Minimum-Entropy-Based Autofocus Algorithm for SAR Data Using ...

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 52, NO. 3, MARCH 2014

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Minimum-Entropy-Based Autofocus Algorithm for SAR Data Using Chebyshev Approximation and Method of Series Reversion, and Its Implementation in a Data Processor Tao Xiong, Mengdao Xing, Member, IEEE, Yong Wang, Shuang Wang, Jialian Sheng, and Liang Guo

Abstract— A novel autofocus method for synthetic aperture radar (SAR) image is studied. Based on a quadratic model for the phase error within each sub-area (narrow strip × subaperture) after a wide range swath is subdivided into narrow range strips and long azimuth aperture into sub-apertures, an objective function for estimation of the error is derived through the principle of minimum entropy. There is only one unknown variable in the function. With the Chebyshev approximation, the function is approximated as a polynomial, and the unknown is then solved using the method of series reversion. Curvefitting methods are applied to estimate phase error for an entire scene of the full-swath by full-aperture. Through simulations, the proposed method is applied to restore the defocused SAR imagery that is well focused. The restored and original images are almost identical qualitatively and quantitatively. Next, the method is implemented into an existing SAR data processor. Two sets of SAR raw data at X- and Ku-bands are processed and two images are formed. Well-focused and high-resolution images from plain and rugged terrain are obtained even without the use of ancillary attitude data of the airborne SAR platform. Thus, the studied method is verified. Index Terms— Autofocus, Chebyshev approximation (CA), method of series reversion (MSR), principle of minimum-entropy, synthetic aperture radar (SAR).

I. I NTRODUCTION

S

YNTHETIC aperture radar (SAR) imagery can be defocused because of the presence of phase error in received signals. Causes for the error include unknown or unpredetermined platform motion or variable propagation delays of

Manuscript received April 29, 2012; revised January 4, 2013; accepted February 22, 2013. Date of publication May 13, 2013; date of current version December 17, 2013. This work was supported in part by the Chinese 973 Program under Grant 2010CB731903 to the Xidian University, Xi’an, Shannxi of China, the National Natural Science Foundation of China under Grant 61173092, Grant 61202176, Grant 61271302, Grant 61107006, and the Program for Cheung Kong Scholars and Innovative Research Team in University under Grant IRT1170. T. Xiong and S. Wang are with the Key Laboratory of Intelligent Perception and Image Understanding of the Ministry of Education of China, Xidian University, Xi’an 710071, China (e-mail: [email protected]; [email protected]). M. Xing and J. Sheng are with the National Key Laboratory for Radar Signal Processing, Xidian University, Xi’an 710071, China. Y. Wang is with the School of Resources and Environment, University of Electronic Science and Technology of China, Chengdu 611731, China and also with the Department of Geography, East Carolina University, Greenville, NC 27858 USA. L. Guo is with the School of Technical Physics, Xidian University, Xi’an 710071, China. Digital Object Identifier 10.1109/TGRS.2013.2253781

signals through different atmospheric conditions. The error is typically modeled as a 1-D function added to Fourier phase of imaging data [1]. A technique to correct the phase aberrations of the defocused imagery is called as an autofocus. Many autofocus algorithms are developed [1]–[14], and a widely used one is the phase gradient autofocus (PGA) algorithm. In the algorithm, inverse filtering, windowing, and averaging process are iteratively employed to estimate the phase error [1], [5]–[8]. An accurate estimation is generally achieved. In addition, with algorithms that optimize image sharpness metrics such as an entropy derived from the defocused images, images are satisfactorily focused [9]–[15]. Compared with the PGA algorithm, the metrics-based autofocus algorithm is more robust under conditions that some of the image samples might be low in signal-to-noise ratio, and be of substantial amount of phase error. Hence, the metrics-based autofocus algorithms [15], [16] can produce superior results in comparison with those using the PGA algorithm. However, the computational load of the metrics-based method is generally high. The reasons include: 1) there is no limitation on what or which analytical expression of the phase error for each method hence the error should be estimated on a pulse-by-pulse basis and 2) the gradient-based linear search method (e.g., a conjugate gradient or quasi-Newton method) is often used for the error estimation. To estimate the error with acceptable level of accuracy, one operates the method iteratively. Thus, the estimation process can be computationally inefficient. To improve the efficiency but to keep the advantages, we present a new metrics-based approach to estimate the phase error. In the approach, the quadratic model of the error [22] is assumed within a sub-aperture. This model is valid for the error that is commonly caused by nonsystematic attitude variations [22] of an airborne platform. With this assumption, the autofocusing becomes to solve coefficient of the quadratic model. Simultaneously, the entropy of an image is introduced to gauge the success of the phase error correction after the autofocus. The smaller the entropy value, the better the quality in focus after the correction [16]. Therefore, coefficient of the phase error model can be derived through the minimization of an entropy-based objective function. By using the Chebyshev approximation (CA), the objective function becomes a polynomial. Then, the method of series reversion (MSR) can be used to solve coefficients of the polynomial.

