filters implemented in fixed or floating point arithmetic. The set of 1-D conditions can be considered the analogous result to practical bounded input bounded output (BIBO) stability for linearm-D systems.     

Reduced-Order H ∞ Filters for Uncertain 2-D Continuous Systems,Via LMIs and Polynomial Matrices

This paper deals with designing H _∞ filters of reduced order for two dimensional (2-D) continuous systems described by Roesser models, with uncertain state space matrices. These filters are characterized in terms of linear matrix inequalities (LMI), to minimize a bound on the H _∞ noise attenuation, by using homogeneous polynomially parameter-dependent matrices of arbitrary degree. The methodology is also particularized for full order and zero order (static) filters, where more simple LMI conditions are derived. Numerical examples are presented to illustrate the proposed methodology.     

Robust H ∞ Control for a Class of 2-D Nonlinear Discrete Stochastic Systems

This paper is concerned with the problem of stability and robust H _∞ control for 2-D stochastic systems with parameter uncertainties and sector nonlinearities. The class of systems under investigation is described by the 2-D state-space Roesser model. Our attention is focused on the design of a state feedback controller for 2-D stochastic system with sector nonlinearity, such that the closed-loop 2-D stochastic system is asymptotically stable and has a prescribed H _∞ disturbance attenuation performance. First, a sufficient condition is established for the 2-D nonlinear stochastic systems to be asymptotically stable. Then, we extend the bounded real lemma for 2-D systems to 2-D stochastic systems with sector nonlinearities. Based on this lemma, solvability conditions for the H _∞ control of 2-D nonlinear stochastic systems in the form of LMIs (linear matrix inequalities) are derived. A numerical example illustrates the effectiveness of the proposed results.     

A novel 5-D depth–velocity filter for enhancing noisy light field videos

A 5-D depth–velocity filter is proposed for enhancing moving objects in noisy light field videos (LFVs) (also known as plenoptic videos). The proposed filter consists of an ultra-low complexity 5-D IIR depth filter and a 5-D FIR velocity filter. The 5-D IIR depth filter is employed to denoise a noisy LFV. The denoised LFV is then utilized to estimate the 3-D apparent velocity of the moving object of interest. The 5-D FIR velocity filter is designed based on the estimated 3-D apparent velocity and is used to enhance the moving object of interest while attenuating other interfering moving objects. Experimental results confirm the effectiveness of the proposed 5-D depth–velocity filter compared to previously reported 5-D depth–velocity filters.     

Transient distortion and nth order filtering in deep level transient spectroscopy (DnLTS)

Problems associated with intrinsic sensitivity and time constant selectivity of capacitive DLTS systems are discussed as well as intrinsic limitations in predominantly deep level doped semiconductors.A previously published figure of merit for the optimized exponential correlator for capacitive DLTS studies is shown to be too high because of improper consideration of effects of level restoration: when correctly compared, a filter with a 12% lower figure of merit can be constructed based purely on gating and a weighted phase inversion before the integrator and a phase sensitive detector has a 20% lower figure of merit. The exponential DLTS correlator is also inadequate for analysis of continuous spectra because of its slow drop off in response (∝ T_S for T_S shorter than the peak response time constant and ∝ T_S^?1 for longer times). Blanking is necessary to achieve more selectivity relative to short time constants. When performed on-line, D²LTS gives a response ∝ T_S^?2 for longer times. Still more selective filters of order n, or DⁿLTS, are considered based on weighted averages over time intervals in geometric progression that are suitable for DLTS and a system with ±1 weighting and suitably chosen time intervals for use with DDLTS. For these filters there is no penalty in figure of merit associated with choice of DDLTS which also appears to be easier to achieve than DLTS. On-line filters with long time constant responses ∝ T_S^?3 or higher order are shown to exact a large penalty in figure of merit. Equivalent filters can, however, be synthesized with much better figure of merit by a software compensation of multichannel data. The channels are then selected so that the responses of successive channels peak a factor of 2 away in T_S and the number of pulses used is decreased by a factor of two. Relative to a multipoint averager, the software compensated analyzer requires a factor of 100 less in measurement time for comparable accuracy when a spectrum must cover a range of 1000:1 in relaxation times. There are also comparable improvements in the holding time specifications and the number of A/D conversions if the system is to be coupled to a computer.When the deep level concentration is comparable to the shallow doping concentration, the peak responses of both DLTS and DDLTS are broadened, the sensitivity increases, the response peaks are shifted to larger relaxation times, and the system becomes essentially nonlinear in response to the individual deep level constituents. This distortion can, in principle, be corrected by going to a constant capacitance mode of operation both during the initial driving pulse and during the recovery transient.     

