ISSN 1751-8784 Sparse representation-based synthetic ...people.sabanciuniv.edu › mcetin › publications › samadi_IET_RSN11.p… · Sparse representation-based synthetic aperture

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Published in IET Radar, Sonar and NavigationReceived on 8th September 2009Revised on 16th April 2010doi: 10.1049/iet-rsn.2009.0235

ISSN 1751-8784

Sparse representation-based synthetic apertureradar imagingS. Samadi1 M. Cetin2 M.A. Masnadi-Shirazi1

1School of Electrical and Computer Engineering, Shiraz University, Zand Street, 71348-51154 Shiraz, Iran2Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla 34956 Istanbul, TurkeyE-mail: [email protected]

Abstract: There is increasing interest in using synthetic aperture radar (SAR) images in automated target recognition anddecision-making tasks. The success of such tasks depends on how well the reconstructed SAR images exhibit certain featuresof the underlying scene. Based on the observation that typical underlying scenes usually exhibit sparsity in terms of suchfeatures, this paper presents an image formation method that formulates the SAR imaging problem as a sparse signalrepresentation problem. For problems of complex-valued nature, such as SAR, a key challenge is how to choose thedictionary and the representation scheme for effective sparse representation. Since features of the SAR reflectivity magnitudeare usually of interest, the approach is designed to sparsely represent the magnitude of the complex-valued scattered field.This turns the image reconstruction problem into a joint optimisation problem over the representation of magnitude and phaseof the underlying field reflectivities. The authors develop the mathematical framework for this method and propose aniterative solution for the corresponding joint optimisation problem. The experimental results demonstrate the superiority ofthis method over previous approaches in terms of both producing high-quality SAR images and exhibiting robustness touncertain or limited data.

1 Introduction

Synthetic aperture radar (SAR) is an active microwave sensorthat is able to produce high-resolution images of the earth’ssurface at any time. With day and night capability andoperation under all weathers, SAR is one of the mostpromising remote sensing modalities. SAR makes use of asensor carried on an airborne or spaceborne platform,transmitting microwave pulses towards an area of intereston the earth, and receives the reflected signal. This signalfirst undergoes some pre-processing tasks including ademodulation process. The SAR image formation problemis the problem of reconstruction of a spatial reflectivitydistribution of the scene from the pre-processed SARreturns. In this paper, we focus on spotlight-mode SAR [1],although the proposed ideas are not limited to a specificmode.

Recently, there has been significant and increasing interestin using SAR images in automated target recognition anddecision-making tasks. The success of such tasks dependson how well the reconstructed SAR images exhibit certainfeatures of the underlying scene. The conventional approachfor spotlight-mode SAR image formation is the polarformat algorithm (PFA) [1], which is based on the premiseof clean, full-aperture, full-bandwidth data. This method hasno explicit mechanism to counter any imperfectionincluding noise in the data, and also suffers from importantshortcomings, such as resolution limitation to systembandwidth, speckle and sidelobe artefacts. These limitations

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make it difficult to use conventionally reconstructed SARimages for robust recognition and decision-making tasks.

Based on the observation that typical underlying scenesusually exhibit sparsity in terms of certain features ofinterest, we develop an image formation method thatformulates the SAR imaging problem as a sparse signalrepresentation problem. Sparse signal representation, whichhas mostly been exploited in real-valued problems, offersmany advantages such as super-resolution and featureenhancement for various reconstruction and recognitiontasks. Recent work on feature-enhanced SAR imageformation based on non-quadratic regularisation [2] has tiesto sparse representation (SR). In particular, oneinterpretation of the technique in [2] involves SR with somefixed and specific dictionaries which makes it useful forscenes containing a limited set of specific feature types.This is only an interpretation as the (fixed) dictionariesinvolved in the approach are only implicit, and theoptimisation problem is formulated over the reflectivitiesrather than the SR coefficients. The method proposed hereextends and generalises the approach in [2] to SR witharbitrary, general dictionaries to handle general scenescontaining any type of features. This is achieved byexplicitly formulating the SAR image formation problem asa SR problem with arbitrary dictionaries.

A key challenge for exploiting SR in complex-valuedproblems such as SAR imaging is how to choose thedictionary and the representation scheme. In this paper, wedevelop a mathematical framework to deal with these

IET Radar Sonar Navig., 2011, Vol. 5, Iss. 2, pp. 182–193doi: 10.1049/iet-rsn.2009.0235

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issues. In particular, we apply SR on the magnitude of thecomplex-valued scattered field, which leads to a jointoptimisation problem over the representation of magnitudeand phase of the underlying field reflectivities. Theframework leads to an iterative algorithm for solving thejoint optimisation problem. To demonstrate our approach,we have used some sample dictionaries, including multi-resolution dictionaries, as well as dictionaries adapted to theshapes of the likely objects in the scene. However, thisframework has the capability of using any appropriatedictionary for the particular application of interest. The restof this paper is organised as follows. In Section 2 details ofSR-based SAR imaging are presented. In Section 3experimental results on various SAR images are presentedto demonstrate the effectiveness of our approach.Eventually, some concluding remarks are made in Section 4.

2 SR framework for SAR image formation

Sparse signal representation has successfully been used forsolving inverse problems in a variety of applications. It hasmany capabilities for various reconstruction and recognitiontasks; however, it has mostly been used in real-valuedproblems. Owing to the complex-valued and potentiallyrandom phase nature of the reflectivities in SAR, our approachis designed to sparsely represent the magnitude of the complex-valued scattered field where we are interested in its features.

2.1 Observation model

We use an observation model for spotlight-mode SARimaging motivated by the tomographic formulation of SAR[3]. Data are collected using a radar sensor traversing aflight path with an antenna boresight realigned continuallyto point at a fixed ground patch. The most commonly usedtransmitting pulse in SAR is a linear FM chirp pulse

s(t) = e j(v0t+mt2) for |t| ≤ Tp/2 and zero otherwise. Here v0

is the carrier frequency and 2m is the so-called chirp rate.SAR transmits pulses at positions of equal angularincrements. The backscattered signal from the scene ismixed with a reference chirp and passed through a low-passfilter. Typically, it is assumed that the distance from radarto the centre of the scene (Ru) is much greater than theground patch radius (L). In this case, the demodulatedobserved signal is given by [1, 2]

sb(u, t) =∫∫

x2+y2≤L2

f (x, y) exp{−jV(t)(xcosu+ ysinu)}dxdy

(1)

where V(t) ¼ 2/c(v0 + 2m(t 2 (2Ru/c))) denotes the radialspatial frequency, f (x, y) is the complex-valued reflectivityfield (the unknown image) and c is the speed of light. Notethat sb(ui, t) is a finite slice through the two-dimensional(2D) Fourier transform of the field reflectivity at angle ui,and is usually referred to as the phase histories. Adiscretised version of (1) can be written as [2]