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To evaluate the CA and MSR algorithm (CA-MSRA), we downloaded well-focused high-resolution SAR images from Sandia National Laboratories, New Mexico, USA (http:// www.sandia.gov/radar/sar-data.html), and added phase error to simulate defocused images. Then, the phase error from the defocused images is estimated and removed. Finally, the proposed method is implemented into an existing SAR data processor. Thus, using the modified processor, one can not only produce SAR imagery from raw data but also assess quality of the formed images. This paper is organized as follows. In Section II, the autofocus issue is briefly discussed. Solution of the objective function is detailed in Section III. In Section IV, implementation of the studied method in an existing SAR data processor is given. Restoration of images after the addition and removal of simulated phase errors and modification of an existing SAR data processor and formation of SAR images from acquired raw data are reported in Section V. This paper is concluded in Section VI.

III. N OVEL AUTOFOCUS A PPROACH U SING CA AND MSR A. Alternative Function Derived by CA Let

k=0

where ymn is value of the (m, n)th image sample and is typically complex. y˜mk and ymn are Fourier transform pairs with 0 ≤ n < N and 0 ≤ k < N. m, n, and k are indexes in range, azimuth, and Doppler frequency, respectively. N is the total number of samples in azimuth direction. Phase error is usually introduced into the data and defocuses imagery. Let φ (k) (k = 0, . . . , N − 1) be a correction function for the error and    N−1 1  2πkn − φ (k) y˜mk exp j (2) z mn = N N k=0

be the corresponding phase-corrected image. With the assumption that the phase error is quadratic (per sub-aperture), the correction function becomes φ (k) = a (k − k0 )2

(3)

where k0 is a known constant representing the nominal center of the aperture. a is the coefficient determining curvature of the quadratic curve and is unknown. One way to solve a is to minimize the entropy of a SAR image. The entropy [10] is defined as follows:    |z mn (a)|2 In |z mn (a)|2 + In E Z (4) ϕ (a) = −1/E Z

(6)

where amax and amin are possible maximum and minimum values of a. Coefficient b varies within [−1, 1]. (When b = −1, a = amin . If b = 1, then a = amax .) Then, the entropy function (4) can be re-expressed as (b). Approximating (b) using the Chebyshev polynomial [18]–[20], one has  (b) ≈

I 

G i Pi (b)

(7)

i=1

with

II. AUTOFOCUS P ROBLEM A sample value of a SAR image after an inverse discrete Fourier transform can be written as follows [10]:   N−1  1 j 2πkn ymn = y˜mk · exp (1) N N

amax + amin amax − amin + b 2 2

a=

⎧ ⎪ ⎪ ⎪ ⎨ G0 =

1 I +1

⎪ ⎪ ⎪ ⎩ Gi =

2 I +1

I

  xp

p=0 I

 

 p+1)iπ  x p cos (2 2I +2

p=0

(8)

Pi (b)(i = 1, 2, . . . , I ) is the i th coefficient of the Chebyshev polynomial and x p ( p = 0, 1, 2, . . . I ) is the pth Chebyshev node. General expressions of Pi (b) and x p can be found in [18]. As a tradeoff between the accuracy and computational efficiency, the approximation up to the fourth-order terms (I = 4) is selected in solving the entropy of (4). Thus, P0 (b) to P4 (b) are ⎧ P0 (b) = 1 ⎪ ⎪ ⎪ ⎪ ⎨ P1 (b) = b P2 (b) = 2b2 − 1 (9) ⎪ ⎪ ⎪ P3 (b) = 4b3 − 3b ⎪ ⎩ P4 (b) = 8b4 − 8b 2 + 1. The pth Chebyshev node is

 2p + 1 π . x p = cos 2I + 2 

(10)

After the substitution of (8) and (9) into (7), (b) becomes      (b) ≈ G 0 + G 1 b + G 2 2b2 − 1 + G 3 4b3 − 3b   +G 4 8b 4 − 8b 2 + 1 = (G 0 − G 2 + G 4 ) + (G 1 − 3G 3 ) b + (2G 2 − 8G 4 ) b 2 + 4G 3 b3 + 8G 4 b4 = μ0 + μ1 b + μ2 b 2 + μ3 b 3 + μ4 b 4

(11)

m,n

where E Z = m,n |z mn (a)|2 . M is the total number samples in range direction. 0 ≤ m < M. It is well known that the smaller the entropy value, the better the image quality in focus [16]. Thus, the goal is to find aopt_r = arg min ϕ (a) or

 (5) ϕ  aopt_r = 0.

with

Unfortunately, it is impossible to solve (5) or (4) analytically. Thus, a function that fits ϕ(a) the best is alternatively sought.

Thus, (b) as the substitute for ϕ(a) is obtained. aopt is derived next.

⎧ μ0 ⎪ ⎪ ⎪ ⎪ μ ⎨ 1 μ2 ⎪ ⎪ μ3 ⎪ ⎪ ⎩ μ4

= G0 − G2 + G4 = G 1 − 3G 3 = 2G 2 − 8G 4 = 4G 3 = 8G 4 .

(12)

XIONG et al.: MINIMUM-ENTROPY-BASED AUTOFOCUS ALGORITHM FOR SAR DATA

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B. Solving aopt Using MSR With (11),  (b) is available. To minimize (b), one needs to find b, hence  (b) = 0 or μ1 + 2μ2 b + 3μ3 b2 + 4μ4 b3 = 0.

