104 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 9, NO. 1, JANUARY 2012

An Autofocus Method for BackprojectionImagery in Synthetic Aperture Radar

Joshua N. Ash, Member, IEEE

Abstract—In this letter, we present an autofocus routine forbackprojection imagery from spotlight-mode synthetic apertureradar data. The approach is based on maximizing image sharpnessand supports the flexible collection and imaging geometries of BP,including wide-angle apertures and the ability to image directlyonto a digital elevation map. While image-quality-based autofocusapproaches can be computationally intensive, in the backprojec-tion setting, we demonstrate a natural geometric interpretationthat allows for optimal single-pulse phase corrections to be derivedin closed form as the solution of a quartic polynomial. The ap-proach is applicable to focusing standard backprojection imagery,as well as providing incremental focusing in sequential imagingapplications based on autoregressive backprojection. An exampledemonstrates the efficacy of the approach applied to real data fora wide-aperture backprojection image.

Index Terms—Autofocus, autoregressive backprojection(ARBP), convolution BP (CBP) imaging, synthetic aperture radar(SAR).

I. INTRODUCTION

M EASUREMENTS in synthetic aperture radar (SAR) aresusceptible to unknown platform motion and varying

atmospheric propagation delays. Both effects result in demod-ulation timing errors that may be modeled as unknown phaseshifts on each pulse of the collected phase history data pθ(r) =ejφpθ(r), where φ is the phase error for the given pulse, pθ(r) isthe range-compressed pulse return from azimuth position θ, andpθ(r) is the phase-corrupted measurement [1]. The phase errorsinduce blurring in subsequent SAR imagery, and it is the roleof an autofocus algorithm to mitigate these effects. Traditionalautofocus routines, such as phase gradient autofocus (PGA) [2],function as a postprocessing step operating on preformed SARimages and make the following assumptions.

1) A Fourier transform relationship exists between the im-age domain znm ∈ C and the range-compressed phasehistory domain Gnk ∈ C

Gnk = Fm(znm) (1)

Manuscript received October 22, 2010; accepted March 14, 2011. Date ofpublication August 18, 2011; date of current version December 23, 2011. Thiswork was supported by the U.S. Air Force Research Laboratory under AwardFA8650-07-D-1220. Any opinions, findings, conclusions, or recommendationsexpressed in this letter are those of the author and do not necessarily reflect theviews of the U.S. Air Force.

The author is with the Department of Electrical and Computer Engi-neering, The Ohio State University, Columbus, OH 43210 USA (e-mail:ashj@ece.osu.edu).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/LGRS.2011.2161456

where znm denotes image domain data with n and m in-dexing range and cross-range, respectively, Gnk denotesthe measured range-compressed sample for the nth rangebin and kth azimuth position, and Fm(·) denotes a 1-Ddiscrete Fourier transform with uniform sampling alongthe cross-range (m) dimension.

2) Phase errors are applied purely in the cross-rangedimension

pθk(rn) ≡ Gnk = Gnkejφk (2)

where θk is the azimuth position of the kth pulse, rndenotes the range value of the nth range bin, Gnk denotesthe uncorrupted range-compressed phase history, and φk

is the phase error applied to pulse k.

These assumptions only hold for axis-aligned imagery collectedover small aperture angles—where all pulses are approximatelyparallel. In this letter, we present a computationally efficientautofocus method compatible with the backprojection (BP)approach to SAR image formation, which directly supports abroader class of collection and imaging geometries, includingwide-angle apertures, near- and far-field collections, arbitraryflight paths, and the ability to image directly onto a digitalelevation map (DEM).