Sb = Cf (2)

where Sb = [stu1

stu2

. . . stuM

] is a vector of sampled phase

histories, C = [ctu1

ctu2· · ·ct

uM]t is a discretised approximation

to the observation kernel in (1) and f a vector of unknownsampled reflectivity image. Using the projection slice


theorem [4], the observed phase histories can also bewritten as [2]

sb(u, t) =∫L

−L

g(u, u) exp{−jV(t)u}du (3)

where g(ui,u) is the projection of f (x, y) at angle ui. Using (1)and (3) a discrete relation between the field f and theprojections g can be obtained as [2]

g = F−1Sb = F−1Cf = Hf (4)

where F21 is a block diagonal matrix with each blockperforming an inverse discrete Fourier transform (IDFT) oneach sui

, and H represents a complex-valued discrete SARprojection operator. The data g in (4) are called rangeprofiles. In the presence of noise, the range profileobservation model becomes

y = Hf +n (5)

where n is the additive observation noise. A similar phasehistory observation model can be obtained based on (2);however, the model in (5) has computational benefits due tothe approximately sparse nature of H [2]. A convolutionallinear model can also be defined by replacing y in (5) witha conventionally reconstructed image, and letting H bereplaced with a matrix with each row containing a spatiallyshifted version of the corresponding point spread functionstacked as a row vector [5]. This form provides somecomputational (especially memory) advantages as thematrix vector operations can be performed throughconvolutions.

2.2 SR scheme

In this subsection we present our SR framework thateffectively deals with the complex-valued nature of theSAR signal and can be used for enhanced SAR imageformation. Since we are usually interested in features of themagnitude of the SAR signal, our new approach is designedto sparsely represent the magnitude of the complex-valuedscattered field. Thus, we consider

| f | = Fa (6)

where F is an appropriate dictionary for our application thatcan sparsely represent the magnitude of the scattered field (orsimply the scene) in terms of the features of interest, and a isthe vector of representation coefficients. For any complex-valued vector f we can write f ¼ P| f |, whereP = diag{ejfi } is a diagonal matrix, and fi represents theunknown phase of image vector element ( f )i. Thus, we canrewrite the observation model as

y = Hf + n = HP| f | + n = HPFa+ n (7)

If we knew P (or phases of the elements of the unknownimage vector), using an atomic decomposition techniquesuch as an extension of basis pursuit denoising [6], wecould find an estimate of a, and hence the magnitude imageof the unknown scene itself, as follows

a = arg mina

‖y − HPFa‖22 + l‖a‖p

p (8)

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where ‖.‖p denotes the ℓp-norm, and ℓ is a positive real scalarparameter. Note that the ℓ2-norm term in (8) is related to theassumption that noise is zero mean white Gaussian (Forcolored noise, a weighted norm can be used in our dataterm). Also note that the perfect sparsity condition termwould involve an ℓ0-norm, which would lead to acombinatorial, hard-to-solve problem. However, it is shownthat for fields that admit a sparse enough representation, theℓp-norm with p ≤ 1, as we use here, also leads to thesparsest of all representations under certain conditions[7–11].

However, the difficulty in solving the optimisation problemin (8) is that we do not know the phase terms of the imagevector elements and hence the matrix P. We propose thefollowing joint optimisation approach to overcome thisproblem:

(a) Start with an initial estimate of f that could be itsconventional reconstruction. Using this f, an initial estimateof the image phase matrix, P can be obtained.(b) Using this estimate of P, the optimisation problem in (8)can be solved and a new estimate of a can be obtained.(c) Using the new estimate of a, the new estimate of | f | canbe produced from (6). Now, a new estimate of the phasematrix P should be found. To do this rewrite theobservation model as

y = HP| f | + n = HBb+ n (9)

where matrix B = diag{|(f )i|} and b is a vector formed bystacking the diagonal elements of matrix P, hence itcontains the unknown phase terms. An estimate of b can beobtained through the following estimator

b = arg minb

‖y − HBb‖22 subject to |(b)i| = 1, ∀i

(10)

where the unconstrained part of (10) is both the maximumlikelihood and the minimum variance unbiased estimator ofb [12]. This is due to the assumption that the observationnoise is independent identically distributed complexGaussian noise, which is the most commonly usedstatistical model for radar measurement noise [13, 14], andalso the linearity of the observation model in (9). However,the prior information on b introduces a constraint to theproblem, which results in the optimisation problem in (10).For most of the SAR scenes phase of the reflectivity at acertain location could be modelled as random, with auniform probability density function, and independent ofthe phase at other locations [15]. Assuming this priorinformation, it is also interesting to investigate the Bayesianestimation approach for this problem. In particular, themaximum a posteriori (MAP) estimator is considered here,which is obtained as follows

bMAP = arg maxb

p(y|b)p(b) (11)

Since the magnitude of any vector element (b)i is a constantindependent of its phase, p(b) can be written as (see

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Appendix 1)

p(b) =∏N2

i=1

d(|(b)i| − 1)U (/(b)i) (12)

where U(.) is a uniform function over [2p, p], d(.) is a deltafunction that can be considered as a uniform function over[21, 1] for a very small value of 1, and N 2 is the numberof elements of vector b for an image of size N × N. Also,p( y|b) is a complex Gaussian pdf. Note that p(b) is equalto a constant when all the values of |(b)i| terms are in thevicinity of 1 and all the phase terms /(b)i are in theinterval of [2p, p]. Otherwise p(b) is equal to zero. Thus,the maximum of (11) occurs when the first term on theright side is maximum with the constraint that all themagnitude values of (b)i terms are in the vicinity of 1 andall their phases are in the interval of [2p, p]. Therefore itcan be easily shown that the constrained estimator in (10) isalso the MAP estimator of b, whenever all phases of thesolution are considered to be in the interval of [2p, p].

Using N 2 Lagrange multipliers the constrained problem in(10) can be replaced with the following unconstrainedproblem

b = arg minb

‖y − HBb‖22 +

∑N2

i=1

l′i(|(b)i| − 1) (13)

Note that the N 2 constraints |(b)i| ¼ 1, ∀i are equivalent to

the one constraint∑N2

i=1 (|(b)i| − 1)2 = 0, since thesummation of some non-negative real terms will be equal tozero if and only if all of them are equal to zero. Thisreduces the optimisation problem in (13) to the followingproblem which is much more tractable

b = arg minb

‖y − HBb‖22 + l′

∑N2

i=1

(|(b)i| − 1)2

= arg minb

‖y − HBb‖22 + l′‖b‖2

2 − 2l′‖b‖1 (14)

Solving this optimisation problem produces a new estimate ofthe phase vector b and hence the matrix P = diag{(b)i}.(d) Go back to step (b) and repeat this loop until(‖| f |n+1 − | f |(n)‖2

2)/(‖| f |n‖22) , d, where d is a small

positive real constant, and | f |(n) is the estimate of | f | instep n.