(13)

Because (13) is a third-order equation, its roots can be solved. However, they are not algebraically solved directly in this paper because finding the roots can be complicate and time consuming for an entire SAR image. In addition, if further improvement in level of accuracy is desired, higher than fourth-order terms should be included in (7). Thus, (13) became an equation of the fourth or higher order, which cannot be solved mathematically. In general, there are two types of methods to solve such high-order equation. One is the numerical (e.g., Newton) method with iteration. The other one is the MSR without iteration [21], which is used. Let β =  (b) or β = μ1 + 2μ2 b + 3μ3 b2 + 4μ4 b3 .

(14)

The series expansion of the inverse function is given by b = A1 (β − μ1 ) + A2 (β − μ1 )2 + A3 (β − μ1 )3 + · · · (15) Solutions of the first three terms are only needed. Then, after the substitution of (15) into (14), one has the following:   β − μ1 ≈ 2μ2 A1 (β − μ1 ) + 3μ3 A21 + 2μ2 A2 (β − μ1 )2   + 4μ4 A31 + 4μ2 A1 A2 + 2μ2 A3 (β − μ1 )3 . (16) Let both sides of (16) equate by the orders of (β − μ1 )i (i = 1, 2, 3). Then ⎧ ⎨ 2μ2 A1 = 1 3μ3 A21 + 2μ2 A2 = 0 (17) ⎩ 4μ4 A31 + 4μ2 A1 A2 + 2μ2 A3 = 0. Thus, coefficients ( A1 , A2 , and A3 ) of the inverse function are ⎧ 1 ⎪ ⎪ ⎪ A1 = 2μ ⎪ ⎪ 2 ⎪ ⎨ 3μ3 A2 = − 3 (18) 8μ2 ⎪ ⎪ ⎪ ⎪ 9μ23 − 4μ2 μ4 ⎪ ⎪ . ⎩ A3 = 16μ52 Let β = 0. Then, (15) becomes b = A1 (−μ1 ) + A2 (−μ1 )2 + A3 (−μ1 )3 + · · · ≈ A1 (−μ1 ) + A2 (−μ1 )2 + A3 (−μ1 )3 = bopt .

(19)

Finally, with (6) amax − amin amax + amin + bopt aopt = (20) 2 2 is obtained. A brief interpretation of the derived CA-MSRA is graphically shown in Fig. 1. ϕ  (a), the actual but unknown function (a ∈ [amin  , amax ]) is shown as a thick solid line. Because ϕ  aopt_r = 0, aopt_r is the true solution for the phase error model (3). Chebyshev polynomial,  (a) used to

Fig. 1.

aopt estimation.

approximate ϕ(a) is illustrated as a dashed line that intercepts a-axis at aopt . As a comparison, the Taylor approximation, a thin solid line is shown. Because the Chebyshev polynomial is the best in approximating a real function among all polynomial approximations at the same order of polynomial terms [18], intuitively aopt_T derived from the Taylor series should be graphically farther away from aopt_r than aopt should be. In short, aopt is a better replacement for aopt_r than aopt_T is. As an integral component of the CA-MSRA, the dataselection operation based on the level of intensity in each range bin along the azimuth direction is included. This operation is similar to that in the quality PGA algorithm [24], and the objective is to ensure the accuracy level of the estimated phase error and to increase computational efficiency. The operation is expressed as follows: γm =

N−1  1 

∗ ymn · ymn N

(21)

n=0

where γm is the intensity at mth range bin, and * denotes a conjugate operator. After the ranking of the intensity values in individual bins, the maximum value of γmax is identified. Then, a threshold, ηγmax is used for the selection. η is a constant that is between 0 and 1 and is empirically determined by return signals. If a range bin whose intensity value is greater than or equal to ηγmax , the bin is selected otherwise excluded. With the data-selection process, the signal-to-noise ratio increases equivalently. In addition, it is highly likely that the number of selected bins is smaller than that of total bins. Thus, the processing time can be shortened. Finally, the selection becomes unavoidable when data acquisition and image formation are needed in mountainous areas where radar shadows exist. The phase error cannot be evaluated in shadowed areas where there are no radar returns. IV. I MPLEMENTATION OF THE CA-MSRA INTO AN E XISTING SAR DATA P ROCESSOR The existing and available SAR data processor is designed to process raw data with a long aperture. The sub-aperture technique is implemented. The sub-aperture is also needed for the CA-MSRA because when the integration time of a synthetic aperture is long, the assumption that the phase error in one full aperture is quadratic (3) can be questionable for

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Fig. 3. Major procedures to generate a SAR image using estimated phase error and SAR raw data.

Fig. 2. Estimation of phase error for raw data in a full aperture through sub-aperture technique. (a) Full aperture and sub-apertures. (b) Value (×) of aopt0 in each sub-aperture. (c) Estimated phase error shown as the best-fit curve for one full aperture.

autofocusing. (Phase errors estimated in the processing of acquired SAR data later for full apertures are not quadratic.) In addition, the CA-MSRA is new to the processor. Thus, processing procedures within the processor are modified to accommodate it. After the modification, there are two major steps below. A. Estimation of Phase Error in Raw Data for a Full Aperture SAR raw data of one full aperture are subdivided Fig. 2(a). The division into sub-apertures is to ensure the validity of the quadratic phase error model. Two factors influence the division. As the size of a sub-aperture measured by the number of azimuth samples or azimuth number for short increases, the accuracy of estimated aopt decreases. However, the more the number of sub-apertures, the longer the processing time. As a tradeoff, the current selection for the azimuth number per sub-aperture is between 500 and 2000. Additionally, an overlap between adjacent sub-apertures is needed, hence there is no gap when a full aperture is formed. The overlap rate is normally 30%–50% of one sub-aperture. Three sub-apertures are shown in Fig. 2(a) as an example. They are slightly displaced in y-direction (range) to show individual ones and overlaps. Then, after deramping operation [22] for each sub-aperture, we use the CA-MSRA to estimate aopt sub-aperture by subaperture.

