IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 51, NO. 9, SEPTEMBER 2013 4885

Measurement of Sharpness and ItsApplication in ISAR Imaging

Junfeng Wang and Xingzhao Liu

Abstract—It is necessary to measure the sharpness of distribu-tions in many situations. A class of functions is investigated inthis paper. First, the relation between this class and sharpnessis clarified, and this justifies this class as sharpness measures.Then, we analyze the performance of different sharpness measuresand present a guide to select the sharpness measure. In addition,the relation of this class to the sparsity measure is addressed,which leads to a deeper understanding about sparsity. Finally, weshow and discuss the application of this class in inverse syntheticaperture radar imaging.

Index Terms—Contrast, entropy, inverse synthetic apertureradar (ISAR), sharpness, sparsity.

I. INTRODUCTION

A NONNEGATIVE function, like an image, is referred toas a distribution in this paper. In many cases, it is neces-sary to measure the sharpness of a distribution. For instance,in synthetic aperture radar (SAR) and inverse SAR (ISAR)imaging, the image is the sharpest when focused. Thus, thefocus phase can be estimated as the one that provides thesharpest image. However, to implement this idea, a measurefor the sharpness of the image must be found. In other fields,like seismic deconvolution and correction of telescope images,there are similar requirements.

Different functions can be used to measure the sharpness ofdistributions. Contrast is a widely used measure. Muller usesit to measure the sharpness of telescope images [1], Wigginsuses it to measure the sharpness of seismic reflectivity functions[2], and Herland uses it to measure the sharpness of SARimages [3]. Negative entropy is another widely used measure.In information theory, entropy is used to measure the averageinformation quantity of a random source [4]. Moreover, itis used to measure the smoothness of a distribution, and itsnegative is used to measure the sharpness of a distribution.De Vries uses the negative entropy to measure the sharpnessof seismic reflectivity functions [5]. Bocker uses the negativeentropy to measure the sharpness of ISAR images [6]. Differentsharpness measures have different performances. For example,in ISAR imaging, if the target has dominant scatterers, the

Manuscript received June 28, 2012; revised February 18, 2013 and May 11,2013; accepted July 2, 2013. Date of publication August 7, 2013; date of currentversion August 30, 2013. This work was supported by the National NaturalScience Foundation of China (61072150), the National High-Technology Re-search and Development Program of China (200812Z108), and the NationalBasic Research Program of China (2010CB731904).

The authors are with the Department of Electronic Engineering, ShanghaiJiao Tong University, Shanghai 200240, China.

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

Digital Object Identifier 10.1109/TGRS.2013.2273554

negative entropy is superior to contrast in the global focusquality of the image [7]. Similar phenomena are also observedin seismic deconvolution [8].

Two questions arise. Can different sharpness measures begeneralized? How should the sharpness measure be chosen in aparticular application? De Vries presents a class of functions forsharpness measurement [5], but there are counterexamples inhis class. Fienup also presents a class of functions for sharpnessmeasurement [9]. However, in analyzing the relation betweensharpness and the class, he only shows that, when the distribu-tion is the smoothest, the measures attain the minima. Fienupalso investigates the performance of different sharpness mea-sures in SAR imaging, but his conclusion is drawn partially byintuition. Schulz derives an optimal sharpness measure in SARimaging but does not verify his conclusion using any data [10].

In this paper, a further investigation is made about Fienup’sclass. First, the relation between this class and sharpness isclarified, and this justifies this class as sharpness measures.Then, independently of particular applications, we analyze theperformance of different sharpness measures and present aguide to select the sharpness measure. A property related tosharpness is sparsity [11]–[14]. In this paper, the relation of thisclass to the sparsity measure is also addressed, which leads toa deeper understanding about sparsity. Finally, we investigatethe application of this class in ISAR imaging. A preliminarydescription of this work has been given in [15].

