In this section, the workflow of the proposed range alignment method is discussed in detail. In general, as mentioned in Section 1, this novel algorithm contains three steps:
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1
Split the full aperture and estimate the target motion parameters in each sub-aperture based on the CDA and the Levenberg-Marquardt (LM) optimization method
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2
Align the envelopes of the average range profiles (ARPs) of every sub-aperture using the MCA
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3
Smooth the estimated deviations by means of locally weighted regression (LOESS)
3.1 Estimating the motion parameters in the sub-apertures
Traditional range alignment methods often focus on the similarity among range profiles without fully considering the target motion information. As a result, these algorithms are vulnerable in low signal-to-noise ratio (SNR) scenarios.
In a relatively short observation time, the target translation motion can be regarded as stable movement, i.e., uniformly accelerated motion; thus, it is rational to model the envelope shift as a second-order polynomial with respect to the slow time. With the adoption of the motion information, the algorithm robustness under a low SNR can be considerably improved. However, in real ISAR imaging, hundreds or even thousands of echoes are accumulated, which leads to a long observation time. To use the motion information, we split the full aperture into a certain number of sub-apertures to obtain a short slow-time span. In each sub-aperture, optimization based on the minimum entropy principle is performed to estimate the velocity and acceleration of the target, which is discussed at length in the following.
Assume every M consecutive echoes of the full aperture are viewed as a sub-aperture and that the echo in the kth sub-aperture is \({x_{k}}\left ({{t_{m}},\hat {t}} \right)\), where tm is the slow time and tn is the fast time (discrete). As mentioned above, the envelope shift in one sub-aperture can be modeled as a second-order polynomial in tm with two unknown parameters, i.e., acceleration a and velocity v:
$$ \begin{aligned} \Phi \left({{t_{m}}} \right) = v{t_{m}} + a{t_{m}}^{2} \end{aligned} $$
(8)
where Φ(tm) represents the envelope shift error.
As (4) indicates, once the optimal values of the unknown parameters are obtained, modulation in the time domain can be carried out to compensate for the range shift:
$$ {}\begin{aligned} {\tilde{x}_{k}}\left({{t_{m}},t_{n}} \right) = {x_{k}}\left({{t_{m}},t_{n}} \right) \cdot \exp \left[ { - j2\pi t_{n}\left({v{t_{m}} + a{t_{m}}^{2}} \right)} \right] \end{aligned} $$
(9)
where \({\tilde {x}_{k}}\) denotes the compensated echo.
To estimate a and v, the minimum entropy method is introduced. By referring to [24], compared with contrast-based methods, this method can attain a good compromise among all kinds of scatters contained in the echo and result in a globally high-quality image. The entropy of the ARP of the current sub-aperture is chosen as the metric, and the unknown parameters are estimated by minimizing the entropy.
The HRRP of one sub-aperture is denoted by
$$ \begin{aligned} HRRP = fft\left\{ {{{\tilde{x}}_{k}}\left({{t_{m}},t_{n}} \right)} \right\} \end{aligned} $$
(10)
where fft{·} represents the FFT operation on tn.
Based on (9) and (10), the ARP of the sub-aperture can be expressed as:
$$ {}\begin{aligned} ARP &= \frac{1}{M}\sum\limits_{m = 0}^{M - 1} {\left|fft\left\{ {{{\tilde{x}}_{k}}\left({{t_{m}},t_{n}} \right)} \right\}\right|}\\ & = \frac{1}{M}\sum\limits_{m = 0}^{M - 1} {\left|fft\! \left\{ {{x_{k}}\left({{t_{m}},t_{n}} \right) \cdot \exp \left[ { \!- j2\pi t_{n}\!\left(\!{v{t_{m}} \,+\, a{t_{m}}^{2}} \right)} \right]} \right\}\right|} \\&\buildrel \Delta \over = \frac{1}{M}\sum\limits_{m = 0}^{M - 1} {|{f_{k}}\left({v,a} \right)|} \end{aligned} $$
(11)
where \({f_{k}}\left ({v,a} \right){\mathrm { = }}fft\left \{ {{x_{k}}\left ({{t_{m}},t_{n}} \right) \cdot \exp \left [ { - j2\pi \hat {t}\left ({v{t_{m}} + a{t_{m}}^{2}} \right)} \right ]} \right \}\) and m is the index of each echo in the sub-aperture. Equation (11) is apparently a one-dimensional real function with a length equal to the number of fast time sampling points N.
