- Research Article
- Open Access

# Subinteger Range-Bin Alignment Method for ISAR Imaging of Noncooperative Targets

- JM Muñoz-Ferreras
^{1}Email author and - F Pérez-Martínez
^{2}

**2010**:438615

https://doi.org/10.1155/2010/438615

© J. M. Mu˜noz-Ferreras and F. Pérez-Martínez 2010

**Received:**17 November 2009**Accepted:**25 March 2010**Published:**31 May 2010

## Abstract

Inverse Synthetic Aperture Radar (ISAR) is a coherent radar technique capable of generating images of noncooperative targets. ISAR may have better performance in adverse meteorological conditions than traditional imaging sensors. Unfortunately, ISAR images are usually blurred because of the relative motion between radar and target. To improve the quality of ISAR products, motion compensation is necessary. In this context, range-bin alignment is the first step for translational motion compensation. In this paper, we propose a subinteger range-bin alignment method based on envelope correlation and reference profiles. The technique, which makes use of a carefully designed optimization stage, is robust against noise, clutter, target scintillation, and error accumulation. It provides us with very fine translational motion compensation. Comparisons with state-of-the-art range-bin alignment methods are included and advantages of the proposal are highlighted. Simulated and live data from a high-resolution linear-frequency-modulated continuous-wave radar are included to perform the pertinent comparisons.

## Keywords

- Radar
- Translational Motion
- Alignment Method
- Inverse Synthetic Aperture Radar
- Optimization Stage

## 1. Introduction

Traditional imaging sensors, such as visible and infrared cameras or laser radar systems, may have a reduced performance in adverse weather conditions, like fog [1–3]. Furthermore, in defense and security scenarios, smoke screens [4] may literally blind these imaging sensors based on very short wavelengths.

The origin of this degradation must be found in the extreme scattering that these wavelengths suffer when interacting with the little particles present in the atmosphere [1–3]. When a high signal attenuation is present, the operation range of these sensors diminishes considerably.

However, in important applications related to defense and security, it is still necessary to obtain images for recognition/identification purposes, regardless of the meteorological and scenario conditions.

Inverse Synthetic Aperture Radar (ISAR) is a coherent radar technique which may obtain images of noncooperative targets [5, 6]. Furthermore, these images may be used for subsequent recognition tasks [7–10]. Although ISAR is usually understood as a complement for electro-optical sensors, it may in fact outperform these traditional sensors in adverse conditions, because it inherits the all-weather feature [11] from the long wavelength radars.

The standard scenario for ISAR consists of a static high-resolution coherent radar which illuminates a moving noncooperative target [12]. In this context, a noncooperative target is a target whose motion is unknown.

*c*is the light speed and is the transmitted bandwidth. The cross-range resolution may be written as

where is the transmitted wavelength and is the variation of the target aspect angle during the illumination (observation) time.

Target motion may be divided into a translational component and a rotational component [13, 14]. The first one is further decomposed into a radial and a tangential component, whereas the second one has three attitude components: yaw, pitch, and roll.

On the one hand, the radial component of the translational motion (i.e., the component along the line-of-sight (LOS)) is undesired, because it does not induce variation of the target aspect angle; that is, it does not generate Doppler gradient among target scatterers situated in the same range bin. Furthermore, this component causes a large blurring in ISAR images.

On the other hand, the rest of motion components may produce the desired Doppler gradient among scatterers, hence obtaining bidimensional information. It is true that the rotational motion (and the tangential component of the translational motion) may also generate blurring effects on the image [15], called Migration Through Resolution Cells (MTRCs), but these effects have minor importance in comparison to the large blurring generated by the radial component of the translational motion, which must always be compensated.

Methods for translational motion compensation work in two steps [16]: range-bin alignment and phase adjustment. For the first stage, which is the motivation of this paper, several methods may be found in the literature, such as the peak tracking approach [5], the centroid tracking algorithm [17], the envelope correlation method [5, 18, 19], the global range alignment approach [20], or the minimum-entropy-based technique [21]. For the second stage regarding phase adjustment, the literature provides us with famous methods such as prominent point processing [22], phase gradient autofocus [23], entropy minimization [16], or contrast maximization [24].