II. CLASS OF SHARPNESS MEASURES

Let an ≥ 0, n = 1, 2, . . . , N be a distribution. The sharpnessof an can be measured by

s =

N∑n=1

φ(anA

)(1)

A =

N∑n=1

an (2)

where φ(x), called the kernel, is convex for 0 ≤ x ≤ 1, i.e.,φ′′(x), the second-order derivative of φ(x), is positive for 0 ≤x ≤ 1 [9] and [10]. Since s does not depend on the order ofthe samples, it is more appropriate to say that s is a measure ofnonuniformity.

Different measures can be obtained from different φ(x)’s.A measure is obtained by letting φ(x) = xβ , β > 1. In par-ticular, when β = 2, the measure is contrast. Another measureis obtained by letting φ(x) = −xγ , 0 < γ < 1. When φ(x) =x ln(x), the measure is the negative entropy.

0196-2892 © 2013 IEEE

4886 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 51, NO. 9, SEPTEMBER 2013

Fig. 1. Relation of s to sharpness.

The aforementioned class of sharpness measures is reason-able, as will be shown in Section III. In addition, [5] presents adifferent class of functions for sharpness measurement, i.e.,

h =

N∑n=1

anA

θ(anA

)(3)

where θ(x) is monotonically increasing over 0 ≤ x ≤ 1. Thisclass, however, is incorrect because there are counterexamples.If θ(x) = −1/x, h = −N . In this example, h cannot be used asa sharpness measure, although θ(x) is monotonically increasingover 0 ≤ x ≤ 1.

III. RELATION OF s TO SHARPNESS

A. Theory

Let us find the bounds of s. The tangent of φ(x) at x = 1/Nhas the equation

φ1(x) = φ′(

1

N

)(x− 1

N

)+ φ

(1

N

)(4)

where φ′(x) is the derivative of φ(x) (Fig. 1). The line passingpoints (0, φ(0)) and (1, φ(1)) has the equation

φ2(x) = [φ(1)− φ(0)]x+ φ(0) (5)

(Fig. 1). Over [0, 1], φ(x) is convex, and thus, φ1(x) ≤ φ(x) ≤φ2(x), i.e.,

φ′(

1

N

)(x− 1

N

)+φ

(1

N

)≤ φ(x)≤ [φ(1)−φ(0)]x+φ(0)

(6)

(see Fig. 1). Letting x = an/A in (6), one obtains

φ′(

1

N

)(anA

− 1N

)+ φ

(1

N

)≤ φ

(anA

)≤ [φ(1)− φ(0)] an

A+ φ(0). (7)

Accumulating (7) for n = 1, 2, . . . , N , one obtains

Nφ

(1

N

)≤ s ≤ φ(1) + (N − 1)φ(0). (8)

Equation (8) gives the minimum and maximum of s. s attainsthe minimum when all an’s are equal, i.e., the distribution is thesmoothest. s attains the maximum when only one an is nonzero,i.e., the distribution is the sharpest. s, the minimum, and themaximum are actually the sums of φ(an/A), φ1(an/A), andφ2(an/A), respectively.

Fig. 2. Distribution.

Fig. 3. Lightly smoothed distribution.

Fig. 4. Heavily smoothed distribution.

Consider the variation of s with sharpness (Fig. 1). When thedistribution is the smoothest, all an/A’s are 1/N . Thus, the sumof φ(an/A) is equal to the sum of φ1(an/A), i.e., s is equalto the minimum. When the distribution becomes sharper, thevalues of an/A spread in the x-axis. Thus, the sum of φ(an/A)leaves the sum of φ1(an/A) and tends to the sum of φ2(an/A),i.e., s leaves the minimum and tends to the maximum. Whenthe distribution is the sharpest, only one an/A is 1, and all otheran/A’s are 0. Thus, the sum of φ(an/A) is equal to the sum ofφ2(an/A), i.e., s is equal to the maximum. As we see, s can beused to measure the sharpness of a distribution.