When a range swath is wide, phase error (in azimuth) can also be range-variant. Then, SAR data are usually subdivided into numerous narrow range strips. Thus, the phase error is considered as range-invariant in each strip. The error model of (3) is valid within every sub-area (narrow strip in range × sub-aperture in azimuth). Then, a set of aopt s using (4)–(20) per sub-aperture is obtained. However, aopt s is not typically implemented in data processing because of the concern of computational efficiency [23]. Instead, with the geometrybased method [23], aopt0 representing all aopt s the best per sub-aperture is derived and shown as a × in Fig. 2(b). To avoid possible discontinuity in the phase error estimation between adjacent sub-apertures or eventually within the full aperture, we do not use aopt0s of sub-apertures directly to model the phase error. Another fitting method (e.g., a polynomial fitting) is used to derive aopt0s as a function of the azimuth number. The function is the best fit to all aopt0s within the full aperture Fig. 2(c). With the fitting function, one is ready to correct the phase error and then to focus the data in one full aperture.

B. Generation of a SAR Image The phase error and raw data are the first input into the motion compensation procedure of step one [23], where the nonsystematic range cell migration caused by the phase error is removed. Next, range compression and correction of the range cell migration are performed in range dimension. In the current SAR data processor, the chirp scaling algorithm [16] is implemented. Then, coupled with the estimated phase error, the data are further corrected for variable phase errors in each range bin. The geometry-based method similar to the one in [23] is used in the second step of the motion compensation. Finally, the data are processed in azimuth with the azimuth compression and correction for the azimuth distortion before the formation of a SAR image (Fig. 3).

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Phase error (radian)

40

Simulated phase error This method

30

20

10

0 0

500

1000

1500

2000

Azimuth number

Fig. 4. image.

Case I. (a) Well-focused SAR image. (b) Simulated and estimated phase errors as functions of azimuth number. (c) Defocused image. (d) Restored

V. R ESULTS AND D ISCUSSIONS A. Addition and Removal of Phase Error to Existing Data Two SAR images are downloaded from Sandia National Laboratories. Each image is 1600 × 2000 (ground range × azimuth) pixels with a pixel resolution of 0.1 × 0.1 m. The SAR is Ku-band. The reasons to analyze the images include that they are publically available and are well focused. After transforming one image into frequency domain in azimuth, we multiply exp[ja(k−k0 )2 ] to the frequency-domain image to add quadratic error. a indicates the degree of defocus caused by the phase error. The larger the value of a, the severer the degree in defocus. Then, the CA-MSRA is used to estimate and remove the error. Finally, after an inverse Fourier transform in azimuth, the image is restored (in azimuth time domain). 1) Restoration of SAR Images: Case I: The identification number of the image is MiniSAR20050519p0003image003. The image in magnitude covered areas around one building inside the Laboratories Fig. 4(a). The building is in the middle, and cars are parked around the building. Roads and trees are clearly identifiable. The image is well focused and its entropy value is 8.3175. Then, the phase error with a = 138.0 and k0 = 1000 is simulated and shown as a dashed line Fig. 4(b). At the pulse number of 0 (the left-most

location) or 2000 (the right-most place), the maximum error is ∼35 rad. After the addition of the error, the image became obscured Fig. 4(c) and has an entropy value of 11.0881. The building, cars, roads, or trees are no longer recognizable. Next, the estimated phase error from the defocused image is 138.2140 · (k − 1000)2 with (0 ≤ k < 2000) using the CA-MSRA. The error is shown as a solid line in Fig. 4(b). Both estimated and simulated errors are almost identical because estimated a is only 0.2140 off the simulated one. After the correction using the estimated phase error curve, the image is restored Fig. 4(d). The image is visually well focused and is very similar to the original one. The entropy value of Fig. 4(d) is 8.3178 that is off only 0.0003 as compared with the original entropy value. It should be noted that because of the presence of strong scatterers (e.g., cars in parking lots, and metallic air vents and air conditioning units on rooftops) in the scene, the scatterers are bright signatures in the defocused image. They should be helpful in image restoration for a nonmetrics-based autofocus approach such as the PGA algorithm. Case II: The second image is MiniSAR20050519p0006 image004. The image is near the clubhouse of the golf course on the Kirtland Air Force Base, New Mexico Fig. 5(a). The house is near the top of the imagery. A teeing area is also near the house and a putting green on the left; both are roughly elliptical in shapes. Trees and canopies are clearly identified

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Phase error (radian)

800

Simulated phase error This method

600

400

200

0 0

500

1000

1500

2000

Azimuth number

Case II. (a) Well-focused SAR image. (b) Simulated and estimated phase errors versus azimuth number. (c) Defocused image. (d) Restored image. 13.32 8.55