In this paper, we concentrate on the range-bin alignment stage, which is fundamental to guarantee a proper translational motion compensation. Concretely, we present a subinteger range-bin alignment approach based on the traditional envelope correlation method and the use of reference profiles. This work was preliminarily presented in the conference paper [19]. Here, deep analyses as well as exhaustive comparisons with other existing methods both for simulated and live data are provided.

The proposed method makes use of reference profiles in order to mitigate the error accumulation phenomenon and the target scintillation effects [18], typical limitations of the earlier range-bin alignment approaches such as the peak and centroid tracking methods. Furthermore, the technique makes a subinteger alignment which provides us with a very fine range profile adjustment. This subinteger alignment approach is based on an optimization stage which has carefully been designed in order to avoid possible convergence to local maxima.

The method is robust against noise, clutter, and target scintillation. Moreover, it properly solves the error accumulation problem. Its performance is similar to the state-of-the-art current methods such as the global range alignment approach [20] and the minimum entropy-based technique [21], although it provides two clear advantages; it properly solves extreme situations with large range shifts from pulse to pulse (unlike the global range alignment algorithm) and it has moreover the ability to produce subinteger fine range adjustments over a wide range of offsets (in contrast to the minimum entropy-based method).

Furthermore, the careful design of the method against local maxima makes it very robust, as shown here for controlled simulated examples for which the state-of-the-art methods have convergence problems.

Simulated and live data from a high-resolution linear-frequency-modulated continuous-wave (LFMCW) radar are used to validate the proposed approach and to make the pertinent comparisons.

## 2. Subinteger Range-Bin Alignment Method

The proposed method uses the cross-correlation of range profiles in order to estimate the misalignment between them. The correlation is not calculated between the current range profile and the previous one, which would generate the undesired error accumulation effect [18]. On the contrary, the cross-correlation is calculated between the current range profile and a reference profile obtained as a combination of the previously aligned range profiles. This reduces the error accumulation effect and provides robustness against noise, clutter, and target scintillation. Moreover, the proposed alignment between the current and the reference profiles may be a fraction of one range bin, providing the method with a subinteger capability. This fine alignment is achieved after an optimization stage which has been designed to minimize possible convergence to local maxima. The following paragraphs describe the method.

Let **p**_{
m
}(*n*) be the *m* th acquired range profile, where
is the total number of range bins and *M* is the number of acquired range profiles. Let us call
as the aligned profile of **p**_{
m
}(*n*), after the alignment process.

**r**

_{m +1}(

*n*) for the alignment of

**p**

_{m +1}(

*n*) as

where **r**_{
m
}(*n*) is the reference profile for the alignment of the *m* th range profile **p**_{
m
}(*n*). Note that the calculation of **r**_{m +1}(*n*) requires the knowledge of the previously aligned range profiles. This clearly demonstrates the iterative nature of the method.

*m*+1)th range profile

**p**

_{m +1}(

*n*) and its associated reference profile

**r**

_{m +1}(

*n*) are available, the objective is to obtain the (

*m*+1)th-aligned range profile . For this purpose, the envelope cross-correlation between

**r**

_{m +1}(

*n*) and a shifted version of

**p**

_{m +1}(

*n*) is defined as

_{m +1}is the range shift applied to the (

*m*+1)th range profile

**p**

_{m +1}(

*n*). The value

_{m +1}is not necessarily an integer, so that

**p**

_{m +1}(

*n*−

*τ*

_{ m }

_{+1}) is calculated by using the shift property of the Fourier transform as follows:

where **n** is the vector [0, 1, …, *N*−1]^{T}.