Note that it is possible to view these two optimisationproblems as coordinate descent stages of an overalloptimisation problem over a and b.

2.3 Solving the joint optimisation problems

Let us call the cost function of the optimisation problem (8)J(a), and use the smooth approximation‖a‖p

p ≃∑M

i=1 (|(a)i|2 + 1)p/2 , where 1 is a small positiveconstant, to avoid the non-differentiability problem of theℓp-norm around the origin. The gradient of J(a) withrespect to a will be

∇aJ (a) = G(a)a− 2(HPF)Hy (15)


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where

G(a) = 2(HPF)H(HPF) + lpC(a) (16)

and the C(.) function is

C(a) = diag{1/(|(a)i|2 + 1)1−p/2} (17)

in which, (a)i’s are the elements of the vector a. Note thatG(a) in (15) is a function of a, therefore this equationgenerally does not yield a closed-form solution for a andrequires numerical optimisation techniques. It has beenshown that the standard methods such as Newton’s methodor quasi-Newton’s method with a conventional Hessianupdate scheme perform poorly for non-quadratic problemsof this form [2, 16]. Using the idea in [2], we use here aquasi-Newton’s method with a Hessian update scheme thatis matched to the structure of our problem. We use G(a) asan approximation to the Hessian and use it in the followingquasi-Newton’s iterative algorithm

a(n+1) = a(n) − g[G(a(n))]−1∇a J (a(n)) (18)

After substituting (15) into (18) and rearranging, we obtainthe following iterative algorithm

G(a(n))a(n+1) = (1 − g)G(a(n))a(n) + 2g(HPF)Hy (19)

The algorithm can be started from an initial estimate of a andrun until ‖a(n+1) − a(n)‖2

2/‖a(n)‖22 , da, where da is a small

positive real constant. Note that G(a(n)) in our problem isapproximately sparse, Hermitian, and positive semidefinite,so the set of linear equations in (19) for finding a(n+1) canitself be solved efficiently using iterative approaches suchas conjugate gradient (CG) [17], which we use in our method.

Although we seek an SR of | f | ¼ Fa, we do not check ifthe representation obtained from the above algorithm alwaysyields a positive-valued signal. In some cases, we may obtainnegative numbers, which would mean we use some part of thephase (actually a phase shift of p) in the magnitude. Thiswould cause an extra redundancy in our model insofar as anegative magnitude can be compensated by a phase shift ofp. Although we have not explored this in detail, thisredundancy may sometimes be helping us obtain bettersparsity. On the other hand, one could try and limit theapproach to guarantee positive numbers in the magnituderepresentation, however, that would lead to additionalcomputational complexity. In our work, we have not felt theneed to limit the solution to be non-negative.

Similarly, we can obtain the following iterative algorithmfor solving the optimisation problem in (14)

G′b(n+1) = (1 − g)G′b

(n) + 2g(HB)Hy + 2gl′w(b(n)

)

(20)

where

(w(b(n)

))i = ej(f((b(n)

)i)) (21)

G′ = 2(HB)H(HB) + 2l′I (22)

in which I is the identity matrix and f((b(n)

)i) is the phase

of (b(n)

)i. The iteration (20) should also be run until

‖b(n+1) − b(n)‖2

2/‖b(n)‖2

2 becomes less than a specified


small positive constant. Note that the properties of G′aresimilar to those of the G(a(n)) mentioned above and

consequently with known b(n)

the set of linear equations in

(20) for finding b(n+1)

can also be efficiently solved usingthe CG algorithm.

2.4 Ties to half-quadratic regularisation andconvergence issues

It is shown in [18] that the quasi-Newton-based algorithm of asimilar type to the one in (19) has ties to half-quadraticregularisation. ‘The main idea in half-quadraticregularisation is to introduce and optimise a new costfunctional, which has the same minimum as the originalcost functional, but one which can be manipulated withlinear algebraic methods. Such a new cost functional isobtained by augmenting the original cost functional with anauxiliary vector’ [18, 19].

Let us consider a new cost function Ka(a, b), which isquadratic in a (hence the name half-quadratic), and b is anauxiliary vector, such that

infb

Ka(a, b) = J (a) (23)

where J(a) is the cost functional in (8). It can be shown thatthe following augmented cost functional Ka(a, b) satisfies(23) (see Appendix 2)

Ka(a, b) = ‖y − HPFa‖22 + l

∑N2

i=1

[bi(|(a)i|2 + 1)

+ p

2bi

( )p/(2−p)

1 − p

2

( )](24)

Note that Ka(a, b) is a quadratic function with respect to a,and can be minimised easily in b. According to (23), Ka(a,b) and J(a) share the same minima in a, so we can use ablock coordinate descent scheme on Ka(a, b) to find the athat minimises J(a)

b(n+1) = arg min

bKa(a(n), b) (25)

a(n+1) = arg mina

Ka(a, b(n+1)

) (26)

Using the results of Appendix 2, we obtain the followingiterative algorithm from (25) and (26)

b(n+1)i = p

2[|(a(n))i|2 + 1]1−p/2(27)

[(HPF)H(HPF) + l diag{b(n+1)i }]a(n+1) = (HPF)Hy (28)

Note that the above iterative algorithm is equivalent to the onein (19) with g ¼ 1. Based on the above half-quadraticinterpretation of the algorithm in (19) we have shown inAppendix 3 that it is convergent in terms of the costfunctional.

