a multiple measurement vector approach to synthetic aperture radar...

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SIAM J. IMAGING SCIENCES

c� 2018 Society for Industrial and Applied Mathematics

Vol. 11, No. 1, pp. 770–801

A Multiple Measurement Vector Approach to Synthetic ApertureRadar Imaging⇤

Liliana Borcea

† and Ilker Kocyigit

†

Abstract. We study a multiple measurement vector (MMV) approach to synthetic aperture radar (SAR) imag-

ing of scenes with direction-dependent reflectivity and with polarization diverse measurements. The

unknown reflectivity is represented by a matrix with row support corresponding to the location

of the scatterers in the scene and columns corresponding to measurements gathered from di↵erent

subapertures or di↵erent polarization of the waves. The MMV methodology is used to estimate the

reflectivity matrix by inverting in an appropriate sense the linear system of equations that models the

SAR data. We introduce a resolution analysis of imaging with MMV, which takes into account the

sparsity of the imaging scene, the separation of the scatterers, and the diversity of the measurements.

The results of the analysis are illustrated with some numerical simulations.

Key words. synthetic aperture radar imaging, convex optimization, multiple measurement vector, simultane-

ously sparse

AMS subject classifications. 35Q93, 58J90, 45Q05

DOI. 10.1137/17M1142065

1. Introduction. Sparsity-promoting optimization [26, 25, 22, 9, 11, 10, 12] is an impor-tant methodology for imaging applications where scenes that are sparse in some representationcan be reconstructed with high resolution. There is a large body of literature on this topic insynthetic aperture radar (SAR) imaging [4, 32, 28], sensor array imaging [13, 14, 7], medicalimaging [30], astronomy [6], geophysics [33], and so on.

We are interested in the application of SAR imaging, where a transmit-receive antennaon a moving platform probes an imaging scene with waves and records the scattered returns[20, 17]. This is a particular inverse problem for the wave equation, where the waves propagatethrough a homogeneous medium, back and forth between the platform and the imaging scene,and the unknown is modeled as a two-dimensional reflectivity function of location on a knownimaging surface. Most SAR imaging is based on a linear model of the data, where the unknownreflectivity is represented by a collection of independent point scatterers [17]. The image isthen formed by inverting approximately this linear relation, using filtered backprojectionor matched filtering [17], also known as Kirchho↵ migration [5]. Such imaging is popularbecause it is robust to noise, it is simple, and it works well when the linear model is a goodapproximation of the data. However, the resolution is limited by the extent of the aperture,

⇤Received by the editors August 4, 2017; accepted for publication (in revised form) November 1, 2017; publishedelectronically March 8, 2018.

http://www.siam.org/journals/siims/11-1/M114206.htmlFunding: The work of the authors was supported by the Air Force O�ce of Scientific Research under award

FA9550-15-1-0118.†Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043 ([email protected],

[email protected]).

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MMV FOR SAR IMAGING 771

the frequency, and the bandwidth of the probing signals emitted by the moving platform[20, 17]. The promise of sparsity-promoting optimization is that these resolution limits canbe overcome when the unknown reflectivity has sparse support [4, 32, 28].

The modeling of the reflectivity as a collection of points that scatter the waves isotropicallymay lead to image artifacts. It is known that even if the scatterers are small, so that theirsupport may be represented by a point and the single scattering approximation (i.e., the lineardata model) can be used, their reflectivity may depend on the frequency and the direction ofillumination [2, Chapters 3, 5]. Moreover, the scatterers have an e↵ective polarization tensorthat describes their response to di↵erent polarizations of the probing electromagnetic waves[2, 3]. Thus, the reflectivity function depends on more variables than assumed in conventionalSAR, and the resulting images may be worse than expected. For example, a scatterer thatreflects only within a narrow cone of incident angles cannot be sensed over most of the syntheticaperture, so its reconstruction with filtered backprojection will have low resolution. Directapplication of sparse optimization methods does not give good results either because of thelarge systematic error in the linear data model that assumes a scalar, constant reflectivityover the entire aperture.

SAR imaging of frequency-dependent reflectivities has been studied in [16, 35, 34], usingeither Doppler e↵ects or data segmentation over frequency subbands. Data segmentationis a natural idea for imaging both frequency- and direction-dependent reflectivities that areregular enough so that they can be approximated as piecewise constant functions over properlychosen frequency subbands and cones of angles of incidence (i.e., subapertures). Images canbe obtained separately from each data set, but the question is how to fuse the informationto achieve better resolution. The study in [8] uses the multiple measurement vector (MMV)methodology [31, 15, 39], also known as simultaneously sparse approximation [38, 37], for thispurpose. The MMV framework fits here because the reflectivity is supported at the samelocations in the imaging scene, for each data set. In the discrete setting, this means that theunknown is represented by a matrix X with row support corresponding to the pixels in theimage that contain scatterers and with columns corresponding to the di↵erent values of thereflectivity for each frequency band, subaperture, and polarization.

The goal of this paper is twofold. First, we introduce a novel resolution theory of imagingwith MMV that applies to a general linear system. We do not pursue the usual question ofexact recovery of the unknown matrixX, which requires stringent assumptions on the imagingscene that are unlikely to hold in practice. Instead, we estimate the neighborhood of the rowsupport of X that contains the largest entries of the MMV reconstruction. The size of thisneighborhood plays the role of resolution limit, and we quantify its dependence on the sparsityof the imaging scene, the separation between the scatterers, the diversity of the data set, andthe noise level. The second goal of the paper is to explain how the theory applies to SARimaging. The study [8] is proof of concept that MMV can be used to image direction-dependentreflectivities from data gathered over multiple subapertures. However, it does not provide aresolution analysis, and it does not demonstrate the advantage of using MMV over imagingwith a single subaperture at a time. In this paper, we quantify the improvement brought bythe MMV approach and assess the results of the resolution theory for the application of SARimaging to both direction- and polarization-dependent reflectivities.

The paper is organized as follows. We begin in section 2 with the theoretical results,stated for a general linear system with unknown matrix X. The application of SAR imagingD

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772 LILIANA BORCEA AND ILKER KOCYIGIT

is discussed in sections 3 and 4. The proofs of the results are in section 5. We end with asummary in section 6.

2. Theory. We state here our main results on the resolution of imaging with MMV. Webegin in section 2.1 with a brief discussion on MMV and then give the results in section 2.2.

We use henceforth the following notation convention. Bold uppercase letters, as in X 2CNy⇥N

v , denote matrices, and bold lowercase letters denote vectors. We also use an arrowindex, as in x

j! 2 C1⇥N

v , to distinguish the rows of X from its column vectors denoted byx

j

2 CNy⇥1.

2.1. Preliminaries. Consider a general linear model of a data matrix D 2 CNr⇥N

v ,

(2.1) GX = D,

where the unknown matrix X 2 CNy⇥N

v is mapped to D by a given sensing matrix G 2CNr⇥Ny . In the context of SAR imaging, X is the unknown reflectivity discretized1 at N

y

points {yj

}1jNy in the imaging region ⌦, a bounded set on a known surface. The matrixD is an aggregate of N

v

data sets or views, each consisting of Nr

measurements of the waveat the moving radar antenna. The column x

v

of X is the reflectivity for the vth view, andthe sensing matrix G is the discretization of the kernel of the integral operator that definesthe single scattering approximation of the wave, as described in section 3.

Denote by S ⇢ {1, . . . , Ny

} the set of indexes of the nonzero rows of X, and supposethat its cardinality |S| is small with respect to N

y

. We call S the row support of X and let⌦S = {y

q

, q 2 S} be the set of associated locations in ⌦.When N

v

= 1, the linear model (2.1) corresponds to the single measurement vector (SMV)problem,

(2.2) Gx = d,

with unknown vector x 2 CNy⇥1 and data vector d 2 CNr⇥1, where we dropped the columnindex 1. This problem has been studied extensively in the context of compressed sensing[26, 25, 22, 9, 11, 10, 12, 29] for the undetermined case N

r

⌧ Ny

. In particular, it is known[23, Corollary 1] that if

(2.3) kxk0 = |S| < spark(G)/2,

where spark(G) is the smallest number of linearly dependent columns of G, then (2.2) hasa unique solution satisfying (2.3), given by the minimizer of the combinatorial optimizationproblem

(2.4) minimize kzk0 subject to Gz = d.

The norm kzk0 equals the number of nonzero entries in z.

1We assume that the Ny points define a fine mesh in ⌦, so we can neglect errors due to scatterer locations

o↵ the mesh.

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This result is generalized in [15, Theorem 2.4] to the MMV problem (2.1) for Nv

> 1. Itstates that when the number of nonzero rows in X, denoted by kXk0, satisfies

(2.5) kXk0 <⇥spark(G) + rank(D)� 1

⇤/2,

the linear system (2.1) has a unique solution satisfying (2.5), given by the minimizer of

(2.6) minimize kZk0 subject to GZ = D.

Thus, if the di↵erent data sets bring new information, so that D has large rank, the MMVproblem is uniquely solvable for less stringent conditions on the row support of X, i.e., forless sparse imaging scenes.

The combinatorial problems (2.4) and (2.6) are not computationally tractable, so they arereplaced by convex relaxations. The minimizer of the convex problem

(2.7) P1 : minimize kzk1 subject to Gz = d,

where k · k1 is the `1 norm, is known to give the exact solution x of (2.2) under variousconditions satisfied by G and x, like the null space property [18], the restricted isometryproperty [10], conditions based on the mutual coherence [24], and the cumulative coherence[36]. Relaxations of (2.6) of the form

(2.8) P1,q : minimize kZk1,q subject to GZ = D, where kZk1,q =NyX

j=1

kzj!k

q

,

are studied in [19, 31, 15, 27, 38, 37, 39]. Conditions of recoverability of X by the minimizerof (2.8) are established in [15, Theorem 3.1] and [38, Theorem 5.1]. However, there are noconclusive results that demonstrate the advantage of the MMV formulation over the SMV onein the convex relaxation form as discussed, for example, in [15, section D], [38, section 5.2]and [39, section 3.2].

These studies make no assumption on the structure of the unknown X, except for sparsityof its row support S, and do not address the case of more general imaging scenes where exactreconstructions of X may not be achieved. Our resolution theory quantifies the error of thereconstruction based on the separation between the points in ⌦S , the correlation of the rowsof X, and the noise level. We show in particular that if X has uncorrelated rows, the MMVformulation may have an advantage over SMV. This is relevant to SAR imaging, as explainedin section 3.