*m*+1)th reference profile

**r**

_{m +1}(

*n*) and the shifted range profile

**p**

_{m +1}(

*n*−

*τ*

_{ m }

_{+1}). Hence, we are interested in obtaining the optimum shift that maximizes the envelope correlation. Mathematically, this optimum shift may be expressed as

where the range shift of the original profile is implemented by using (5), because this optimum shift may be not an integer.

The envelope correlation shown in Figure 1 shows a standard example which we have found typical for both simulated and real data. It can easily be seen that the objective cost function suffers from local maxima.

As a consequence, it is obvious that we have to pay great attention to the correct initialization of the optimization algorithm in order to guarantee the desired convergence to the global maximum. That is, if the initial guess for the optimization method is near to the global maximum, a correct convergence to it is achievable.

Since the problem under study is an optimization problem with local maxima, it is clear that we could employ blind approaches to solve it, such as genetic algorithms or exhaustive procedures (like a grid method or a random walk).

Nevertheless, according to our observations of our available simulated and real data, we have noticed that the peak corresponding to the global maximum of the cost function is quite wide, as seen in Figure 1. This means that a proper initial guess for the optimum shift is simply the integer range shift for which the cross-correlation is maximum. Hence, we evaluate the cross-correlation for the possible integer shifts , and select the one for which the cost function is maximum as the initial guess for the subsequent standard optimization algorithm. This process lets us finally converge to the desired global maximum. In this paper, a zero-order optimization algorithm (the Nelder-Mead algorithm [25]) has been utilized to obtain the desired subinteger refinement for .

We would like to highlight that the commented method to obtain the initial guess properly worked in all the simulated and real data available to the authors. On the other hand, it is true that other optimization approaches, like a gradient-based method or the Newton method, may have been used for the refinement stage. Our experience is that the zero-order Nelder-Mead algorithm worked properly for the analyzed examples.

As a summary, the following steps implement the proposed technique for ISAR subinteger range-bin alignment.

Step 1 (*m* = 0).

Step 2.

Calculate the (*m*+1)th reference profile **r**_{m +1}(*n*) using (3).

Step 3.

Obtain the envelope correlation (see (4)) between **r**_{m +1}(*n*) and the (*m*+1)th-shifted range profile
, with
_{m +1} being an integer in the interval [
] and *N* the number of range bins.

Step 4.

Calculate the integer value of , from the possible set [0, 1, …, ], that maximizes the result of the previous step. Call this initial value .

Step 5.

Solve (6) with the Nelder-Mead algorithm, taking as the initial guess for the iterative technique. As a result, obtain the optimum range shift .

Step 6.

Obtain using (7). Use (5) if the optimum shift is not an integer.

If , where is number of range profiles, go to Step 2 to align the next range profile.

## 3. Simulation of an LFMCW Radar

In this section, we provide the tools to simulate targets illuminated by an LFMCW radar, because the simulated and real data used in this paper correspond to this type of radar. Next section will detail the properties of the proposed alignment method for this kind of data. Although the results detailed in the paper only refer to LFMCW radars, it is important to highlight that the proposed method is applicable to any coherent imaging radar.

*T*

_{ C }represents the necessary time for the radar circuits to guarantee the signal coherence from ramp to ramp.

*s*

_{ T }(

*t*) may be written as

where
, *t* is the time, *m* is the number of range profile (i.e., the number of ramp), and
is the chirp rate. Note that the received signal in the intervals corresponding to *T* _{
C
} is not processed.

*R*

_{ tk }from the radar, the received signal from the scatterer is

where is a complex value associated to the scatterer, whose amplitude represents the scatterer backscattering and the propagation losses, whereas its phase models a possible phase change inserted by the scatterer.

*K*scatterers, the beat signal after the deramping processing may be written as

*m*. The beat frequency

*f*

_{ tk }for a scatterer situated at a range

*R*

_{ tk }may be written, from (10), as

A correct sampling of (10) provides us with the possibility of simulating complex scenes. This is simply made by calculating the ranges from the radar to all target scatterers for all ramps. For each ramp, the corresponding range profile may be obtained by applying a Fourier transform to (10).

