Document Sample

Compressed Sensing Phase Retrieval Matthew L. Moravecr , Justin K. Rombergg , and Richard G. Baraniukr r Department of Electrical and Computer Engineering, Rice University g School of Electrical and Computer Engineering, Georgia Tech ABSTRACT The theory of Compressed sensing (CS) allows for signal reconstruction from a number of mea- surements dictated by the signal’s structure, rather than its bandwidth. The information in signals which have only a few nonzero values can be captured in only a few measurements, such as random Fourier coeﬃcients. However, there exist scenarios where we can only observe the magnitude of these coeﬃcients. It is natural to ask if such measurements could still capture the signal’s information, and if so, how many are needed. We have found an upper bound of O(k 2 log(N ) random Fourier modulus measurements needed to uniquely specify k-sparse sig- nals. In addition to using a signal’s structure to observe it with fewer measurements, we also propose a method to use this structure to aid in recovery from the Fourier transform modulus. Existing methods to solve this phase retrieval (PR) problem with complex signals require a priori signal assumptions on the signal’s support. We have shown that a constraint based upon signal structure, the 1 norm, is also eﬀective for PR. Not only does it help recover structured signals from their Fourier transform modulus, but it can do so with fewer measurements than PR traditionally requires. Keywords: Compressed sensing, phase retrieval, projection algorithms 1. INTRODUCTION In many physical scenarios for signal processing, the measurement process can be time consuming or expensive. High frequency radar is beginning to test the limits of analog to digital conversion. Sensors for terahertz electromagnetic frequencies cost many times times more than those for the visible spectrum. MRI scans can take over an hour. The irony of these situations is that in each case the wide-band signal in question is undeniably structured, yet aside from general bandwidth considerations, no assumptions about this structure are made to ease the sensing process. The ﬁeld of compressed sensing (CS) [1–3] takes the logical step of making measurements that take a signal’s structure into account, and taking only as many measurements as the structure requires. CS has already seen practical success such as imaging with a single pixel [4] and reduction in MRI scan time [5]. Since random samples of a signal’s Fourier transform are one way to make CS measurements, it would seem the beneﬁts of CS can apply to when we have access to the an object’s diﬀraction pattern, since it closely approximates the object’s Fourier Email: {moravec, richb}@rice.edu, jrom@ece.gatech.edu Transform. Taking the object’s structure into account, we could randomly sub-sample the diﬀraction pattern, rather than needlessly observe the entire thing. Signal reconstruction from less measurements could aﬀect such ﬁelds as crystallography, astronomy, and wavefront sensing. Making the jump to applying CS in these cases introduces a new challenge–the measurements of the complex-valued diﬀraction pattern are in practice made with sensors that can only observe its intensity. In this paper we present both theoretical and practical results to demonstrate that we can indeed take random Fourier measurements and reap their beneﬁts though we cannot observe their phase. We show that the number of magnitude-only measurements suﬃcient for perfect recovery is of the order of k 2 log N for sparse signals, where N is the length of the signal and k is the number of nonzero elements. In the process of ﬁnding a way to recover signals from sub-sampled Fourier transform mod- ulus, we developed a new method of phase retrieval that is based on a signal’s structure. Phase retrieval (PR) is the process of recovering the phase, given just the magnitude, of a signal’s Fourier Transform, thereby recovering the signal itself. It amounts to ﬁnding a reconstruction candidate signal that has the same bandwidth and Fourier modulus as the original. Since the Fourier modulus constraint set is non-convex, there is not a straightforward method of perform- ing PR. Alternating projection strategies work if the signal constraints are stringent enough. Positivity is an eﬀective constraint for real signals, and exact support constraints perform well for complex-valued data. Such a support constraint is somewhat unrealistic, and