Our autofocus approach is based on maximizing imagesharpness

s(φ) =∑i

Ψ(vi(φ)

)(3)

where vi(φ), is the intensity of the ith pixel of the imageresulting from a particular set of phase error estimates φ, andΨ(x) is a convex function. In (3) and the remainder of thisletter, it is more convenient to use a single index i for thepixels of an image z. There are many possible forms for Ψ(x);however, the intensity-squared metric Ψ(x) = x2 is known toapply equal emphasis on brightening bright points as it doeson darkening shadows and has been empirically shown to workwell on a variety of scene types [3]. Furthermore, this metrichas been shown to produce phase error estimates equivalent tomaximum likelihood estimates under particular conditions [4].In this letter, we demonstrate that, for backprojection imagery,the optimal sharpness-maximizing phase may be determined ona per-pulse basis in closed form. The computational efficiencyis achieved by identifying the phase optimization problem witha basic ellipsoidal geometry problem solvable by finding theminimum root of a quartic polynomial. This solution may beused within a coordinate descent framework to focus a standardbackprojection image with respect to all pulses, or it may be

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ASH: AUTOFOCUS METHOD FOR BACKPROJECTION IMAGERY IN SYNTHETIC APERTURE RADAR 105

used to directly focus update frames in sequential imageryprovided by autoregressive backprojection (ARBP) [5].

While autofocus for SAR under the two earlier stated as-sumptions has been considered extensively, to our knowledge,[6] is the only work considering autofocus under more generalcollections and with BP imaging. The method in [6] estimatesphase corrections by employing PGA on an intermediate range-bearing grid, but the approach is only applicable for a restrictedset of blurring kernels. Similar to this letter, joint imaging andfocusing is considered in [7]—although based on �1-regularizedimaging, not backprojection.

II. AUTOFOCUS IN BACKPROJECTION IMAGING

A. Standard Convolution Backprojection

In the convolution backprojection (CBP) approach to SARimage formation, the image value for pixel i corresponding toground-plane coordinates qi = [xi yi] is computed as [8]

zi =

θmax∫θmin

ωmax∫ωmin

pθ(ω)ejωd(qi)|ω|Wr(ω)Wx(θ) dω dθ (4)

where [θmin, θmax] and [ωmin, ωmax] denote the azimuth andfrequency support of the collection, respectively, Wr(ω) andWx(θ) are apodization windows to control sidelobe levels inthe range and cross-range dimensions, respectively, and pθ(ω)is the Fourier transform of the range profile at angle θ. Thequantity d(qi) denotes the position that point qi appears in therange profile. When the platform is in the far field, d(qi) =xi cos θk + yi sin θk for pulse k. Other formulations of d(qi)account for near-field imaging or imaging to a DEM [9]. Ascollected phase history data are discrete, in practice, (4) iswritten as a discrete sum over the number of pulses K

Qk(r) =F−1 {pθk(ω)|ω|Wr(ω)} (5)

zi =

K∑k=1

Qk (d(qi))Wx(θk) (6)

where the inverse discrete Fourier transform in (5) yields thefiltered range profile for pulse k, and the backprojection step isperformed in (6).

Let the vector bk = {Qk(d(qi))Wx(θk)}i denote the filteredbackprojection of pulse k over all pixels i of interest. In theabsence of phase errors, the focused (vectorized) image iscomputed as

z =∑k

bk. (7)

When the pulses are subject to phase errors, we see from(5) that the backprojections become phase corrupted as wellbk → bk = bke

jφk , and∑

bk does not form a focused image.The objective of the autofocus algorithm is to produce phaseestimates φ = {φ1 . . . φK} such that the image quality of

z =∑k

bke−jφk (8)

is maximized over a given metric. As motivated earlier, weconsider the sharpness metric s(φ) =

∑i v

2i , where vi = ziz

∗i

is the intensity of the ith pixel. Thus, the optimal correction is

φ = argmaxφ

s(φ). (9)

As the optimization in (9) has no closed-form solution, weresort to an iterative method based on coordinate descent. Asimilar approach for polar-formatted images was considered in[10]. In the backprojection case, we will demonstrate that theoptimal phase error correction for each pulse may be deter-mined in closed form within the coordinate descent framework.