B. Example

Fig. 2 shows a distribution. Fig. 3 shows the distributionsmoothed by a mean filter of length 3. Fig. 4 shows the distri-bution smoothed by a mean filter of length 5. Table I shows the

WANG AND LIU: MEASUREMENT OF SHARPNESS AND ITS APPLICATION IN ISAR IMAGING 4887

TABLE IVALUES OF SHARPNESS MEASURES FOR DISTRIBUTIONS IN FIGS. 2–4

values of some sharpness measures for the three distributions. Itcan be seen that, when the distribution becomes smoother andsmoother, the sharpness measures become smaller and smaller.This indicates that the sharpness measures are reasonable.

IV. SELECTION OF KERNEL

A. Theory

Different sharpness measures have different performances[7]–[10]. In a particular application, one may be superior toanother. Thus, it is significant to analyze the effect of the kernelon the sharpness measure and to find a guide to select the kernel.

Consider the normalized distribution an/A. Its variation canbe decomposed into the mass transfers between its samples.Let ai/A and aj/A be samples of an/A. In the mass transferfrom aj/A to ai/A, aj/A decreases, ai/A increases, but thesum of aj/A and ai/A is a constant, denoted by c. s can bewritten as

s =φ(aiA

)+ φ

(ajA

)+

∑n�=i,j

φ(anA

)

=φ(aiA

)+ φ

(c− ai

A

)+

∑n�=i,j

φ(anA

). (9)

We are interested in the sensitivity of s to the mass transferfrom aj/A to ai/A. It can be measured by the derivative of swith respect to ai/A, i.e.,

ds

d(ai/A)=φ′

(aiA

)− φ′

(c− ai

A

)

=φ′(aiA

)− φ′

(ajA

)=

ai/A∫aj/A

φ′′(x)dx. (10)

Equation (10) shows that the sensitivity of s to the mass transferfrom aj/A to ai/A depends on the integral of φ′′(x) fromaj/A to ai/A. The absolute value of this integral is determinedby the absolute difference between aj/A and ai/A and φ′′(x)over [aj/A, ai/A]. When the absolute difference between aj/Aand ai/A is larger and φ′′(x) is larger over [aj/A, ai/A], theintegral of φ′′(x) from aj/A to ai/A has a larger absolute value.This means that s is more sensitive to the mass transfer fromaj/A to ai/A. This gives a guide to select φ(x).

Fig. 5. Selection of φ(x).

Assume that the normalized distribution an/A consists ofmultiple objects, such as T1 and T2 in Fig. 5, and they corre-spond to different value ranges, such as R1 and R2 in Fig. 5.Then, s has different average sensitivities to the mass transfersin different objects. The average sensitivity of s to the masstransfers in an object is determined by the absolute differencesbetween samples in this object and φ′′(x) over the value rangeof this object. If the absolute differences between samples arelarger in this object and φ′′(x) is larger over the value rangeof this object, s has a larger average sensitivity to the masstransfers in this object. Therefore, φ(x) should be selected suchthat φ′′(x) has a proper shape to adjust the average sensitivitiesof s to the mass transfers in different objects in a particularapplication.

Here is an example. In a distribution, the absolute differencesbetween samples may be different for different objects. Thisdetermines the proportions between the average sensitivities ofs to the mass transfers in different objects if φ(x) is selectedsuch that φ′′(x) is a constant over [0, 1], like φ(x) = x2. If theaverage sensitivity of s to the mass transfers in weak objectsneeds to have an increased proportion, then φ(x) should beselected such that φ′′(x) is decreasing over [0, 1], like φ(x) =xβ with 1 < β < 2, φ(x) = −xγ with 0 < γ < 1, or φ(x) =x ln(x). On the contrary, if the average sensitivity of s to themass transfers in strong objects needs to have an increasedproportion, then φ(x) should be selected such that φ′′(x) isincreasing over [0, 1], like φ(x) = xβ with β > 2.