Estimated Actual

13.31

Image entropy

and they are scattered around the house and along the sides of fairways. The entropy of the original image is 13.2638. Phase error with a = 2768.0 and k0 = 1000 is added and shown as a dashed line in Fig. 5(b). At azimuth number of 0 or 2000, the phase error is near 700 rad. (The error is much larger than that in Case I.) With the added error, the image became severely degraded and has an entropy value of 14.1916. Nothing is recognizable. Then, the estimated phase error shown as a solid line Fig. 5(b) using the CA-MSRA is 2767.8531 · (k − 1000)2 with (0 ≤ k < 2000). As the assumed and estimated as are almost the same, the simulated and estimated phase errors as functions of the azimuth number are nearly identical. Finally, the restored image is obtained Fig. 5(d). Its entropy value is 13.2639, which is only 0.0001 off that of the original image. 2) Impact of Coarsely Estimated amin and amax on aopt Measured by Entropy Values: aopt is the approximation of aopt_r (Fig. 1). In the search for aopt through b in (6), amin and amax are introduced. Both are needed to be determined. Generally, amin and amax could be coarsely estimated from the SAR imaging geometry [4]. In Case I, amin and amax are 128 and 152, respectively. Using the derived entropy values, we next investigated the impact of the roughly estimated amin and amax on aopt and ultimately on aopt_r . As aopt_r is unknown, one brute force way to find it for an image is to compute entropy values within [amin , amax ] using a small incremental step. The step is 0.1. The calculated values

Image entropy

Fig. 5.

8.5 8.45 8.4 8.35 8.3 125

13.3 13.29 13.28 13.27

130

135

140

a

Fig. 6.

Estimated Actual

145

150

13.26 2750

2760

2770

2780

a

Image entropy values versus a in (a) Case I and (b) Case II.

using (4) as a function of a is shown as a dotted line Fig. 6(a). The entropy value reached the minimum of 8.3172 at a = 138. Thus, aopt_r could be interpreted as being 138. For each a, a corresponding b is obtained (6). Thus, values of (b) derived from (6) and (11) are computed. The relationship between the entropy values and as is shown as a solid line Fig. 6(a). Also at a = aopt = 138, the minimum value of 8.3172 occurred. In addition, there is a general agreement Fig. 6(a) in comparison with the actual and estimated entropy values. Especially, both dotted and solid lines are almost identical when a ∈ [128, 152]. Although discrepancy between two lines might exist when a is between amin and 128 or between 152 and amax , the minimization process of (b) eventually converges when a or aopt is near 138 (Table I). Therefore,

XIONG et al.: MINIMUM-ENTROPY-BASED AUTOFOCUS ALGORITHM FOR SAR DATA

TABLE I E NTROPY VALUES N EAR E ACH aopt T HAT I S S HOWN IN B OLD F ONT Case I a 137.8 137.9

TABLE II C OMPARISONS OF CA-MSRA AND PGA A LGORITHM Image Entropy Values

Case II

Entropy Value

a

Entropy Value

8.3173

2767.8

13.2630

8.3172

1725

2767.9

Time (s)

Iteration in Phase Error Estimation

Case I

Case II

13.2630

CA-MSRA

8.3178

13.2639

1.15

0

PGA algorithm

8.3181

13.2679

15.87

8

138.0

8.3172

2768.0

13.2630

138.1

8.3172

2768.1

13.2630

138.2

8.3173

2768.2

13.2630

coarse estimation of amin or amax had little influence on the search and find aopt in the CA-MSRA. In Case II, amin is 2752 and amax 2778. Similar to Case I, entropy values are computed between 2752 and 2778 with 0.1 as an incremental step and shown as a dotted line Fig. 6(b). At a = aopt_r = 2768, the smallest entropy value of 13.2630 is observed. Also, entropy values of (b) are calculated using (6) and (11). The entropy reached its smallest value of 13.2630 at a = 2768. Relationship between the entropy values and as is plotted as a solid line. In comparison with both dotted and solid lines, they are closely in agreement when a ∈ [2755, 2775]. Difference existed when a ∈ [2750, 2755) or a ∈ (2775, 2780]. Again, aopt could be found through the minimization process of the entropy values of (b) (Table I), and is not influenced by the approximately estimated amin or amax . It should be caution that the phase error in a full aperture is simulated as being quadratic. If the third- or higher-order terms in phase error exist in a full aperture or even a subaperture, the search for the entropy that is global minimum could be adversely impacted. The search in the CA-MSRA could be trapped in a local minimum caused by terms higher than the second order. The chance of such occurrence should be low because of the dominant influence of the second-order term on the phase error. As a precaution to avoid the trap, additional entropy values on both sides of aopt are computed. All entropy values are compared, and aopt is updated to a with the smallest entropy value. B. Effectiveness and Computational Efficiency In assessment of the effectiveness of the CA-MSRA, results using the PGA algorithm [7] are studied because the PGA algorithm is one of the most and widely used autofocus methods. Entropy values computed from the restored image in Case I using both CA-MSRA and PGA algorithm are tabulated (Table II). The entropy value from the CA-MSRA is 0.0003 less than that of the PGA algorithm. The performance of either method measured by the entropy value is almost the same. Thus, both methods are nearly effective. Similar analysis for Case II is done. Assessed by entropy values, the performance of the CA-MSRA is only marginally better than that of the PGA algorithm. Both methods should be effective and acceptable. Of an image with 1600 × 2000 pixels (Case I or II) and with the Intel Celeron 2.20 GHz processor, it took 1.15 s for the CA-MSRA to complete the restoration, and 15.87 s for