Now, let us consider the algorithm in (20). Below, we showthat it also has a half-quadratic interpretation. The costfunction of the optimisation problem of b in (14) can be

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written as

J (b) = ‖y − HBb‖22 + l′

∑N2

i=1

(|(b)i| − 1)2 (29)

Let us consider the augmented function Kb(b, s), which isquadratic in b such that

infs

Kb(b, s) = J (b) (30)

It is shown in Appendix 4 that the following augmented costfunction satisfies (30)

Kb(b, s) = ‖y − HBb‖22 + l′

∑N2

i=1

|(Sb)i − 1|2 (31)

where S ¼ diag{exp(2jsi)}, with si the ith element of thevector s. Similar to (25) and (26), a block coordinatedescent scheme on Kb(b, s) can be used to find theoptimum b that minimises J(b)

s(n+1) = arg mins

Kb(b(n)

, s) (32)

b(n+1) = arg min

bKb(b, s(n+1)) (33)

According to the results of Appendix 4, we can obtain thefollowing iterative algorithm from (32) and (33)

s(n+1)i = f[(b

(n))i] (34)

[(HB)H(HB) + l′(S(n+1)

)HS(n+1)

]b(n+1)

= (HB)Hy + l′(S(n+1)

)H1 (35)

where 1 is a vector of ones with the same size as b. It can beeasily shown that the above algorithm is equivalent to the onein (20) with g ¼ 1. Convergence of the algorithm in terms ofcost functional can then be easily established in a similar wayto that in Appendix 3.

Based on the convergence results of the two optimisationproblems, Appendix 5 proves the convergence of theoverall image reconstruction algorithm presented in Section2.2 in terms of the cost functional. In general, the algorithmconverges to a minimum (may be local or global) and sincewe use a conventional reconstruction as our initialisation wealways come to an improvement over the conventionalreconstruction in terms of the cost functional. For a typical256 × 256 pixel image that we have used in ourexperiments, using wavelet dictionary and non-optimisedMATLAB code, convergence time of the algorithm is onthe order of few minutes on a Pentium IV 3 GHz computer.

2.5 Dictionary selection

Selection of the proper dictionary F is an important part ofthis method. This dictionary should sparsely represent themagnitude of the complex-valued image which contains thefeatures of interest in the scene and so it depends on theapplication and the type of objects or features of interest inour image.

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2.5.1 Overcomplete shape-based dictionaries: If theunderlying scene can be represented as a combination ofsome limited simple shapes such as points, lines andsquares of different sizes and so on, then a powerfuldictionary can be constructed by gathering all possiblepositions of these fundamental elements in an overcompletedictionary. This type of dictionary provides a convenienttool to demonstrate the capabilities of the proposedframework of image formation for synthetic or simple realscenes (e.g. man-made targets in smooth low reflectivitybackgrounds). For example, if we are interested in pointscatterers along with smooth regions with simple shapes ina SAR image, use of this type of dictionary can lead toexcellent results. The use of such an overcompletedictionary is demonstrated in Section 3 for some syntheticscenes. Although this kind of dictionary has interestingproperties, in large-scale problems it may lead tocomputational problems because of required large numberof dictionary atoms. Thus, it is required to provide moreefficient dictionaries for practical applications. It is possibleto use more general dictionaries that are well known for SRof 2D signals (images).

2.5.2 Wavelet dictionary: Standard multiresolutiondictionaries, such as those based on wavelets, are one ofthe effective options that can be used in our framework.Previous works have established that wavelet transformcan sparsely represent natural scene images [20, 21]. Theapplication of wavelet transforms to image compressionleads to impressive results over other representations, sodepending on the scene this dictionary may have theability to sparsely represent complicated SARmagnitude images [22]. It is also shown in [23] thatthe wavelet transform can sparsely represent features ofman-made targets with stronger reflectivities than thebackground.

2.5.3 Other dictionaries: Depending on our application,we may need to use other dictionaries with propertiesmatched to that application. For example, curvelettransform enables directional analysis of an image indifferent scales [20, 24], so it is well suited for enhancingfeatures such as edges and smooth curves in an image. Ifwe have textures with periodic patterns in our magnitudeimage then one appropriate dictionary to sparsely representthese patterns could be based on the discrete cosinetransform [20].

There exist many other popular dictionaries, which we donot mention here for the sake of brevity. We should justpoint out that any such dictionary could be used in ourframework, if it is appropriate for the particular applicationof interest.

3 Experimental results

Here we present the results of applying the proposedalgorithm in experiments with various synthetic and realSAR scenes and compare them with those of theconventional PFA and the non-quadratic regularisationapproach of [2] to demonstrate the achieved improvementsin performance. We also present results with differentdictionaries to show the effect of dictionary selection andthe capability of the proposed framework in using anyappropriate dictionary.


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3.1 Parameter selection and initialisationof the algorithm

We have used the conventional polar format reconstruction asan initial estimate of f to be used in both our method and thenon-quadratic regularisation method, which provides aninitial estimate for both a and b. Based on our discussionson convergence in Section 2.4, for the iterative algorithmsin (19) and (20) we have used a fixed step size of g ¼ 1,which guarantees convergence in terms of the costfunctional. The smoothing parameter, 1, in the definition ofthe approximate lp-norm, is set to 1025, which is smallenough not to affect the results given the normalised (tomaximum magnitude of 1) input data to our algorithm.

For synthetic image experiments, we have used parametersof a spotlight mode SAR of 10 GHz centre frequency and0.375 m range and cross-range resolution. For the resultsthat show super-resolution capability, the resolution valuesare doubled so that the system resolution is a multiple ofimage pixel size and we can put multiple scatterers in oneresolution cell and observe the results. All synthetic scenesconsist of 32 × 32 complex-valued pixels.

The parameters l, l′ and p reflect the degree of emphasison the data against the constraints, as well as the nature ofour prior information about the scene leading to the type ofsparsity constraint used. The value of p could be less thanor equal to 1 and smaller values of p usually producesparser results. Selection of l and l′ involves a trade-offbetween relying on data or the prior information. Forexample, higher values of l usually produce sparser results.Thus, in general if we have enough information of thescene or we are interested only in special features ofthe scene, such that a proper dictionary with a good SR ofthe scene can be chosen, higher values of l and lowervalues of p will produce better results. Considering theseguidelines, we have used l ¼ 10, p ¼ 0.6 and l′ ¼ 2 forthe synthetic scene experiments, and l ¼ 1, p ¼ 0.7 andl′ ¼ 2 for experiments on the AFRL backhoe data. Ourexperience as well as the reported experience in [2] showthat a set of parameters chosen on a single image of adatabase can usually be used for other similar images ofthat database as long as the observation quality does notchange. There are also automatic parameter selectionmethods [25] developed for similar problems that could bean area of research in continuation of this work.

3.2 Quality metrics for evaluation ofreconstructed images

To provide a quantitative evaluation for the synthetic sceneexperiments, we consider metrics that directly use theground truth image. First idea could be signal-to-noise ratio(SNR) or mean squared error, which are defined as [26]

MSE = 1

N 2‖| f | − | f |‖2

2

SNR(dB) = 10 log10

s2| f |

MSE

( )(36)

where | f | and | f | are true and reconstructed magnitudeimages, and s2

| f | is the variance of the true magnitudeimage. Another idea could be performing an adaptivethreshold on the images to separate the target and thebackground regions and then counting the number of


matched pixels in the binary results [27]. This is a kind ofsegmentation metric, and could be viewed as a targetlocalisation metric (TLM).

