Compressive Sensing Using Iterative Hard Thresholding with Low Precision Data Representation: Theory and Applications

Modern scientific instruments produce vast amounts of data, which can overwhelm the processing ability of computer systems. Lossy compression of data is an intriguing solution, but comes with its own drawbacks, such as potential signal loss, and the need for careful optimization of the compression ratio. In this work, we focus on a setting where this problem is especially acute: compressive sensing frameworks for interferometry and medical imaging. We ask the following question: can the precision of the data representation be lowered for all inputs, with recovery guarantees and practical performance? Our first contribution is a theoretical analysis of the normalized Iterative Hard Thresholding (IHT) algorithm when all input data, meaning both the measurement matrix and the observation vector are quantized aggressively. We present a variant of low precision normalized {IHT} that, under mild conditions, can still provide recovery guarantees. The second contribution is the application of our quantization framework to radio astronomy and magnetic resonance imaging. We show that lowering the precision of the data can significantly accelerate image recovery. We evaluate our approach on telescope data and samples of brain images using CPU and FPGA implementations achieving up to a 9x speed-up with negligible loss of recovery quality.Comment: 19 pages, 5 figures, 1 table, in IEEE Transactions on Signal Processin

Optimal Quantization for Compressive Sensing under Message Passing Reconstruction

We consider the optimal quantization of compressive sensing measurements following the work on generalization of relaxed belief propagation (BP) for arbitrary measurement channels. Relaxed BP is an iterative reconstruction scheme inspired by message passing algorithms on bipartite graphs. Its asymptotic error performance can be accurately predicted and tracked through the state evolution formalism. We utilize these results to design mean-square optimal scalar quantizers for relaxed BP signal reconstruction and empirically demonstrate the superior error performance of the resulting quantizers.Comment: 5 pages, 3 figures, submitted to IEEE International Symposium on Information Theory (ISIT) 2011; minor corrections in v

Modulated Unit-Norm Tight Frames for Compressed Sensing

In this paper, we propose a compressed sensing (CS) framework that consists of three parts: a unit-norm tight frame (UTF), a random diagonal matrix and a column-wise orthonormal matrix. We prove that this structure satisfies the restricted isometry property (RIP) with high probability if the number of measurements $m = O(s \log^2s \log^2n)$ for $s$-sparse signals of length $n$ and if the column-wise orthonormal matrix is bounded. Some existing structured sensing models can be studied under this framework, which then gives tighter bounds on the required number of measurements to satisfy the RIP. More importantly, we propose several structured sensing models by appealing to this unified framework, such as a general sensing model with arbitrary/determinisic subsamplers, a fast and efficient block compressed sensing scheme, and structured sensing matrices with deterministic phase modulations, all of which can lead to improvements on practical applications. In particular, one of the constructions is applied to simplify the transceiver design of CS-based channel estimation for orthogonal frequency division multiplexing (OFDM) systems.Comment: submitted to IEEE Transactions on Signal Processin