We propose a sparse signal representation-based method for complex-valued imaging. Many coherent imaging systems such as synthetic aperture radar (SAR) have an inherent random phase, complex-valued nature. On the other hand sparse signal representation, which has mostly been exploited in real-valued problems, has many capabilities such as superresolution and feature enhancement for various reconstruction and recognition tasks. For complex-valued problems, the key challenge is how to choose the dictionary and the representation scheme for effective sparse representation. We propose a mathematical framework and an associated optimization algorithm for a sparse signal representation-based imaging method that can deal with these issues. Simulation results show that this method offers improved results compared to existing powerful imaging techniques

Cetin, Mujdat

Masnadi-Shirazi, Mohammad Ali

Samadi, Sadegh

Çetin, Müjdat

English

Sabanci University Research Database

SPARSE SIGNAL REPRESENTATION FOR COMPLEX-VALUED IMAGINGSadegh Samadi 1, M üjdat Çetin 2, Mohammad Ali Masnadi-Shirazi 11. Shiraz University, Shiraz, Iran ,   2. Sabanci University, Istanbul, Turkeyssamadi@shirazu.ac.ir , mcetin@sabanciuniv.edu , masnadi@shirazu.ac.irAbstract: We propose a sparse signalrepresentation-based method for complex-valued imaging. Many coherent imagingsystems such as synthetic aperture radar (SAR) have an inherent random phase, complex-valued nature. On the other hand sparse signal representation, which has mostly beenexploited in real-valued problems, has manycapabilities such as superresolution and feature enhancement for various reconstruction andrecognition tasks. For complex-valuedproblems, the key challenge is how to choose the dictionary and the representation schemefor effective sparse representation. We propose a mathematical framework and an associatedoptimization algorithm for a sparse signalrepresentation-based imaging method that candeal with these issues. Simulation results showthat this method offers improved resultscompared to existing powerful imagingtechniques.Index Terms: sparse signal representation, complex-valued imaging, image reconstruction, coherentimaging, synthetic aperture radar.1. INTRODUCTIONThis paper presents a new image reconstructiontechnique based on sparse signal representationthat can be used for complex-valued imagingsystems. Many coherent imaging systems such assynthetic aperture radar (SAR), holography,magnetic resonance imaging, laser imaging, sonarimaging, and so on, have an inherent complex-valued nature [1]. Also, phase of the imaged field might be random and spatially uncorrelated insome applications such as SAR [2].In these systems, straightforward application ofimage reconstruction techniques developedoriginally for incoherent real-valued imagingsystems will not produce high quality images. Oneof these techniques that have mostly beenexploited in real-valued problems is sparse signalrepresentation, which has many capabilities suchas superresolution and feature enhancement forvarious reconstruction and recognition tasks.Recent work on feature-enhanced complex-valuedimaging [3] has ties to sparse representation. Inparticular, one interpretation of this techniqueinvolves sparse representation with some fixed,specific dictionaries. Our work generalizes thatwork to sparse representation with arbitraryovercomplete dictionaries.The key challenge for exploiting sparserepresentation in complex-valued problems is how to choose the dictionary and the representationscheme for effective sparse representation.In the next sections we present a mathematicalframework that deals with these issues. Aspresented in Section 2, we have used a generallinear observation model, and applied sparserepresentation on the magnitude of the complex-valued scattered field. This then turns the imagereconstruction problem into a joint optimizationproblem over the magnitude and the phase of the underlying field reflectivities. We present aniterative algorithm to solve this joint optimizationproblem. In the end of Section 2, a discussion on dictionary selection is presented. In Section 3 our experimental results are presented to show theeffectiveness of this new method.2. SPARSE REPRESENTATION FOR COMPLEX-VALUED PROBLEMSThis section contains a description of our sparserepresentation technique for complex-valuedproblems. We start with describing our general365978-1-4244-3677-4/09/$25.00 ©2009 IEEEAuthorized licensed use limited to: ULAKBIM UASL - SABANCI UNIVERSITY. Downloaded on December 4, 2009 at 04:09 from IEEE Xplore.  