unfortunately, PR of complex-valued data is notoriously diﬃcult apart from a strict support constraint. We introduce a new constraint of structure on the signal and use this to recover the signals. To enforce this structure we require that the 1 norm of the reconstruction candidate match that of the true signal, in addition to matching the Fourier modulus. With this constraint we have been able to recover signals that normally would only be recoverable if their support were known. In addition to aiding in reconstruction enforcing signal structure matches with random Fourier measurements to result in needing fewer measurements to reconstruct. To illustrate the eﬀectiveness of a structure constraint for PR we consider a terahertz imaging example (Figure 1). Terahertz (THz) imaging oﬀers the beneﬁts of x-ray imaging without ionizing exposure. In [6] the diﬀraction pattern of an object illuminated by a collimated THz beam is scanned, and its magnitude recorded. Since it is not a perfect plane wave hitting the object, its diﬀreaction pattern is the Fourier transform of a complex-valued signal. As such, loose support constraints (25% of the diﬀraction pattern size, to ensure there is no aliasing( are not eﬀective in recovering the signal. To enforce structure of the reconstructed object, we estimate and use the 1 norm as a constraint. As a side beneﬁt, we are able to perform reconstruction with fewer measurements than traditionally needed for PR. This can be useful because existing methods for terahertz imaging perform a raster scan of an object, recording the intensity of a focused beam at each location where it passes through the object. This can be a time-consuming task. An array of THz detectors could be made to speed up acquisition time, but would be very expensive. By using CSPR, we can save time and expense by randomly sampling the diﬀraction pattern. This paper is organized as follows. Section 2 provides background on CS and PR. In Section 3 we introduce the compressibility constraint for PR and show how it both helps recovery Object THz transmitter THz receiver 6cm 12cm Fourier Transform Modulus compressibility constraint loose support constraint randomly sub-sample compressibility constraint Figure 1. A THz trasmitter illuminates an object, in this case a T-shaped aperture. The far-ﬁeld diﬀraction pattern is focuesed in to a reasonable distance with a lens, and is recureded with a raster- scanning receiver. The 64 × 64 pxel diﬀraction pattern is approximately the modulus of the signal’s Fourier transform. We perform PR with a compressibility constraint and retrieve the object. A loose support constraint is not eﬀective, as the algorithm cannot ﬁnd an intersection between the support and Fourier modulus constraint sets. We also use the compressibility constraint with only 2.5% of the measurements and still nearly recover the object. The method of recovery is explained in Section 3.3 of this paper. and allows for less measurements, and describe the algorithm we use to implement CSPR. We conclude the paper with numerical results, both simulated and from physical application. 2. BACKGROUND 2.1. Compressed Sensing Traditionally, digital signal processing has meant ﬁrst observing a signal, then processing it in some way that enhances it and/or prepares it for storage or transmission. The sampling part of this process is governed by the Nyquist rate for band-limited signals. Shannon sampling is the best that can be done to recover a sampled signal if all that is known about a signal is that it is band-limited. However in many cases the signal class can be more speciﬁed. We expect a natural band- limited signal to be structured and hence “sparse” in some basis. This means that a large signal can be represented well with only a few of its elements or a few of the coeﬃcients of its transform. Smooth signals are sparse in the Fourier basis and piecewise smooth signals are sparse in a wavelet basis. Knowing that a signal is sparse gives much more information than just knowing it is band-limited. Transform coding uses this knowledge to eﬃciently store or represent this signal, however it was not until the last two years that this type of structure was exploited in sampling the signal. Fully sampling a compressible signal is wasteful. Just as we only save the “important parts” when storing or transmitting signals we ought to strive to sense only the signiﬁcant content of a sparse signal. CS aims to eﬃciently observe such signals. 