B. Example

The distribution in Fig. 6 consists of a weak object and astrong object. Fig. 7 shows the distribution with the weak objectsmoothed. Fig. 8 shows the distribution with the strong objectsmoothed. In each case, a mean filter of length 3 is used. Table IIshows the values of some sharpness measures for the threedistributions.

Here, the weak object and the strong object have the samestates about the absolute differences between samples. There-fore, depending on φ′′(x), s has different average sensitivitiesto the mass transfers in different objects. When φ(x) = x1.5,−x0.5, or x ln(x), φ′′(x) is decreasing over [0, 1]. Hence, the

4888 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 51, NO. 9, SEPTEMBER 2013

Fig. 6. Distribution.

Fig. 7. Distribution with weak object smoothed.

Fig. 8. Distribution with strong object smoothed.

sharpness measure is more sensitive to the weak object than tothe strong object. Table II confirms this. This table shows thatthe sharpness measures with φ(x) = x1.5, φ(x) = −x0.5, andφ(x) = x ln(x) change more when the weak object is smoothedthan when the strong object is smoothed. When φ(x) = x3,φ′′(x) is increasing over [0, 1]. Hence, the sharpness measure ismore sensitive to the strong object than to the weak object. Thisis also confirmed by Table II. This table shows that the sharp-ness measure with φ(x) = x3 changes more when the strongobject is smoothed than when the weak object is smoothed.When φ(x) = x2, φ′′(x) is a constant over [0, 1]. Hence, thesharpness measure is equally sensitive to different objects. Thisjudgment is also confirmed by Table II. This table shows thatthe sharpness measure with φ(x) = x2 changes equally whendifferent objects are smoothed.

TABLE IIVALUES OF SHARPNESS MEASURES FOR DISTRIBUTIONS IN FIGS. 6–8

V. RELATION OF s TO SPARSITY MEASURE

Let un, n = 1, 2, . . . , N , be a sequence. According to (1) and(2), the sharpness of |un|2 can be measured by

s =

N∑n=1

φ

(|un|2E

)(11)

E =

N∑n=1

|un|2 (12)

where φ(x) is convex for 0 ≤ x ≤ 1, i.e., φ′′(x) > 0 for 0 ≤x ≤ 1. If φ(x) = −√x, then

s =

−N∑

n=1|un|

√E

. (13)

In sparse signal processing, the numerator in (13) is used tomeasure the sparsity of un [11]–[14]. Thus, the sharpnessmeasure of |un|2 with φ(x) = −

√x is the sparsity measure

of un divided by√E, and the sparsity measure of un is

the sharpness measure of |un|2 with φ(x) = −√x multiplied

by√E.

The negative of a sharpness measure can be used as asmoothness measure. Therefore, the smoothness of |un|2 canbe measured by the negative of (13), i.e.,

t =

N∑n=1

|un|√E

. (14)

In sparse signal processing, the numerator in (14) is usedto measure the density of un [11]–[14]. Thus, in fact, thissmoothness measure is the density measure of un divided by√E, the density measure of un is this smoothness measure

multiplied by√E, and minimizing the density measure of

un is equal to minimizing this smoothness measure multipliedby

√E.

VI. APPLICATION IN ISAR IMAGING

The aforementioned class of sharpness measures can be ap-plied to the sharpest image phase adjustment in ISAR imaging.

WANG AND LIU: MEASUREMENT OF SHARPNESS AND ITS APPLICATION IN ISAR IMAGING 4889

The addressed ideas and methods can also be extended to SARimaging and other fields.

A. Overview

ISAR uses the motion between the radar and the target toattain a fine resolution in azimuth. The radar may be groundbased, airborne, or spaceborne. The platform may be stationaryor moving. The beam tracks moving targets of interest. Thetargets may be man-made objects like ships, airplanes, andsatellites or natural objects like moons and planets.