Restrictions apply. observation model. Next we present our sparserepresentation scheme and the relatedoptimization problem. Then we present ouralgorithm to solve this optimization problem.Finally a discussion about proper dictionaryselection is presented.2.1 Observation modelWe use the following general observation model:y = T f + n  (1)where y  is the sampled measured data, f is theunknown scene or object reflectivity image, and nis noise; all are complex and column-reordered asvectors. T represents a complex-valued discreteobservation kernel. T could be, for example, aband-limited Fourier transform operator matrix, ora projection operator matrix as in SAR or other tomographic imaging system [3,4].2.2 Sparse representation schemeIn many applications of imaging systemsintroduced in Section 1, such as SAR, we could have a sparse representation in a proper domainfor magnitude of complex scattered field (image).This means that we can write:αf    (2)Where  is an appropriate dictionary for ourapplication, and α is the new domain coefficientvector with which the magnitude of scattered fieldcan be sparsely represented.We can write fPf  , where  ijediagP φ  , and φi's are the unknown phases of each imagevector element fi. So we can rewrite ourobservation model as: nPTnfPTnfTy  α   (3)If we knew P (the unknown image vectorelements phases), using a sparsity regularizationmethod such as an extension of basis pursuit [5] as follows, we could find an estimate of α  andhence of the image itself:ppTPy αλααα 22minargˆ   (4)wherep  denotes the pl -norm and λ is a scalar parameter. However we don't know the imagephases and hence P. We use the followingapproach to overcome this problem:a. First we start with an initial estimate for f that could be a conventional reconstruction of f, so we can also produce an initial estimate of the imagephase matrix P. In this way, we can solve theoptimization problem in (4) and find a newestimate forα .b. Now we have a new estimate of α  and so f ,and we should now find a new estimate for P. Wecan rewrite our observation model as:nBTnfPTy  β      (5)where  ifdiagB  and β  is vector of unknown phases, hence a vector containing the diagonalelements of P. We can find an estimate of β asfollows:itosubjectBTy i  ,1minargˆ 22 βββ β(6)We replace the constrained optimization problemin (6), with the following unconstrained problemthat incorporates a penalty term on the magnitudes of si 'β :NiqiBTy1222)1(minargˆ βλβββqqqqBTy βλβλββ  2minarg2222     (7)Solving this optimization problem will give us anew estimate for the phase vector β  and so thematrix  βdiagP  .366Authorized licensed use limited to: ULAKBIM UASL - SABANCI UNIVERSITY. Downloaded on December 4, 2009 at 04:09 from IEEE Xplore.  Restrictions apply. c. We are now back to step (a) and we repeat theabove steps until the algorithm converges to itsfinal solution.2.3 Solving the optimization problemsIf we call the cost functio n of optimizationproblem (4) J(α) and use the smoothapproximation Mipipp12/2 )( εαα , where   is a small positive constant, to avoidnondifferentiability problems of the pl -normaround the origin, then the gradient of J(α) willbe:yPTHJ H)(2)()(  ααα    ( 8)where)()()(2)( αλα  pPTPTH H  2/12 )(1)(pidiagεαα (9)Using the quasi-Newton iterative methoddeveloped in [3], which is well suited for this kind of nonquadratic optimization problems, withapproximate Hessian H(α) : )ˆ()]ˆ([ˆˆ )(1)()()1( nnnn JH ααγαα     (10)And after substituting (8) in (10) and rearrangingwe obtain the iterative algorithm:yPTHH Hnnn )(2)ˆ()1(ˆ)ˆ( )()1()(  γαγαα(11)The algorithm can be started from an initialestimate of α and run until convergence occurs.Solution of the optimization problem (7) can bederived in a similar manner:yBTHH Hnnn )(2)ˆ()1(ˆ)ˆ( )()1()( γβγββ  (12)where)(2)(2)()(2)( 21 βλβλβ  qqBTBTH H (13) qidiag121 )(1)(εββ  2/122 )(1)(qidiagεββ   (14)2.4 Dictionary selectionSelection of the proper dictionary  is animportant part of this method. This dictionaryshould sparsely represent the magnitude of thecomplex-valued image, and so it depends on theapplication and the type of objects or features ofinterest in our image.2.4.1 Overcomplete shape-based dictionariesIf our scene can be represented as combination ofsome limited simple shapes such as points, lines,squares, and so on, then an efficient