2.1.1. Eﬃciency of Compressed Sensing Consider a signal x ∈ RN . Let x be compressible, meaning that if the magnitude of its elements were ordered from greatest to smallest they would decay like n−1/p . Many natural signals satisfy this condition when represented as a linear combination of vectors from a basis Ψ that sparsiﬁes the signal. The best k term approximation of x is deﬁned as xk , and is equal to x at the k largest entries and is 0 otherwise. The k term approximation error σk (x) is deﬁned as x − xk 2 and is bounded by Ck 1/2−1/p . Let y be linear measurements of x where y = Φx. The “miracle” of CS is that x can be reconstructed with error O(σk (x)) with only n measurements, where n is O(k log(N ) and is much smaller than N [1–3]. This is to say that the CS recovery of a given signal has optimal error and is accomplished not by sampling the whole signal but rather only as much as the structure of the signal requires. 2.1.2. Measuring and Reconstruction Accurate signal recovery from limited measurements requires more than the sparsity of a signal. The measurement matrix Φ must satisfy certain properties so that no matter what the sparsifying matrix Ψ may be, Φ will capture all the signiﬁcant information. In [2] it is shown that if Φ is a random Fourier ensemble, it can capture information in a structured signal with high probability, and furthermore this recovery is attained via 1 minimization: x = arg min x 1 s.t. y = Φx. (1) x CS properties holding for a random Fourier ensemble and 1 minimization being the key to recover are the inspiration for our PR method which uses random Fourier modulus values and an 1 constraint. 2.2. Phase Retrieval In many areas of science and engineering, such as crystallography, astronomy, and wavefront sensing, measurements of a complex-valued signal must be made with sensors that can only observe its intensity. These magnitude-only measurements are acceptable in instances like pho- tography, since our eyes also only observe intensity of a light ﬁeld. However in certain appli- cations, the phase of a signal is very important information to have. An object’s diﬀraction pattern, like from a crystal illuminated by an x-ray, or a landscape illuminated by a laser [7], closely approximates the object’s Fourier transform. It would be useful in these circumstances to recover the object from its transform, but the phase of the diﬀraction pattern would have to be known in addition to the magnitude. The task of PR is to recover the Fourier phase information in order to perform this seemingly hopeless inverse problem. Methods for PR are introduced by Fienup in [8, 9]. His concern, as ours, is PR from the intensity of an unknown object’s Fourier transform.∗ In these papers he shows a way to accurately recover a discrete signal from its transform modulus, and ﬁnds surprising his algorithms’ ability to recover the phase from a variety of random starting points. 2.2.1. Unique Recovery The theory behind this uniqueness of recovered results is explained by Hayes in [10]. Observing the magnitude (squared) of a signal’s Fourier transform is equivalent to observing the autocor- relation, as these are Fourier transform pairs. Having a signal’s autocorrelation is equivalent to knowing the z-transform of its autocorrelation. This polynomial is the product of the signal’s z-transform, and the transform of it’s ﬂipped version. Therefore if a signal’s z-transform is irreducible, then it is the only signal which will yield it’s autocorrelation (excepting a shifted or ﬂipped version, since the absolute position and orientation of the signal are irretrievably lost with the phase). Since virtually all polynomials in two or more dimensions are irreducible, one can be sure that if a recovered signal has ﬁnite support, and has the same autocorrelation as the original signal, it is indeed equal to the original signal. Hayes also explains the conditions needed to guarantee that a recovered signal has the same autocorrelation as the original signal. To do this he considers how one would know if the z- transform of a signal’s autocorrelation were equivalent to that of the original signal’s. For a signal of support N , the z-transform of the autocorrelation is a polynomial of degree 2N − 1. In order for two polynomial functions of degree 2N − 1 to be equal, they must have the same evaluations at 2N locations. These 2N locations may be chosen as distinct locations anywhere on the complex plane, but it is convenient to consider equally spaced positions about the complex unit circle. The z-transform of the autocorrelation evaluated at these points is the discrete Fourier transform (DFT) of the autocorrelation at these points, which is simply the magnitude-squared ∗ The phase problem, in general, refers to the task of recovering the phase of an object’s Fourier transform, given its magnitude, and the magnitude of the object itself. of the original signal’s Fourier transform. Through these relations he shows a signal in two or more dimensions is uniquely speciﬁed by its DFT modulus. 