2.2. Resolution theory. Let us consider the modification of the linear system (2.1)

(2.9) D

W

= GX +W ,

which accounts for data D

W

2 CNr⇥N

v contaminated by the noise matrix W 2 CNr⇥N

v . Weestimate X by the minimizer X" of the convex problem

(2.10) P"

1,2 : minimize kZk1,2 subject to kGZ �D

W

kF

",

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where k · kF

is the Frobenius norm and " is a chosen tolerance, satisfying

(2.11) kW kF

< ".

Our goal is to quantify the approximation of X by X

" by taking into account the sep-aration of the points in ⌦S and the correlation of the rows of X. These determine how theunknowns interact with each other, as described by the X dependent “multiple view inter-action coe�cient” I

N

v

defined in section 2.2.1. The smaller IN

v

is, the better the imagingresults are, as stated by the estimates in sections 2.2.2–2.2.4. We also study in section 2.2.5the case of clusters of points in ⌦S , where I

N

v

is large and the previous estimates are notuseful. We introduce a new interaction coe�cient for the cluster, which is much smaller thanI

N

v

, and show that when this is small, the MMV reconstruction is supported in the vicinityof ⌦S .

2.2.1. The multiple view interaction coe�cient. The interaction between the unknownsis quantified by the X dependent multiple view interaction coe�cient defined by

(2.12) IN

v

= max1jNy

supv!2C1⇥N

v

X

q2S\{n(j)}

|µ(gj

, gq

)||µ(v!,xq!)|,

using the correlation of the columns of G,

(2.13) µ(gj

, gq

) = hgj

, gq

i , 1 j, q Ny

,

where hgj

, gq

i = g

?

j

g

q

is the Hermitian inner product and ? denotes complex conjugate andtranspose. These columns are normalized, so that

(2.14) kgj

k2 = hgj

, gj

i1/2 = 1, 1 j Ny

,

and we suppose that

(2.15) |µ(gj

, gq

)| < 1 8 j 6= q, 1 j, q Ny

.

This assumption holds in the SAR imaging application, and it allows us to quantify thedistance between the points using the semimetric

(2.16) D : {1, . . . , Ny

}⇥ {1, . . . , Ny

} ! [0, 1], D(j, q) = 1� |µ(gj

, gq

)|.

We will see in section 3 that |µ(gj

, gq

)| is approximately a function of yj

� y

q

, which peaksat the origin, i.e., for y

j

= y

q

, and decreases monotonically in the vicinity of the peak. Thus,points at small distance with respect to D are also close in the Euclidian distance.

We use the semimetricD in definition (2.12) to select the closest point to y

j

in ⌦S , indexedby n(j) 2 S. If this point is not unique, we just pick one and let n(j) be its index. In anabuse of notation, we also let µ(·, ·) be the correlation of the rows of X with v!, defined by

(2.17) µ(v!,xq!) =

hv!,xq!i

kv!k2kxq!k2,

where hv!,xq!i = v!x

?

q! is the Hermitian inner product of row vectors and k · k2 is theinduced `2 norm.

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Note that (2.17) has absolute value equal to 1 in the SMV setting, where Nv

= 1 and v!and x

k! are complex numbers. Then (2.12) reduces to the single view interaction coe�cientI1 used in [7, section 4] to quantify the quality of imaging with `1 optimization. As shown in[7], I1 is small if the points in ⌦S are su�ciently far apart. Here we consider N

v

> 1 and notethat since |µ(v!,x

k!)| 1, we have IN

v

I1. In section 2.2.4, we show that depending onthe correlation of the rows of X, we may have I

N

v

⌧ I1. The resolution estimates belowshow an advantage of using MMV in such cases.

2.2.2. Estimation of the support of X. The next theorem, proved in section 5.2, showsthat when I

N

v

and the noise level " are small, the large entries in X

" are supported at pointsnear ⌦S .

Theorem 2.1. Consider the matrix W

" = G(X" � X), defined in terms of the unknownX and its reconstruction X

", the minimizer of (2.10). This matrix cannot be computed, butit is guaranteed to satisfy

(2.18) kW "kF

2".

Suppose that there exists r 2 (0, 1) so that 2IN

v

< r < 1, and define the set

Br

(S) = {1 j Ny

such that 9 q 2 S satisfying D(j, q) < r},

called the r-vicinity of S with respect to the semimetric D. If we decompose the reconstructionin two parts,

(2.19) X

" = X

",r +E

",r,

with X

",r row supported in Br

(S) and E

",r row supported in the complement {1, . . . , Ny

} \B

r

(S), we have

kE",rk1,2 2I

N

v

rkX"k1,2 +

1

r

��G

?

W

"

�S!��1,2

2IN

v

rkX"k1,2 +

2"|S|r

,(2.20)

where G

? 2 CNy⇥Nr is the Hermitian adjoint of G and�G

?

W

"

�S! 2 C|S|⇥N

v is the restric-tion of the matrix G

?

W

" to the rows indexed by the entries in S.We may think of E",r as an error in the reconstruction because its rows are supported away

from S. The theorem says that this error is small when the multiple interaction coe�cientand the noise level are small. The estimate of the noise e↵ect in the second bound in (2.20) ispessimistic. In the numerical simulations, we found that

��G

?

W

"

�S!��1,2

is typically much

smaller than 2"|S|.

2.2.3. Quantitative estimation of X. Theorem 2.1 says that if we threshold the entriesin X

" at a value commensurate to the right-hand side in (2.20), we obtain the approximationX

",r with row support S" ⇢ Br

(S). Here we quantify how wellX",r approximatesX. BecauseS and S" are di↵erent sets in general, an estimate of some norm of X",r � X is not useful.Instead, we decompose X",r in one part supported in S that we compare with X in Theorem2.2 and a residual.

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Let GS = (gj

)j2S be the N

r

⇥ |S| matrix obtained by restricting the columns of G to theindexes in S. Suppose that GS has linearly independent columns, as otherwise it is impossibleto recover X even with noiseless data, and introduce its pseudoinverse

(2.21) G

†S = (G?

SGS)�1

G

⇤S .

Decompose X

",r in two parts,

(2.22) X

",r = X",r + E",r,

where X",r has row support in S and its restriction to the rows indexed by S satisfies

(2.23) X",r

S! = G

†SGX

",r.

This definition gives that

G

†SGX

",r = (G?

SGS)�1

G

⇤S

⇣GSX

",r

S! +GE",r

⌘= X",r

S! + (G?

SGS)�1

G

⇤SGE",r,(2.24)

so the residual E",r satisfies

(2.25) G

⇤SGE",r = 0.

That is to say, the columns of GE",r are orthogonal to the range of GS . Note that E",r hasrow support in S[S". If S" were the same as S, then (2.25) would imply that E",r = 0. Thus,E",r is a residual that accounts for X",r not having the exact support S.

The next theorem, proved in section 5.3, shows that under the same conditions as inTheorem 2.1, the matrix X",r is a good approximation of the unknown X. However, X",r

cannot be computed directly, so we need to relate it to X

",r. To do so, we introduce an“e↵ective matrix” supported in S, obtained by local aggregation of the rows of X",r. We showthat X",r is close to to this matrix if the single view interaction coe�cient I1 is small. Thisreveals the fact that while I

N

v

⌧ I1 brings an improved support of the MMV reconstructionvs. that of SMV, the quantitative estimate of X cannot be expected to be better.

Theorem 2.2. Let X",r and X",r be defined as in (2.19) and (2.22). Then

(2.26) kX",r �Xk1,2 2I

N

v

rkX"k1,2 +

6"|S|r

.

Moreover, if the support of X",r is decomposed in |S| disjoint parts, each corresponding to apoint in S,

(2.27) S" =[

j2SS"

j

, S"

j

= {q 2 S" such that D(q, j) D(q, j0) 8 j0 2 S}, j 2 S,

and we define the e↵ective matrix X

",r with row support in S and entries

(2.28) X

",r

j,v

=

8<

:

X

l2S"

j

µ(gj

, gl

)X",r

l,v

, if j 2 S,

0, otherwise,

for 1 j Ny

, 1 v Nv

,

we have the estimate

(2.29) (1� I1)kX",r �X

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Note that because µ(gj

, gl

) are complex valued, there may be cancellations in the localaggregation (2.28) of the entries of X",r. Only if the set S"

j

is small, so that D(j, l) ⌧ 1 forl 2 S"

j

, do we have µ(gj

, gl

) ⇡ 1, and (2.28) is approximately the local sum of the entriesin X

",r.

2.2.4. Matrices X with orthogonal rows. We now show that if the unknown matrix X

has orthogonal rows2 (i.e., uncorrelated), then the multiple view interaction coe�cient IN

v

may be much smaller than the interaction coe�cient I1. By Theorem 2.1, this means that theMMV approach can give improved estimates of the row support S of X, under less stringentconditions than in the SMV formulation.

Proposition 2.3. Suppose that X 2 CNy⇥N

v has row support in the set S with cardinality1 < |S| N

v

and that its nonzero rows are orthogonal. Then the multiple view interactioncoe�cient (2.12) is given by

(2.30) IN

v

= max1jNy

s X

q2S\{n(j)}

|µ(gj

, gq

)|2.

This proposition, proved in section 5.4, gives a simpler expression of IN

v

that we cancompare with

(2.31) I1 = max1jNy

X

q2S\{n(j)}

|µ(gj

, gq

)|

to understand when IN

v

⌧ I1. For this purpose, let us define the vector �

(j) 2 R1⇥(|S|�1)

with entries |µ(gj

, gq

)| for q 2 S \ {n(j)} and rewrite (2.30) and (2.31) as

(2.32) IN

v

= max1jNy

k�(j)k2, I1 = max1jNy

k�(j)k1,

using the `2 and `1 vector norms. Suppose that the maximizer in the definition of IN

v

is atindex j = m. Basic vector norm inequalities give the general relation

IN

v

= k�(m)k2 k�(m)k1 I1,

which is nothing new than was discussed previously. However, if we assume further that theentries in �

(m) are of the same order, meaning that there exist positive numbers �� and �+,ordered as �� + and satisfying �+/�� = O(1), such that

(2.33) �� |µ(gm

, gq

)| �+ 8 q 2 S \ {n(m)},

then we have

IN

v

�+p|S|� 1 =

�+⇥��(|S|� 1)

⇤

��p|S|� 1

�+k�(m)k1��p|S|� 1

�+I1

��p|S|� 1

= O

I1p|S|� 1

!.