In coordinate descent optimization, each parameter is op-timized, in turn, while holding all other parameters fixed.Because the parameters are interdependent, optimization overthe entire set of parameters must be performed a number oftimes—called iterations. Let φi

k denote the kth coordinate (kthpulse phase correction) of the ith iteration; then, the coordinatedescent estimate at the next iteration i+ 1 is defined as

φi+1k = argmax

φs(φi+11 , . . . , φi+1

k−1, φ, φik+1, . . . , φ

iK

),

k = 1, . . . ,K. (10)

Consider the reflectivity image z(φ) at the ith iterationparameterized by the unknown phase of pulse k while holdingall other phases constant

z(φ) =

k−1∑p=1

e−jφi+1p bp +

K∑p=k+1

e−jφip bp + e−jφbk (11)

=x+ e−jφy (12)

where x comprises the backprojection sum of all pulses (withcurrent corrections) other than pulse k and y is the uncorrectedbackprojection of pulse k. The associated intensity of the ithpixel of z(φ) is

vi = ziz∗i = (xi + e−jφyi)

(x∗i + ejφy∗i

)(13)

= |xi|2 + |yi|2︸ ︷︷ ︸(v0)i

+2�(y∗ixie

jφ)︸ ︷︷ ︸

(vφ)i

. (14)

In vector form, the “intensity image” may be written as v =v0 + vφ, where v0 is constant and vφ depends on the unknownphase of pulse k. The sharpness metric over all pixels may bewritten as ∑

i

v2i = ‖v‖2 (15)

and interpreted as the squared length of v. Hence, the optimalphase correction for pulse k at iteration i is the value of φ thatmaximizes the length of v.

We observe that the set of backprojections {bk} for all pulses{k} only needs to be computed once, irrespective of the numberof coordinate descent iterations. However, bk is only neededwhen optimizing for φk. Thus, in memory-limited cases, bk canbe computed as needed without storing the backprojections ofall pulses.

106 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 9, NO. 1, JANUARY 2012

B. Autoregressive Backprojection

ARBP is a recently proposed image formation method ap-plicable to persistent sensing applications where an airborneplatform continuously circles a scene of interest and a sequenceof SAR images is to be produced, as in, e.g., a SAR video [5].While standard CBP (7) sums the backprojections from alarge number of previous pulses, in sequential imaging, ARBPexploits previously generated images in order to reduce thememory and computational requirements of generating thecurrent image

Ik =M∑

m=1

αmIk−m + βbk (16)

where M is the autoregression order (typically M ≤ 3), Ik isthe vectorized image at aperture position k, bk is the currentpulse filtered and backprojected over the pixels of interest, and{αk} and β are constants controlling the coherent azimuthintegration angle and the cross-range sidelobe structure. Whenthe current pulse contains a phase error and the previous imagesIk−1, Ik−2, . . . have been focused, the current image may beparameterized by a phase correction φ of the current pulseIk =

∑Mm=1 αmIk−m + e−jφβbk and optimized for maximal

sharpness. Letting x =∑M

m=1 αmIk−m and y = βbk, the cur-rent image may be written as z = x+ e−jφy. Thus, given thatprevious images have been focused, focusing the current imagein an ARBP update is mathematically equivalent to maximizingthe sharpness over a single coordinate in the coordinate descentapproach to sharpness maximization in standard BP.

III. GEOMETRIC INTERPRETATION

The previous section demonstrated that candidate per-pulse-corrected BP images and ARBP images may be written asz = x+ e−jφy and that elements of the intensity image maybe written as vi = (v0)i + (vφ)i, where (vφ)i = 2�(y∗ixie

jφ).By defining the vectors a and b with elements ai = 2�(y∗ixi)and bi = −2I(y∗ixi), vφ may be written as

vφ = a cosφ+ b sinφ (17)

which has the interpretation of an ellipse, even though a andb are not necessarily orthogonal. Hence, for an N ×N image,v = v0 + vφ traces out an ellipse on a 2-D plane embedded inR

N2. The plane is spanned by a, b ∈ R

N2and offset from the

origin by v0 ∈ RN2

. As shown in Fig. 1, the phase φ indexesthe ellipse, and the optimal phase φ maximizes the length of v.

