SPARSE SIGNAL REPRESENTATION FOR COMPLEX-VALUED IMAGING Sadegh Samadi 1, M üjdat Çetin 2, Mohammad Ali Masnadi-Shirazi 1

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1 SPARSE SIGNAL REPRESENTATION FOR COMPLEX-VALUED IMAGING Sadegh Samadi 1, M üjdat Çetin, Mohammad Ali Masnadi-Shirazi 1 1. Shiraz University, Shiraz, Iran,. Sabanci University, Istanbul, Turkey ssamadi@shirazu.ac.ir, mcetin@sabanciuniv.edu, masnadi@shirazu.ac.ir Abstract: We roose a sarse signal reresentation-based method for comlexvalued imaging. Many coherent imaging systems such as synthetic aerture radar (SAR) have an inherent random hase, comlexvalued nature. On the other hand sarse signal reresentation, which has mostly been exloited in real-valued roblems, has many caabilities such as suerresolution and feature enhancement for various reconstruction and recognition tasks. For comlex-valued roblems, the key challenge is how to choose the dictionary and the reresentation scheme for effective sarse reresentation. We roose a mathematical framework and an associated otimization algorithm for a sarse signal reresentation-based imaging method that can deal with these issues. Simulation results show that this method offers imroved results comared to existing owerful imaging techniques. Index Terms: sarse signal reresentation, comlexvalued imaging, image reconstruction, coherent imaging, synthetic aerture radar. 1. INTRODUCTION This aer resents a new image reconstruction technique based on sarse signal reresentation that can be used for comlex-valued imaging systems. Many coherent imaging systems such as synthetic aerture radar (SAR), holograhy, magnetic resonance imaging, laser imaging, sonar imaging, and so on, have an inherent comlexvalued nature [1]. Also, hase of the imaged field might be random and satially uncorrelated in some alications such as SAR []. In these systems, straightforward alication of image reconstruction techniques develoed originally for incoherent real-valued imaging systems will not roduce high quality images. One of these techniques that have mostly been exloited in real-valued roblems is sarse signal reresentation, which has many caabilities such as suerresolution and feature enhancement for various reconstruction and recognition tasks. Recent work on feature-enhanced comlex-valued imaging [3] has ties to sarse reresentation. In articular, one interretation of this technique involves sarse reresentation with some fixed, secific dictionaries. Our work generalizes that work to sarse reresentation with arbitrary overcomlete dictionaries. The key challenge for exloiting sarse reresentation in comlex-valued roblems is how to choose the dictionary and the reresentation scheme for effective sarse reresentation. In the next sections we resent a mathematical framework that deals with these issues. As resented in Section, we have used a general linear observation model, and alied sarse reresentation on the magnitude of the comlexvalued scattered field. This then turns the image reconstruction roblem into a joint otimization roblem over the magnitude and the hase of the underlying field reflectivities. We resent an iterative algorithm to solve this joint otimization roblem. In the end of Section, a discussion on dictionary selection is resented. In Section 3 our exerimental results are resented to show the effectiveness of this new method.. SPARSE REPRESENTATION FOR COMPLEX-VALUED PROBLEMS This section contains a descrition of our sarse reresentation technique for comlex-valued roblems. We start with describing our general

2 observation model. Next we resent our sarse reresentation scheme and the related otimization roblem. Then we resent our algorithm to solve this otimization roblem. Finally a discussion about roer dictionary selection is resented..1 Observation model We use the following general observation model: y = T f + n (1) where y is the samled measured data, f is the unknown scene or object reflectivity image, and n is noise; all are comlex and column-reordered as vectors. T reresents a comlex-valued discrete observation kernel. T could be, for examle, a band-limited Fourier transform oerator matrix, or a rojection oerator matrix as in SAR or other tomograhic imaging system [3,4].. Sarse reresentation scheme In many alications of imaging systems introduced in Section 1, such as SAR, we could have a sarse reresentation in a roer domain for magnitude of comlex scattered field (image). This means that we can write: f = Φα () Where Φ is an aroriate dictionary for our alication, and α is the new domain coefficient vector with which the magnitude of scattered field can be sarsely reresented. jφ We can write P f P = diag e i, and φ i 's are the unknown hases of each image vector element f i. So we can rewrite our observation model as: f =, where { } y = T f + n= T P f + n= T PΦα + n (3) If we knew P (the unknown image vector elements hases), using a sarsity regularization method such as an extension of basis ursuit [5] as follows, we could find an estimate of α and hence of the image itself: αˆ = arg min y TPΦα + λ α (4) α where denotes the l -norm and λ is a scalar arameter. owever we don't know the image hases and hence P. We use the following aroach to overcome this roblem: a. First we start with an initial estimate for f that could be a conventional reconstruction of f, so we can also roduce an initial estimate of the image hase matrix P. In this way, we can solve the otimization roblem in (4) and find a new estimate for α. b. Now we have a new estimate of α and so f, and we should now find a new estimate for P. We can rewrite our observation model as: where diag{ } y = T P f + n = T B + n (5) B= f i and is vector of unknown hases, hence a vector containing the diagonal elements of P. We can find an estimate of as follows: ˆ = arg min y T B subject to i = 1, i (6) We relace the constrained otimization roblem in (6), with the following unconstrained roblem that incororates a enalty term on the magnitudes of ' i s : N q ˆ = arg min y T B + λ ( i 1) q q i= 1 = arg min y T B + λ λ q q (7) Solving this otimization roblem will give us a new estimate for the hase vector and so the matrix P = diag { }.