(2.34)

2The results extend to nearly orthogonal rows, but to simplify the proof, we assume orthogonality.

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Recalling the discussion below definition (2.16) of the semimetric D and that |µ(gm

, gq

)| =1 � D(m, q), we can interpret (2.33) as having points in ⌦S evenly distributed, at similarspacing. If this condition holds, then I

N

v

is smaller than I1, by orderp|S|. In practice, it

may be di�cult to have a large number |S| of points at similar distance in the imaging planein order to see the improvement predicted by (2.34). However, this is just a bound, and thenumerical simulations in section 3.4.1 show that a significant reduction of I

N

v

/I1 is achievedeven when the imaging region is reduced to a line.

2.2.5. Clusters of unknowns. The multiple view interaction coe�cient IN

v

may be largefor arbitrary distributions of points in ⌦S , so we cannot conclude from the estimates abovethat the reconstruction X

" approximates X. However, if the points are clustered around afew locations, indexed by the elements in the set C ⇢ {1, . . . , N

y

} of cardinality |C| ⌧ |S|, thereconstruction is still useful, as we now show.

The result follows by recasting Theorem 2.1 for the new linear system

(2.35) GU +W = D

W

with cluster unknown matrix U and redefined “noise” W = W +GR with R = X �U . Thematrix U is defined by projection of X on the set of matrices with row support in C such thatits restriction to the rows indexed by C satisfies

(2.36) UC! = G

†CGX.

Here G

†C is the pseudoinverse of GC = (g

j

)j2C , the restriction of the sensing matrix to the

columns indexed in C, assumed to have full column rank. A similar calculation to that in(2.24) implies that the “residual” R satisfies G

?

CGR = 0, meaning that the columns of GR

are orthogonal to the range of GC . In other words, R accounts for the row support S of Xbeing di↵erent from C. The magnitude of this residual depends on how close the points areclustered together, as stated in the next lemma proved in section 5.5.

Lemma 2.4. Decompose the set S in |C| disjoint parts, called “cluster sets,” indexed by theentries in C:

(2.37) S =[

j2CS

j

, Sj

= {q 2 S such that D(q, j) < D(q, j0) 8j0 2 C, j0 6= j}, j 2 C.

Suppose that each cluster set Sj

is supported within a ball of radius rC around the point j 2 Cwith respect to the semimetric D for all j 2 C and that D(j, j0) > rC for all distinct j0, j 2 C.Then

(2.38) kGRkF

p2rCkXT k2,1,

where the index T denotes the transpose.

The next theorem, proved in section 5.5, is the extension of Theorem 2.1. It says that ifthe cluster radius rC and the cluster multiple view interaction coe�cient

(2.39) I U

N

v

= max1jNy

supv!2C1⇥N

v

X

q2C\{n(j)}

|µ(gj

, gq

)||µ(v!,uq!)|

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are small, the MMV reconstruction is row supported near C. This is an improvement overthe estimate in Theorem 2.1 because I U

N

v

is much smaller than IN

v

when the points in C arewell separated.

Theorem 2.5. Let X" be the minimizer of (2.10) with " chosen large enough to satisfy

(2.40) kWkF

= kW +GRkF

< ".

Decompose X

" it in two parts,

(2.41) X

" = U

",r +E

",r,

where U

",r is row supported in the set Br

(C), the r vicinity of C with respect to the semimetricD, and E

",r is the error supported in the complement {1, . . . , Ny

} \Br

(C). This satisfies theestimate

(2.42) kE",rk1,2 2I U

N

v

rkX"k1,2 +

1

r

��G

?

W

"

�C!��1,2

with W

" = G(X" �X) defined as in Theorem 2.1.

An extension of the quantitative estimate in Theorem 2.2 is possible, but we omit it herefor brevity. The result says that we should expect a good qualitative agreement between X

"

and a local aggregate of X over the cluster sets if the single view interaction coe�cient I U

1

is small, meaning that the points in C are su�ciently far apart.

3. SAR imaging of direction-dependent reflectivity. In this section, we consider theapplication of SAR imaging of direction-dependent reflectivities. We begin with the datamodel in section 3.1 and then derive in section 3.2 the linear system (2.1). The discussion inthese two sections is similar to that in [8], so we keep it short and give only the informationthat is needed to connect to the theory in section 2.2. We explore in section 3.3 the conditionof orthogonality of the rows of X, assumed in Proposition 2.3, and use numerical simulationsin section 3.4 to illustrate the theoretical results.

3.1. The SAR data model. Consider the setup illustrated in Figure 1, where we displaya piece of the synthetic aperture spanned by the moving transmit-receive antenna, called asubaperture. We approximate the subaperture by a line segment along the unit vector ⌧

with center at location r and length a. The imaging region ⌦ lies on a plane surface andis centered at location y, at distance L = |r � y| from the aperture center r. The antennaemits periodically the signal f(t) and measures the back-scattered waves. The waves propagatemuch faster than the antenna, so we assume that the emission and reception occur at the samelocation. The antenna moves by a small increment �r = a

(Nr�1)⌧ between two emissions, so

the measurements are at locations rj

= r � a⌧

2 + (j � 1)�r, for j = 1, . . . , Nr

.In the single scattering (Born) approximation and neglecting for now polarization e↵ects,

the scattered wave at rj

is given by

p(rj

, t; r,!) =

Zd!

2⇡e�i!t bf(!)k2(!)

NyX

q=1

⇢q

(r,!)exp

⇥2ik(!)|r

j

� y

q

|⇤

�4⇡|r

j

� y

q

|�2 .(3.1)

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L

y

m

r

r

y

⌦

Figure 1. Setup for SAR imaging using a linear synthetic aperture centered at r, at distance L from thecenter y of the imaging region ⌦. The antenna locations r span the aperture of length a, and y denotes a pointin ⌦. The unit vector m = (r � y)/L pointing from y to r defines the range direction.

Here the hat denotes Fourier transform with respect to time, ! is the frequency, and ⇢q

(r,!)is the reflectivity3 of the scatterer at y

q

2 ⌦. This depends on the subaperture center r andthe central frequency ! of the signal f . The integral over ! is over the support |!�!| . b ofbf , where b is the bandwidth. The propagation of the waves between the antenna location r

j

and y

q

is modeled with Green’s function for Helmoltz’s equation in the medium with constantwave speed c, and the wavenumber is k(!) = !/c.

In SAR imaging, the wave field (3.1) is convolved with the time-reversed emitted pulse,delayed by the round-trip travel time of the waves between the antenna and the center pointy in ⌦. This data processing is called down-ramping [20], and we denote the result by

d(rj

, t; r,!) = p(rj

, t; r,!) ?t

f?

�� t� 2|r

j

� y|/c�,(3.2)

where f? denotes the complex conjugate of f . The convolution f(t)?t

f?(�t) is called the pulsecompressed signal. We denote it by '(bt) = f(t) ?

t

f?(�t) with function ' of dimensionlessargument. This is supported at t = O(1/b).

Let us define the unit vectorm = (r�y)/L, which determines the so-called range directionin imaging, and the orthogonal projection P = I�mm

T in the cross-range plane, orthogonalto m. The size of the imaging region in the range and cross-range direction is given by thelength scales

Y = supy2⌦

|(y � y) ·m|, Y ? = supy2⌦

|P(y � y)|.

We assume a typical imaging regime defined by the scale order L � a > Y ? � � and Fresnelnumbers

(3.3)a2

�L& (Y ?)2

�L& 1,

3The reflectivity is assumed slowly changing, so it can be approximated by a constant over this sub-aperture

and bandwidth.

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where � = 2⇡/k is the central wavelength and k = k(!) = !/c. These inequalities meanphysically that the wave front observed at the subaperture or in ⌦ is not planar. If this werenot the case, it would be impossible to localize the scatterers in cross-range.

Since the cross-range resolution of classic SAR imaging [17] equals �L/a, the inequalities(3.3) ensure that Y ? is larger than this limit, so image focusing can be observed. The rangeresolution is determined by the accuracy of travel time estimation from the down-rampeddata (3.2). It is of the order c/b, so typically Y & c/b. In most imaging systems, b ⌧ !. Tosimplify the data model, we assume a bandwidth and aperture segmentation in small enoughsubbands b and subaperture sizes a so that

(3.4)b

!

aY ?

�L⌧ 1,

a2Y

�L2 ⌧ 1,

a2Y ?

�L2 ⌧ 1.

Under these scaling assumptions and using the approximations described in [8, section 3.1],we can write (3.2) in the form

(3.5) Dj

(r) =

NyX

q=1

exph� 2ik�r

j

·P�y

q

L

i

pN

r

Xq

(r)

with the notation �r

j

= r

j

� r and �y

q

= y

q

� y. Here Dj

(r) are the down-ramped data(3.2), up to some scaling factor, and evaluated at a fixed time t,

(3.6) Dj

(r) = d(rj

, t; r,!)eik t

✓4⇡L

k

◆2

,

whereas

(3.7) Xq

(r) = ⇢q

(r,!)pN

r

'hb⇣t+

2m ·�y

q

c

⌘iexp

h� 2ik

⇣m ·�y

q

� �y

q

· P�y

q

2L

⌘i.

We suppressed all the constant variables in the arguments of Dj

. By fixing the time t, welimit the sum in (3.5) to the set of points with range coordinates m ·�y

q

= �t+O(c/b). Thisset is called a range bin in the SAR literature [20]. We consider a single range bin and studythe estimation in the cross-range direction of the reflectivity for the single frequency subbandcentered at !.

3.2. The MMV formulation. The multiple views correspond to di↵erent subaperturesof size a, dividing a larger aperture of size A. The subapertures are centered at r

v

forv = 1, . . . , N

v

. The noiseless data model for the vth view is (3.5) with r replaced by r

v

, Lreplaced by L

v

= |rv

�y|, m replaced bym

v

= |rv

�y|/Lv

, and P replaced by Pv

= I�m

v

m

T

v

.We assume for simplicity that the large aperture is linear along the unit vector ⌧ .