On the practical realization of higher-order filters with fractional stepping

We propose the use of a compact integer-order transfer function approximation of the fractional-order Laplacian operator s^α to realize fractional-step filters. Lowpass and bandpass filters of orders (n+α) and 2(n+α), where n is an integer and 0<α<1, can, respectively, be designed. A 5th-order lowpass filter with fractional steps from 0.1 to 0.9 (i.e. 5.1→5.9) is given as an example with its characteristics compared to 5th- and 6th-order Butterworth filters. Spice simulations and experimental results are shown.     

Direct, prediction- and smoothing-based Kalman and particle filter algorithms

We address the recursive computation of the filtering probability density function (pdf) p_n|n in a hidden Markov chain (HMC) model. We first observe that the classical path p_n−1|n−1→p_n|n−1→p_n|n is not the only possible one that enables to compute p_n|n recursively, and we explore the direct, prediction-based (P-based) and smoothing-based (S-based) recursive loops for computing p_n|n. We next propose a common methodology for computing these equations in practice. Since each path can be decomposed into an updating step and a propagation step, in the linear Gaussian case these two steps are implemented by Gaussian transforms, and in the general case by elementary simulation techniques. By proceeding this way we routinely obtain in parallel, for each filtering path, one set of Kalman filter (KF) equations and one generic sequential Monte Carlo (SMC) algorithm. Finally we classify in a common framework four KF (two of which are original), which themselves can be associated to four generic SMC algorithms (two of which are original). We finally compare our algorithms via simulations. S-based filters behave better than P-based ones, and within each class of filters better results are obtained when updating precedes propagation.     

Application of Inductive Topological Transfer Function Generation for a Cascaded Low-Pass Filter

This work presents applications of an inductive topological approach in the calculation of the transfer function of a cascading network control system. The method provides a very versatile and effective way of counting all the graphs that contribute and of translating them into their algebraic contribution. The contributing graphs are very simple and reflect the morphology of the original control system signal flow graph. Application of the method and results are presented for an n-cell RC network. We derive an analytical formula relating ω(3 dB) to the filter size (n) and R _j C _j (j=1,2,…,n). This formula can be used for designing preset frequency range self-loaded LP filters by varying their size. We also derive general expressions for the sensitivities which we use to observe a sensitivity-insensitivity filter transition for various frequency ranges, relaxation times, and, most importantly, filter sizes.     

Distributed stabilisation of spatially invariant systems: positive polynomial approach

The paper gives a computationally feasible characterisation of spatially distributed controllers stabilising a linear spatially invariant system, that is, a system described by linear partial differential equations with coefficients independent on time and location. With one spatial and one temporal variable such a system can be modelled by a 2-D transfer function. Stabilising distributed feedback controllers are then parametrised as a solution to the Diophantine equation ax + by = c for a given stable bi-variate polynomial c. The paper is built on the relationship between stability of a 2-D polynomial and positiveness of a related polynomial matrix on the unit circle. Such matrices are usually bilinear in the coefficients of the original polynomials. For low-order discrete-time systems it is shown that a linearising factorisation of the polynomial Schur-Cohn matrix exists. For higher order plants and/or controllers such factorisation is not possible as the solution set is non-convex and one has to resort to some relaxation. For continuous-time systems, an analogue factorisation of the polynomial Hermite-Fujiwara matrix is not known. However, for low-order systems and/or controller, positivity conditions on the closed-loop polynomial coefficients can be invoked. Then the computational framework of linear matrix inequalities can be used to describe the stability regions in the parameter space using a convex constraint.     

Design of 3-D Noncausal Filters with Small Roundoff Noise and No Overflow Oscillations

The contribution of this paper consists of two individual parts. First, an invertible mapping technique is presented for 3-D digital system design, and it is applied to approximate 3-D noncausal filters in the spatial domain. Secondly, an algorithm is proposed for obtaining a structure for 3-D IIR filters with small roundoff noise and no overflow oscillations. The design of noncausal filters can be carried out by three steps: 1), a given noncausal impulse response is transformed into the first octant using the proposed 3-D invertible mapping technique; 2), the transformed impulse response in the first octant is approximated by balanced model reduction of 3-D separable denominator systems;3), the resultant 3-D IIR filter is transformed back to the original coordinates.     

