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Compressed Sensing Reconstruction

via Belief Propagation

Shriram Sarvotham, Dror Baron and Richard G. Baraniuk

Department of Electrical and Computer Engineering

Rice University, Houston, TX 77005, USA

July 14, 2006

Abstract

Compressed sensing is an emerging field that enables to reconstruct sparse or com- pressible signals from a small number of linear projections. We describe a specific measurement scheme using an LDPC-like measurement matrix, which is a real-valued analogue to LDPC techniques over a finite alphabet. We then describe the reconstruc- tion details for mixture Gaussian signals. The technique can be extended to additional compressible signal models.

1 Introduction

In many signal processing applications the focus is often on identifying and estimating a few significant coefficients from a high dimension vector. The wisdom behind this is the ubiquitous compressibility of signals: most of the information contained in a signal resides in a few large coefficients. Traditional sensing and processing first acquires the entire data, only to throw away most of the coefficients and retaining the small number of significant coefficients. Clearly, it is wasteful to sense or compute all of the coefficients when most of it will be discarded at a later stage. This naturally suggests the question: can we sense compressible signals in a compressible way? In other words, can we sense only that part of the signal that will not be thrown away? The ground-breaking work in compressed sensing (CS) pioneered by Candés et al. [1] and Donoho [2] answers the above question in the affirmative. They demonstrate that the information contained in the few significant coefficients can be captured (encoded) by a small number of random linear projections. The original signal can then be reconstructed (decoded) from these random projections using an appropriate decoding scheme.

The initial discovery has led to a vibrant activity in the area of CS research, opening many intriguing questions both in theoretical and practical aspects of CS. Two fundamental questions naturally emerge in compressed sensing. The first question concerns efficiency: what is the minimum number of projections required to capture the information contained in the signal (either losslessly or with a certain fidelity)? Clearly, the number of measurements

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needed depends on the signal model as well as the measurement model. The second question relates to algorithmic achievability: can we construct practical CS coding schemes that approach the performance limits? Questions such as these have been addressed in various other contexts. The vision of our work is to leverage the insights from information theory [3] to obtain new CS coding algorithms. In particular, we draw insights from low density parity check (LDPC) codes.

1.1 Information theory

The main problem that information theory deals with is reliable transmission of information over communication channels. A fundamental result by Shannon states that the upper-limit on the rate at which we can send information over a channel is given by the channel capac- ity [4]. Since Shannon’s seminal work, several approaches to building practical codes have been proposed. The emphasis has been on imposing structure to the codewords; for example, explicit algebraic construction of very good codes were designed for some channels. Although the decoding of such codes has polynomial complexity (and thus practical), most of these codes fared poorly in the asymptotic regime – they achieved arbitrarily small probabilities of decoding errors only by decreasing the information rate to zero. The grand breakthrough occurred quite recently, with the advent of turbo codes [5] and the rediscovery of LDPC codes [6]. These codes belong to a class of linear error correcting codes that use sparse par- ity check matrices and achieve information rates close to the Shannon limit. In addition to their excellent performance, turbo and LDPC codes lend themselves to simple and practical decoding algorithms, thanks to the sparse structure of the parity check matrices.

1.2 Compressed sensing

Consider a discrete-time signal x ∈ RN that has only K � N non-zero coefficients. The core tenet of CS is that it is unnecessary to measure all the N values of x; rather, we can recover x from a small number of projections onto an incoherent basis [1, 2]. To measure (encode) x, we compute the measurement vector y ∈ RM as M linear projections of x via the matrix-vector multiplication y = Φx. Our goal is to reconstruct (decode) x – either accurately or approximately – given y and Φ using M � N measurements.

Although the recovery of the signal x from the measurements y = Φx appears to be a severely ill-posed inverse problem, the strong prior knowledge of sparsity in x gives us hope to reconstruct x using M � N measurements. In fact the signal recovery can be achieved using optimization by searching for the sparsest signal that agrees with the M observed measurements in y. The key observation is that the signal x is the solution to the `0 minimization

x̂ = arg min ‖x‖0 s.t. y = Φx (1)

with overwhelming probability as long as we have sufficiently many measurements, where ‖ · ‖0 denotes the `0 “norm” that counts the number of non-zero elements. Unfortunately, solving the `0 optimization is known to be an NP-complete problem [7]. In order to recover the signal, the decoder needs to perform combinatorial enumeration of all the

( N K

) possible

sparse subspaces.