The displacement vector v0 may be decomposed into aportion in the a, b plane and a portion perpendicular tothis plane. The perpendicular component contributes a fixedφ-independent amount to the length ‖v‖ and may be ignoredfor optimization purposes. As such, we may work completelyin the 2-D plane S spanned by a, b and define a new 2-DS-coordinate system with the ellipse at the origin (see Fig. 1).We are free to choose any basis {e1, e2} for S , but we use anorthonormalization of a, b for convenience

e1 =a

‖a‖ e2 =b− e1e

T1 b∥∥b− e1eT1 b

∥∥ . (18)

Fig. 1. Geometry of the autofocus problem. Candidate phase corrections mapout a displaced ellipse, and the optimal correction maximizes the length of v.

In the new coordinate system, a and b become

a ≡[a1a2

]= ETa b ≡

[b1b2

]= ET b (19)

where E = [e1 e2]. Finally, the origin of RN2

must be pro-jected onto S . Relative to the origin of S , the origin of RN2

is at−v0. After projecting and reparameterizing, this point becomes

x0 = ET (EET )(−v0) = −ETv0. (20)

Hence, the final optimization problem reduces to a basic ge-ometry problem in R

2 where one has to find the point x,on an origin-centered ellipse, that lies farthest from the givenpoint x0.

IV. PHASE OPTIMIZATION

Having reduced the phase estimation problem to one ofbasic geometry, we now present a closed-form solution for theoptimal phase correction. The approach is similar to the workin [11], extended for maximal distance and to support rotatedellipses defined by oblique axes. At the farthest point x, thevector x0 − x will be parallel to the normal n(x) of the ellipse

x0 − x = α n(x) (21)

for some real constant α. The ellipse in S may be written para-metrically x(φ) = [x1 x2]

T = a cosφ+ b sinφ or implicitly

f(x) ≡ xTRx = 1 (22)

with

R =

[r1 r3r3 r2

]. (23)

ASH: AUTOFOCUS METHOD FOR BACKPROJECTION IMAGERY IN SYNTHETIC APERTURE RADAR 107

The elements of R equating the implicit ellipse to the paramet-ric form can be shown to be

r1 =(a22 + b22

)/c (24)

r2 =(a21 + b21

)/c (25)

r3 = −(a1a2 + b1b2)/c (26)

where c = (a2b1 − a1b2)2. Using the implicit form, the normal

is computed from the gradient of f(x), n(x) ≡ ∇f(x) =2Rx. Thus, absorbing the factor of two into the unknown α,x0 − x = αRx and

x = (αR+ I)−1x0. (27)

Let R = V ΛV T denote the eigendecomposition of R; then,we have

(αR+ I) = V (αΛ+ I)V T (28)

for the eigendecomposition of (αR+ I). Substituting (27) and(28) into (22), we obtain

1 =x0(αR+ I)−1R(αR+ I)−1x0 (29)

=xT0 V (αΛ+ I)−1V T (V ΛV T )V (αΛ+ I)−1V Tx0 (30)

=xT0 V (αΛ+ I)−1Λ(αΛ+ I)−1V Tx0 (31)

=xT0 V

[λ1

(αλ1+1)2 0

0 λ2

(αλ2+1)2

]V Tx0 (32)

where λ1 and λ2 are the eigenvalues of R, and Λ =diag({λ1, λ2}).

