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Leaky integrate-and-fire (LIF) encoding is a model of neuron transfer function in biology that has recently attracted the attention of the signal processing and neuromorphic computing communities as a technique of event-based sampling for data acquisition. While LIF enables the implementation of analog-circuit signal samplers of lower complexity and higher accuracy simultaneously, the core difficulty of this technique is the retrieval of an input from its LIF-encoded output. In this article, we study this problem in the context of bandlimited inputs, by extracting the most abstract features of an LIF encoder as a generalized nonuniform sampler. In this view, the LIF output is seen as the transformation of the input by a known linear operator. We show that the signal reconstruction method of projection onto convex sets (POCS) converges to a weighted pseudo-inverse of this operator. This allows perfect recovery under uniqueness of reconstruction, minimum-norm reconstruction under incomplete sampling, as well as a noise shaping of time quantization that outperforms standard pseudo-inversion. On the practical side, a single iteration of the POCS method can be used to improve any estimate whose LIF samples are not consistent with those of the input, and a rigorous discrete-time implementation of this iteration is proposed that does not require a Nyquist-rate representation of the signals.