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The practical revelation that supports the CS theory is that it is not necessary to resort to combinatorial search to recover the set of non-zero coefficients of x from the ill-posed inverse problem y = Φx. A much easier problem yields an equivalent solution. We need only solve for the `1-sparsest coefficients that agree with the measurements y [1, 2]

x̂ = arg min ‖x‖1 s.t. y = Φx, (2)

as long as Φ satisfies the restricted isometry (RIP) condition [1]. The RIP condition is shown to be satisfied by a measurement strategy using independent and identically distributed (iid) Gaussian entries for Φ. Furthermore, the `1 optimization problem, also known as Basis Pursuit [8], is significantly more approachable and can be solved with linear programming techniques. The decoder based on linear programming requires cK projections for signal reconstruction where c ≈ log2(1+N/K) [9] and reconstruction complexity is Ω(N

3) [7, 10, 11].

1.3 Compressed sensing reconstruction algorithms

While linear programming techniques figure prominently in the design of tractable CS de- coders, their Ω(N3) complexity still renders them impractical for many applications. We often encounter sparse signals with large N ; for example, current digital cameras acquire images with the number of pixels N of the order of 106 or more. For such applications, the need for faster decoding algorithms is critical. Furthermore, the encoding also has high com- plexity; encoding with a full Gaussian Φ requires Θ(MN) computations.1 This realization has spawned a large number of decoding schemes in the CS research community in search of new measurement strategies and accompanying low-complexity decoders. We briefly review some of the previous work.

At the expense of slightly more measurements, iterative greedy algorithms have been developed to recover the signal x from the measurements y. Examples include the iterative Orthogonal Matching Pursuit (OMP) [12], matching pursuit (MP), and tree matching pursuit (TMP) [13] algorithms. In CS applications, OMP requires c ≈ 2 ln(N) [12] to succeed with high probability; decoding complexity is Θ(NK2). Unfortunately, Θ(NK2) is cubic in N and K, and therefore OMP is also impractical for large K and N .

Donoho et al. recently proposed the Stage-wise Orthogonal Matching Pursuit (StOMP) [14]. StOMP is an enhanced version of OMP where multiple coefficients are re- solved at each stage of the greedy algorithm, as opposed to only one in the case of OMP. Moreover, StOMP takes a fixed number of stages while OMP takes many stages to recover the large coefficients of x. The authors show that StOMP with fast operators for Φ (such as permuted FFT’s) can recover the signal in N log N complexity. Therefore StOMP runs much faster than OMP or `1 minimization and can be used for solving large-scale problems.

While the CS algorithms discussed above use a full Φ (all the entries of Φ are non-zero in general), a class of techniques has emerged that employ sparse Φ and use group testing to decode x. Cormode and Muthukrishnan proposed a fast algorithm based on group testing [15, 16]. Their scheme considers subsets of the signal coefficients in which we expect at most

1For two functions f(n) and g(n), f(n) = O(g(n)) if ∃c, n0 ∈ R+, 0 ≤ f(n) ≤ cg(n) for all n > n0. Similarly, f(n) = Ω(g(n)) if g(n) = O(f(n)) and f(n) = Θ(g(n)) if ∃c1, c2, n0 ∈ R+, 0 ≤ c1g(n) ≤ f(n) ≤ c2g(n) for all n > n0.

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Table 1: CS schemes

Scheme Setting Mmin Complexity `0 Optimization noise-free K + 1 NP complete

measurements `1 Optimization noise-free K log(1 + N/K) Ω(N

3) measurements

`1 regularization noisy K log(1 + N/K) Ω(N 3)

measurements OMP noiseless 2K log N NK2

measurements StOMP noiseless K log N N log N

measurements

Cormode-Muthu noiseless K log2 N K log2 N measurements

Chaining Pursuit noiseless K log2 N K log2 N log2 K measurements

Sudocodes [18] noiseless K log N K(log K)(log N) measurements

CS-LDPC noisy K log N N log N measurements