3 c. We are now back to ste (a) and we reeat the above stes until the algorithm converges to its final solution..3 Solving the otimization roblems If we call the cost functio n of otimization roblem (4) J(α) and use the smooth aroximation α M i= 1 i ( α + ε ) /, where ε is a small ositive constant, to avoid nondifferentiability roblems of the l -norm around the origin, then the gradient of J(α) will be: J ( α ) = ( α ) α ( T P Φ) y ( 8) where ( α ) = ( T P Φ) ( T PΦ) + λ Λ( α) 1 Λ( α ) = diag (9) 1 / ( αi + ε ) Using the quasi-newton iterative method develoed in [3], which is well suited for this kind of nonquadratic otimization roblems, with aroximate essian (α) : ( n n) ˆ α + 1) ( n ) ( n) 1 ( = αˆ γ [ ( αˆ )] J ( αˆ ) () And after substituting (8) in () and rearranging we obtain the iterative algorithm: ( n) ( n+ 1) ( n) ( α ˆ ) αˆ = (1 γ ) ( αˆ ) + γ ( T PΦ) y (11) The algorithm can be started from an initial estimate of α and run until convergence occurs. Solution of the otimization roblem (7) can be derived in a similar manner: ˆ ( n) ˆ ( n+ 1) n (1 ) ( ˆ ( ) ( ) = γ ) + γ ( T B) y (1) where ( ) = ( T B) ( T B) + λ q Λ1( ) λ q Λ ( ) (13) 1 Λ1( ) = diag 1 q ( i + ε ) 1 Λ ( ) = diag (14) 1 q/ ( i + ε ).4 Dictionary selection Selection of the roer dictionary Φ is an imortant art of this method. This dictionary should sarsely reresent the magnitude of the comlex-valued image, and so it deends on the alication and the tye of objects or features of interest in our image..4.1 Overcomlete shae-based dictionaries If our scene can be reresented as combination of some limited simle shaes such as oints, lines, squares, and so on, then an efficient dictionary can be constructed by gathering all ossible ositions of these fundamental elements in an overcomlete dictionary. The use of such an overcomlete dictionary is demonstrated in Section 3 on some synthetic scenes. It is also ossible to use more general, ossibly multiresolution dictionaries that are well known for sarse reresentation of two dimensional signals (images). We resent some ossible choices below:.4. Biorthogonal wavelet transforms Previous works have established that wavelet transform can sarsely reresent natural scene images [6,7]. The alication of wavelet transforms to image comression leads to imressive results over Fourier-based reresentations such as the discrete cosine transform (DCT). owever, wavelet transform offers only a fixed number of directional elements indeendent of scale..4.3 Curvelet Transform Curvelet transform enables directional analysis of an image in different scales. This dictionary is well suited for enhancing features like edges and smooth curves in an image.

