506 JOURNAL OF DISPLAY TECHNOLOGY, VOL. 6, NO. 10, OCTOBER 2010 Compressive Fresnel Holography Yair Rivenson, Student Member, IEEE, Adrian Stern, Member, IEEE, and Bahram Javidi, Fellow, IEEE Abstract—Compressive sensing is a relatively new measurement paradigm which seeks to capture the “essential” aspects of a high-dimensional object using as few measurements as possible. In this work we demonstrate successful application of compressive sensing framework to digital Fresnel holography. It is shown that when applying compressive sensing approach to Fresnel fields a special sampling scheme should be adopted for improved results. Index Terms—Compressive imaging, Fresnel digital holography, variable density sampling. I. INTRODUCTION T HE recently introduced theory of compressive sensing (CS) [1], [2] has attracted the interest of theoreticians and practitioners. CS seeks to minimize the collection of redundant data in the acquisition step, in contrast to common framework of first collecting as much data as possible and then discarding the redundant data by digital compression techniques. One of the fields that benefits from CS theory is compres- sive imaging. Compressive imaging is motivated by the fact that most natural images are highly redundant; as experienced with digital compression (e.g., JPEG and JPEG2000). Several works were done already in the field, amongst them we might note [3]–[6]. Recently, CS was introduced in the realm of holography [7]. In [7] reconstruction techniques used in CS were applied on data captured with Gabor holography to reconstruct three-di- mensional objects that are sparse in the spatial domain. In this paper, we will demonstrate the application of CS framework with Fresnel digital holography. Objects that are sparse in ar- bitrary domain (e.g., some wavelet domain) will be considered. In Section III it is explained explain why Fresnel holography fits well to CS framework. However, as shown in Section IV, in order implement optimized compressive digital holography, the sampling scheme needs to be adapted to the special mechanism of Fresnel holography. II. COMPRESSIVE SENSING In order to describe CS, let us think of an object described by an -dimensional real valued vector (in case that the object Manuscript received December 03, 2009; revised January 26, 2010; accepted January 26, 2010. Date of publication May 03, 2010; date of current version September 10, 2010. Y. Rivenson is with the Department of Electrical and Computer Engi- neering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel (email: rivenson@ee.bgu.ac.il). A. Stern is with the Electro-Optics Engineering Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel (email: stern@bgu.ac.il). B. Javidi is with the Department of Electrical and Computer Engineering, University of Connecticut, Storrs, CT 06269-2157 USA (e-mail: bahram@engr. uconn.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JDT.2010.2042276 represents an image of pixels, is a one-dimensional vector obtained by rearranging the image in a lexicographic order) being projected (imaged), to an -dimensional vector. One can also think of as the number of detector pixels. We are in- terested in the case of , meaning that the captured image is undersampled in the conventional sense. Often, the subsam- pling is done by picking out of measurements uniformly at random [1], [2]. The sensing process can be written as (1) (1) where is an by matrix. CS relies on two guiding princi- ples. The first one is signal sparsity; we assume that the signal can be sparsely represented in some arbitrary orthonormal basis (e.g., wavelet, or DCT). Thus, is the -sparse representa- tion of image projected on , meaning, that has only nonzero terms. Another principle that CS requires is the incoherence (dis- similarity) between the sensing and sparsifying operators. This measurement is known as the mutual coherence, , and is defined as (2) where , denote the column vector of and , respectively, and is the length of the column vector. The mutual coherence is bounded by [1]. Many pairs of bases which exhibit low coherence and were reported extensively in the lit- erature [2]. Amongst them we wish to note the Fourier-Delta impulses ( ), Fourier-wavelet ( ) and Gaussian random projections, which are universally incoherent with any sparsifying basis (with ). CS theory suggests that a signal (image) measured with sensing operator, , can actually be recovered by -norm minimization; the estimated coefficients vector is the solution of the convex optimization program [2]: subject to (3) where is the norm which measures the number of nonzero entries of the coefficient vector . One way of guaranteeing recovery via the -norm minimization of a -sparse signal is by taking measurements satisfying (4) where is some small positive constant. III. COMPRESSIVE FRESNEL HOLOGRAPHY Let us consider the one-dimensional free-space propagation formula in the Fresnel approximation, which relates the com- plex values of a propagating wave, measured in a plane perpen- 1551-319X/$26.00 © 2010 IEEE