the PGA algorithm. No iteration in the phase error estimation using the CA-MSRA is attributed to the major cause for a short processing time. In contrast, eight iterations are used in the PGA algorithm to achieve the same level of accuracy. In short, the CA-MSRA and PGA algorithm have almost the same level of accuracy. However, the CA-MSRA is faster than the PGA algorithm. C. Analysis of Acquired SAR Raw Data With promising results in simulations, we then implement the CA-MSRA into an existing SAR processor. To evaluate CA-MSRA after its implementation, we analyze two SAR raw datasets acquired in strip-map mode. As the data cover a large spatial extent, the imaged area is divided into narrow strips in range and sub-apertures in azimuth. Each strip consists of 256 bins. There are 1024 samples in each supaperture. Thus, M is 256 and N 1024. Each sub-area is 256 (range) × 1024 (azimuth). The first analyzed dataset is collected by an X-band SAR onboard a Y-7 transport aircraft (http://en.wikipedia. org/wiki/Xian_Y-7). The area imaged is the countryside of Shannxi Province, China. The area is predominantly covered by agricultural fields with villages and patches of forested areas scattered around. A highway roughly from left-to-right went through the area. An onramp is near the upper left corner. The scene of raw data is divided into 33 narrow strips and 83 sub-apertures in processing. (The overlap between adjacent sub-apertures is 50% of one sub-aperture.) After estimation of the phase error in each sub-area, and curve fittings for errors of sub-areas (in range) per sub-aperture and for errors of sub-apertures of an entire aperture (in azimuth) or scene, the estimated phase errors of the raw data are shown in Fig. 7. The azimuth number is 42 496. The error varied in azimuth, and the magnitude of the variation is ∼350 rad. Then, with the estimated error, an SAR image is created. Fig. 8 shows the image after a two-look operation in azimuth to reduce speckle. The image is 8448 (slant range) × 21 248 (azimuth) with a resolution of 0.3 × 0.3 m. The image is well focused because targets such as agricultural fields, villages, forests, and roads are clearly delineated. To further study the degree in focus with the estimated phase error, we extracted a small portion of the entire image outlined by a rectangle (Fig. 8). Fig. 9(a) shows the area. Individual trees and their shadows are identifiable and are scattered near upper left of the image. Trees, villages, and a segment of the highway are evidently delineated near middleto-bottom part of the image. As a comparison, the SAR image covering the same area without the use of estimated

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200

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Processed SAR image of countryside, Shannxi Province of China. Fig. 11.

Fig. 9. Images (a) with and (b) without use of phase error estimated by CA-MSRA.

phase error in the current SAR data processor is produced Fig. 9(b). The CA-MSRA is effective in focusing because the defocusing is obviously in Fig. 9(b). The entropy value of Fig. 9(a) is 15.8691, whereas the value of Fig. 9(b) is 16.1012. The entropy value increased 0.2321 due to defocusing. The increase would be larger if the areas near the lower right corner (Fig. 9) are extracted and then entropy value computed. The severest defocusing could occur there. It is well known that topographic variations (e.g., elevation values, slope and aspect facing) in mountains coupled with the side-looking imaging geometry of a SAR sensor make the formation of a SAR image more challenging from the rugged terrain than plain. Layover and foreshortening distorts the imaging geometry, and no radar echoes return from shadowed areas. Thus, the estimation of the phase error caused by motion error is difficulty or even impossible. In addition, the atmospheric turbulence is typically strong near or above the peak of a high mountain. Once an airborne SAR is not

Image from an UAV SAR in rugged terrain.

flying far above the peak, the turbulence can negatively affect the flight attitude greatly. Motion error can be additionally introduced. Raw data collected by a SAR sensor onboard an unmanned aerial vehicle (UAV) is analyzed next. The wavelength of the SAR is Ku-band. The range gate and platform speed are provided. Both parameters are needed in forming a focused SAR image. Other attitude parameters such as accelerations of the UAV in air (x, y, and z) are not available. The imaged area of Shannxi Province of China consists of mountains, plains at feet of the mountains, and a river that flows through the area. In data processing, the raw data are divided into 24 narrow strips in range and 120 sub-apertures in azimuth because of the large spatial extent. With the CA-MSRA, the phase error for the full aperture is estimated (Fig. 10). The error varied by ∼1000 rad. Then, the motion error is compensated for with the estimated error. A SAR image with 6144 (slant range) × 10 240 (azimuth) is shown (Fig. 11) after a six-look operation in azimuth. The image resolution is 1 × 1 m. The image is well focused. From top to bottom, the mountains, the valleys with farmland, forested areas and villages, and the riverbank, stream channel and riverbank are clearly identified. As the final comparison, Fig. 12 is a close look of a small mountainous area outlined by a rectangle (Fig. 11). Visually, there is significant deterioration in focus without phase error compensation (Fig. 12). The entropy values increased from 14.8744 in Fig. 12(a) to 15.2032 in Fig. 12(b). Before concluding this paper, we briefly make three remarks. First, the phase error varies ∼1000 rad from the UAV