For experiments with AFRL backhoe data we do not havethe ground truth, however we have a rich data of differentbandwidths. We can produce a reference image using thehighest bandwidths and use it for evaluation of lowerbandwidth results. To do this we apply the adaptivethresholding on the highest bandwidth (2 GHz)conventional reconstruction, which is the most reasonableand fair one for evaluating our method, to obtain theapproximate target region. We then use the 2 GHzconventional image in this region as the ground truth forlower bandwidth reconstructions. Therefore we can computethe above-mentioned metrics for backhoe data too.

We can also use the following quality metrics that are usedin the literature to evaluate the quality of reconstructed imagesof unknown scenes. For these metrics we consider a rectanglesurrounding the target as the target region (T ) and out of it asthe background region (B). We can limit the backgroundregion to a region with the same area as the target region [28].

1. Target-to-background ratio (TBR) [27, 29]: as a measureof accentuation of the target pixels with respect to thebackground which is defined as

TBR = 20 log10

maxi[T (|( f )i|)(1/NB)

∑j[B |(f )j|

( )(37)

where NB denotes the number of pixels in the backgroundregion B.2. Mainlobe width (MLW) [27]: as a measure of the effectiveresolution. We obtain an estimate of it by averaging the 3-dBlobe width of the strong scatterers. In each row and column inthe target region, we find the nearest point below 3 dB of themaximum value. A better estimate of the 3-dB distance canthen be estimated using a linear interpolation betweenpixels. Finally, the distances obtained from each row andcolumn are averaged to find an overall estimate of the 3-dBlobe width [27].3. Entropy of the full image (ENT) [30]: as a measure ofimage sharpness. A sharper image has a smaller entropyvalue. Let p(i) denote the normalised frequency ofoccurrence of each gray level, that is, the pixel intensityhistogram of the image.

ENT = −∑G

i=0

p(i) log2(p(i)) (38)

where G is the number of levels in the histogram.4. Target to background entropy difference (TBED) [28]: asa measure of extractability of the target from its background.It is defined as the absolute difference between the entropy oftarget region and the entropy of background region,normalised to the entropy of the full image. This measure,when low, indicates that the target and background regionhave very similar entropy levels, making the target moredifficult to extract from its local background. Clark andVelten [28] have examined nine different image qualitymeasures and found that this measure is significantly relatedto the performance of an automatic target recognition algorithm.

The above metrics have been exploited to evaluate the resultsof the experiments in the following subsections.

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3.3 Synthetic scene reconstruction

Experiment 1: First we show an important capability of thismethod that is super-resolution, which means that it canreconstruct image details under bandwidth limitations. Todemonstrate this property, we apply our method on asynthetic scene composed of eight point scatterers with unitreflectivity magnitude and random (uniform) uncorrelatedphase. The magnitude field for this scene is shown inFig. 1a. We set our system parameters so that the systemresolution is twice the image pixel size, so we seek super-resolution reconstructions. Fig. 1b shows conventionalspotlight mode SAR reconstruction using the PFA [1] thatcannot resolve closely spaced scatterers and suffers fromhigh sidelobes. Fig. 1c shows the result of non-quadraticregularisation reconstruction method [2] by usingregularisers for both point and region feature enhancement.Note that this method also has super-resolution capabilitywhen we just use the point enhancement regulariser.However, as we usually do not have enough priorknowledge about the scene, here we consider the scene maycontain both point and region-based features. As it is seen,this technique fails to reconstruct the scene accurately.

Fig. 1d shows the result of the proposed SR method inwhich we observe that the reconstructed image is very closeto the true image. To be comparable with the result of non-quadratic regularisation method, here we have exploited anovercomplete shape-based (SB) dictionary that consists ofpoints as well as squares of various sizes at every possiblelocation in the scene. Note that such a dictionary can beused to sparsely represent many scenes containing point-like targets as well as smooth regions.Experiment 2: To demonstrate the capabilities of the proposedframework and contrast it with existing methods, in thisexperiment we consider a more general synthetic scenecomposed of point scatterers as well as a smooth distributedregion as shown in Fig. 2a. The image consists of 32 × 32complex-valued pixels, and we show only the magnitude

Fig. 1 Super-resolving a scene with isolated point scatterers

a True sceneb Conventional reconstructionc Point-region-enhanced non-quadratic regularisationd SR-based reconstruction

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field in Fig. 2a. Phase is randomly distributed with auniform density function in [2p, p].

The conventional image reconstruction based on PFA isshown in Fig. 2b, which is a poor result. Fig. 2c shows thenon-quadratic regularisation method with regularisers forboth point and region enhancement. Note that this methodcannot enhance both types of features simultaneously in thereconstructed image due to applying two contraryregularisers on the whole image. Whether the points or theregions are better preserved is a trade-off that could beadjusted through the regularisation parameters.

Fig. 2d through Fig. 2f show the reconstructed images withthe proposed SR method using dictionaries described inSubsections 2.5.1 and 2.5.2. The SB dictionary here, whichconsists of points as well as squares of various sizes atevery possible location of the scene, has a good SR for themagnitude of this synthetic image, so the reconstructedimage with this dictionary in Fig. 2d is very close to theperfect reconstruction. Note that for better comparison of allresults, we have not shown the values below 50 dB of themaximum value of the image.

The multi-resolution wavelet dictionary is a much moregeneral dictionary that can be used for unknown complicatedscenes. Here we have used the Haar wavelet. Its result,which is shown in Fig. 2e, shows an interesting andrelatively good agreement with the true scene. As this figureshows, it seems that using wavelet dictionary alone in theproposed framework is not so powerful to reconstruct thepoint scatterers. To overcome this problem, we propose touse an overcomplete dictionary composed of point dictionary(i.e. spikes) and the wavelet. The result obtained by usingthis overcomplete dictionary is shown in Fig. 2f, where itclearly represents both the smooth region and point targets.Consider that while the SB dictionary appears to be verygood in terms of reconstruction quality, computationally it isthe most demanding one as it is based on a highly redundantovercomplete dictionary. Evaluation results using qualitymetrics defined in Section 3.2 are shown in Table 1, whichdemonstrate the superiority of the proposed SR-based methods.Experiment 3: To be more realistic, in this experiment we usea synthetic image, constructed from the MIT LincolnLaboratory Advanced Detection Technology Sensor