2.2.2. Recovery Methods While this work addresses the uniqueness of a recovered solution, it provides no guaranteed method of ﬁnding a candidate solution with the necessary support size and DFT modulus. This problem is an optimization in which a reconstruction candidate’s distance from these constraint sets must be minimized. Since the Fourier modulus constraint set is non-convex, it is a non- convex optimization. Though problems of this nature are diﬃcult because only exhaustive algorithms can guarantee convergence, there are a variety heuristics that do well in solving this particular optimization [11]. To help ﬁnd a solution these heuristics should preferably have signiﬁcant a priori signal information in addition to the Fourier modulus and maximum support size constraints. This information could be positivity, explicit support, or a histogram of signal values. 3. COMPRESSED SENSING PHASE RETRIEVAL 3.1. 1 Constraint for Phase Retrieval Though the Fourier modulus and bandwidth constraints are suﬃcient to guarantee a unique solution, ﬁnding the intersection of these two sets is a diﬃcult problem, since the Fourier modulus constraint set is non-convex. Were both sets convex, alternating projections would ﬁnd the intersection. Despite the non-convexity, projection algorithms can be used to solve the problem. One in particular, Fienup’s hybrid input-output algorithm† , can ﬁnd the intersection even in the presence of local minima. However, if a signal is complex-valued, the loose suﬃciency constraints of maximal bandwidth (which may be derived from the autocorrelation) which can guarantee the existence of a unique solution are not suﬃcient in practice to ﬁnd the solution. A typical algorithm may search indeﬁnitely for a global minimum, getting trapped by local minima in a condition known as stagnation [12]. Exact support constraints are needed to aid in ﬁnding a solution. The reason they work is they reduce the size of the constraint set, ostensibly making it an easier process for iterative projection procedures to ﬁnd the solution. Other strict constraints can also reduce the set size. A histogram constraint deﬁnes a dis- tribution of pixel values that the image must satisfy. A sparsity constraint (referred to as a number of non-zero constraint in [13]) forces the image to have a certain number of non-zero elements. A drawback of these constraints is that they may be unknown quantities which are diﬃcult to estimate. We propose an 1 constraint. The 1 norm of a signal is the sum of the absolute values of its components or its coeﬃcients in a transform basis. It is similar to both sparsity and histogram constraints but is more forgiving: an exact distribution is not needed, nor an exactly sparse signal. Rather the value of the signal’s 1 norm is needed, and this could be estimated and then † Fienupand others refer to these as algorithms, but strictly speaking they are not, since the running length to ﬁnd a solution is indeﬁnite. optimized over a single degree of freedom. We have found that this constraint is just as eﬀective as a strict support constraint. Section 4.1 provides empirical support for our assertion. This constraint can be useful with all measurements, but is especially helpful in that it can be used with a structured signal to recover it with less measurements than bandwidth would require, bringing us to the second contribution of this paper. 