LFMCW radar parameters for the simulated example in Figure 3.

For this simulated example, we have considered that
and
is uniformly distributed between 0 and 2*π*. Moreover, a signal-to-noise ratio of 10 dB has been considered, with the noise being additive, white, and Gaussian.

*v*

_{ r }= 0 m/s. By applying an FFT in each range bin, we obtain the ISAR image in the conventional range-Doppler coordinates (Figure 4(b)). In the context of this paper, Figure 4(b) must be considered as the optimum ISAR image for this simulation, because there is no blurring (i.e., the radial component of the translational motion is zero).

*v*

_{ r }= 10 m/s. The leaning observed in the range profiles and the large blurring in the ISAR image are characteristic effects due to the radial component of the translational motion.

Because the dynamics for noncooperative targets are unknown, the objective of the blind motion compensation techniques consists of focusing the ISAR images without additional information. For our simulated example, the blind techniques should obtain an image similar to Figure 4(b) from processing the data in Figure 5.

Next sections detail the performance of the proposed alignment algorithm in comparison with other existing methods for compensating the translational motion. In order to make fair comparisons among the diverse range-bin alignment techniques, in this paper we always use the method in [16] for the phase adjustment step.

## 4. Properties of the Proposed Method

This section addresses the performance of the proposed range-bin alignment technique in relation to important features: robustness against target scintillation, against clutter, and so forth. Both simulated and real data are used to verify the good performance of the proposed method.

### 4.1. Robustness against Target Scintillation

The signal received by the radar is the coherent sum of many contributions from target scatterers. This implies that the power in each range bin is not constant during the illumination time. This effect is known as target scintillation.

Figure 7(b) shows the motion-compensated ISAR image obtained with the proposed technique for the range-bin alignment stage and with the method in [16] for the phase adjustment stage. Hence, Figure 7(b) is the reconstructed ISAR image after using the proposed technique. Note that this reconstruction is a very good approximation to the optimum ISAR image shown in Figure 4(b).

### 4.2. Robustness against Noise and Clutter

Thermal noise is always present in real systems. The simulated examples shown in this paper include thermal noise. On the other hand, clutter may exist depending on the acquisition scenario. For example, in maritime scenarios, the clutter due to echoes from the sea may be a problem.

Real radar parameters for the acquired data corresponding to the sailboat in Figure 8.

Hence, the proposed range-bin alignment method is robust against high levels of noise and clutter. So far, the drawn conclusions are based on visual inspection. However, we can use focusing indicators to quantify the quality improvement observed in the ISAR image of Figure 11(b). In this context and for this kind of examples, we can use the entropy [16] and the contrast [24], whose mathematical definitions are given in the corresponding references. The lower the entropy, the more focused the ISAR image is [16]. And, the greater the contrast, the more focused the ISAR image is [24].

### 4.3. Robustness against Error Accumulation

The error accumulation phenomenon is an effect observed when using some range-bin alignment methods [18]. The proposed method tries to mitigate this phenomenon by using reference profiles, whose calculation is based on the previously aligned range profiles.

For example, for envelope correlation-based methods, the error accumulation effect takes importance when the alignment of the current range profile is only based on the previously aligned range profile, as shown next.

**r**

_{m +1}(

*n*) for the alignment of the (

*m*+1)th range profile

**p**

_{m +1}(

*n*) as the previously aligned range profile, that is,

**r**

_{m +1}(

*n*) = , according to the nomenclature used in Section 2. In Figure 12(a), some misalignment error accumulates, as clearly shown if we concentrate, for example, on the range history for the scatterers on sailboat stern. These range variations are not observed in Figure 7(a).

The error accumulation effect in Figure 12(a) has minor incidence on the results of Figure 7(a). Hence, the proposed range-bin alignment method based on the use of reference profiles is robust against this phenomenon.