Fig. 2 Synthetic scene reconstruction

a Synthetic sceneb Conventional reconstructionc Point-region-enhanced non-quadratic regularisationd– f SR based reconstructionsd SB dictionarye Wavelet dictionaryf Spike-wavelet overcomplete dictionary


IET Radar Sonar Ndoi: 10.1049/iet-rsn

www.ietdl.org

Table 1 Evaluation results of experiments 2 and 3

Conventional Non-quadratic SR-SB SR-wavelet SR-spkwav

Experiment 2 SNR, dB 11.50 16.22 27.76 22.07 27.95

TLM, % 85.25 92.57 98.14 96.87 99.70

Experiment 3 SNR, dB 13.87 23.01 29.72 28.31 30.03

TLM, % 78.22 98.82 99.80 99.31 100

(ADTS) data set [31] by segmentation techniques, as well asaddition of some point scatterers (with random phase), asshown in Fig. 3a. The conventional and non-quadraticregularisation reconstructions are shown in Figs. 3b and c,respectively. Fig. 3d through Fig. 3f show the reconstructedimages with the SR method with different dictionaries. Wehave used the same SB dictionary described in the previousexperiment. Because of non-zero background and arbitrarydistributed regions, the representation of this image is not assparse as the one in the previous synthetic scene; however,the resultant reconstructed image still is in good agreementwith the true image. We can see in Fig. 3f that for thismore realistic scene, the overcomplete spike-waveletdictionary has excellent result. We have used the Haarwavelet for this experiment. Evaluation results usingpreviously defined quality metrics are depicted in Table 1,which show the improvements achieved using the proposedSR-based methods.

3.4 AFRL backhoe data

Experiment 4: We now present our experimental results basedon the AFRL backhoe data [32], which is a wideband, fullpolarisation, complex-valued backscattered data from abackhoe vehicle (shown in Fig. 4) in free space. In ourexperiment, we use VV polarisation data centred at 10 GHzwith three available bandwidths of 500 MHz, 1 GHz and2 GHz, with an azimuthal span of 1108 (centred at 458).Fig. 5a shows the conventional composite reconstructionsof these data as described in [33] for bandwidths of500 MHz, 1 GHz and 2 GHz. We obtain the compositeimages through combination of 19 subapertures, in each of

Fig. 3 Synthetic ADTS scene reconstruction

a Synthetic sceneb Conventional reconstructionc Point-region-enhanced non-quadratic regularisationd– f SR based reconstructionsd SB dictionarye Wavelet dictionaryf Spike-wavelet overcomplete dictionary

avig., 2011, Vol. 5, Iss. 2, pp. 182–193.2009.0235

which we use the spotlight-mode SAR formulation. Also, apoint-enhanced composite reconstruction method usingnon-quadratic regularisation has been recently presentedin [5] for backhoe data. Fig. 5b shows the reconstructedimages based on this technique. Fig. 5c shows the resultsbased on our proposed method using the wavelet dictionary.We should expect to obtain the sparsest representation ofthe important/interested features (according to the selecteddictionary) in the reconstructed image. For bettercomparison, in all results of Fig. 5, the values below 55 dBof the image maximum value are not shown. As it isexpected, for this complicated scene using the waveletdictionary, the SR method produces very good results withvery little artefacts. Also based on our knowledge of theshape and structure of the scene, there are very little falsetargets/features out of the expected zone of the backhoevehicle. We have used the Daubechies 2 (db2) wavelet forthis experiment. Quantitative evaluation results depicted inTables 2 and 3 demonstrate the improved quality ofSR-based reconstructions in terms of all quality metricsexcept the MLW, for which the non-quadratic regularisationmethod is the best with the proposed SR-based methodfollowing close behind. Note that the point-enhanced non-quadratic method is optimised for enhancing spike-liketargets and therefore it is expected to have better MLW.Experiment 5: Finally, in the last experiment we show theimproved robustness of the SR method in data limitationscenarios. One important data limitation scenario that mayoccur in many SAR applications is frequency bandomission that may be encountered in several situations suchas jamming and data dropouts or in VHF/UHF frequencyband systems such as foliage penetration radar in which itis likely that we will not be able to use an uninterruptedfrequency band [5]. In this experiment for each of the threeavailable bandwidths of backhoe data, we consider the caseof frequency-band omissions where 20% of the spectraldata within that bandwidth are available. We have selectedthe available band randomly, with a preference forcontiguous bands (expressed through a parameter used inrandom band generation). The corresponding results for this

Fig. 4 Backhoe vehicle model

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Fig. 5 Results with the AFRL Backhoe data of 2 GHz, 1 GHz and 500 MHz bandwidth

a Conventional composite reconstructionb Point-enhanced (non-quadratic regularisation) composite reconstructionc SR-based (with wavelet dictionary) composite reconstruction

Table 2 Evaluation results of experiments 4 and 5 (true scene-dependent metrics)

1 GHz 500 MHz

Conventional Non-quadratic SR-wav Conventional Non-quadratic SR-wav

Experiment 4 SNR, dB 28.01 28.65 31.33 28.19 28.55 30.87

TLM, % 80.71 85.11 91.02 80.88 85.89 91.01

Experiment 5 SNR, dB 26.27 27.39 30.34 26.03 26.50 29.87

TLM, % 77.76 86.36 89.24 77.60 84.90 89.85

Table 3 Evaluation results of experiments 4 and 5 (unknown scene metrics)

2 GHz 1 GHz 500 MHz

Conventional Non-quadratic SR-wav Conventional Non-quadratic SR-wav Conventional Non-quadratic SR-wav

Experiment 4 TBR, dB 50.01 55.31 71.99 43.07 47.42 60.47 42.47 47.86 58.42

MLW, m 0.061 0.016 0.019 0.141 0.034 0.040 0.293 0.074 0.082

ENT 1.485 1.003 0.475 2.324 1.789 1.026 2.138 1.865 1.140

TBED 1.633 2.546 3.890 0.930 1.735 3.362 0.973 1.779 3.230

Experiment 5 TBR, dB 36.48 45.88 60.04 29.49 40.88 55.87 27.66 33.36 51.55

MLW, m 0.099 0.028 0.020 0.185 0.040 0.040 0.456 0.091 0.084

ENT 3.250 2.118 0.804 4.048 2.690 1.271 4.403 3.657 1.323

TBED 0.576 1.396 3.445 0.337 1.244 2.979 0.287 0.814 2.877

190 IET Radar Sonar Navig., 2011, Vol. 5, Iss. 2, pp. 182–193

& The Institution of Engineering and Technology 2011 doi: 10.1049/iet-rsn.2009.0235

www.ietdl.org

Fig. 6 Results with the AFRL Backhoe data of 2 GHz, 1 GHz, and 500 MHz bandwidth with frequency band omissions (20% of the full banddata available)

a Conventional composite reconstructionb Point-enhanced (non-quadratic regularisation) composite reconstructionc SR-based (with wavelet dictionary) composite reconstruction

case are shown in Fig. 6. In these results the top 50 dB part ofall images are shown for better comparison. Robustness of theproposed SR framework to bandwidth limitations as well asfrequency band data limitations is clearly revealed in theseresults. In Tables 2 and 3 computed quality metrics aredepicted, which show the improvements provided by theproposed SR-based reconstructions in terms of all qualitymetrics. It is interesting that here MLW of SR-based arebetter than that of non-quadratic method, which shows itsexcellent robustness to data limitation scenarios.