There are various algorithms for ISAR imaging. In this paper,we only discuss the range-Doppler algorithm. First, the scatter-ers with different ranges are resolved using their differences intime delay. Then, translation compensation is used to removethe effect of the translation between the radar and the targetin range. It is usually done in two steps: range alignment andphase adjustment. In range alignment, the signals from the samescatterer are aligned in range by shifting the echoes. In phaseadjustment, the translational Doppler phase is removed. Finally,in each range bin, the scatterers with different azimuths areresolved using their differences in Doppler frequency.

A lot of attention is paid to the sharpest image phase adjust-ment owing to its good image quality and robustness againstnoise and target scintillation. It assumes that the image isthe sharpest when focused. Therefore, the adjustment phasecan be estimated as the one that provides the sharpest image.This is reasonable intuitively and theoretically [16]. Differentalgorithms are used to implement the sharpest image phase ad-justment. In the parametric algorithms, the adjustment phase isderived by parametric modeling [3] and [17]–[19]. Dependingon the relative motion between the radar and the target, theadjustment phase may take any form. If the adjustment phasedoes not fit the assumed model, the parametric algorithm cannotwork well. In order to remove this limitation, a nonparametricalgorithm is used to implement the sharpest image phase ad-justment [20]. This algorithm uses no parametric model for theadjustment phase and therefore applies universally. However, itachieves the optimization by a simple trial-and-error method,which is computationally inefficient. In order to improve thecomputational efficiency, two different nonparametric algo-rithms, the steepest ascent algorithm [9], [21] and the fixed-point algorithm [7], are presented to implement the sharpestimage phase adjustment. Both algorithms are computationallymuch more efficient than the algorithm in [20]. In this paper,we use the steepest ascent algorithm to carry out the sharpestimage phase adjustment.

B. Steepest Ascent Algorithm

In ISAR imaging, the complex image is written as

g(k, n) =

M−1∑m=0

f(m,n) exp [jϕ(m)] exp

(−j 2π

Mkm

)(15)

where m, n, and k are the indices of echoes, range bins, andDoppler frequencies, respectively, f(m,n) is the signal re-solved and aligned in range, and ϕ(m) is the adjustment phase.

In (15), f(m,n) is multiplied by exp[jϕ(m)] to implementphase adjustment, and azimuth resolving is implemented bytaking the discrete Fourier transform of f(m,n) exp[jϕ(m)]with respect to m. In order to carry out phase adjustment, ϕ(m)has to be estimated. In the sharpest image phase adjustment,ϕ(m) is estimated as the one which provides the sharpest|g(k, n)|2. According to (1) and (2), the sharpness of |g(k, n)|2is measured by

s =

N−1∑n=0

M−1∑k=0

φ

(|g(k, n)|2

E

)(16)

E =

N−1∑n=0

M−1∑k=0

|g(k, n)|2 (17)

where φ(x) is convex for 0 ≤ x ≤ 1, i.e., φ′′(x) > 0 for 0 ≤x ≤ 1. Thus, the sharpest image phase adjustment can beformulated as finding ϕ(m) to maximize s.

In the steepest ascent algorithm, the search is made in thedirection of the gradient [9], [21]. That is, in each iteration

ϕ̂(m) =ϕ(m) +1

L

∂s

∂ϕ(m)d (18)

L =

√√√√M−1∑m=0

[∂s

∂ϕ(m)

]2(19)

where ϕ̂(m) and ϕ(m) are the next value and the current valueof ϕ(m), respectively, and d is the step size. ∂s/∂ϕ(m) iscalculated by

∂s

∂ϕ(m)=

2M

EIm {exp [−jϕ(m)] z(m)} (20)

where

z(m) =

N−1∑n=0

f ∗(m,n)1

M

×M−1∑k=0

φ′

[|g(k, n)|2

E

]g(k, n) exp

(j2π

Mkm

). (21)

In the calculation, constant factors of ∂s/∂ϕ(m) can be ignoredbecause ∂s/∂ϕ(m) will be normalized by L. In our implemen-tation, the search is first made with a large step size until s ismaximized. Then, smaller step sizes are used to continue thesearch until s is maximized.

