Abstract-In this paper, the class of morphological/rank/linear (MRL)-filters is presented as a general nonlinear tool for image processing. They consist of a linear combination between a morphological/rank filter and a linear filter. A gradient steepest descent method is proposed to optimally design these filters, using the averaged least mean squares (LMS) algorithm. The filter design is viewed as a learning process, and convergence issues are theoretically and experimentally investigated. A systematic approach is proposed to overcome the problem of nondifferentiability of the nonlinear filter component and to improve the numerical robustness of the training algorithm, which results in simple training equations. Image processing applications in system identification and image restoration are also presented, illustrating the simplicity of training MRL-filters and their effectiveness for image/signal processing.Index Terms-Adaptive filtering, image restoration, LMS algorithm, nonlinear systems, optimal filter design, system identification.

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In a multicarrier modulation system, a time domain equalizer (TEQ)

A time domain equalizer (TEQ) is a finite impulse response Elter that shortens the channel impulse response (CIR) to mitigate inter-symbol interference (ISI). Melsa, Younce, and Rohrs minimized ISI in the time domain by m.uximizing the shortening signd-tonoise ratio (SSNR) of the energy in a target window to the energy outside the target window in the shortened CIR. Infinite SSNR means zero ISI. A4elsa, Younce, and Rohrs also developed a joint channel shortening method to design a single TEQ to shorten a channel and a near-end echo impulse response. I n this paper, we extend the joint SSNR method to design a single TEQ to shorten multiple channels by mmimizing the composite SSNR. The composite SSNR is a weighted sum of normalized channel SSNRs. The normalized SSNR is the ratio of the energy in the target window samples to the energy of all samples in the shortened CIR and has a range of [O, 11 so that it is better suited for numerical stability and bed-point implementation. Our proposed method outperforms the joint channel shortening method. because it achieves higher weighted sum of SSNRs of the used channels.