## 5. Comparison with State-of-the-Art Methods

In previous section, we exposed the properties of the proposed alignment method by using both simulated and real data. Here, it is our intention to compare the proposed approach with state-of-the-art methods recently proposed in the literature: concretely, the global range alignment algorithm [20] and the minimum entropy-based approach [21].

The performance of the proposed subinteger range-bin alignment technique is similar to the one for these state-of-the-art methods, as shown next. However, the proposed approach can deal with extreme situations with large-range shifts from range profile to range profile. Moreover, its subinteger alignment capability is also noticeable and, unlike the other methods, the careful design of the optimization stage increases its robustness against possible convergence to local maxima.

Real radar parameters for the acquired data corresponding to the vessel in Figure 13.

### 5.1. Comparison with the Global Range Alignment Algorithm

Hence, the global range alignment algorithm provides a good alignment. In fact, the focusing indicators for Figure 16(b) are almost the same as the ones for Figure 15(b). The entropy and contrast for Figure 16(b) are 7.36 and 39.9, respectively.

However, when we are faced with situations in which large range shifts from range profile to range profile may arise, the global range alignment algorithm may fail, as also indicated in [21].

*a*,

*b*,

*c,*and

*d*are the absolute values of the echoes in the corresponding range bins. An intermediate step in the optimization process of the global range alignment method is depicted in Figure 20(b). Figure 20(c) indicates the correct alignment of the two range profiles.

*ad*>

*ac*+

*db*, then the global range alignment algorithm will not converge to the correct result in Figure 20(c). Instead of that, it will try to refine the situation in Figure 20(a). As a proof of this fact, Figure 21 shows the two range profiles aligned after using the proposed method and the global range alignment algorithm, when

*a*= 3,

*b*= 0.25,

*c*= 0.5, and

*d*= 1. The global range alignment method is clearly affected by the local maximum and does not converge to the situation in Figure 20(c). On the contrary, the proposed method can deal with these cases because of the careful design of its optimization stage.

### 5.2. Comparison with the Minimum Entropy-Based Approach

The minimum entropy-based approach [21] for range-bin alignment is an iterative approach which is based on integer shifts of the range profiles, unlike the proposed method and the global range alignment algorithm.

These discontinuities appearing in the range profiles induce an amplitude modulation in Doppler. This is the reason why spurious images are clearly visible in Figure 22(b). Hence, the ISAR image obtained with the minimum entropy-based approach has a quality poorer than the one in Figure 15(b). The entropy and contrast for the ISAR image in Figure 22(b) (8.01 and 35.3, resp.) are also indicative of this quality decrease. Please refer to Table 6.

We have tried to implement an extension of the minimum entropy-based method in order to consider subinteger range-bin alignments. This has been done in a way similar to the optimization stage given in the proposed method. Unfortunately, the commented jumps still appear when applying this extension.

*a*, ,

*b,*and are the absolute values of the echoes in the corresponding range bins. Let us also assume that we have a little target scintillation, in such a way that we assume and .

*a*= 3.1,

*b*= 1, = 3, and = 1.1. These values satisfy (13). Figure 25(b) shows the two range profiles after applying the minimum entropy-based approach for this case. As predicted, the method is unable to align the two range profiles. On the contrary, the proposed method can align the two range profiles, as shown in Figure 25(a). Again, we would like to highlight that the optimization stage of the proposed method has carefully been designed.

## 6. Conclusions

The traditional imaging sensors, such as cameras or laser radars, may have a reduced performance in adverse meteorological conditions or in difficult scenarios where, for example, smoke screens are present. ISAR is an all-weather radar technique which may provide images of noncooperative targets in such adverse environments. Hence, such images are interesting for defense and security applications. Furthermore, the ISAR images may be exploited for subsequent recognition/identification tasks.

Unfortunately, the standard ISAR images are usually blurred because of the target motion. Motion compensation techniques should be applied in order to have focused ISAR images. Generally, it is at least necessary to compensate the radial component of the translational motion. To achieve this, the methods for translational motion compensation work in two stages: range-bin alignment and phase adjustment.