4 Conclusions

In this paper we have proposed a new approach for SARimage formation based on sparse signal representation.Owing to the complex-valued nature of the SARreflectivities, we have designed our approach to sparselyrepresent the magnitude of the complex-valued scatteredfield in terms of the features of interest. We haveformulated the mathematical framework for this method andproposed an iterative algorithm to solve the correspondingjoint optimisation problem over the representation ofmagnitude and phase of the underlying field reflectivities.In various experiments, we have demonstrated theperformance of this approach that produces high-qualitySAR images with enhanced features and very littleartefacts, which is ideal for automatic recognition tasks.


Selection of the dictionary depends on the application. ForSAR images of natural scenes wavelet dictionary seems tobe a good choice and for images of man-made targets withknown-shape parts, SB dictionaries could result in betterreconstructions than the standard dictionaries at the expenseof higher computational load. Also as we havedemonstrated, this approach exhibits interesting featuresincluding super-resolution, and robustness to bandwidthlimitations as well as to uncertain and limited data. Thus, itcould be a good choice in such scenarios. In additionto these characteristics, the proposed approach hasthe potential to provide enhanced image quality innon-conventional data collection scenarios, for example,those involving sparse apertures.

5 Acknowledgment

This work was partially supported by an Iranian Ministry ofScience, Research and Technology Scholarship, and by theScientific and Technological Research Council of Turkeyunder Grant 10SE090, and by a Turkish Academy ofSciences Distinguished Young Scientist Award.

6 References

1 Carrara, W.G., Goodman, R.S., Majewski, R.M.: ‘Spotlight syntheticaperture radar: signal processing algorithms’ (Artech House, Boston,MA, 1995)

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2 Cetin, M., Karl, W.C.: ‘Feature-enhanced synthetic aperture radar imageformation based on nonquadratic regularization’, IEEE Trans. ImageProcess., 2001, 10, (4), pp. 623–631

3 Munson, Jr D.C., O’Brien, J.D., Jenkins, W.K.: ‘A tomographicformulation of spotlight-mode synthetic aperture radar’, Proc. IEEE,1983, 71, pp. 917–925

4 Kak, A.C., Slaney, M.: ‘Principles of computerized tomographicimaging’ (IEEE Press, New York, 1988)

5 Cetin, M., Moses, R.L.: ‘SAR imaging from partial-aperture data withfrequency-band omissions’. SPIE Defense and Security Symposium,Algorithms for Synthetic Aperture Radar Imagery XII, Orlando, FL,USA, March 2005, pp. 32–43

6 Chen, S.S., Donoho, D.L., Saunders, M.A.: ‘Atomic decomposition bybasis pursuit’, SIAM J. Sci. Comput., 1998, 20, pp. 33–61

7 Donoho, D.L., Huo, X.: ‘Uncertainty principles and ideal atomicdecomposition’, IEEE Trans. Inf. Theory, 2001, 47, (7), pp. 2845–2862

8 Donoho, D.L., Elad, M.: ‘Maximal sparsity representation via l1minimization’, Proc. Natl. Acad. Sci., 2003, 100, pp. 2197–2202

9 Elad, M., Bruckstein, A.: ‘A generalized uncertainty principle and sparserepresentation in pairs of bases’, IEEE Trans. Inf. Theory, 2002, 48, (9),pp. 2558–2567

10 Chartrand, R.: ‘Exact reconstruction of sparse signals via nonconvexminimization’, IEEE Signal Process. Lett., 2007, 14, (10), pp. 707–710

11 Malioutov, D.M., Cetin, M., Willsky, A.S.: ‘Optimal sparserepresentations in general overcomplete bases’. IEEE Int. Conf.Acoustics, Speech, and Signal Processing (ICASSP), May 2004,pp. 793–796

12 Kay, S.M.: ‘Fundamentals of statistical signal processing: estimationtheory’ (Prentice-Hall, Englewood Cliffs, NJ, 1993)

13 Borden, B.: ‘Maximum entropy regularization in inverse syntheticaperture radar imagery’, IEEE Trans. Signal Process., 1992, 40, (4),pp. 969–973

14 Mensa, D.L.: ‘High resolution radar imaging’ (Artech House, Dedham,MA, 1981)

15 Munson, Jr D.C., Sanz, J.L.C.: ‘Image reconstruction from frequency-offset Fourier data’, Proc. IEEE, 1984, 72, pp. 661–669

16 Vogel, C.R., Oman, M.E.: ‘Iterative methods for total variationdenoising’, SIAM J. Sci. Comput., 1996, 17, (1), pp. 227–238

17 Golub, G.H., Van Loan, C.F.: ‘Matrix computations’ (The JohnsHopkins University Press, Baltimore, MD, 1996)

18 Cetin, M., Karl, W.C., Willsky, A.S.: ‘Feature-preserving regularizationmethod for complex-valued inverse problems with application tocoherent imaging’, Opt. Eng., 2006, 45, (1), p. 017003

19 Geman, D., Yang, C.: ‘Nonlinear image recovery with half-quadraticregularization’, IEEE Trans. Image Process., 1995, 4, (7), pp. 932–946

20 Starck, J.L., Elad, M., Donoho, D.L.: ‘Image decomposition via thecombination of sparse representations and a variational approach’,IEEE Trans. Image Process., 2005, 14, (10), pp. 1570–1582

21 Donoho, D.L., Johnstone, I.: ‘Ideal spatial adaptation via waveletshrinkage’, Biometrika, 1994, 81, pp. 425–455

22 Zeng, Z., Cumming, I.G.: ‘SAR image data compression using a tree-structured wavelet transform’, IEEE Trans. Geosci. Remote Sens.,2001, 39, (3), pp. 546–552

