1bit Compressive Sensing studies how Compressive Sensing interacts with a particularly interesting quantizer. Specifically, it examines the most severe form of scalar quantization, which preserves only the sign of each measurement as one bit of information. I first introduced this line of work in [1] and followed up in [2, 3].
There are several reasons to study the performance of such a system, some of which theoretical, some very practical:
 The analysis shifts the focus to bits instead of measurements; we discuss ratedistortion properties of CS systems.
 A 1bit quantizer is an extremely simple device—just a comparator to zero—that is very cheap and can operate extremely fast.
 A fast quantizer allows the compressive acquisition system to take many more measurements, providing robustness to measurement noise, while still achieving compressive acquisition.
 The reconstruction algorithms and the theory are very useful in recovering signals from nonlinearly distorted measurements even if the nonlinear distortion is not known, as long as the distortion is monotonic (see [4] for details).
 The theoretical properties of such acquisition have very interesting connections with JohnsonLindenstrauss embeddings, locality sensitive hashing (LSH), and the (sparse) logistic regression.
In few words, 1bit CS acquires a sparse signal using
i.e., measures the signals and only preserves the sign of the measurements. It should be clear that from such measurements the signal can only be reconstructed within a scaling factor . All signal scalings produce the same sign pattern since .
Geometrically, each measurement vector defines a hyperplane through the origin in the signal space, as shown in the figure on the right. The sign of the measurements determines which sign of the hyperplane the signal lies in.
The goal is to obtain a sufficient number of measurements to accurately reconstruct the signal, using the knowledge that the signal is sparse in some basis. Since we cannot recover the scaling factor , it suffices to reconstruct any signal in the same direction as the original signal, such as a signal on the unit sphere.
The red area in the figure shows all the signals that could produce the same measurement pattern. The sparsest signal in that area would lie along the seam of the sphere—colored magenta—i.e., would be 2sparse. The ambiguity is significantly reduced thanks to our sparse signal model.
Since the introduction of this topic, a number of methods, algorithms and theoretical results have appeared. The 1bit CS page is a very good resource on the progress in this area and includes related papers, code and presentations from a variety of groups.
[1] 
P. T. Boufounos and R. G. Baraniuk, “1Bit Compressive Sensing,” Proc. Conf. Inform. Science and Systems (CISS), Princeton, NJ, March 1921, 2008.

[2] 
P. T. Boufounos, “Greedy sparse signal reconstruction from sign measurements,” Proc. Asilomar Conf. on Signals Systems and Comput., pp. 13051309, Pacific Grove, CA, November 14, 2009.

[3] 
L. Jacques, J. N. Laska, P. T. Boufounos, and R. G. Baraniuk, “Robust 1Bit Compressive Sensing via Binary Stable Embeddings of Sparse Vectors,” IEEE Trans. Info. Theory, v. 59, no. 4, April, 2013.

[4] 
P. T. Boufounos, “Reconstruction of sparse signals from distorted randomized measurements,” Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing (ICASSP), Dallas, TX, March 1419, 2010.
