SPL-06333-2008 .R1 1 Abstract— Compressive imaging (CI) is a natural branch of compressed sensing (CS). Although a number of CI implementations have started to appear, the design of efficient CI system still remains a challenging problem. One of the main difficulties in implementing CI is that it involves huge amounts of data, which has far-reaching implications for the complexity of the optical design, calibration, data storage and computational burden. In this paper, we solve these problems by using a two- dimensional separable sensing operator. By so doing, we reduce the complexity by factor of 10 6 for megapixel images. We show that applying this method requires only a reasonable amount of additional samples. Index Terms— Compressed Sensing, Compressive Imaging, Separable Operator, Kronecker Product, Mutual Coherence I. INTRODUCTION he recently introduced theory of compressed sensing (CS) [1]-[4] has attracted the interest of theoreticians and practitioners alike and has initiated a fast emerging research field. CS suggests a new framework for simultaneous sampling and compression of signals. In contrast to the common framework of first collecting as much data as possible and then discarding the redundant data by digital compression techniques, CS seeks to minimize the collection of redundant data in the acquisition step. A natural branch of CS is compressive imaging (CI). A block diagram for CI with random projections is shown in Fig. 1. The object f consisting of N pixels is imaged by taking a set, g, of M random projections. One can also think of M as the number of detector pixels. We are interested in the case of M<N, meaning that the captured image is undersampled in the conventional sense. Ψ is an arbitrary orthonormal signal sparsifying basis (i.e., Fourier, wavelet, or DCT), and ∈ℜ M g is the acquired image. ∈ N α C is the K-sparse representation of image f projected on T Ψ (Fig. 1), meaning, that α has only K non-zero terms, and M ∈ℜ MxN Φ is the sensing matrix. Manuscript submitted December 8, 2008, revised February 17, 2009. Yair Rivenson is with the Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel. (email: rivenson@ee.bgu.ac.il ) Adrian Stern is with the Electro-Optics Engineering Department, Ben- Gurion University of the Negev, Beer-Sheva 84105, Israel. (email: stern@bgu.ac.il) In CS, the measurement operator Φ M is typically chosen to be a random projection matrix, expressed as M = Φ RΦ , where Φ is an N N × orthogonal matrix and R samples M rows uniformly at random from Φ [10] (e.g., matrices with random Gaussian or Bernoulli entries). This choice of sensing measurement operator is preferred for a wide range of applications, because it simplifies theoretical analysis and implementation of CI. The properties of random projection operator have been studied extensively [1]-[10]. One property of random Φ that is of particular interest here is that it is incoherent for almost all possible choices of the sparsifying operator Ψ . One practical suggestion of the CS theory is that a signal (image) measured with Φ can actually be recovered by l 1 -norm minimization; the estimated coefficients vector α  is the solution of the convex optimization program: 1 ˆ min M subject to = = α α α g Φ Ψα (1) where 1 ˆ i i = ∑ α α . If the signal is sufficiently sparse and given that some technical condition on the number of measurements M and the measurement operator Φ M are available, l 1 minimization (1) yields exact signal estimation with overwhelming probability. One way of guaranteeing recovery via the l 1 -norm minimization of a K-sparse signal f is by taking M measurements satisfying: ( ) 2 , log( ) M C K N  ≥ ⋅ ⋅ ⋅ ΦΨ (2) where C is some small positive constant. It can be shown [10] that the solution of (1) is exact with overwhelming probability (in addition, the result is guaranteed only for nearly all sign sequences with a fixed support [10]). μ is known as the bases mutual coherence, defined as: max , i j ij N  = < > φ ψ (3) where , N i j ∈ℜ φ ψ denote the column vector of Φ and Ψ respectively, and N is the length of the column vector. The mutual coherence is bounded by 1 N  ≤ ≤ [10], where the lower bound is for the completely incoherent case (e.g., , ΦΨ are Fourier and spikes bases pair [2],[10]), and the upper bound is for completely coherent bases. The size of the imaging operator Φ M creates several challenging implementation issues: 1) Computational – For large M, N (the typical size of an image is N=10 6 pixels) storage of Φ M and computational solving (1) are hardly possible. 2) Optical implementation - Realization of random Φ M requires the design of an imaging system with a space Yair Rivenson and Adrian Stern Compressed Imaging with a Separable Sensing Operator T