Under technical scaling assumptions described in detail in [8], which mean physically thatthe imaging points remain within the same classic SAR resolution limits for all the views, weobtain from (3.5) the linear system (2.1) for matrices D, X, and G with entries

(3.8) Dj,v

= Dj

(rv

), Xq,v

= Xq

(rj

), Gj,q

=1pN

r

exph� 2ik

�r

j

· P1�y

q

L1

i.

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Note that the sensing matrix G is defined relative to the first subaperture. Its columns g

q

,for q = 1, . . . , N

y

, have norm one, as assumed in (2.14), and their correlation

(3.9) µ(gq

, gl

) =N

rX

j=1

G?

j,q

Gj,l

=1

Nr

N

rX

j=1

exph� 2ik

�r

j

· P1(yq

� y

l

)

L1

i

is a function of yq

� y

l

, as stated below (2.16). We can approximate further this correlationby replacing the sum with the integral over the subaperture:

(3.10) µ(gq

, gl

) ⇡ 1

a

Za/2

�a/2dr exp

h� 2ikr

⌧ · P1(yq

� y

l

)

L1

i= sinc

hka⌧ · P1(yq

� y

l

)

L1

i.

This attains its maximum, equal to 1, when q = l, and satisfies |µ(gq

, gl

) < 1 for all q 6= l, asassumed in (2.15). Moreover, |µ(g

q

, gl

)| decays monotonically in the vicinity of its peak, sowe can relate the Euclidian distance between the points to the semimetric D(q, l), as pointedout below (2.16).

3.3. Orthogonality of the rows. To use the results in section 2.2.4, we now study underwhich conditions the rows x

q! of X are approximately orthogonal. For this purpose, weassume that ⇢

q

(rv

,!) changes slowly with r

v

on a length scale larger than a. This is consis-tent with the MMV formulation, which approximates the reflectivity by a constant for eachsubaperture. We also suppose that the subapertures overlap, with two consecutive centersseparated by a small distance with respect to a. This allows us to approximate the sums inthe correlations of the rows by integrals over the large aperture of linear size A, centered at r

o

.

Proposition 3.1. There exists a constant Cq,l

that depends on how fast the reflectivities atpoints y

q

and y

l

change with direction, such that

��µ(xq!,x

l!)�� min{1, C

q,l

/|Q|} for q 6= l, q, l = q, . . . , Ny

, Q = 4⇡A⌧ · P

o

(yq

� y

l

)

�|ro

� y|,

(3.11)

where m

o

= (ro

� y)/|ro

� y| and Po

= I �m

o

m

T

o

.

This proposition, proved in Appendix A, shows that the correlation of the rows of theunknown matrix X is small for points that are separated in cross-range by distances largerthan �|r

o

� y|/A. This length scale is the cross-range resolution of SAR imaging over thelarge aperture A. It is also the distance at which isotropic scatterers must be separated inorder to guarantee unique recovery of their reflectivity with `1 (SMV) optimization over thelarge aperture, as follows from [28, 13, 14, 7].

In the linear system (2.1) with matrices (3.8), we use multiple views from subapertures ofsize a ⌧ A. Each single view corresponds to an SMV problem, and the condition of uniquerecovery for that problem is known to be that the scatterers should be much farther apart atdistance of order �|r

o

� y|/a. In MMV, we use the entire large aperture, segmented in Nv

smaller subapertures.When the scatterers are approximately isotropic, the constant in (3.11) is C

q,l

⇡ 2. In thiscase, there is no need to segment the aperture, so it is natural to ask if the MMV reconstruction

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is similar to the SMV one over the large aperture. This is a di�cult question, but we can sayfrom the results in section 2.2.4 that MMV will work better4 than SMV over one subaperturebecause the rows of the unknown matrix X are approximately orthogonal when the points inits support are at distances of order �|r

o

� y|/A ⌧ �|ro

� y|/a. The numerical simulations inthe next section demonstrate that this is the case as well.

When the scatterers have a stronger dependence on direction, the SMV approach over thelarge aperture does not work well. Aperture segmentation is needed to avoid systematic model-ing errors in the optimization. While we may apply the SMV approach for a single subaperture,Proposition 3.1 and the results in section 2.2.4 show that the MMV method performs better.

3.4. Numerical results. We present here numerical results that illustrate the theory pre-sented in section 2.2. We begin in section 3.4.1 with a computational assessment of thereduction of the multiple view interaction coe�cient with respect to the single view one in thecase of orthogonal rows of the unknown matrix X. Then we present in section 3.4.2 imagingresults, using the parameters of the X-band GOTCHA SAR data set [1]: The receive-transmitplatform moves on a linear aperture A = 1.5 km at altitude 8 km and with center r

o

at 7 kmwest of y. The platform emits and receives signals every meter. The central frequency is 10GHz, and since we only present imaging in cross-range, the bandwidth plays no role. Thewaves propagate at speed c = 3 · 108m/s.

The data are generated numerically using the single scattering approximation. The addi-tive noise matrix W has mean zero and independent complex Gaussian entries with standarddeviation � given as a percent of the largest entry in D. The optimization problem (2.10) issolved using the software package CVX [21].

3.4.1. Numerical illustration of e↵ects of orthogonality of rows of X. The discussionin section 2.2.4 says that if the points in ⌦S are distributed evenly in the imaging window ⌦and the rows of X are orthogonal, then the multiple view interaction coe�cient I

N

v

is smallerthan I1 by a factor of order

p|S|. Here we focus attention on imaging in the cross-range

direction, so the imaging region is reduced to a line segment. We cannot have a large numberof points with similar mutual separation on a line. Nevertheless, we show that the numericallycomputed ratio I1/IN

v

increases with |S| at a slightly slower rate thanp|S|.

We display in Figure 2 the ratio I1/IN

v

computed for imaging scenes with |S| rangingfrom 4 to 50 and cross-range separation of nearby neighbors chosen randomly, uniformlydistributed in the interval

⇥�L

o

/A, 3�Lo

/A⇤, where L

o

= |ro

� y|. The large aperture A isdivided in subapertures of size a = A/20. The rows of X have length 50 and are orthogonalto stay within the setting of section 2.2.4.

The left plot in Figure 2 shows the ratio I1/�I

N

v

p|S|�computed for one realization of

the imaging scene. We note that the increase of I1/IN

v

with |S| is slightly slower thanp|S|.

The histograms in Figure 2, computed for 2500 realizations of the imaging scene, also showthat the ratio is slightly less than

p|S|.

3.4.2. Imaging results. We begin with a comparison of imaging results obtained with theMMV optimization formulation (2.10) for N

v

= 24, the SMV formulation for Nv

= 1, and theconventional SAR image. The latter is given by the superposition of the down-ramped data

4As shown in section 2.2.4, the improvement is dependent on the distribution of the scatterers in the imaging

region.

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0 10 20 30 40 500

0.5

1

1.5

2.65 2.7 2.75 2.8 2.85 2.9 2.95 3 3.050

50

100

150

200

250

3009 Points

3.5 3.55 3.6 3.65 3.7 3.75 3.80

50

100

150

200

250

300

35016 Points

4.3 4.4 4.5 4.6 4.7 4.8 4.9 50

50

100

150

200

250

300

35036 Points

Figure 2. Left plot: The ratio I1

INv

p|S|

vs. |S| in the abscissa. The other plots: Histograms of the ratio

I1/INv for 2500 realizations of the imaging scene. From left to right, |S| equals 9, 16, and 36. The ordinateshows the number of realizations, and the abscissa is the value of I1/INv .

-4 -2 0 2 40

500

1000

1500

0

0.2

0.4

0.6

0.8

1

-4 -2 0 2 40

500

1000

1500

0

0.2

0.4

0.6

0.8

1

-4 -2 0 2 40

500

1000

1500

0

0.2

0.4

0.6

0.8

1

-4 -2 0 2 40

500

1000

1500

0

0.2

0.4

0.6

0.8

1

Figure 3. From left to right: (1) Exact reflectivity function as viewed from the location on the flight path(ordinate, in meters) vs. the cross-range location in the imaging scene (abscissa, in units �L

o

/A). (2) Theconventional SAR image (3.12) calculated over the entire aperture. (3) The MMV reconstruction. (4) TheSMV reconstruction.

(3.2), synchronized using the round-trip travel time of the waves from the radar platform tothe imaging point

(3.12) ISAR(y; r) =N

rX

j=1

d�r

j

, t = 2|rj

� y|/c; r,!�.

The superposition may be over the entire aperture centered at r = r

o

, in which caseNr

= 1500,or over a subaperture, centered at r = r

v

for v = 1, . . . , Nv

, in which case Nr

= 300. Thesubaperture length is a = A/6 = 300 m, and the spacing between the subapertures is 50 m,center to center. The results in Figures 3 and 4 are for noiseless data, and in Figure 5 weconsider noise with standard deviation � = 10%.

The images in Figure 3 are obtained for a scene with six small scatterers at cross-rangelocations spaced by distances of approximately �L

o

/A. The exact reflectivity is shown in theleft plot. The SAR image (3.12) computed over the entire aperture A = 1.5 km is shown inthe second plot. Note that this treats the reflectivity as isotropic (i.e., constant along theordinate). It does not resolve well the location of the five scatterers that are visible only onabout a sixth of A, but it obtains a large peak for the one scatterer with reflectivity that variesless with direction. The MMV image recovers exactly the support of the scatterers, whereasthe SMV method has many spurious peaks. This is an illustration of the result in section 2.2.3,which says that MMVmay give a better estimate of the support of the scatterers. However, theestimate of the value of the reflectivity is not accurate unless the scatterers are farther apart.D

ownl

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-15 -10 -5 0 5 10 150

500

1000

1500

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-15 -10 -5 0 5 10 15

0

500

1000

1500

0

0.2

0.4

0.6

0.8

1

-15 -10 -5 0 5 10 150

500

1000

1500

0

0.2

0.4

0.6

0.8

1

-15 -10 -5 0 5 10 150

500

1000

1500

0

0.2

0.4

0.6

0.8

1

Figure 4. Left plot: Exact reflectivity function as viewed from the location on the flight path (ordinate, inmeters) vs. the cross-range location in the imaging scene (abscissa, in units �L

o

/A). Other plots: The MMVreconstruction for apertures a = 50 m, 70 m, and 100 m, from left to right.