On General Factorizations for n-D Polynomial Matrices

Multivariate (n-D) polynomial matrix factorizations are basic research subjects in multidimensional (n-D) systems and signal processing. In this paper, several results on general matrix factorizations are provided for extracting a matrix factor from a given n-D polynomial matrix whose lower order minors satisfy certain conditions. These results are further generalizations of previous results in (Lin et al. in Circuits Syst. Signal Process. 20(6):601–618, 2001). As a consequence, the application range of the constructive algorithm in (Lin et al. in Circuits Syst. Signal Process. 20(6):601–618, 2001) has been greatly extended. Three examples are worked out in detail to show the practical value of the proposed method for obtaining general factorizations for a class of n-D polynomial matrices.     

The synthesis of two-dimensional passive n-ports containing lumped elements

Two-dimensional (2-D) passive networks are of interest e.g. for use as reference filters for two-dimensional wave digital filters. Necessary properties of the impedance matrix and scattering matrix, respectively, of such n-ports have been established, but not yet been shown to be also sufficient for a given two-variable rational matrix to be the impedance matrix or scattering matrix, respectively, of a passive network containing lumped elements. In the design of 2-D passive n-ports it will be however of great interest whether this mentioned feature can be used as a basis for ageneral synthesis procedure.In this paper it is shown that this is the case. The method presented for the synthesis of 2-D multiports is based mainly on a paraunitary bordering of the given scattering matrix of the desired network in order to obtain the scattering matrix of alossless 2-D multiport, which can be realized by using known procedures. The socalled spectral factorization of a two-variable para-Hermitian polynomial matrix which is nonnegative definite forp =j w plays a crucial role in the design approach presented. No restrictions are made concerning the coefficients of the given rational scattering matrix; they may be either real or complex, so as to include even complex networks which are of special interest for multidimensional systems.     

A direct-construction approach to multidimensional realization and LFR uncertainty modeling

This article proposes a direct-construction realization procedure that simultaneously treats all the involved variables and/or uncertain parameters and directly generates an overall multidimensional (n-D) Roesser model realization or linear fractional representation (LFR) model for a given n-D polynomial or causal rational transfer matrix. It is shown for the first time that the realization problem for an n-D transfer matrix G(z ₁, . . . , z _n ), which is assumed without loss of generality to be strictly causal and given in the form of G(z ₁, . . . , z _n )=N _r (z ₁, . . . , z _n )D _r ^?1 (z ₁,..., z _n ) with D _r (0, . . . , 0)=I and N _r (0, . . . , 0)  = 0, can be essentially reduced to the construction of an admissible n-D polynomial matrix Ψ for which there exist real matrices A, B, C such that N _r (z ₁, . . . , z _n ) = CZΨ and Ψ D _r ^?1 (z ₁, . . . , z _n ) = (I ? AZ)^?1 B with Z being the corresponding variable and/or uncertainty block structure, i.e., ${Z={\rm diag} \{z_1I_{r_1},\ldots,z_nI_{r_n} \}}$ . This important fact reveals a substantial difference between the 1-D and n-D (n ≥  2) realization problems as in the 1-D case Ψ can only be a monomial matrix and never a polynomial one. Necessary and sufficient conditions for Ψ to satisfy the above restrictions are given and algorithms are proposed for the construction of such an admissible n-D polynomial matrix Ψ with low order (for an arbitrary but fixed field of coefficients) and the corresponding realization. Symbolic and numerical examples are presented to illustrate the basic ideas as well as the effectiveness of the proposed procedure.     

电流模式n阶CCCII-C的多输入单输出滤波器设计

该文提出了一种结构简单的基于MCCCII的n阶多输入单输出电流模式滤波器电路。该滤波器电路包含n个有源器件、n个接地电容元件,无需接任何电阻元件,可以产生n阶低通、带通、高通、带阻及全通电流模式滤波。由于仅依靠改变外部输入电流信号的接入数目和方式来实现不同功能的滤波器,而电路内部结构及器件数目不变,所以该电路便于单片集成。文中对滤波器进行了PSPICE模拟。     

Analytical synthesis of general high-order voltage/current transfer functions using CCIIs

This paper proposes two circuit structures to realize general high-order voltage and current transfer functions using second-generation current conveyors (CCIIs). The proposed general high-order voltage/current-mode circuit structures are designed based on an analytical synthesis method. The nth-order voltage-mode transfer function requires 2n+1 CCIIs, 2n+2 grounded resistors and n grounded capacitors. The nth-order current-mode transfer function requires 2n+2 CCIIs, 2n+2 grounded resistors and n grounded capacitors. The voltage-mode third-order allpass, lowpass and highpass filters and the current-mode third-order notch, lowpass and highpass filters that are derived out from the proposed high-order voltage and current-mode circuit structures are simulated to verify the synthesis methods.