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Compressed sensing, viewed as a type of random undersampling, considers the acquisition and reconstruction of sparse or compressible signals at a rate significantly lower than that of Nyquist. Exact reconstruction from incompletely acquired random measurements is, under certain constraints, achievable with high probability. However, randomness may not always be desirable in certain applications. Taking a nonrandom approach using deterministic chaos and following closely a recently proposed novel efficient structure of chaos filters, we propose a chaos filter structure by exploring the use of chaotic deterministic processes in designing the filter taps. By numerical performance, we show that, chaos filters generated by the logistic map, while being possible to exactly reconstruct original time-sparse signals from their incompletely acquired measurements, outperforms random filters.