According to [18], the entropy of the ARP can be written as
$$ \begin{aligned} E\left(ARP \right) = - \frac{1}{{{S_{{arp}}}}}\sum\limits_{n = 0}^{N - 1} {|{ARP}{|^{2}}\ln |{ARP}{|^{2}}} {\mathrm{ + }}\ln {S_{{arp}}} \end{aligned} $$
(12)
where n is the index of the fast-time sampling points and Sarp is the intensity of the ARP, namely,
$$ \begin{aligned} {S_{{arp}}} = \sum\limits_{n = 0}^{N - 1} {|{ARP}{|^{2}}} \end{aligned} $$
(13)
Combining (11) and (12), it can be seen that the entropy E is a function of the unknown parameters a and v. Therefore, the problem of estimating these parameters can be abstracted to the following form:
$$ \begin{aligned} \left\langle {\hat{a},\hat{v}} \right\rangle = \underset{a,v}{\arg \min } E(a,v) \end{aligned} $$
(14)
where \(\hat {a}\) and \(\hat {v}\) are the estimated values that minimize the entropy.
Equation (14) is a two-dimensional optimization. In the proposed algorithm, the CDA is implemented as the optimization solver, which is an iterative method with outer and inner iterations. The inner ones, with the same number of unknown parameters, are accomplished by minimizing the objective along a certain dimension while fixing the remaining components of the vector of the parameters at their current values. The outer one is not to be terminated until the criteria on the tolerance of the change in the cost function or the preset maximum loop times are met [21].
According to [25], by using a proximal point update technique, the CDA can achieve better robustness in solving one-dimensional sub-problems. Suppose the vector of the unknown parameters is θ and that the CDA procedure is in the pth outer loop and implemented to update the ipth parameter. The CDA updating scheme with the proximal point update can be written as
$$ {}\begin{aligned} \theta_{{i_{p}}}^{p} = \underset{{\theta_{{i_{p}}}}}{\arg \min } \!\left[\! {E\left({{\theta_{{i_{p}}}},\theta_{\ne {i_{p}}}^{p - 1}} \right) \,+\, \frac{1}{{2a_{{i_{p}}}^{p - 1}}}\left|\left|{\theta_{{i_{p}}}} - \theta_{{i_{p}}}^{p - 1}\right|\right|_{2}^{2}}\! \right] \end{aligned} $$
(15)
where \({\frac {1}{{2a_{{i_{p}}}^{p - 1}}}\left |\left |{\theta _{{i_{p}}}} - \theta _{{i_{p}}}^{p - 1}\right |\right |_{2}^{2}}\) is the so-called quadratic proximal term and \(a_{{i_{p}}}^{p - 1}\) serves as a step size and can be any bounded positive number. The addition of the quadratic proximal term makes the function of each sub-problem dominate the original objective around the current iteration and therefore produces increased stability and better convergence properties, especially in the case of non-smooth optimization [25].
For the one-dimensional search in the CDA, the LM algorithm, which has been the de facto standard for most optimization problems [26], is utilized. In the LM method, the cost function in the neighborhood of the current iteration θi can be approximated as
$$ {}\begin{aligned} E\left({{\theta_{i}} + \Delta} \right) \approx L\left(\Delta \right) = E\left({{\theta_{i}}} \right) + \frac{{\partial E\left({{\theta_{i}}} \right)}}{{\partial {\theta_{i}}}} \cdot \Delta + \frac{1}{2}\frac{{\partial {E^{2}}\left({{\theta_{i}}} \right)}}{{\partial \theta_{i}^{2}}}{\Delta^{2}} \end{aligned} $$
(16)
where Δ is the update value and L(Δ) represents the approximation of E(θi+Δ).
