Compressed Sensing using Chaos Filters

Abstract

Abstract-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 timesparse signals from their incompletely acquired measurements, outperforms random filters.Index Terms-Compressed sensing, random undersampling, random filters, chaos filters, chaotic undersampling. I. NONRANDOMNESS IN COMPRESSED SENSING?Compressed sensing (CS), recently been introduced by Candes and Tao [1] and Donoho [2] as a type of random undersampling, allows for the acquisition and reconstruction of sparse/compressible signals at a rate lower than that of Nyquist. First, random linear projection is used to acquire efficient representations of the signals directly. Then, nonlinear techniques, such as l 1 optimization-based algorithms or sparse approximation algorithms, are used to faithfully reconstructed the acquired compressed data under certain constraints.CS framework has attracted an overwhelming research attention, due to several advantages. In data compression, significant reduction of data storage can be obtained thanks to the incomplete measurements. In data acquisition, much less power consumption is used at the sensing device since simple and less signal acquisition is performed while the computational load is pushed toward the reconstruction side. In addition, it can trade for fundamental limits of physical systems, such as the limited sampling clock frequency of ADCs used in sensing extremely high frequency spectrum holes in ultra-wideband cognitive radio communications [3].The use of randomness, which leads to the so-called incoherence property in CS, facilitates the faithful reconstruction with high probability. In practice, however, the use of purely random undersampling is expensive in hardware design. Concerned with the undesirable randomness in such situations, a question naturally arises: Can we use a nonrandom structure that still mimics or approximates the incoherence property for compressed sensing framework? Our approach to find an answer to this question is to explore the use of deterministic chaos. A chaos system is a nonlinear system that has a very unstable structure so that, under specific initial and control conditions, the output of the system behaves as random in just a few steps. Chaos have been studied in various scientific and engineering contexts [4], and recently found various...

Compressive sampling with chaotic dynamical systems

“…In [5] the authors continue on the work in [6] on random filters in CS, and examine the use of chaos filters in CS with filter taps calculated from the Logistic map. The authors claim that their numerical simulations indicate that chaos filters generated by the Logistic map outperform random filters.…”

Spread spectrum in chaotic compressed sensing with application to MRI

“…As we have mentioned in the introduction, chaotic design for a deterministic measurement matrix in CS has been pro posed in [5]. In this method, a sampled logistic sequence is generated by a deterministic chaotic system called Logistic Map.…”

“…We convert these obtained chaotic values by an appropriate formula (see [5] for more details) so that the elements of P becomes i.i.d Gaussian-like, approximating the power decay law in the k-space. The resulting values are used to determine which values of kx and ky are to be obtained from sampling …”

“…One method is to generate the element of the measurement matrix by deterministic chaotic sequences [5].…”

The Chaotic Measurement Matrix for Compressed Sensing

“…The sequences generated by chaotic system have outstanding pseudo-randomness and are easy to generate and reproduce. Therefore, chaotic sequence [8,9] can be used to construct measurement matrix in CS. So this paper makes use of strong pseudorandomness of chaotic system, introduces the concept of the incoherence factors and rotation based on the discrete chaotic sequence, and adopts QR decomposition to obtain a measurement matrix suited for sparse reconstruction.…”