In order to increase the quality of the ISAR images, the range-bin alignment step must properly be designed. In this paper, we have proposed a range-bin alignment method based on the envelope correlation between the range profiles and their corresponding reference profiles, calculated as a combination of the previously aligned range profiles. Furthermore, the method achieves an accurate subinteger refinement for the range profile alignment. This subinteger adjustment is based on an optimization stage which has carefully been designed in order to avoid convergence to undesired local maxima.

The paper addresses the performance of the proposed algorithm in an exhaustive manner, by using both simulated and real data from LFMCW radars. In this context, it has been shown that the method is robust against target scintillation, noise, and clutter. Its robustness against the error accumulation effect has also been verified.

On the other hand, the proposed method has also been compared with recently proposed state-of-the-art range-bin alignment methods, such as the global range alignment algorithm and the minimum entropy-based approach. We have verified that the subinteger feature of the proposed method provides us with extremely accurate range-bin alignments, in contrast to the minimum entropy-based approach. It has also been shown that the method may deal with large range shifts from range profile to range profile, unlike the global range alignment algorithm. Finally, the careful design of the proposed optimization stage has been highlighted. We have addressed simple simulated examples in which both the global range alignment algorithm and the minimum entropy-based technique have problems with local maxima.

The proposed algorithm is robust in many scenarios and is hence a very interesting alternative for the range-bin alignment stage in the task of ISAR translational motion compensation. The improved obtained ISAR images may be of interest for subsequent automatic target recognition methods.

## Declarations

### Acknowledgments

This work was financially supported by the Spanish National Board of Scientific and Technology Research under Project TEC2008-02148/TEC. The authors thank Dr. A. Blanco-del-Campo, Dr. A. Asensio-López, and Dr. B. P. Dorta-Naranjo for providing the live data of the sailboat and the vessel.