4 .4.4 Discrete cosine transform (DCT) Both of above general dictionaries are suitable for iecewise smooth content images, however for sarse reresentation of eriodic atterns, as in some textures, they may not be good choices and other dictionaries such as DCT may be referred. DCT is known to be well suited for first order Markov stationary signals [6]. For nonstationary signals it could be alied in blocks. There exist many other oular dictionaries, which we don't mention here for the sake of brevity. We should just oint out that any such dictionary could be used in our framework, if it is aroriate for the articular alication of interest. 3. EXPERIMENTAL RESULTS We resent here our results of exeriments with various synthetic images to demonstrate different caabilities of this new method. In our exeriments we have selected a sotlight-mode synthetic aerture radar (SAR) modality as our comlex-valued sensing system. In all resented results we have used arameters of an X-band SAR of Gz center frequency and m range and azimuth resolution, excet for results that show suerresolution caability, in which resolution values are doubled so that system resolution is a multile of image ixel size and we can ut multile scatterers in one resolution cell and observe the results. All images consist of 3 3 comlex-valued ixels. To be able to comare the results, we also resent results of conventional reconstruction as well as nonquadratic regularization method [3] that have recently been develoed for oint and region enhanced SAR image reconstruction. First we show an imortant caability of this method that is suerresolution, which means that it can reconstruct image details under bandwidth limitations. To show this rooerty, we aly our method to a synthetic scene comosed of eight oint scatterers with unit reflectivity magnitude and random (uniform) uncorrelated hase. We set our system arameters so that system resolution is two times of image ixel size, so we seek suerresolution reconstructions. Fig. 1 (b) shows conventional sotlight mode SAR reconstruction using olar format algorithm [8] that cannot resolve closely-saced scatterers and suffers from high sidelobes. Fig. 1(c) shows the result of nonquadratic regularization reconstruction method with enalties for both oint and region feature enhancement. Note that this method also has suerresolution caability when we just use oint enhancement enalty, however in this case we allow the ossibility of both oint and regionbased features in the scene, e.g. due to lack of more secific rior knowledge on the scene. As a result, this technique fails to reconstruct the scene accurately. Fig. 1(d) shows the result of the sarse reresentation method we roose in this aer, with arameters =0.7, λ=3, q=0.3, and λ =0.07. We have used the conventional reconstruction as the initial estimate of f. The dictionary used here is the same as described in the next exeriment. (a) (c) (b) (d) Fig. 1. Suerresolving a scene with isolated oint scatterers. (a) True scene (b) Conventional reconstruction (c) Point-regionenhanced nonquadratic regularization (d) Sarse reresentation-based reconstruction. In order to demonstrate the caabilities of this new method and contrast it with existing methods, we now consider a more general scene comosed

5 of both oint targets and distributed targets, shown in Fig. (a). Parameters used in this result are =0.9, λ=3, q=1, and λ =0.. ere we use an overcomlete shae-based dictionary. In articular our dictionary consists of satial scales. Parameters used in this exeriment are =0.6, λ=17, q=1, and λ =. We have used aar wavelet as our dictionary in this result. Again we see near erfect reconstruction with our sarse signal reresentation-based method. (a) (b) (a) (b) (c) (d) Fig.. Scene comosed of oint and distributed targets. (a) True scene (b) Conventional reconstruction (c) Point-region-enhanced nonquadratic regularization (d) Sarse reresentation-based reconstruction. oints as well as squares of various sizes at every ossible location in the scene. We note that such a dictionary can be used to sarsely reresent many interesting scenes. Sueriority of sarse reresentation method over others can be clearly observed in this figure. To be more realistic, in the next exeriment we use a synthetic image, constructed from the MIT Lincoln Laboratory Advanced Detection Technology Sensor (ADTS) data set [9] by segmentation techniques, as shown in Fig. 3(a). In these tye real scenes, we may not have enough rior information for using overcomlete shaebased dictionaries and/or we may need more general dictionaries that can sarsely reresent many robable different scenes. ere we use the wavelet transform which is a multiresolution dictionary and can sarsely reresent many natural scenes containing information at multile (c) (d) Fig. 3. Synthetic ADTS hantom image reconstruction (a) Synthetic scene (b) Conventional reconstruction (c) Point-region-enhanced nonquadratic regularization (d) Sarse reresentation-based reconstruction. 4. CONCLUSIONS In this aer we have resented a new method for comlex-valued image reconstruction based on sarse signal reresentation, which brings many caabilities of sarse signal reresentation to the content of comlex-valued imaging systems. We have develoed a mathematical framework for this urose and shown its effectiveness on various synthetic scenes. REFERENCES: [1] M. Çetin, W. C. Karl, and A. S. Willsky, "Featurereserving regularization method for comlex-valued inverse roblems with alication to coherent imaging", Otical Engineering, 45(1) : , January 06. [] D. C. Munson, Jr. and J. L. C. Sanz, Image reconstruction from frequency-offset Fourier data, Proc. IEEE, vol. 7, , June [3] M. Çetin and W.C. Karl, "Feature-enhanced synthetic aerture radar image formation based on

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