-15 -10 -5 0 5 10 150

500

1000

1500

0

0.2

0.4

0.6

0.8

1

-15 -10 -5 0 5 10 150

500

1000

1500

0

0.2

0.4

0.6

0.8

1

Figure 5. MMV reconstructions with noiseless data (left) and noisy data (right). The noise is additive,complex Gaussian, with mean zero independent entries and standard deviation � = 10% of the largest entry inD. The axes are as in Figure 3.

In Figure 4, we consider reflectivities that vary more rapidly over directions and comparethe e↵ect of the size of the subaperture on the quality of the reconstructions with the MMVapproach. The images show that the best reconstruction is for a = 70 m, which correspondsroughly with the scale of variation of the true reflectivity in the top plot. For the smalleraperture a = 40 m (left, bottom plot), the reconstructed support is close but not exact,whereas for the larger aperture a = 100 m (right, bottom plot), the image has spuriouspeaks caused by the systematic error due to the reflectivity varying on a smaller scale thanthe subaperture. Thus, we conclude that in order to image successfully direction-dependentreflectivities, it is necessary to have a good estimate of their scale of variation, so that theaperture is properly segmented.

In Figure 4, we display the e↵ect of additive noise with standard deviation � = 10% onthe MMV reconstruction of the reflectivity for subaperture size a = 70 m. We note that forsuch noise, the support of the reconstruction is basically unchanged, and the values of thereflectivity are only slightly di↵erent. Naturally, at higher noise levels, the reconstruction willbe worse.

4. SAR imaging with polarization diverse measurements. In this section, we describebriefly the application of SAR imaging with polarization. We begin in section 4.1 with thederivation of the data model (2.1) used in the MMV formulation and then show numericalresults in section 4.2.

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4.1. Data model. Consider a collection of |S| penetrable scatterers, with volume smaller

than �3by a factor ↵ ⌧ 1, so that the scattered electric field at the SAR platform can be

modeled by [3]

(4.1) E (rj

, t; r,f) =

Zd!

2⇡e�i!tik3(!)

rµ

✏

X

q2S

bG (!, rj

,yq

)⇢q

(r)bG (!,yq

, rj

) bf(!) +O(↵4),

where � is the central wavelength and µ and ✏ are the magnetic permeability and the electricpermittivity in the medium. These define the wave speed c = 1/

pµ✏ and the wavenumber

k(!) = !/c. The scatterers are represented in (4.1) by their center location y

q

and theirreflectivity tensor assumed constant over the subaperture centered at r,

(4.2) ⇢

q

(r) = ↵3⇣✏

q

✏� 1⌘M

q

(r),

where ✏q

is the electric permittivity in the scatterer and M

q

is its ↵-independent polarizationtensor. We refer to [2] for details on M

q

, which depends on the shape of the scatterer.Here we assume that it is a real valued 3 ⇥ 3 symmetric matrix. Since we consider a fixedcentral frequency !, we suppress in the notation the dependence of ⇢

q

on !. We also neglectthe variation of the magnetic permeability in the scatterer, although this can be taken intoaccount, as shown in [2].

The wave propagation from the radar platform to the scatterers and back is modeled in(4.1) by the dyadic Green’s tensor

(4.3) bG (!, r,y) =

✓I +

rrT

k2(!)

◆exp[ik(!)|r � y|]

4⇡|r � y| ,

where I is the 3⇥3 identity matrix. The wave excitation is modeled by the vector bf . To avoida lengthy discussion,5 suppose that the radar emits and receives all possible polarizations, sothat we have access to the 3⇥ 3 frequency-dependent data matrix

(4.4) bD(rj

,!; r) ⇡NyX

q=1

bG (!, rj

,yq

)⇢q

(r)bG (!,yq

, rj

)

with the approximation due to the neglected O(↵4) residual. Here we sum over all the Ny

points in the imaging region with the convention that ⇢q

= 0 for q /2 S.As in the previous section, we focus attention on imaging in the cross-range direction.

This is why it is su�cient to consider a single frequency, equal to the central one !. The wavenumber at this frequency is denoted by k, as in the previous section.

The subaperture centered at r is linear, of length a, like before, and we assume for sim-plicity that it is at constant altitude h, as shown in Figure 6. We let u3 be the unit vector inthe vertical direction and introduce the unit vector u1 = ⌧ ⇥ u3, where ⌧ is the unit tangent

5In fact, only the transverse components of the electric field, in the plane orthogonal to the range direction

r � y, play a role in the end, as discussed at the end of this section.

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Lh

a

r

Y ?

u2u1

u3

y

Figure 6. Geometry of the data acquisition. The radar platform flies at elevation h from the plane surfacecontaining the imaging region ⌦, centered at y. The distance L from the center r of the aperture to y is orderh. The drawing is not up to scale, as the aperture a and side Y ? of the imaging region are much smallerthan L.

to the aperture, orthogonal to u3. The imaging region ⌦ is in the plane spanned by u1 and ⌧ .We are interested in its cross section in the direction of the aperture, which is the cross-rangeinterval centered at y of length Y ?.

In the system of coordinates with center at y and orthonormal basis {uj

}1j3 withu2 = ⌧ , we have r = r1u1 + r2u2 + hu3 and y = y2u2 for all r in the aperture and y in thecross-range imaging interval. We also represent the symmetric 3⇥ 3 reflectivity tensor ⇢

q

(r)by the 1⇥ 6 row-vector formed with the entries in its upper-tridiagonal part

⇢

q! = (⇢q,11, ⇢q,22, ⇢q,33, ⇢q,12, ⇢q,13, ⇢q,23), ⇢

q,jl

= u

T

j

⇢

q

u

l

.

The scaling regime is as in the previous section, with length scales ordered as �⌧ Y ? .a ⌧ h, satisfying L = |r| = O(h) and |r

j

| = O(L) for j = 1, 2. Green’s tensor (4.3) has thefollowing approximation in this regime

bG (!, rj

,yq

) ⇡ exp[ik|rj

� y

q

|]4⇡L

0

@1� ⌘21 �⌘1⌘2 �⌘1��⌘1⌘2 1� ⌘22 �⌘2��⌘1� �⌘2� 1� �2

1

A , ⌘j

= rj

/L, j = 1, 2, � = h/L.

(4.5)

Substituting it in (4.4) and representing the symmetric matrix (4⇡L)2/pN

r

bD(rj

,!; r) by the1⇥6 row vector formed with the entries in its upper-triangular part, we obtain the data model

d

j!(r) =

NyX

q=1

exp[2ik|rj

� y

q

|]pN

r

⇢

q!(r)�(r), j = 1, . . . , Nr

,(4.6)

with r = r1u1 + r2u2 + hu3 and constant matrix �(r) given in Appendix B. This is a linearsystem of form (2.1) for N

v

= 6, data matrix D 2 CN

r

⇥6 with rows d

j!, unknown matrixX 2 CNy⇥6 with rows

(4.7) x

q! = ⇢

q!�,

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and sensing matrix G with normalized columns

g

q

=1pN

r

(exp[2ik|r1 � y

q

|], . . . , exp[2ik|rN

r

� y

q

|])T .

The system (4.6) is for a single subaperture. More subapertures, centered at rv

, can be takeninto account as explained in the previous section, with the only di↵erence being that insteadof having a scalar unknown, we now have the unknown 1 ⇥ 6 row vector ⇢

q

(rv

)�(rv

). Thelinear system that fuses the data from all the subapertures is obtained as in section 3.2, andthe unknown matrix X has six times more columns than in the acoustic case.

Note that the approximation (4.5) of Green’s tensor bG (!, rj

,yq

) for the subaperture cen-tered at r has the one-dimensional null space span{r}. This implies that the matrix �(r) isalso singular, so we cannot determine uniquely the reflectivity vectors ⇢

q! from (4.7). To bemore explicit, we can represent the reflectivity tensor ⇢

q

in (4.4) in the subaperture-dependentorthonormal basis {v

j

}j=1,2,3 of eigenvectors of the matrix in (4.5) with v3 = r/|r|. Then we

obtain that the components {vT

j

⇢

q

v3}j=1,2,3 play no role in the data model (4.4), so we can

only estimate (vT

j

⇢

q

v

l

)j,l=1,2. This ambiguity is due to the scaling relation a/|r| ⌧ 1, and

it implies that only the transverse components of the electric field are needed in imaging, asthe longitudinal component along v3 adds no information. If the reflectivity tensor does notchange over directions or it changes slowly, then the ambiguity can be overcome by taking intoconsideration the multiple subapertures because r changes orientation from one subapertureto another.

4.2. Numerical results. The setup for the numerical results is the same as in section 3.4.The data are generated using the single scattering model (4.1) for a reflectivity function thatchanges with the direction of illumination and is supported at two points at distance of order�L

o

/A, where Lo

= |ro

� y|.We display in Figure 7 the six entries of the row vectors x

q! defined in (4.7), as r variesin the large aperture, and for points in ⌦ indexed by q, separated by distances �L

o

/A incross-range. The plots in the bottom line of Figure 7 show that the MMV method gives goodestimates of these row vectors.

In Figure 8, we display the components (vT

j

⇢

q

v

l

)j,l=1,2 of the reflectivity matrix ⇢

q

and itsreconstruction for each subaperture centered at r. As in the note at the end of the previoussection, we let {v

j

}j=1,2,3 be the orthonormal basis of eigenvectors of the approximation (4.5)

of Green’s tensor with v3 along r. The reconstruction displayed in Figure 8 is calculated asfollows: With the estimated vectors x

q! displayed in Figure 7, we calculate the minimum `2norm solution of (4.7), using the truncated SVD of the singular matrix �(r). This correspondsto setting to zero the components vT

j

⇢

q

v

l

of the estimated ⇢

q

for either j or l equal to 3. Theother components are displayed in the figure, and they are well reconstructed.

5. Proofs. Here we prove the results stated in section 2.2. We begin with a lemmain section 5.1, which we then use in sections 5.2 and 5.3 to prove Theorems 2.1 and 2.2.Proposition 2.3 is proved in section 5.4, and the results for the clusters are proved in section 5.5.

