Ji-Nan Lin

Canonical piecewise-linear approximations

The canonical representation of piecewise-linear functions is considered as a universal approximation scheme of multivariate functions. Meanwhile, two universal approximation schemes in terms of combinations of univariate canonical piecewise-linear functions are proposed. The discussion supports the application of these schemes in mapping networks, e.g. neural networks or adaptive nonlinear filters. >

Adaptive nonlinear digital filter with canonical piecewise-linear structure

A novel adaptive nonlinear filter with the least-mean-square (LMS) error criterion is presented. It is based on the so-called canonical piecewise-linear structure. As an alternative to approaches based on the Wiener-Volterra series which have so far been widely employed for adaptive nonlinear filtering, the proposed approach can exhibit adaptive performance, especially in strongly nonlinear cases, while saving computation and implementation cost. The performance of this adaptive nonlinear filter is illustrated by computer simulation results. >

A generalization of canonical piecewise-linear functions

This letter proves the availability of the so-called high-level canonical representation, which is a generalization of the useful canonical representation, in the set of all continuous piecewise-linear (PWL) functions. The result is of interest in connection with research and applications where PWL functions are involved, e.g., about neural networks. >

Bias-remedy least mean square equation error algorithm for IIR parameter recursive estimation

In the area of infinite impulse response (IIR) system identification and adaptive filtering the equation error algorithms used for recursive estimation of the plant parameters are well known for their good convergence properties. However, these algorithms give biased parameter estimates in the presence of measurement noise. A new algorithm is proposed on the basis of the least mean square equation error (LMSEE) algorithm, which manages to remedy the bias while retaining the parameter stability. The so-called bias-remedy least mean square equation error (BRLE) algorithm has a simple form. The compatibility of the concept of bias remedy with the stability requirement for the convergence procedure is supported by a practically meaningful theorem. The behavior of the BRLE has been examined extensively in a series of computer simulations. >

Canonical representation: from piecewise-linear function to piecewise-smooth functions

The canonical representation of piecewise-linear (PWL) functions provides a global compact formulation of continuous PWL functions, which has significant advantages in the research and applications concerning nonlinear systems. This work studies the generalization of the canonical representation from PWL functions to piecewise-smooth (PWS) functions. First a class of PWS functions, called the regular PWS functions, is defined as a generalization of the continuous PWL functions. An important example of the regular PWS functions is the continuous piecewise-polynomial function. The continuous PWL function with a PWL partition is also covered by the regular PWS function. Then the canonical representation of the PWS function is defined and the existence conditions are discussed. The PWS generalization of the canonical representation is significant in applications where a PWS scheme can improve the performance of a PWL scheme in the approximation of a nonlinear function, i.e., in approximating the input/output (I/O) relation of a nonlinear system or a mapping neural network or in nonlinear signal processing. >

Unification of order-statistics based filters to piecewise-linear filters

Order-statistics based filters that were originally provided by the robust estimation theory have proved to be efficient in image/signal filtering in the presence of additive white noise or impulsive noise. Their algorithms are simple and easy to implement. Their analysis, however, is not straightforward. In this paper, we show that filters based on order statistics can be explained by using the theory of piecewise-linear (PWL) functions which was established originally for circuit analysis and has recently been applied to nonlinear filtering. We also prove that an L-filter is a PWL filter defined on IR/sup n/ and a median filter by threshold decomposition is a piecewise-constant (PWC) filter on [0,M-1]/sup n/. The main results lead to the unification of order-statistics based filters with the PWL filter class. Based on the fact that PWL functions are a general class of approximation functions which are uniformly dense in the domain concerned, it is expected that the results obtained can provide a new way to the extension, as well as further study of, order-statistics based filters.

Canonical representation of piecewise-polynomial functions with nondegenerate linear-domain partitions

Piecewise-linear (PWL) functions are a widely used class of nonlinear approximate functions with applications in both mathematics and engineering. As an extension of this function class, piecewise-polynomial (PWP) functions with a linear-domain partition represents a more general function class than PWL functions, in that all function pieces in a partitioned domain are (instead of hyperplanes) hypersurfaces described by polynomials. Just like PWL functions, the global expression of PWP functions requires a so-called canonical representation, which is meaningful for practical applications. However, such a canonical representation is still unknown. Our study showed that it is not a straightforward extension of the canonical representation of PWL functions; instead, it has a more general form than the latter. In this paper, we discuss the canonical representation of PWP functions with nondegenerate linear-domain partitions. A canonical representation formula is derived and a sufficient condition for its existence is given. We show that under some degree constraints, the derived canonical formula reduces to the canonical formula of PWL functions. The consistent variation property, which is a sufficient and necessary condition for the canonical representation of PWL functions, is found to be less important for PWP functions.

2-D adaptive CPWQ filter for image enhancement

While the least mean square (LMS) adaptive filtering technique is extensively used in 1-D signal processing, its 2-D version achieves only limited success in image processing due to its linear inherence. A 2-D adaptive nonlinear scheme is developed for enhancing images from broadband noise. It is based on a generalization of the conventional concept of optimal linear filtering to the nonlinear case. A new type of adaptive nonlinear filter called the adaptive canonical piecewise-linear (CPWQ) filter is introduced. Simulation examples with an artificial image and a natural image are presented to show the superior performance of the proposed approach in contrast to the 2-D LMS adaptive filter and two other types of nonlinear ones, i.e., the 2-D adaptive Volterra filter and the 2-D adaptive CPWL filter.<<ETX>>

On the analysis of sorting networks from the viewpoint of circuit theory

A sorting network is the kernel of an order statistics-based filter and contains all information about the nonlinearity. Its implementation is simple; its analysis, however, is relatively difficult due to the hiding of the nonlinearity. In this short paper, we analyze a sorting network from the viewpoint of classical circuit theory, and reveal its relation to a nonlinear lossless n-port. We show that a sorting network is in fact the wave digital filter (WDF) realization of an n-port memoryless nonlinear classical network. Hence, it can be unified to the group of nonlinear WDF networks. Useful properties such as passivity and losslessness are hence inherent properties of such networks.

Wavelet based nonlinear image enhancement for Gaussian and uniform noise

We consider image enhancement by using a wavelet based nonlinear filtering scheme. It is well-known that the wavelet transform decomposes an image into a finite number of resolution scales that is very suitable for image analysis and image compression. Such a resolution decomposition is also useful for noisy image enhancement, in view of that the visible noise can be suitably separated in the low-frequency (LF) component and the high-frequency (HF) components of a wavelet transform. While the wavelet transform is used for the decomposition and reconstruction of images, we apply nonlinear piecewise-quadratic (PWQ) filters for the processing of the decomposed progressive wavelet HF components before reconstruction. We show by simulations that such a filtering scheme can achieve a considerable improvement in noise smoothing, details-preserving and the visual quality of images.