C. Results

The field data of a Boeing-727 aircraft [22], providedby Prof. B. D. Steinberg of the University of Pennsylvania,Philadelphia, PA, USA, are used to test our ideas. The aircraftwas 2.7 km from the radar and flew at a speed of 147 m/s. Theradar transmitted short pulses at a wavelength of 3.123 cm anda width of 7 ns. The echoes were sampled at an interval of 5 ns.The pulse repetition frequency was 400 Hz. Here, 512 echoeswith 120 range bins each were recorded. The echoes are di-vided into four segments, and each segment is processed using

4890 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 51, NO. 9, SEPTEMBER 2013

Fig. 9. Sharpest image phase adjustment with φ(x) = x ln(x).

Fig. 10. Sharpest image phase adjustment with φ(x) = −x0.5.

the range-Doppler algorithm. In the range-Doppler algorithm,range alignment is carried out by the improved global algorithm[23], and phase adjustment is carried out by the steepest ascentalgorithm. Different sharpness measures are used in the steepestascent algorithm.

Figs. 9–13 show the resulting images. Note that thegrayscales of these images are reversed in order that weakdetails can be seen more clearly. Thus, in these images, weakscatterers have high grayscales, and strong scatterers have lowgrayscales. As we see, all of these images are acceptable. Thisindicates the effectiveness of the sharpness measures.

Consider the normalized image |g(k, n)|2/E. Assume that itconsists of regions with different intensity ranges. If a Gaussian

Fig. 11. Sharpest image phase adjustment with φ(x) = x1.5.

Fig. 12. Sharpest image phase adjustment with φ(x) = x2.

distribution is assumed for real and imaginary parts in eachregion, the intensity I of a region has an exponential probabilitydensity function

p(I) =1

μexp

(− Iμ

), I ≥ 0 (22)

where μ is the mean of I [24]. μ is also the standard deviation ofI and thus reflects the absolute differences between samples inthis region. In fact, it needs different ϕ(m)’s to make differentregions the sharpest, and the estimated ϕ(m) is a compromiseof these ϕ(m)’s. In the estimation, different regions have dif-ferent importance. The importance of a region is proportionalto the sensitivity of this region to ϕ(m) and the sensitivity of

WANG AND LIU: MEASUREMENT OF SHARPNESS AND ITS APPLICATION IN ISAR IMAGING 4891

Fig. 13. Sharpest image phase adjustment with φ(x) = x3.

s to this region. The former is roughly proportional to μ, andthe latter is roughly proportional to μ and φ′′(μ). Therefore,in the estimation of ϕ(m), the importance of a region isroughly proportional to μ2φ′′(μ). In a particular application,φ(x) should be selected such that μ2φ′′(μ) has a desired shapeto adjust the importance of different regions in the estimationof ϕ(m).

When φ(x) = x ln(x), φ′′(x) = 1/x, and thus, μ2φ′′(μ) =μ. This means that, in the estimation of ϕ(m), the importanceof a region is roughly proportional to μ. The sharpest imagephase adjustment usually works well in this case, as shown inFig. 9.

If φ(x) = −x0.5, φ′′(x) = 0.25x−1.5. Therefore, μ2φ′′(μ) =0.25 μ0.5. Thus, in the estimation of ϕ(m), the importance of aregion is roughly proportional to μ0.5. Compared with φ(x) =x ln(x), φ(x) = −x0.5 increases the importance of weak re-gions. This may produce local maxima of s and may causeϕ(m) to converge to a local maximizer, as seen from the top-left image in Fig. 10. It should be mentioned that, since E isa constant, this sharpness measure is actually equivalent to thesparsity measure.