With two parameters, i.e., the damping parameter λ and the division factor γ, the LM procedure can be summarized as in Algorithm 1, where the initial values of λ and γ are empirically obtained.
It can be seen from Algorithm 1 that the first and second derivatives of the cost functions are needed to complete the LM method, as illustrated in the Appendix.
Assume that the number of sub-apertures is SN; therefore, there are M·SN echoes in total. The envelope deviations obtained by parameter estimation can be written as
$$ {}\begin{aligned} {\Delta_{\text{sub}}} = {\left[{v_{1}}{\mathbf{t_{s1}}} + {a_{1}}{\mathbf{t_{s1}}}^{2},\ldots,{v_{SN}}{{\mathbf{t_{sSN}}}} + {a_{SN}}{\mathbf{t_{sSN}}}^{2}\right]^{T}} \end{aligned} $$
(17)
where vi and ai (i=1,2,…,SN) denote the estimated parameters and tsm (m=1,2,…,M) represents the slow-time vector with length M of each sub-aperture. Therefore, the total length of vector Δsub is M·SN.
3.2 Aligning the ARPs of the sub-apertures
After the parameter estimation based on the method mentioned in the previous subsection and compensation according to the estimated results, the envelopes in each sub-aperture have been aligned. However, due to the following two reasons, some fine-tuned techniques are required to achieve better alignment.
On the one hand, the parameter estimation method focuses mainly on the alignment in the sub-apertures, and as a result, there exist envelope fluctuations among different sub-apertures; on the other hand, after the misalignment compensation in each sub-aperture, the processing gain of non-coherent integration can be obtained by means of averaging all envelopes in a sub-aperture, which provides useful information for performance enhancement in low SNR scenarios.
To make full use of the non-coherent integration gain and improve the effect of alignment between sub-apertures, we implement the ACM (5) on the ARPs of all sub-apertures. The framework of this fine-tuned technique is demonstrated below.
Suppose the estimated motion parameters of the kth sub-aperture are \(\hat {v}\) and \(\hat {a}\); therefore, the compensated echoes of this sub-aperture can be written as
$$ \begin{aligned} {x_{ck}}\left({{t_{m}},t_{n}} \right) = {\tilde{x}_{k}}\left({{t_{m}},t_{n}} \right) \cdot \exp \left[ {j2\pi \left({\hat{v}{t_{m}} + \hat{at}_{m}^{2}} \right)} \right] \end{aligned} $$
(18)
where xck(tm,tn) denotes the aligned echoes of the kth sub-aperture. The ARP of the kth sub-aperture can be expressed as
$$ \begin{aligned} h = \frac{1}{M}\sum\limits_{m = 0}^{M - 1} {|fft\left\{ {{x_{ck}}\left({{t_{m}},t_{n}} \right)} \right\}|} \end{aligned} $$
(19)
Again, M is the number of echoes in one sub-aperture, and fft{·} represents application of the FFT along the fast-time direction.
Assume that the number of sub-apertures is SN. After the ACM, the SN values of the envelope deviations are obtained, i.e.,
$$ \begin{aligned} {\Delta_{\text{ave}}} = {[{\Delta_{1}},{\Delta_{2}},\ldots,{\Delta_{SN}}]^{T}} \end{aligned} $$
(20)
where Δi(i=1,2,…,SN) represents the envelope deviation of each sub-aperture’s ARP and Δave denotes the vector of all deviations.