## Authors’ Affiliations

## References

- Hovanessian SA:
*Introduction to Sensor Systems*. Artech House, Boston, Mass, USA; 1988.Google Scholar - Jelalian AV:
*Laser Radar Systems*. Artech House, Boston, Mass, USA; 1992.Google Scholar - Osche GR, Young DS: Imaging laser radar in the near and far infrared.
*Proceedings of the IEEE*1996, 84(2):103-125. 10.1109/5.482225View ArticleGoogle Scholar - Chen H-Y, Tarn I-Y, Hwang Y-J: Infrared extinction of the powder of brass 70Cu/30Zn calculated by the FDTD and turning bands methods.
*IEEE Transactions on Geoscience and Remote Sensing*1995, 33(6):1321-1324. 10.1109/36.477188View ArticleGoogle Scholar - Chen C-C, Andrews HC: Target motion induced radar imaging.
*IEEE Transactions on Aerospace and Electronic Systems*1980, 16(1):2-14.View ArticleGoogle Scholar - Ausherman DA, Kozma A, Walker JL, Jones HM, Poggio EC: Developments in radar imaging.
*IEEE Transactions on Aerospace and Electronic Systems*1984, 20(4):363-400.View ArticleGoogle Scholar - Kim K-T, Seo D-K, Kim H-T: Efficient classification of ISAR images.
*IEEE Transactions on Antennas and Propagation*2005, 53(5):1611-1621.View ArticleGoogle Scholar - Shreyamsha Kumar BK, Prabhakar B, Suryanarayana K, Thilagavathi V, Rajagopal R: Target identification using harmonic wavelet based ISAR imaging.
*EURASIP Journal on Applied Signal Processing*2006, 2006:-13.Google Scholar - Radoi E, Quinquis A, Totir F: Supervised self-organizing classification of superresolution ISAR images: an anechoic chamber experiment.
*EURASIP Journal on Applied Signal Processing*2006, 2006:-14.Google Scholar - Musman S, Kerr D, Bachmann C: Automatic recognition of ISAR ship images.
*IEEE Transactions on Aerospace and Electronic Systems*1996, 32(4):1392-1404. 10.1109/7.543860View ArticleGoogle Scholar - Skolnik MI:
*Introduction to Radar Systems, McGraw Hill Higher Education*. 3rd edition. McGraw Hill, New York, NY, USA; 2001.Google Scholar - Wehner DR:
*High Resolution Radar*. 2nd edition. Artech House, Boston, Mass, USA; 1995.Google Scholar - Chen VC, Miceli WJ: Simulation of ISAR imaging of moving targets.
*IEE Proceedings: Radar, Sonar and Navigation*2001, 148(3):160-166. 10.1049/ip-rsn:20010384View ArticleGoogle Scholar - Chen VC:
*Time-Frequency Transforms for Radar Imaging and Signal Analysis*. Artech House, Boston, Mass, USA; 2002.MATHGoogle Scholar - Walker JL: Range-Doppler imaging of rotating objects.
*IEEE Transactions on Aerospace and Electronic Systems*1980, 16(1):23-52.View ArticleGoogle Scholar - Xi LI, Liu G, Jinlin N: Autofocusing of ISAR images based on entropy minimization.
*IEEE Transactions on Aerospace and Electronic Systems*1999, 35(4):1240-1252. 10.1109/7.805442View ArticleGoogle Scholar - Muñoz-Ferreras JM, Calvo-Gallego J, Pérez-Martínez F, Blanco-del-Campo A, Asensio-López A, Dorta-Naranjo BP: Motion compensation for ISAR based on the shift-and-convolution algorithm.
*Proceedings of the IEEE Conference on Radar, April 2006, Verona, NY, USA*366-370.Google Scholar - Delisle GY, Wu H: Moving target imaging and trajectory computation using ISAR.
*IEEE Transactions on Aerospace and Electronic Systems*1994, 30(3):887-899. 10.1109/7.303757View ArticleGoogle Scholar - Muñoz-Ferreras JM, Pérez-Martínez F: Extended envelope correlation for range bin alignment in ISAR.
*Proceedings of the IET International Conference on Radar Systems, October 2007, Edinburg, UK*65-68.Google Scholar - Wang J, Kasilingam D: Global range alignment for ISAR.
*IEEE Transactions on Aerospace and Electronic Systems*2003, 39(1):351-357. 10.1109/TAES.2003.1188917View ArticleGoogle Scholar - Zhu D, Wang L, Yu Y, Tao Q, Zhu Z: Robust ISAR range alignment via minimizing the entropy of the average range profile.
*IEEE Geoscience and Remote Sensing Letters*2009, 6(2):204-208.View ArticleGoogle Scholar - Steinberg BD: Microwave imaging of aircraft.
*Proceedings of the IEEE*1988, 76(12):1578-1592. 10.1109/5.16351View ArticleGoogle Scholar - Wahl DE, Eichel PH, Ghiglia DC, Jakowatz CV Jr.: Phase gradient autofocus—a robust tool for high resolution SAR phase correction.
*IEEE Transactions on Aerospace and Electronic Systems*1994, 30(3):827-835. 10.1109/7.303752View ArticleGoogle Scholar - Martorella M, Berizzi F, Haywood B: Contrast maximisation based technique for 2-D ISAR autofocusing.
*IEE Proceedings: Radar, Sonar and Navigation*2005, 152(4):253-262. 10.1049/ip-rsn:20045123Google Scholar - Nelder JA, Mead R: A simplex method for function minimization.
*The Computer Journal*1965, 7: 308-313.View ArticleMATHGoogle Scholar - Blanco-del-Campo A, Asensio-López A, Dorta-Naranjo BP,
*et al*.: Millimeter-wave radar demonstrator for high resolution imaging.*Proceedings of the 1st European Radar Conference (EuRAD '04), October 2004, Amsterdam, The Netherlands*65-68.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.