XIONG et al.: MINIMUM-ENTROPY-BASED AUTOFOCUS ALGORITHM FOR SAR DATA

Fig. 12. Close look of small area (a) after and (b) before compensation for phase error.

data (Fig. 10), which is about three times larger than that of the Y-7 data (Fig. 7). The curve in the former (Fig. 10) is more complicated than that of the latter (Fig. 7). The UAV that is usually less stable than the fixed-wing airplane and a full aperture of 61 440 in azimuth for the UAV data versus that of 42 496 in azimuth for the Y-7 data, as well as possible strong air turbulence in mountainous areas may be attributed as major causes for the differences. Second, no attitude data are used in forming both images (Figs. 8 and 11). Because of sizes and weights of the global positioning system and inertial navigation system units, none is onboard the UAV. Although attitude data of the SAR sensor onboard the Y-7 are collected, they are not provided. Finally, the unstable UAV (as well as aircraft sometimes) can lead to antenna vibration and unsystematic maneuver, which are other sources for phase error. A severe vibration and/or unsystematic maneuver can potentially invalid the assumption that the phase error in one sub-area is quadratic. Fortunately, it is not the case for the SAR raw data because a quadratic phase error is assumed and used, and two well-focused images are obtained. Finally, the quadratic phase error assumption was used in [22] and [23]. The authors reported satisfactory results as well. VI. C ONCLUSION A new autofocus approach was developed to compensate for phase error caused by the motion error in SAR raw data. With the phase error modeled in quadratic form in each subaperture, the phase error was estimated through the minimization of an objective function that was based on entropy values. The function was approximated as a Chebyshev polynomial equation. With the MSR, coefficients of the equation were solved. Then with a curve-fitting method for phase errors of all sub-apertures, the phase error was estimated in a full aperture. It should be noted that the sub-aperture technique was needed in the CA-MSRA, hence, the validity of the quadratic phase error model in each sub-aperture can be guaranteed. In addition, if a wide range swath was imaged, division of the swath into narrow range strips was required. Therefore, the range-variant phase error can be processed as range-invariant in each strip. As shown in this paper, one advantage of the CA-MSRA was to use the CA to mimic the objective function because at the same order of polynomial terms, the approximation was the most accurate when compared with other polynomial approximations. The use of the MSR to solve the polynomial

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coefficients of the objective function consisting of high-order terms was another advantage because the procedure was computationally efficient and the solution was accurate. The accuracy level could further increase by including additional higher-terms in the CA and MSR. In simulation studies, the accuracy level of the CA-MSRA was slightly better than that of the PGA algorithm based on entropy values. Because no iteration was required in phase error estimation, the method is faster than the PGA algorithm. Thus, the method should be very attractive in terms of computational efficiency and satisfactory products in accuracy. After the implementation of the CA-MSRA into an existing SAR data processor, SAR imagery was produced using collected SAR raw data. The imaged area included plain or mountain. The estimated phase error caused by the motion error was ∼ 1000 rad. The spatial extent was over 8000 (slant range) × 60 000 (azimuth). After the compensation for phase error, the high-resolution (up to 0.3 × 0.3 m) images were well focused. Detailed delineation of individual targets (e.g., trees, villages, agricultural fields, highways, roads, and mountain ranges) was achieved. R EFERENCES [1] C. V. Jakowatz, Jr., D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach. Boston, MA, USA: Kluwer, 1996. [2] W. D. Brown and D. C. Ghiglia, “Some methods for reducing propagation-induced phase errors in coherent imaging systems—I: Formalism,” J. Opt. Soc. Amer. A, vol. 5, no. 6, pp. 924–942, Jun. 1988. [3] D. C. Ghiglia and W. D. Brown, “Some methods for reducing propagation-induced phase errors in coherent imaging systems—II: Numerical results,” J. Opt. Soc. Amer. A, vol. 5, no. 6, pp. 943–957, 1988. [4] C. E. Mancill and J. M. Swiger, “A map drift autofocus technique for correcting higher-order SAR phase errors,” in Proc. 27th Annu. TriService Radar Symp., Jun. 1981, pp. 391–400. [5] C. V. Jakowatz and D. E. Wahl, “Eigenvector method for maximumlikelihood estimation of phase errors in synthetic-aperture-radar imagery,” J. Opt. Soc. Amer. A, vol. 10, no. 12, pp. 2539–2546, Dec. 1993. [6] P. H. Eichel, D. C. Ghiglia, and C. V. Jakowatz, Jr., “Speckle processing method for synthetic-aperture-radar phase correction,” Opt. Lett., vol. 14, no. 1, pp. 1101–1103, Jan. 1989. [7] D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and C. V. Jakowatz, Jr., “Phase gradient autofocus-A robust tool for high resolution SAR phase correction,” IEEE Trans. Aerosp. Electron. Syst., vol. 30, no. 7, pp. 827–835, Jul. 1994. [8] P. Tsakalides and C. L. Nikias, “High-resolution autofocus techniques for SAR imaging based on fractional lower-order statistics,” Proc. Inst. Electr. Eng. Radar, Sonar, Navigat., vol. 148, no. 5, pp. 267–276, Oct. 2001. [9] J. R. Fienup and J. J. Miller, “Aberration correction by maximizing generalized sharpness metrics,” J. Opt. Soc. Amer. A, vol. 20, no. 4, pp. 609–620, Apr. 2003. [10] T. J. Kragh, “Monotonic iterative algorithm for minimum-entropy autofocus,” in Proc. 14th Annu. ASAP Workshop, Jun. 2006, pp. 1–38. [11] L. Xi, L. Guosui, and J. Ni, “Autofocusing of ISAR images based on entropy minimization,” IEEE Trans. Aerosp. Electron. Syst., vol. 35, no. 10, pp. 1240–1252, Oct. 1999. [12] M. P. Hayes and S. A. Fortune, “Recursive phase estimation for image sharpening,” in Proc. Present. Image Vis. Comput., 2005, pp. 1–6. [13] R. G. Paxman and J. C. Marron, “Aberration correction of speckled imagery with an image sharpness criterion,” Proc. SPIE, vol. 0976, pp. 37–47, 1988. [14] F. Berizzi and G. Corsini, “Autofocusing of inverse synthetic aperture radar images using contrast optimization,” IEEE Trans. Aerosp. Electron. Syst., vol. 32, no. 7, pp. 1185–1191, Jul. 1996.