5.1. A basic lemma. Let us denote by cX the matrix obtained by normalizing the nonzerorows in X, the unknown in the inverse problem, bx

q! = x

q!kx

q!k2 for q 2 S. Introduce the linearoperator

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Figure 7. Top line: From left to right we display all six components of the row vector x

q!(r) defined in(4.7) as a function of location along the aperture (the ordinate in meters) and cross-range location indexed byq in the imaging region (the abscissa, in units of �L

o

/A). Bottom line: The MMV reconstruction.

Figure 8. Top line: The components v

T

j

⇢

q

v

l

of the reflectivity matrix for j = l = 1 (left plot), j = l = 2

(middle plot), and j = 1, l = 2 (right plot). The orthonormal basis (v

j

)

j=1,2,3 depends on the center locationr of the subaperture (the ordinate in meters). The abscissa is the cross-range location indexed by q in units of�L

o

/A. Bottom line: The reconstruction.

(5.1) L : CNr⇥N

v ! C, L(V ) = trh(GcX)?V

i8V 2 CNr⇥N

v ,

where tr[·] denotes the trace. We have the following result.

Lemma 5.1. The linear operator L defined in (5.1) satisfies the inequality

(5.2)��L(V )

�� k(G?

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for any V 2 CNr⇥N

v . The matrix X satisfies the inequality

(5.3) kXk1,2�1� I

N

v

��L(GX)

��,

and with r, X",r, and E

",r defined as in Theorem 2.1, we have��L(GX

",r)��

�1 + I

N

v

�kX",rk1,2(5.4) ��L(GE

",r)��

�1� r + I

N

v

�kE",rk1,2.(5.5)

Proof. We start with definition (5.1) and use the invariance of the trace under cyclicpermutations and the row support S of X to obtain

L(GX) = trh(GcX)?GX

i= tr

hX

cX

?

G

?

G

i=X

j,q2S(Xc

X

?)j,q

(G?

G)q,j

=X

j,q2Shx

j!, bxq!i hg

q

, gj

i .

We rewrite this further with the normalization condition (2.14) and definition (2.12) and usethe triangle inequality to obtain the bound

��L(GX)�� =

��X

q2S

hhx

q!, bxq!i hg

q

, gq

i+X

j2S\{q}

hxj!, bx

q!i hgq

, gj

ii��

=��X

q2Skx

q!k2h1 +

X

j2S\{q}

µ(xj!,x

q!)µ(gq

, gj

)i��

�X

q2Skx

q!k2h1�

X

j2S\{q}

|µ(xj!,x

q!)||µ(gq

, gj

)|i��

�X

q2Skx

q!k2(1� IN

v

).

The result (5.3) follows from definition of the matrix norm kXk1,2.Similarly,

L(GX

",r) =X

j2S"

X

q2S

Dx

",r

j!, bxq!Ehg

q

, gj

i =X

j2S"

kx",r

j!k2X

q2Sµ(x",r

j!, bxq!)µ(g

q

, gj

),

where x

",r

j! denotes the jth row of X",r. Using the decomposition (2.27) of the row supportS" of X",r,

��L(GX

",r)�� =

��X

i2S

X

j2S"

i

kx",r

j!k2X

q2Sµ(x",r

j!, bxq!)µ(g

q

, gj

)��

=��X

i2S

X

j2S"

i

kx",r

j!k2hµ(x",r

j!, bxi!)µ(g

i

, gj

) +X

q2S\{i}

µ(x",r

j!, bxq!)µ(g

q

, gj

)i��.

By the construction in (2.27) for any j 2 S"

i

, the index n(j) 2 S of the nearest point to y

j

is n(j) = i, so the sum in q is over the set S \ {n(j)}. Using the triangle inequality and thedefinition (2.12) of I

N

v

, we get

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��L(GX

",r)��

X

i2S

X

j2S"

i

kx",r

j!k2h|µ(x",r

j!, bxi!)µ(g

i

, gj

)|+X

q2S\{n(j)}

|µ(x",r

j!, bxq!)|µ(g

q

, gj

)|i

X

i2S

X

j2S"

i

kx",r

j!k2(1 + IN

v

) =X

j2S"

kx",r

j!k2(1 + IN

v

).

Since S" is the row support of X",r, we can extend the sum to 1 j Ny

, and the result(5.4) follows from the definition of the k · k1,2 norm.

To prove (5.5), recall that E",r is supported by definition in the set Bc

r

(S) = {1, . . . , Ny

}\B

r

(S). Then, if we denote by e

",r

j! the rows of E",r, we have

L(GE

",r) =X

j2Bc

r

(S)

X

q2S

De

",r

j!, bxq!Ehg

q

, gj

i

=X

j2Bc

r

(S)

ke",rj!k2

X

q2Sµ(e",r

j!, bxq!)µ(g

q

, gj

)

=X

j2Bc

r

(S)

ke",rj!k2

hµ(e",r

j!, bxn(j)!)µ(g

n(j), gj) +X

q2S\{n(j)}

µ(e",rj!, bx

q!)µ(gq

, gj

)i.

Taking the absolute value and using the triangle inequality and definition (2.12) of IN

v

, weobtain the bound

��L(GE

",r)��

X

j2Bc

r

(S)

ke",rj!k2

h|µ(e",r

j!, bxn(j)!)||µ(g

n(j), gj)|+ IN

v

i.

But |µ(e",rj!, bx

n(j)!)| 1 and |µ(gn(j), gj)| = 1 � D(j, n(j)) with D(j, n(j)) � r for any

j 2 Bc

r

(S), so the bound becomes

��L(GE

",r)��

X

j2Bc

r

(S)

ke",rj!k2(1� r + I

N

v

).

We can extend the sum to 1 j Ny

because E

",r is supported in Bc

r

(S), and the result(5.5) follows from the definition of the k · k1,2 norm.

Finally, for any V 2 CNr⇥N

v , we obtain, using the invariance of the trace to cyclicpermutations, that

L(V ) = trh(GcX)?V

i= tr

hG

?

V

cX

?

i=

NyX

j=1

⌦�G

?

V )j!, bx

j!↵=X

j2S

⌦�G

?

V )j!, bx

j!↵,

where the last equality is because X is row supported in S. Taking the absolute value andusing the triangle and Cauchy–Schwartz inequalities, we get

��L(V )��

X

j2S

�� ⌦�G

?

V )j!, bx

j!↵ ��

X

j2Sk�G

?

V )j!k2 = k

�G

?

V )S!k1,2.

This is the result (5.2) in the lemma.

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5.2. Proof of Theorem 2.1. The bound (2.18) follows from the definition of W " and thetriangle inequality,

kW "kF

= kDW

�GX

" �W kF

kDW

�GX

"kF

+ kW kF

2",

where we used the assumption (2.11) and that X" is the minimizer of (2.10).Using again the definition of W " and the linearity of the operator (5.1), we write

L(GX) + L(W ") = L(GX +W

") = L(GX

") = L(GX

",r) + L(GE

",r),

where the last equality is by the decomposition (2.19). The result (5.3) in Lemma 5.1 gives

kXk1,2(1� IN

v

) ��L(GX)| =

��L(GX

",r) + L(GE

",r)� L(W ")��,

and using the triangle inequality and the estimates (5.2), (5.4), and (5.5), we get

kXk1,2(1� IN

v

) ��L(GX

",r)��+��L(GE

",r)|+��L(W ")

��

(1 + IN

v

)kX",rk1,2 + (1� r + IN

v

)kE",rk1,2 + k(G?

W

")S!k1,2.(5.6)

Note that by (2.9) and (2.11),

kGX �D

W

kF

= kW kF

< ",

so since X

" is the minimizer of (2.10), we must have kX"k1,2 kXk1,2. We also obtain fromthe decomposition (2.19) of X" in the matrices X",r and E

",r with disjoint row support that

kX"k1,2 = kX",r +E

",rk1,2 = kX",rk1,2 + kE",rk1,2.

Substituting in (5.6), we get

kX"k1,2(1� IN

v

) (1 + IN

v

)(kX"k1,2 � kE",rk1,2)+ (1� r + I

N

v

)kE",rk1,2 + k(G?

W

")S!k1,2= (1 + I

N

v

)kX"k1,2 � rkE",rk1,2 + k(G?

W

")S!k1,2.

We also have from the definition of k · k1,2, the normalization of the columns of G and (2.18),that

��G

?

W

"

�S!��1,2

=X

j2Skg?

j

W

"k2 =X

j2S

"N

vX

v=1

|g?

j

w"

v

|2#1/2

X

j2S

"N

vX

v=1

kw"

v

k22

#1/2(5.7)

= |S|kW "kF

2"|S|,

where w"

v

are the columns of W ". The result (2.20) stated in the theorem follows.

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5.3. Proof of Theorem 2.2. Let us start with the definition of the matrices W

", X",r,and E

",r given in Theorem 2.1 and write

GX

" = G(X",r +E

",r) = GX +W

".

With the decomposition (2.22) of X",r, we get

G(X �X",r) = GE

",r +GE",r �W

",(5.8)

and we prove next the analogue of the result (5.6) for X replaced by the matrix X � X",r

and X

",r replaced by 0. Looking at the proof of (5.3) in section 5.1, we note that we onlyused that X has row support in S. The same holds for the matrix X �X",r, so we can writedirectly the analogue of (5.3):

kX �X",rk1,2(1� IN

v

) ��L⇣G(X �X",r)

⌘��.(5.9)

The right-hand side in this equation can be estimated using (5.8) and the linearity of theoperator L,��L⇣G(X �X",r)

⌘�� =��L(GE

",r) + L(GE",r �W

")��

��L(GE

",r)��+��L(GE",r �W

")��.

Substituting in (5.9) and using the estimates (5.5) and (5.2) with V replaced by GE",r�W

",we obtain

kX �X",rk1,2(1� IN

v

) (1� r + IN

v

)kE",rk1,2 +��⇣G

?(GE",r �W

")⌘

S!

��1,2

.

But, by (2.25),(G?

GE",r)S! = G

?

SGE",r = 0,

and the desired estimate is

kX �X",rk1,2(1� IN

v

) (1� r + IN

v

)kE",rk1,2 +��⇣G

?

W

")⌘

S!