dictionary can be constructed by gathering all possible positionsof these fundamental elements in an overcompletedictionary. The use of such an overcompletedictionary is demonstrated in Section 3 on somesynthetic scenes. It is also possible to use more general, possiblymultiresolution dictionaries that are well knownfor sparse representation of two dimensionalsignals (images). We present some possiblechoices below:2.4.2 Biorthogonal wavelet transformsPrevious works have established that wavelettransform can sparsely represent natural sceneimages [6,7]. The application of wavelettransforms to image compression leads toimpressive results over Fourier-basedrepresentations such as the discrete cosinetransform (DCT). However, wavelet transformoffers only a fixed number of directional elementsindependent of scale.2.4.3 Curvelet TransformCurvelet transform enables directional analysis ofan image in different scales. This dictionary iswell suited for enhancing features like edges andsmooth curves in an image.367Authorized licensed use limited to: ULAKBIM UASL - SABANCI UNIVERSITY. Downloaded on December 4, 2009 at 04:09 from IEEE Xplore.  Restrictions apply. 2.4.4 Discrete cosine transform (DCT)Both of above general dictionaries are suitable forpiecewise smooth content images, however forsparse representation of periodic patterns, as insome textures, they may not be good choices and other dictionaries such as DCT may be preferred.DCT is known to be well suited for first orderMarkov stationary signals [6]. For nonstationarysignals it could be applied in blocks.There exist many other popular dictionaries,which we don't mention here for the sake ofbrevity. We should just point out that any suchdictionary could be used in our framework, if it isappropriate for the particular application ofinterest.3. EXPERIMENTAL RESULTSWe present here our results of experiments withvarious synthetic images to demonstrate differentcapabilities of this new method. In ourexperiments we have selected a spotlight-modesynthetic aperture radar (SAR) modality as ourcomplex-valued sensing system. In all presentedresults we have used parameters of an X-bandSAR of 10 GHz center frequency and 0.375 mrange and azimuth resolution, except for resultsthat show superresolution capability, in whichresolution values are doubled so that systemresolution is a multiple of image pixel size and we can put multiple scatterers in one resolution celland observe the results. All images consist of3232 complex-valued pixels. To be able tocompare the results, we also present results ofconventional reconstruction as well asnonquadratic regularization method [3] that haverecently been developed for point and regionenhanced SAR image reconstruction.First we show an important capability of thismethod that is superresolution, which means thatit can reconstruct image details under bandwidthlimitations. To show this propoerty, we apply ourmethod to a synthetic scene composed of eightpoint scatterers with unit reflectivity magnitudeand random (uniform) uncorrelated phase. We setour system parameters so that system resolution istwo times of image pixel size, so we seeksuperresolution reconstructions. Fig. 1 (b) showsconventional spotlight mode SAR reconstructionusing polar format algorithm [8] that cannotresolve closely-spaced scatterers and suffers fromhigh sidelobes. Fig. 1(c) shows the result of nonquadraticregularization reconstruction method withpenalties for both point and region featureenhancement. Note that this method also hassuperresolution capability when we just use point enhancement penalty, however in this case weallow the possibility of both point and region-based features in the scene, e.g. due to lack ofmore specific prior knowledge  on the scene. As a result, this technique fails to reconstruct the scene accurately.Fig. 1(d) shows the result of the sparserepresentation method we propose in this paper,with parameters p=0.7, λ=3, q=0.3, and λ=0.07.We have used the conventional reconstruction asthe initial estimate of f. The dictionary used hereis the same as described in the next experiment.(a)10 20 30102030(b)10 20 30102030(c)10 20 30102030(d)10 20 30102030Fig. 1. Superresolving a scene with isolated pointscatterers. (a) True scene(b) Conventional reconstruction (c) Point-region-enhanced nonquadratic regularization (d) Sparse representation-based reconstruction.In order to demonstrate the capabilities of thisnew method and contrast it with existing methods, we now consider a more general scene composed 368Authorized licensed use limited to: ULAKBIM UASL - SABANCI UNIVERSITY. Downloaded on December 4, 2009 at 04:09 from IEEE Xplore.  