3.2. Theoretical Recovery It has been shown by others, and discussed in the introduction, that the number of measurements needed to guarantee uniqueness in PR is a function of the bandwidth: the Fourier modulus must be sampled twice as much in each dimension as the bandwidth of the original. However we know more about the signal than this. We know that natural signals will probably have a relatively small number of coeﬃcients in a sparsifying basis, compared to the total size of the signal. Intuition says that if only a few pieces of information of the signal suﬃce to represent it well, then only a few measurements should be needed to capture this information. We have access to linear measurements of the signals autocorrelation, as the intensity of a signal’s Fourier transform is equivalent to the Fourier transform of its autocorrelation. Each Fourier magnitude measurement is a linear projection of the signal’s autocorrelation onto a complex sinusoid. The good news for us is that a random collection of these projections is suﬃcient to specify a sparse autocorrelation, and which is suﬃcient to uniquely specify the signal. Lemma 1. Suppose x[n] is a two-dimensional sequence of complex-numbers of support N1 × N2 , and x has a z-transform which, except for trivial factors, is irreducible and nonsymmetric. Then x[n] is uniquely speciﬁed by its Fourier transform modulus, and an M (≥ 2N1 × 2N2 ) point DFT is suﬃcient for this unique speciﬁcation. Proof sketch: Hayes proves in theorem 9 of [10] that sequences of real numbers are uniquely speciﬁed by their DFT modulus, as a consequence of their z transforms being irreducible. These arguments also follow for complex sequences as long as their z-transforms are also irreducible. This means that if x ∈ CN1 ×N2 has an irreducible z-transform, and there exists an x ∈ CN1 ×N2 such that |F x| = |Fx| on a 2N1 × 2N2 lattice, then x ∼ x (the two signals are equal within a ﬂip, shift, and/or a constant phase factor). Since the autocorrelation is a Fourier transform pair with a signal’s Fourier Transform intensity (its modulus-squared), then the Lemma implies that a signal’s autocorrelation is suﬃcient to uniquely specify it. Hayes notes that the irreducible requirement is not strict in two or more dimensions for complex signals, since they correspond to a set of measure zero [14]. Since an arbitrary complex signal is speciﬁed by its autocorrelation, we must now consider how many measurements of a signal’s autocorrelation are needed to specify it. We consider the case in which the signal has only a few non-zero complex entries, whose values and locations are unknown. Theorem 1. Suppose x[n1 , n2 ] ∈ CN1 ×N2 has an irreducible z-transform, and is k sparse. Let N = N1 N2 . Then with probability of at least 1 − O(N −ρ/α ) for some ﬁxed ρ > 0, O(k 2 log(N )) random Fourier modulus measurements of x are suﬃcient to uniquely specify it. Proof: Since x is k-sparse in Ψ, its autocorrelation is at most k 2 sparse in some Ψ. For a k 2 - sparse signal in some basis Ψ, it was shown in [3] that due to the Uniform Uncertainty Principle of a random Fourier ensemble, only n = O(k 2 log(N )) are needed to specify a signal, with the probability stated above in the theorem. This means that, with overwhelming probability, if autocorrelation Rx matches Rx at n random Fourier locations, and both Rx and Rx are k 2 sparse, then Rx = Rx . From the lemma, this implies that x ∼ x. This gives a new paradigm for deﬁning the number of Fourier modulus measurements needed to capture the information in a signal, and is signiﬁcant because it scales with the signal’s structure, rather than its bandwidth. Though the theorem applies to signals sparse in space, we have seen in practice the same results for a signal sparse in a diﬀerent basis, such as wavelets. We have also observed that the order of measurements needed scales more like k log N than k 2 log N . 3.3. Practical Recovery As with the case of regular PR, our goal is to ﬁnd a signal x that lies in the intersection of two constraint sets. For us the constraint sets are slightly diﬀerent. One set is all signals which have the same Fourier modulus values as x at speciﬁed random locations. The other set is all the signals which have the same 1 norm as x, or whose coeﬃcients in some transform basis have the same 1 norm as the coeﬃcients for x. Like with regular PR, we would like to ﬁnd an x that minimizes the distance between the two sets (since the two sets intersect this distance will be zero). In our case, as with regular PR, the Fourier constraint set is non-convex. For us the 1 constraint set is also not convex, where a support constraint is convex. The presence of many local minima in the distance between the two constraint sets precludes a direct optimization procedure. Rather we use the same kind of iterative projection scheme that is used for regular PR. There are many variants of Fienup’s original error reduction and Hybrid Input-Output