��1,2

(5.10)

with the last term bounded as in (5.7).Next, we substitute the bound (2.20) on the error term E

",r in (5.10) and obtain aftersimple algebraic manipulations that

kX �X",rk1,2 2I

N

v

(1� r + IN

v

)

r(1� IN

v

)kX"k1,2 +

(1 + IN

v

)

r(1� IN

v

)

��⇣G

?

W

")⌘

S!

��1,2

.(5.11)

The assumption 2IN

v

< r < 1 implies that

1� r + IN

v

1� 2IN

v

+ IN

v

= 1� IN

v

and1 + I

N

v

1� IN

v

<1 + I

N

v

1� r/2< 2(1 + I

N

v

) < 3.

Substituting in (5.11), we obtain the result (2.26) of Theorem 2.2.It remains to prove the estimate (2.29). We begin with the identity

X",r �X

",r = X

",r �X

",r � E",r

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and use equation (2.25) to conclude that

G

?

SG(X",r �X

",r) = G

?

SG(X",r �X

",r).

By construction, bothX",r andX

",r are row supported in S, so we can rewrite this equation as

(X",r �X

",r)S! � (I �G

?

SGS)(X",r �X

",r)S! = G

?

SG(X",r �X

",r),(5.12)

where I is the |S|⇥ |S| identity matrix. We now estimate each term in this equation.For the right-hand side in (5.12), we have

kG?

SG(X",r�X

",r)k1,1 =X

q2S

N

vX

v=1

��NyX

j=1

(G?

SG)q,j

(X",r �X

",r)j,v

��

=X

q2S

N

vX

v=1

��X

j2S[S"

µ(gq

, gj

)(X",r �X

",r)j,v

��

=X

q2S

N

vX

v=1

��X

j2S[S"\S"

q

µ(gq

, gj

)(X",r �X

",r)j,v

+X

j2(S[S")\S"

q

µ(gq

, gj

)(X",r �X

",r)j,v

��,(5.13)

where the first two equalities are by the definition of the norm and of the matrix product andthe third equality uses the definition (2.13) and the row support S [ S" of X",r �X

",r. Nowlet us recall the definition (2.28) of X",r and the decomposition (2.27) of the support S" ofX

",r, to obtain

X

j2(S[S")\S"

q

µ(gq

, gj

)X",r

j,v

=X

j2S"

q

µ(gq

, gj

)X",r

j,v

= X

",r

q,v

and X

",r

j,v

= X

",r

q,v

�j,q

, 8 j 2 S"

q

,

where �j,q

is the Kronecker delta symbol. Since µ(gq

, gq

) = 1, we conclude that the secondterm in (5.13) vanishes, and the result becomes

kG?

SG(X",r �X

",r)k1,1 =X

q2S

N

vX

v=1

��X

j2S[S"\S"

q

µ(gq

, gj

)(X",r �X

",r)j,v

��

N

vX

v=1

X

q2S

X

j2S[S"\S"

q

|µ(gq

, gj

)|��(X",r �X

",r)j,v

��.(5.14)

Note that the set {(j, q) : j 2 S[S" \S"

q

, q 2 S} is the same as the set {(j, q) : j 2 S[S", q 2S \ {n(j)}}, so we can rewrite (5.14) as

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kG?

SG(X",r �X

",r)k1,1 N

vX

v=1

X

j2S[S"

��(X",r �X

",r)j,v

��X

q2S\{n(j)}

|µ(gq

, gj

)|.

The last sum in this equation is bounded above by the interaction coe�cient I1, and usingthe definition of the k · k1,1 norm, we get

(5.15) kG?

SG(X",r �X

",r)k1,1 I1kX",r �X

",rk1,1.

With a similar calculation, we obtain

��(I �G

?

SGS)(X",r �X

",r)��1,1

=X

q2S

N

vX

v=1

��X

j2S(G?

SGS � I)q,j

(X",r �X

",r)j,v

��

X

j2S

N

vX

v=1

|(X",r �X

",r)j,v

|X

q2S|µ(g

q

, gj

)� �q,j

|

=X

j2S

N

vX

v=1

|(X",r �X

",r)j,v

|X

q2S\{j}

|µ(gq

, gj

)|,

where we used the triangle inequality, the identity (G?

SGS)q,j = µ(gq

, gj

), and µ(gq

, gq

) = 1.The last sum is bounded above by the interaction coe�cient I1, and using that X",r �X

",r

is row supported in S and the definition of the k · k1,1 norm, we get

(5.16)��(I �G

?

SGS)(X",r �X

",r)��1,1

I1kX",r �X

",r

��1,1

.

Gathering the results (5.12), (5.15), and (5.16) and using the triangle inequality, we obtainthe bound

(5.17) (1� I1)kX",r �X

",rk1,1 I1kX",r �X

",rk1,1 I1

⇣kX",rk1,1 + kX",rk1,1

⌘.

We also have from the definition (2.28) and the inequality |µ(gj

, gl

)| 1 for all j, l = 1, . . . , Ny

,that

kX",rk1,1 kX",rk1,1.The estimate (2.29) in Theorem 2.2 follows by substituting this in (5.17).

5.4. Proof of Proposition 2.3. Recall from section 5.1 the definition of the unit rowvectors bx

q!. Because the rows of X are assumed orthogonal in the proposition, {bxq!, q 2 S}

is an orthonormal subset of C1⇥N

v , and we conclude from Bessel’s inequality that

X

q2S\{n(j)}

| hv!, bxq!i |2 kv!k22 8 v! 2 C1⇥N

v and j = 1, . . . , Ny

.

Dividing both sides in this equation by kv!k22 and recalling definition (2.17), we obtain

(5.18)X

q2S\{n(j)}

|µ(v!, bxq!)|2 1 8 v! 2 C1⇥N

v and j = 1, . . . , Ny

.

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For a given j and v, we define the vector ⌫(j,v!) 2 R1⇥(|S|�1) with entries |µ(v!, bxq!)|. Recall

also from section 2.2.4 the vector �(j) 2 R1⇥(|S|�1) with entries |µ(gj

, gq

)| for q 2 S \ {n(j)},which is a set with cardinality |S|� 1. Using these vectors, we have

supv!2C1⇥N

v

X

q2S\{n(j)}

|µ(gj

, gq

)||µ(v!, bxq!)| = sup

⌫

(j,v!)2R1⇥|S|�1,k⌫(j,v!)k1

⇣⌫

(j,v!),�(j)⌘

= k�(j)k2,

where (·, ·) is the Euclidian inner product in R1⇥|S|�1 and we used inequality (5.18) to concludethat ⌫(j) lies in the unit ball in R1⇥|S|�1. The last equality is because the sup is achieved for⌫

(j,v!) = �

(j)/k�(j)k2. Substituting in the definition (2.12), we obtain the result (2.30).

5.5. Proof of cluster results. The proof of Theorem 2.5 is the same as in section 5.2,with X replaced by U , S replaced by C, and W replaced by W . This leads to the estimate

kE",rk1,2 2I U

N

v

rkX"k1,2 +

1

r

��G

?

G(X" �X +R)�C!��1,2

=2I U

N

v

rkX"k1,2 +

1

r

��G

?

W

"

�C!��1,2

,

where we used that U = X �R, the definition of W " in Theorem 2.1, and G

?

CGR = 0.It remains to prove Lemma 2.4. The projection (2.36) that defines U induces the linear

operator T : CN

r

⇥N

v ! CN

r

⇥N

v that maps GU = TGX. Note that GU = TGU , and since

0 = G

?

CGR = G

?

CG(X �U) = G

?

C(GX � TGX),

T is the orthogonal projection onto the range of GC . To estimate

(5.19) kGRk2F

= kG(X �U)k2F

=N

vX

v=1

kG(xv

� u

v

)k22 =N

vX

v=1

kGx

v

� TGx

v

k22,

we note that since T is the orthogonal projection on range(GC),

(5.20) kGx

v

� TGx

v

k2 kGx

v

� z)k2 8 z 2 range(GC).

Now let us define the “e↵ective cluster matrix” X with entries

(5.21) X

j,v

=

8<

:

X

l2Sj

X

l,v

µ(gj

, gl

), j 2 C,

0, otherwise,for 1 j N

y

, 1 v Nv

.

We use the inequality (5.20) for z = GX = GCXC! and obtain

kGx

v

� TGx

v

k2 kGx

v

�Gx

v

)k2 =��X

j2SX

j,v

g

j

�X

j2CX

j,v

g

j

��2

(5.22)

because X is row supported in S and X is row supported in C. Next, using the decomposition(2.37) of S and the definition (5.21) of X, we have

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��X

j2SX

j,v

g

j

�X

j2CX

j,v

g

j

��2=��X

j2C

X

l2Sj

Xl,v

g

l

�X

j2C

h X

l2Sj

Xl,v

µ(gj

, gl

)ig

j

��2

=��X

j2C

X

l2Sj

Xl,v

⇥g

l

� µ(gj

, gl

)gj

⇤��2.(5.23)

We can bound this using the triangle inequality and

��g

l

� µ(gj

, gl

)gj

��22= hg

l

� µ(gj

, gl

)gj

, gl

� µ(gj

, gl

)gj

i = 1� |µ(j, l)|2 2D(j, l),(5.24)

where we used the definition of the semimetric D and of µ. Since Sj

is contained within aball of radius rC centered at j 2 C, we have D(j, l) rC in (5.24), and gathering the results(5.22)–(5.24), we get

(5.25) kGx

v

� TGx

v

k2 p2rCX

j2C

X

l2Sj

|Xl,v

| =p2rCkxv

k1.

Finally, substituting in (5.19),

kGRk2F

2rC

N

vX

v=1

�kx

v

k1)2 = 2rCkXT k22,1.

6. Summary. We presented a novel resolution theory for SAR imaging using the MMVapproach, also known as simultaneously sparse optimization. This seeks to find an unknownmatrix X with sparse row support by inverting a linear system of equations using sparsity-promoting convex optimization. In the SAR imaging application, X models the unknownreflectivity of a scattering scene. The rows of X are indexed by the points in the imagingregion, and the columns correspond to its values for multiple views of the imaging scene fromdi↵erent subapertures and polarization diverse measurements.