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[15] R. L. Morrison and D. C. Munson, “An experimental study of a new entropy-based SAR autofocus technique,” in Proc. IEEE Int. Conf. Image Process., vol. 2. Jan. 2002, pp. 441–444. [16] H. Erdogan and J. A. Fessler, “Monotonic algorithms for transmission tomography,” IEEE Trans. Med. Imag., vol. 18, no. 9, pp. 801–814, Sep. 1999. [17] W. Y. Yang, W. Cao, T.-S. Chung, and J. Morris, Applied Numerical Methods Using MATLAB. New York, NY, USA: Wiley, 2005. [18] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, the Art of Scientific Computing. Cambridge, U.K.: Cambridge Univ. Press, 1992. [19] G. A. Watson, Approximation Theory and Numerical Methods. New York, NY, USA: Wiley, 1980. [20] S. D. Conte and C. Boor, Elementary Numerical Analysis: An Algorithm Approach. New York, NY, USA: McGraw-Hill, 1980. [21] P. M. Morse and H. Feshbch, Method of Theoretical Physics, 1st ed. New York, NY, USA: McGraw-Hill, 1953. [22] W. G. Carrara, R. S. Goodman, and R. M. Majewski, Spotlight Synthetic Aperture Radar: Signal Processing Algorithm. Boston, MA, USA: Artech House, 1995. [23] M. Xing, X. Jiang, R. Wu, F. Zhou, and Z. Bao, “Motion compensation for UAV SAR based on raw radar data,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 8, pp. 2870–2883, Aug. 2009. [24] H. L. Chan and T. S. Yeo, “Non-iterative quality phase-gradient autofocus (QPGA) algorithm for spotlight SAR imagery,” IEEE Trans. Geosci. Remote Sens., vol. 36, no. 5, pp. 1531–1539, Sep. 1998.

Yong Wang received the Ph.D. degree from the University of California, Santa Barbara, CA, USA, in 1992, focus on synthetic aperture radar (SAR) and its application in forested environments. He is currently a Faculty Member with the University of Science and Technology of China, Hefei, China, and East Carolina University, NC, USA. His current research interests include the application of remotely sensed and geo-spatial datasets to environments, natural hazards and air pollution, and is firmly couched within coastal areas of the U.S. and China, and in the Sichuan basin, China.

Tao Xiong was born in Hubei, China, in January 1984. He received the Masters degree in control technology and instrument from Xidian University, Xi’an, China, in 2006. He is currently with the Key Laboratory of Intelligent Perception and Image Understanding of the Ministry of Education of China, Xidian University. His current research interests include imaging of several SAR modes and autofocus.

Jialian Sheng received the B.S. degree in biomedical engineering from Xidian University, Xi’an, China, in 2010. She is currently pursuing the Ph.D. degree with the National Key Laboratory of Signal Processing, Xidian University. Her current research interests include inverse synthetic aperture radar (ISAR) imaging and sparse optimization with applications in the field of ISAR super-resolution imaging.

Mengdao Xing (M’04) was born in Zhejiang, China, in November 1975. He received the B.S. and Ph.D. degrees in electrical engineering from Xidian University, Xi’an, China, in 1997 and 2002, respectively. He is currently a Full Professor with the National Laboratory of Radar Signal Processing, Xidian University. He is also with the National Key Laboratory of Microwave Imaging Technology, Institute of Electronics, Chinese Academy of Sciences, Beijing, China. He has authored or co-authored two books and published over 200 papers. His current research interests include synthetic aperture radar (SAR), inversed SAR, and sparse signal processing.

Shuang Wang was born in Shannxi, China, in 1978. She received the B.S. and M.S. degrees from Xidian University, Xi’an, China, in 2000 and 2003, respectively, and the Ph.D. degree in circuits and systems from Xidian University in 2007. She is currently a Professor with the Key Laboratory of Intelligent Perception and Image Understanding of the Ministry of Education of China, Xidian University. Her current research interests include sparse representation, image processing, and high-resolution SAR image processing.

Liang Guo was born on April 9, 1983. He received the B.S. and Ph.D. degrees in electronic engineering from Xidian University, Xi’an, China, in 2005 and 2009, respectively. He is currently with the School of Technical Physics, Xidian University, as an Associate Professor. His current research interests include signal processing techniques in radar and lidar systems.