Letting [β1 β2]T = V Tx0, (32) may be written as

q(α) = β21

λ1

(αλ1 + 1)2+ β2

2

λ2

(αλ2 + 1)2− 1 = 0 (33)

which has the same roots as the fourth-order polynomial

p(α) ≡4∑

i=0

γiαi = 0 (34)

where

γ0 =λ1β21 + λ2β

22 − 1 (35)

γ1 =2λ1

(λ2β

22 − 1

)+ 2λ2

(λ1β

21 − 1

)(36)

γ2 =(λ1β

21 − 1

)λ22 +

(λ2β

22 − 1

)λ21 − 4λ1λ2 (37)

γ3 = −2λ1λ2(λ1 + λ2) (38)

γ4 = −(λ1λ2)2. (39)

The roots of p(α) may be found from the eigenvalues of thepolynomial’s 4 × 4 companion matrix [12] or any other com-putationally efficient rooting algorithm. In general, p(α) willhave up to four real roots, and it is straightforward to show thatthe closest and farthest points on the ellipse correspond to thelargest and smallest purely real roots, respectively [11]. Letting

α denote the smallest real root, the optimal phase correction φmay be computed as

φ = tan−1

(sin φ

cos φ

)(40)

where

[cos φ

sin φ

]= [a b]−1(αR+ I)−1x0 (41)

and a four-quadrant inverse tangent is used in (40).

V. EXAMPLE

In this section, we demonstrate results on real data fromthe Gotcha Public Release Dataset [13]—an X-band collectionwith 9.6-Hz center frequency and 640-MHz bandwidth. Wenote that, for this bandwidth, a 3◦ aperture provides approxi-mately equal range and cross-range support in the transducedphase history data. The data were collected in circular spotlightmode with an approximate standoff of 10 km. The released datahave an azimuth sample rate of approximately 117 pulses perdegree and have a phase history that is already focused. Wetherefore artificially introduce phase errors into the pulse his-tory for this example. In contrast to the traditional autofocus lit-erature, we induce phase errors directly on the pulses making upthe annular aperture—not on range-compressed data obtainedby a 1-D inverse fast Fourier transform applied to a preformedimage. The phase error applied across the annular aperture waswhite Gaussian noise with zero mean and a standard deviationof π radians. Fig. 2 shows the resulting CBP images obtainedfor a 25◦ aperture without and with the autofocus methodapplied. The focused image resulted from four iterations ofcoordinate descent using an initial phase estimate of zero forall pulses. Fig. 2(c) shows the resulting sharpness after eachiteration, relative to the initial unfocused image in Fig. 2(a).

VI. CONCLUSION

In this letter, we have considered an autofocus method forbackprojection imagery from spotlight-mode SAR data anddemonstrated that the optimal per-pulse phase correction un-der the image sharpness metric has an appealing ellipsoidalgeometric interpretation in a reduced subspace. Exploiting thisgeometry enabled us to solve for the optimal per-pulse phasecorrection in closed form. This method may be used in a co-ordinate descent framework for efficient focusing over multiplepulses in a CBP image, or it may be integrated in an ARBPframework for in situ focusing of sequentially updated images.

Finally, we note that, as the method herein was developedunder more general conditions than the assumptions describedin Section I, it naturally supports the more restrictive scenariodescribed there, even for imagery produced via the polar formatalgorithm (PFA). A phase-corrected PFA image may be writ-ten as znm =

∑k Gnke

j(2π/N)kme−jφk , where Gnk represents

108 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 9, NO. 1, JANUARY 2012

Fig. 2. Example backprojection SAR imagery of a parking lot using a wide25◦ aperture, (a) without and (b) with autofocus. (c) Relative sharpness versusiteration count for the algorithm.

measured data after polar formatting and range compression. Inmatrix form, the image Z = {znm} may be written as

Z =∑k

Bke−jφk (42)

where the elements of the matrix Bk are equal to (Bk)nm =

Gnkej(2π/N)km. As the form of (42) is identical to that of

(8), the geometric interpretation and phase optimization methoddeveloped for backprojection may be directly applied in a polarformat setting as well.

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