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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
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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
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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
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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.
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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
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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
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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.
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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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