algorithms, and one that we found eﬀective and implemented is Relaxed Averaged Alternating Reﬂections (RAAR) [15]. Starting with x(0) , an initial guess of random values, each successive step is calculated as a combination of projections and reﬂections: 1 x(n+1) = [ β(R1 Rm + I) + (1 − β)Pm ]x(n) . (2) 2 Pm x refers to taking the current iterate x and projecting it onto Fourier modulus space, by taking its Fourier transform, ﬁxing the magnitudes of a deﬁned subset to match the known values while keeping the phases the same, and then performing an inverse transform. Rm x is a reﬂection deﬁned as Rm x = 2Pm x − x. The reﬂection R1 = 2P1 x − x where P1 x is the projection of x onto a known 1 norm. This is accomplished by uniformly adding or subtraction a constant value to the magnitudes of the entries of x until x 1 reaches the desired value. We stop the applications of RAAR once the diﬀerence between a modulus projection, and a subsequent 1 projection, P1 Pm x(n) − Pm x(n) 2 E= , (3) Pm x(n) 2 reaches a negligible value, implying that we have a candidate solution which lies in both con- straint sets. 4. PERFORMANCE There are two issues to consider as we evaluate the performance of these PR methods for diﬀerent signal sizes (N ), number of nonzero values (k), and number of measurements (n). One is the rate of convergence, how often a particular kind of signal will converge in a deﬁned amount of time. The other is the error of reconstructed signals. For the signal sizes we have considered, simulation suggests that CSPR performs just as well as PR in terms of convergence, and as well as CS in terms of accuracy. For converged signals, the simulation aﬃrms that k 2 log N is indeed an upper bound on the number of measurements needed for perfect reconstruction. 4.1. Convergence To evaluate the eﬀectiveness of the 1 constraint for regular PR, we compare it in diﬀerent kinds of simulations, against exact and loose support constraints (see Table 4.1). In each test the second constraint is the modulus of the input signal’s Fourier transform. For each combination of test and constraint we perform the RAAR algorithm on 500 randomly generated signals, recording the number of times a convergent solution is found within 1000 iterations. As expected, we ﬁnd the exact support constraint almost always converges for signals which are sparse in space. We also ﬁnd that loose support constraints, which ensure that the solution does not alias the Fourier transform modulus, are not eﬀective. As discussed in the introduction, this has been known for some time. The 1 constraint performs fairly well for sparse signals, but appears to be eﬀected more than the exact support constraint is for growing signal sizes. This may suggest that the 1 constraint is more sensitive to local minima when searching for a solution. The 1 is better than the strict support constraint when a signal is structured, but takes a full amount of bandwidth. This is illustrated in the test in which the 64 pixel image has every pixel as non-zero, but is composed of only 5 wavelets. The exact support constraint does not really the search space very much, since there are many non-zero pixels compared to the number of Fourier measurements. However the 1 constraint, does restrict the search space more and converges at a higher rate. The ﬁndings support the idea that a smaller search space results in more convergent solutions. The dimensionality of exact support constraint set is much smaller for sparse signals, which is why this constraint performs better when the ratio of non-zero pixels to image size is small. The 1 constraint has a similar property, except its behavior is related not to the number of non-zero pixels, but to the number of non-zero coeﬃcients in an arbitrary basis. For sensing scenarios in which the signal has a large bandwidth but is very structured, the 1 oﬀers more hope for ﬁnding a solution than an exact support constraint alone. Table 1. The convergence properties of PR with diﬀerent constraint sets are compared under diﬀerent tests. In two cases, “exact support” and “loose support” (anti-aliasing) are equivalent, because the number of non-zero pixels is at the anti-aliasing bandwidth limit of N/4. Percent Convergence Test strict support constraint 1 constraint loose support constraint N = 64, k = 5 99.8% 59.6% 5.4% N = 64, k = 5 (wavelets) .6% 38.4% – N = 64, k = 64 2.2% 2% – N = 256, k = 20 100% 16.8% .2% 4.2. Accuracy To determine how accurate