Restrictions apply. of both point targets and distributed targets,shown in Fig. 2(a). Parameters used in this resultare p=0.9, λ=3, q=1, and λ=0.2.Here we use an overcomplete shape-baseddictionary. In  particular our dictionary consists of(a)10 20 30102030(b)10 20 30102030(c)10 20 30102030(d)10 20 30102030Fig. 2. Scene composed of point and distributed targets. (a) True scene  (b) Conventional reconstruction(c) Point-region-enhanced nonquadratic regularization(d) Sparse representation-based reconstruction.points as well as squares of various sizes at everypossible location in the scene. We note that such a dictionary can be used to sparsely represent manyinteresting scenes. Superiority of sparserepresentation method over others can be clearlyobserved in this figure.To be more realistic, in the next experiment weuse a synthetic image, constructed from the MITLincoln Laboratory Advanced DetectionTechnology Sensor (ADTS) data set [9] bysegmentation techniques, as shown in Fig. 3(a). Inthese type real scenes, we may not have enoughprior information for using overcomplete shape-based dictionaries and/or we may need moregeneral dictionaries that can sparsely representmany probable different scenes. Here we use thewavelet transform which is a multiresolutiondictionary and can sparsely represent manynatural scenes containing information at multiplespatial scales. Parameters used in this experimentare p=0.6, λ=17, q=1, and λ =10. We have used Haar wavelet as our dictionary in this result.Again we see near perfect reconstruction with oursparse signal representation-based method.(a)10 20 30102030(b)10 20 30102030(c)10 20 30102030(d)10 20 30102030Fig. 3. Synthetic ADTS phantom image reconstruction(a) Synthetic scene (b) Conventional reconstruction (c) Point-region-enhanced nonquadratic regularization(d) Sparse representation-based reconstruction.4. CONCLUSIONSIn this paper we have presented a new method for complex-valued image reconstruction based onsparse signal representation, which brings manycapabilities of sparse signal representation to thecontent of complex-valued imaging systems. Wehave developed a mathematical framework forthis purpose and shown its effectiveness onvarious synthetic scenes.REFERENCES:[1] M. Çetin, W. C. Karl, and A. S. Willsky, "Feature-preserving regularization method for complex-valuedinverse problems with application to coherentimaging", Optical Engineering, 45(1) : 017003,January 2006.[2] D. C. Munson, Jr. and J. L. C. Sanz, “Image reconstruction from frequency-offset Fourier data,”Proc. IEEE, vol. 72, pp. 661–669, June 1984.[3] M. Çetin and W.C. Karl, "Feature-enhancedsynthetic aperture radar image formation based on369Authorized licensed use limited to: ULAKBIM UASL - SABANCI UNIVERSITY. Downloaded on December 4, 2009 at 04:09 from IEEE Xplore.  Restrictions apply. nonquadratic regularization", IEEE Trans. ImageProcessing, vol. 10, pp. 623–631, April 2001.[4] D. C. Munson Jr., J. D. O’Brien, and W. K.Jenkins, "A tomographic formulation of spotlight-mode synthetic aperture radar", Proceedings of the IEEE, 71:917–925, August 1983.[5] S. S. Chen, D. L. Donoho, and M. A. Saunders,"Atomic decomposition by basis pursuit", SIAM J. Sci.Comput., Vol. 20, pp. 33-61, August 1998.[6] J. L. Starck, M. Elad, and D. L. Donoho, "Image decomposition via the combination of sparserepresentations and a variational Approach", IEEETrans. Image Processing, Vol. 14, pp. 1570-1582,October 2005.[7] D. L. Donoho and I. Johnstone, "Ideal spatialadaptation via wavelet shrinkage", Biometrika, vol. 81, pp. 425-455, 1994.[8] W. G. Carrara, R. S. Goodman, and R. M.Majewski, Spotlight Synthetic Aperture Radar: Signal Processing Algorithms . Boston, MA: ArtechHouse, 1995.[9] Air Force Research Laboratory, Model BasedVision Laboratory, Sensor Data Management System ADTS. [Online] available:http://www.mbvlab.wpafb.af.mil/public/sdms/datasets/adts/.370Authorized licensed use limited to: ULAKBIM UASL - SABANCI UNIVERSITY. Downloaded on December 4, 2009 at 04:09 from IEEE Xplore.  Restrictions apply. 

Sparse signal representation for complex-valued imaging

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Sparse signal representation for complex-valued imaging

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