If φ(x) = x1.5, x2, or x3, then φ′′(x) = 0.75x−0.5, 2, or6x. Therefore, μ2φ′′(μ) = 0.75 μ1.5, 2 μ2, or 6 μ3. Thus, inthe estimation of ϕ(m), the importance of a region is roughlyproportional to μ1.5, μ2, or μ3. Compared with φ(x) = x ln(x),φ(x) = x1.5, x2, or x3 increases the importance of strongregions. In the result, the sharpest image phase adjustment maybe sensitive to a few dominant scatterers, and most scatterersmay not be focused as well as these dominant scatterers, as wesee from the top images in Figs. 11–13.

VII. CONCLUSION

From (1) and (2), we have justified s as a sharpness measureand have presented a guide to select φ(x). Assume that the

normalized distribution an/A is made up of multiple objectsand they correspond to different value ranges. Then, s hasdifferent average sensitivities to the mass transfers in differentobjects. The average sensitivity of s to the mass transfers inan object is determined by the absolute differences betweensamples in this object and φ′′(x) over the value range of thisobject. If the absolute differences between samples are largerin this object and φ′′(x) is larger over the value range of thisobject, s has a larger average sensitivity to the mass transfers inthis object. Therefore, φ(x) should be chosen such that φ′′(x)has a proper shape to adjust the average sensitivities of s to themass transfers in different objects in a particular application.

In addition, as an example, we have shown and discussed theapplication of the aforementioned theory in ISAR imaging. Theaddressed ideas and methods can be extended to SAR imagingand other fields.

REFERENCES

[1] R. A. Muller and A. Buffington, “Real-time correction of atmosphericallydegraded telescope images through image sharpening,” J. Opt. Soc. Amer.,vol. 64, no. 9, pp. 1200–1210, Sep. 1974.

[2] R. A. Wiggins, “Minimum entropy deconvolution,” Geoexploration,vol. 16, no. 1/2, pp. 21–35, Apr. 1978.

[3] E. A. Herland, “Seasat SAR processing at the Norwegian defense researchestablishment,” in Proc. EARSeL-ESA Symp., 1981, pp. 247–253.

[4] C. E. Shannon, “A mathematical theory of communication,” Bell Syst.Tech. J., vol. 27, no. 3, pp. 379–423, Jul. 1948.

[5] D. De Vries and A. J. Berkhout, “Velocity analysis based on minimumentropy,” Geophysics, vol. 49, no. 12, pp. 2132–2142, Dec. 1984.

[6] R. P. Bocker, T. B. Henderson, S. A. Jones, and B. R. Frieden, “A newinverse synthetic aperture radar algorithm for translational motion com-pensation,” in Proc. SPIE, 1991, vol. 1569, pp. 298–310.

[7] J. Wang, X. Liu, and Z. Zhou, “Minimum-entropy phase adjustment forISAR,” Proc. Inst. Elect. Eng.—Radar, Sonar Navig., vol. 151, no. 4,pp. 203–209, Aug. 2004.

[8] M. D. Sacchi, D. R. Velis, and A. H. Cominguez, “Minimum entropydeconvolution with frequency-domain constraints,” Geophysics, vol. 59,no. 6, pp. 938–945, Jun. 1994.

[9] J. R. Fienup and J. J. Miller, “Aberration correction by maximizing gener-alized sharpness metrics,” J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 20,no. 4, pp. 609–620, Apr. 2003.

[10] T. J. Schulz, “Optimal sharpness function for SAR autofocus,” IEEESignal Process. Lett., vol. 14, no. 1, pp. 27–30, Jan. 2007.

[11] E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Ex-act signal reconstruction from highly incomplete frequency information,”IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006.

[12] D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory, vol. 52,no. 4, pp. 1289–1306, Apr. 2006.

[13] J. A. Tropp and A. C. Gilbert, “Signal recovery from random measure-ments via orthogonal matching pursuit,” IEEE Trans. Inf. Theory, vol. 53,no. 12, pp. 4655–4666, Dec. 2007.