the algorithm is for various measurement rates, we consider k-sparse signals and empirically determine how many Fourier modulus measurements are needed for diﬀerent signal sizes and sparsity rates for consistent (95 %) exact recovery of 100 converged solutions. We hold N constant and vary k, and also hold k constant and vary N , in order to empirically understand the dependence of the number of measurements on these values. We compare these results with those found via regular CS if the phases were known, using the SPGL1 solver ‡ . For convergent solutions we ﬁnd that the number of measurements needed does not appear to follow a k 2 log N trend but appears to be closer to the k log N , as Figure 2 shows. When the signal size is held constant, the number of measurements needed increases linearly. The slope is the same as for CS with the known phases, though more measurements are needed for CSPR. When k is held constant, the number of measurements follows a sub-linear trend, just as CS does, though as in the other case loosing the phase results in needed more measurements. These results support the theorem and give conﬁdence that the number of Fourier mea- surements truly is a function of the structure of the signal, rather than its bandwidth. This knowledge, along with the eﬀectiveness of the 1 constraint, lead us to believe that the princi- ples of CS apply even when only the modulus of measurements is observed, and that the same structure which allows for less measurements aids in PR reconstruction. 5. ACKNOWLEDGEMENTS We would like to thank Wai Lam Chan and Dan Mittleman for the terahertz data. ‡ TheSPGL1 solver can perform 1 minimization for complex signals and measurements, and is free for download at http://www.cs.ubc.ca/labs/scl/index.php/Main/Spgl1. 100 100 CSPR CSPR CS CS 80 80 60 60 n n 40 40 20 20 0 0 0 2 4 6 8 10 12 0 20 40 60 80 100 k N (a) (b) Figure 2. For a given value of N and k, enough trials of CSPR are performed on randomly generated signals until 100 convergent solutions have been found. The number of measurements n recorded is the smallest value needed so that at least 95 are perfectly reconstructed. In (a) we hold N constant at 64 and vary k. The number of measurements needed increases linearly. In (b) we hold k constant at 5 and vary the signal size. The increase in the number of measurements needed is sub-linear. In each case the number of measurements needed has a trend similar to that of CS with the phases known, though clearly more measurements are required. REFERENCES 1. D. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory 52(4), pp. 1289– 1306, 2006. e 2. E. Cand`s, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory 52(2), pp. 489–509, 2006. e 3. E. Cand`s and T. Tao, “Near optimal signal recovery from random projections and universal encoding strategies,” IEEE Transactions on Information Theory 52(12), pp. 5406–5425, 2006. 4. D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. of Computational Imaging IV at SPIE Electronic Imaging, San Jose, CA , January 2006. 5. M. Lustig, J. Santos, J. Lee, D. Donoho, and J. Pauly, “Compressed sensing for rapid mr imaging,” Proc. of SPARS , 2005. 6. W. Chan, M. Moravec, R. Baraniuk, and D. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” to appear in CLEO-2007 , May 2007. 7. J. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Optics Express 14(2), pp. 498–508, 2006. 8. J. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Optics Letters 3(1), pp. 27–29, July 1978. 9. J. Fienup, “Phase retrieval algorithms: a comparison,” Applied Optics 21(15), pp. 2758–2769, August 1982. 10. M. H. Hayes, “The reconstructionn of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP- 30(2), pp. 140–154, 1982. 11. S. Marchesini, “A uniﬁed evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum 78, 2007. 12. J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, pp. 1897–1907, 1986. 13. H. He, “Simple constraint for phase retrieval with high eﬃciency,” J. Opt. Soc. Am. A 23, pp. 550– 556, 2006. 14. M. H. Hayes and J. H. McClellan, “Reducible polynomials in more than one variable,” Proceedings of the IEEE 70(2), pp. 197–198, 1982. 15. D. R. Luke, “Relaxed averaged alternating reﬂections for diﬀraction imaging,” Inverse Problems 21, pp. 37–50, 2005.

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 3 |

posted: | 12/28/2011 |

language: | |

pages: | 12 |

OTHER DOCS BY yurtgc548

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.