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This paper presents a novel class of cellular neural networks (CNNs), where output of a cell in the CNN is given by the piecewise-linear (PWL) function having multiple constant regions or a quantization function. CNNs with one of these output functions allow us to extend CNNs to image processing with multiple gray levels. Since each cell of the original CNN has the PWL output function with two saturation regions, the image-processing tasks are mainly developed for black and white output images. Hence, the proposed architecture will extend the promising nature of the CNN further. Moreover, the hysteresis characteristics are introduced for these functions, which make tolerance to a noise robust. It is demonstrated mathematically that under a mild assumption, the stability of the CNN, which has an output function with hysteresis characteristics, is guaranteed, and the impressive simulation results are also presented.