The resolution theory does not pursue the question of exact recovery but seeks to estimatethe neighborhood of the support of X where the largest entries in the reconstruction lie. Theradius of this neighborhood represents the resolution limit, and it depends on the noise level.We introduced a quantifier of how the unknowns influence each other in imaging, called themultiple view interaction coe�cient, and showed that the smaller this is and the weaker thenoise, the better the estimate of the support of X. We also quantified the error of the recon-struction and studied the advantage of having multiple views. The existing literature showsthat the MMV method does not always perform better than sparsity-promoting optimizationwith a single view, the so-called SMV formulation. We showed that if the rows of X areorthogonal, then the MMV approach is expected to perform better, depending on how theunknowns are distributed in the imaging scene. We quantified this advantage and explainedhow the condition of orthogonality of the rows of X arises in the application of SAR imagingof direction-dependent reflectivity.

We also studied imaging of well-separated clusters of scatterers and showed that the MMVapproach gives a reconstruction supported near these clusters.

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Appendix A. Proof of Proposition 3.1. Let us introduce the notation

(A.1) ⇠q,v

= ⇢q

(rv

,!)pN

r

'hb⇣t+

2m1 ·�y

q

c

⌘i.

Assuming that ' is smooth and using that the spacing between the centers of consecutivesubapertures is small, we approximate the correlation of the rows of X by

��µ(xq!,x

l!)�� ⇡

��

ZA/2

�A/2dr

q,l

(r) exph2ik

(ro

+ r⌧ � y)

|ro

+ r⌧ � y| · (yq

� y

l

)i��

k q,q

k1/2L1(�A/2,A/2)k l,l

k1/2L1(�A/2,A/2)

.(A.2)

Recall that ro

is the center of the large linear aperture along ⌧ . We parametrize this apertureby the arclength r 2 [�A/2, A/2], and

q,l

(r) is the smooth kernel satisfying the interpolationconditions

(A.3) q,l

✓r =

⇣ v � 1

Nv

� 1� 1

2

⌘A

◆= ⇠

q,v

⇠?l,v

.

To estimate (A.2), we expand the exponent in r,

(A.4) k(r

o

+ r⌧ � y)

|ro

+ r⌧ � y| · (yq

� y

l

) = kmo

· (yq

� y

l

) + kr⌧ · P

o

(yq

� y

l

)

|ro

� y| + . . . ,

with m

o

and Po

defined as in Proposition 3.1. Suppose that A and the cross-range o↵setbetween y

q

and y

l

are small enough so we can neglect the higher terms6 in (A.4). Then usingQ defined in Proposition 3.1 and integrating by parts in (A.2), we obtain

|µ(xq!,x

l!)| ⇡A��

q,l

(A/2)eiQ/2 � q,l

(�A/2)e�iQ/2 �RA/2�A/2 dr

0q,l

(r)eirQ/A

��

|Q|k q,q

k1/2L1(�A/2,A/2)k l,l

k1/2L1(�A/2,A/2)

.(A.5)

If the reflectivities are independent of direction, then (A.5) becomes |µ(xq!,x

l!)| ⇡|sinc(Q/2)|. This attains its maximum at Q = 0, i.e., at q = l, and decays as 1/|Q|, asstated in the proposition. It remains to show that the result extends to reflectivities that varysmoothly with direction. We obtain from (A.5), using the triangle inequality, that

|µ(xq!,x

l!)| A⇥|

q,l

(A/2)|+ | q,l

(�A/2)|+ k 0q,l

kL1(�A/2,A/2)

⇤

|Q||k q,q

k1/2L1(�A/2,A/2)|k l,l

k1/2L1(�A/2,A/2)

,(A.6)

and we estimate next the three terms in the numerator. We begin with

| q,l

(A/2)| | q,l

(s)|+��Z

A/2

s

dr 0q,l

(r)��,

6The results are qualitatively the same if we include quadratic terms in r and neglect cubic and higher-order

terms. The discussion is simpler if we consider only the shown terms in (A.4).

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where we used the fundamental theorem of calculus and the triangle inequality. Therefore,

A| q,l

(A/2)| =Z

A/2

�A/2ds |

q,l

(A/2)| Z

A/2

�A/2ds |

q,l

(s)|+Z

A/2

�A/2ds��Z

A/2

s

dr 0q,l

(r)��

k q,l

kL1(�A/2,A/2) +

ZA/2

�A/2ds

ZA/2

�A/2dr�� 0

q,l

(r)��

= k q,l

kL1(�A/2,A/2) +Ak 0

q,l

kL1(�A/2,A/2).(A.7)

The first term in this equation can be bound using the Cauchy–Schwartz inequality once werecall the definition (A.3) of

q,l

. We rewrite this definition as

(A.8) q,l

(r) = ⇠q

(ro

+ r⌧ )⇠?l

(ro

+ r⌧ )

in an abuse of notation, so that ⇠q,v

= ⇠q

(rv

) for rv

= r

o

+ ( v�1N

v

�1 � 12)A⌧ . We obtain that

L2k q,l

kL1(�A/2,A/2) =

ZA/2

�A/2dr |⇠

q

(ro

+ r⌧ )⇠?l

(ro

+ r⌧ )|

"Z

A/2

�A/2dr |⇠

q

(ro

+ r⌧ )|2#1/2 "Z

A/2

�A/2dr |⇠

l

(ro

+ r⌧ )|2#1/2

= k q,q

k1/2L1(�A/2,A/2)k l,l

k1/2L1(�A/2,A/2).

We also have from (A.8) that

0q,l

(r) = ⌧ ·r⇠q

(ro

+ r⌧ )⇠?l

(ro

+ r⌧ ) + ⇠q

(ro

+ r⌧ )⌧ ·r⇠?l

(ro

+ r⌧ ),

and from the Cauchy–Schwartz and triangle inequalities, we get

k 0q,l

kL1(�A/2,A/2) k⌧ ·r⇠

q

kL2(�A/2,A/2)k⇠lkL2(�A/2,A/2)

+ k⇠q

kL2(�A/2,A/2)k⌧ ·r⇠

l

kL2(�A/2,A/2)

kr⇠q

kL2(�A/2,A/2)k⇠lkL2(�A/2,A/2) + k⇠

q

kL2(�A/2,A/2)kr⇠lkL2(�A/2,A/2).

To estimate this further, let us introduce the constant Kq

, which depends on the scale ofvariation of the reflectivity ⇠

q

, such that

kr⇠q

kL2(�A/2,A/2)

Kq

Ak⇠

q

kL2(�A/2,A/2).

Since k⇠q

kL2(�A/2,A/2) = k

q,q

k1/2L1(�A/2,A/2) by definition (A.8), we obtain that

Ak 0q,l

kL1(�A/2,A/2) (K

q

+Kl

)k q,q

k1/2L1(�A/2,A/2)k l,l

k1/2L1(�A/2,A/2).

The estimate (A.7) becomes

A| q,l

(A/2)| (1 +Kq

+Kl

)k q,q

k1/2L1(�A/2,A/2)k l,l

k1/2L1(�A/2,A/2),D

ownl

oade

d 01

/11/

19 to

141

.211

.60.

248.

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istrib

utio

n su

bjec

t to

SIA

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ttp://

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and a similar bound applies to A| q,l

(�A/2)|. Gathering the results and substituting in (A.6),we obtain the statement of the proposition with C

q,l

= 12⇡(1 +Kq

+Kq

).

Appendix B. Expression of matrix �(r). The 6 ⇥ 6 matrix �(r) that enters the datamodel (4.6) can be written as

�(r) = �diag(r) + �o↵-diag(r),

where

�diag(r) = diagonal⇣(1� ⌘21)

2, (1� ⌘2)2, (1� �2)2, (1� ⌘21)(1� ⌘22) + (⌘1⌘2)

2,

(1� ⌘21)(1� �2) + (⌘1�)2, (1� ⌘22)(1� �2) + (⌘2�)

2⌘,

is the diagonal part of �(r) and

�o↵-diag(r) =

0

BBBBBB@

0 (⌘1⌘2)2

(⌘1�)2 ⌘1⌘2(⌘

21 � 1) ⌘1�(⌘

21 � 1) ⌘2

1⌘2�(⌘1⌘2)

20 (⌘2�)

2 ⌘1⌘2(⌘22 � 1) ⌘1⌘

22� ⌘2�(⌘

22 � 1)

(⌘1�)2

(⌘2�)2

0 ⌘1⌘2�2 ⌘1�(�

2 � 1) ⌘2�(�2 � 1)

2⌘1⌘2(⌘21 � 1) 2⌘1⌘2(⌘

22 � 1) 2⌘1⌘2�

20 ⌘2�(2⌘

21 � 1) ⌘1�(2⌘

22 � 1)

2⌘1�(⌘21 � 1) 2⌘1⌘

22� 2⌘1�(�

2 � 1) ⌘2�(2⌘21 � 1) 0 ⌘1⌘2(2�

2 � 1)

2⌘21⌘2� 2⌘2�(⌘

22 � 1) 2⌘2�(�

2 � 1) ⌘1�(2⌘22 � 1) ⌘1⌘2(2�

2 � 1) 0

1

CCCCCCA

is its o↵-diagonal part.

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[8] L. Borcea, M. Moscoso, G. C. Papanicolaou, and C. Tsogka, Synthetic aperture imaging ofdirection-and frequency-dependent reflectivities, SIAM J. Imaging Sci., 9 (2016), pp. 52–81.

[9] A. M. Bruckstein, D. L. Donoho, and M. Elad, From sparse solutions of systems of equations tosparse modeling of signals and images, SIAM Rev., 51 (2009), pp. 34–81.

[10] E. J. Candes, J. Romberg, and T. Tao, Robust uncertainty principles: Exact signal reconstructionfrom highly incomplete frequency information, IEEE Trans. Inform. Theory, 52 (2006), pp. 489–509.

[11] E. J. Candes and T. Tao, Decoding by linear programming, IEEE Trans. Inform. Theory, 51 (2005),

pp. 4203–4215.

[12] E. J. Candes and T. Tao, Near-optimal signal recovery from random projections: Universal encodingstrategies?, IEEE Trans. Inform. Theory, 52 (2006), pp. 5406–5425.

[13] A. Chai, M. Moscoso, and G. Papanicolaou, Robust imaging of localized scatterers using the singularvalue decomposition and l1 minimization, Inverse Problems, 29 (2013), 025016.

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a multiple measurement vector approach to synthetic aperture radar...

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