23 Rilling, G., Davies, M., Mulgrew, B.: ‘Compressed sensing basedcompression of SAR raw data’. Signal Processing with Adaptive SparseStructured Representations Workshop, Saint-Malo, France, April 2009

24 Starck, J.-L., Candes, E., Donoho, D.L.: ‘The curvelet transform forimage denoising’, IEEE Trans. Image Process., 2002, 11, (6),pp. 131–141

25 Batu, O., Cetin, M.: ‘Hyper-parameter selection in non-quadraticregularization-based radar image formation’. SPIE Defense andSecurity Symp., Algorithms for Synthetic Aperture Radar ImageryXV, Orlando, FL, USA, March 2008

26 Argenti, F., Alparone, L.: ‘Speckle removal from SAR images in theundecimated wavelet domain’, IEEE Trans. Geosci. Remote Sens.,2002, 40, (11), pp. 2363–2374

27 Cetin, M., Karl, W.C., Castanon, D.A.: ‘Feature enhancement and ATRperformance using non-quadratic optimization-based SAR imaging’,IEEE Trans. Aerospace Electr. Syst., 2003, 39, (4), pp. 1375–1395

28 Clark, L.G., Velten, W.J.: ‘Image characterization for automatic targetrecognition algorithm evaluations’, Opt. Eng., 1991, 30, (2),pp. 147–153

29 Benitz, G.R.: ‘High-definition vector imaging’, Lincoln Lab. J., 1997,10, (2), pp. 147–170

30 Wang, J., Liu, X.: ‘SAR minimum-entropy autofocus using an adaptive-order polynomial model’, IEEE Trans. Geosci. Remote Sens., 2006, 3,(4), pp. 512–516

31 Air Force Research Laboratory, Model Based Vision Laboratory, SensorData Management System ADTS: http://www.mbvlab.wpafb.af.mil/public/sdms/datasets/adts/

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32 Backhoe Data Sample & Visual-D Challenge Problem, Air ForceResearch Laboratory, Sensor Data Management System: https://www.sdms.afrl.af.mil/main.htm

33 Moses, R.L., Potter, L.C., Cetin, M.: ‘Wide angle SAR imaging’. Proc.SPIE, Algorithms for Synthetic Aperture Radar Imagery XI, Orlando,FL, USA, April 2004, pp. 164–175

34 Papoulis, A.: ‘Probability, random variables, and stochastic processes’(McGraw-Hill, 1991, 3rd edn.)

7 Appendix 1

7.1 Derivation of the PDF of b

Consider the complex random variable(b)i = ui + j vi = ri ejfi . Here, we know that ri ¼ 1independent of fi, and fi possesses a uniform pdf over|fi| , p which we denote it by U(fi). The pdf of thecomplex-valued random variable (b)i can be defined as [34]

p((b)i) = p(ui, vi) =p(ri, fi)

|J (ri, fi)|(39)

where the Jacobian J(ri,fi) ¼ ri. Since ri and fi areindependent

p(ri, fi) = p(ri)p(fi) = d(ri − 1)U (fi) (40)

Thus, p((b)i) can be written as

p((b)i) = p(ui, vi) =p(ri, fi)

|J (ri, fi)|= 1

ri

d(ri − 1)U (fi)

= d(ri − 1)U (fi) = d(|(b)i| − 1)U (/(b)i) (41)

Since |(b)i| ¼ 1, ∀ i independent of each other and the phaseterms, and also since all fi random variables are assumed tobe independent, (12) follows.

8 Appendix 2

Let us differentiate Ka(a, b) with respect to bi and set it tozero. We obtain the following relationship for the optimum(minimising) b

bi =p

2[|(a)i|2 + 1]1−p/2 (42)

Substituting (42) into (24) we obtain

infb

Ka(a, b) = ‖y−HPFa‖22 +l

∑N2

i=1

[|(a)i|2 + 1]p/2 = J (a)

(43)

9 Appendix 3

Let us define the sequence Kn = Ka(a(n), b(n+1)

) and showthat it is convergent. From (25) and (26) we can deduce that

Ka(a(n), b(n+1)

) ≤ Ka(a(n), b(n)

) ∀n (44)

Ka(a(n+1), b(n+1)

) ≤ Ka(a(n), b(n+1)

) ∀n (45)


www.ietdl.org

The difference Kn 2 Kn21 can be written as

Kn − Kn−1 = [Ka(a(n), b(n+1)

) − Ka(a(n), b(n)

)]

+ [Ka(a(n), b(n)

) − Ka(a(n−1), b(n)

)] (46)

Using (44) and (43) we obtain

Kn − Kn−1 ≤ 0 ∀n (47)

which means that the sequence Kn is decreasing. Since it isbounded below and decreasing, the sequence converges.Hence the algorithm is convergent in terms of the costfunctional.

10 Appendix 4

We want to find s that minimises Kb(b, s) of (31). The portionof Kb(b, s) that depends on s is its second term, as follows

∑N2

i=1

|(Sb)i − 1|2 (48)

|(Sb)i − 1|2 = | exp(−jsi)(b)i − 1|2

= |(b)i|2 + 1 − 2<[|(b)i| exp( j{f[(b)i] − si}]

(49)

where f[(b)i] denotes the phase of the complex number (b)i.The sum in (48) takes its minimum value when the terminside the bracket in (49) has a zero imaginary part for all i.Therefore the s that minimises the sum in (48) and hence(31) satisfies

si = f[(b)i] ∀i (50)


With this s, we have Sb ¼ |b|, and hence

infs

Kb(b, s) = ‖y − HBb‖22 + l′

∑N2

i=1

(|(b)i| − 1)2 = J (b)

(51)

11 Appendix 5

Considering that B ¼ diag{Fa} and P ¼ diag{b}, the costfunctions of both (8) and (14) are actually functions of aand b. Note that the first part (ℓ2-norm data dependentterm) of both cost functions are equal for given a and b, sowe denote the data term by J1(a, b). Based on that, let usdefine a total cost function as below

Jt(a, b) = J1(a, b) + l‖a‖pp + l′

∑N2

i=1

(|(b)i| − 1)2 (52)

Note that finding the minimising a for Jt(a, b) reduces to theoptimisation problem in (8), and finding the minimising b forJt(a, b) reduces to the optimisation problem in (14). Thus, theoverall image reconstruction algorithm of Section 2.2 isactually a block coordinate descent approach as follows

a(l+1) = arg mina

Jt(a, b(l)

) (53)

b(l+1) = arg min

bJt(a

(l+1), b) (54)

where l denotes the iteration number of the overall algorithm.Using (53) and (54), convergence of the overall algorithm interms of cost functional can be easily established in a similarway to the one in Appendix 3.

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