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A method based on an underlying quantization-based model is introduced for performing near-optimal estimation of a process observed through a nonlinear mapping and distorted by noise. This quantization-based estimation (QBE) framework generalizes previously suggested special cases. We distinguish between QBE processing based on hard-inverse quantization, which requires sequence detection, and soft-inverse quantization, which requires a soft-in/soft-out processor. It is shown that, under the quantization-based model, the optimal processing is soft-inverse quantization with source-sensitive decoding, which is the exploitation of memory in the model of the desired process. With these techniques, we show that QBE can significantly outperform an extended Kalman smoother in estimating a Gaussian phase process.