[14] L. C. Potter, E. Ertin, J. T. Parker, and M. Cetin, “Sparsity and compressedsensing in radar imaging,” Proc. IEEE, vol. 98, no. 6, pp. 1006–1020,Jun. 2010.

[15] J. Wang and X. Liu, “A class of sharpness measures,” in Proc. Asia-Pac.Conf. Synth. Aperture Radar, 2009, pp. 697–700.

[16] R. L. Morrison, M. N. Do, and D. C. Munson, “SAR image autofocus bysharpness optimization: A theoretical study,” IEEE Trans. Image Process.,vol. 16, no. 9, pp. 2309–2321, Sep. 2007.

[17] F. Berizzi and G. Corsini, “Autofocusing of inverse synthetic apertureradar images using contrast optimization,” IEEE Trans. Aerosp. Electron.Syst., vol. 32, no. 3, pp. 1185–1191, Jul. 1996.

[18] J. Wang and X. Liu, “SAR minimum-entropy autofocus using an adaptive-order polynomial model,” IEEE Geosci. Remote Sens. Lett., vol. 3, no. 4,pp. 512–516, Oct. 2006.

[19] T. Xiong, M. Xing, Y. Wang, S. Wang, J. Sheng, and L. Guo, “Minimum-entropy-based autofocus algorithm for SAR data using Chebyshev ap-proximation and method of series reversion, and its implementation ina data processor,” IEEE Trans. Geosci. Remote Sens., to be published.[Online]. Availabe: http://ieeexplore.ieee.org

4892 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 51, NO. 9, SEPTEMBER 2013

[20] X. Li, G. Liu, and J. Ni, “Autofocusing of ISAR images based on en-tropy minimization,” IEEE Trans. Aerosp. Electron. Syst., vol. 35, no. 4,pp. 1240–1251, Oct. 1999.

[21] J. R. Fienup, “Synthetic-aperture radar autofocus by maximizing sharp-ness,” Opt. Lett., vol. 25, no. 4, pp. 221–223, Feb. 2000.

[22] B. D. Steinberg and H. M. Subbaram, Microwave Imaging Techniques.Hoboken, NJ, USA: Wiley, 1991.

[23] J. Wang and X. Liu, “Improved global range alignment for ISAR,” IEEETrans. Aerosp. Electron. Syst., vol. 43, no. 3, pp. 12–17, Jul. 2007.

[24] C. Oliver and S. Quegan, Understanding Synthetic Aperture RadarImages. Norwood, MA, USA: Artech House, 1998.

Junfeng Wang received the B.S. degree in electricalengineering from the Beijing University of Technol-ogy, Beijing, China, in 1993, the M.S. degree inelectrical engineering from the Institute of Electron-ics, Chinese Academy of Sciences, Beijing, in 1996,and the Ph.D. degree in electrical engineering fromthe University of Massachusetts at Dartmouth, NorthDartmouth, MA, USA, in 2002.

He was with the Institute of Electronics, ChineseAcademy of Sciences, from 1996 to 1998. He wasa Postdoctoral Research Fellow with the University

of Michigan, Ann Arbor, MI, USA, from 2002 to 2003. He is currently anAssociate Professor with Shanghai Jiao Tong University, Shanghai, China. Hisresearch interests include signal and image processing in radar, medical, andastronomical imaging.

Xingzhao Liu received the B.S. and M.S. degreesin electrical engineering from the Harbin Insti-tute of Technology, Harbin, China, in 1984 and1992, respectively, and the Ph.D. degree in electri-cal engineering from the University of Tokushima,Tokushima, Japan, in 1995.

He was an Assistant Professor, an Associate Pro-fessor, and a Professor, successively, at the HarbinInstitute of Technology from 1984 to 1998. Since1998, he has been a Professor with Shanghai JiaoTong University, Shanghai, China. His research in-

terests include radar signal processing and related fields.

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