The length of vector Δave should be extended to M when carrying out compensation for each echo. The longer version of Δave can be expressed as
$$ \begin{aligned} {\Delta_{\text{ave}}} = {[\overbrace {\underbrace{\Delta_{1},\ldots,\Delta_{1}}_{M},\ldots,\underbrace{\Delta_{SN},\ldots,\Delta_{SN}}_{M}}^{M \cdot SN}]^{T}} \end{aligned} $$
(21)
3.3 Total error fitting using locally weighted regression
In the previous two subsections, the envelope deviations in each sub-aperture and between every two sub-apertures were obtained through optimization and the ACM, respectively. With the combination of (17) and (21), the total envelope deviations can be expressed as
$$ \begin{aligned} {\Delta_{\text{total}}} = {\Delta_{\text{sub}}} + {\Delta_{\text{ave}}} \end{aligned} $$
(22)
where Δtotal denotes the total deviations of the envelopes.
After aligning the ARPs of all sub-apertures, in general, the misaligned envelopes can be calibrated well. To achieve a higher performance, some fine tuning is still required. Figure 1 shows an estimation result for a full aperture’s envelope misalignment error, where relatively accurate error estimation could be achieved; however, according to the enlarged error estimation curve, there exist step changes between the two sub-apertures, which can undermine the imaging quality. Because each sub-aperture is aligned as a whole by the method proposed in the previous subsection, the step changes are inevitable.
These step changes can be easily smoothed and eliminated by some curve-fitting techniques. By referring to [27] and [11], we use locally weighted regression (LOESS) to smooth the step changes between every two adjacent sub-apertures. The procedure of LOESS is briefly introduced in the following:
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Suppose that there are N points to be fitted, i.e., [x1,…,xN]T. For the ith point xi, put N·fr(0<fr≤1) points into its neighborhood Ωi.
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Determine the weight wk(xi),k=1,2,…,N·fr for the weighted least squares (WLS) in the neighborhood of xi using tricube functions.
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Because of the short length of the neighborhood, it is reasonable to model the points in the neighborhood as quadratic. After conducting WLS, the ith fitted value can be expressed as \({y_{i}} = {\beta _{0}} + {\beta _{1}}{x_{i}} + {\beta _{2}}x_{i}^{2}\), where βi,i=0,1,2 are the estimated coefficients of the quadratic polynomial.
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Repeat steps 1–3 for all N points to obtain N fitted values.
After LOESS, accurate envelope deviations can be obtained, and good alignment can be achieved. The whole framework is shown in Fig. 2.
3.4 Optimal selection and computational complexity
This subsection discusses how to choose the number of sub-apertures and the computational complexity of the proposed algorithm.
By referring to [28] and [29], we develop the following adaptive selection method:
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Initialize SN. The principle of initializing SN is that the envelope deviation in one sub-aperture should not exceed half the range unit, i.e., c/4Fs, where c is the velocity of light and Fs is the sampling rate.
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Implement the minimum entropy optimization. The estimated error of the pth sub-aperture is ΔRp(tm), where tm represents the slow time of the current aperture.
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Double SN and implement the minimum entropy optimization. The pth aperture in (2) is split into two equal sub-apertures, and the estimated envelope errors in each one can be expressed as ΔRp1(tm) and ΔRp2(tm). They can be denoted as ΔRpNew(tm)=[ΔRp1(tm),ΔRp2(tm)], and the length of tm is equal to that in (2).
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If the following condition is satisfied, the initialized SN can be used; otherwise, return to (3) and repeat.
$$ \begin{aligned} &|\max \left(\Delta {R_{p}}({t_{m}}) - \Delta {R_{pNew}}({t_{m}})\right) \\&- \min \left(\Delta {R_{p}}({t_{m}}) - \Delta {R_{pNew}}({t_{m}})\right)| \le c/4{F_{s}} \end{aligned} $$
In addition to the selection scheme above, in real cases, it is also important to jointly consider prior information, such as the motion parameters, the expected gain through accumulated echoes, and the computational burden (a larger number of apertures means a higher computational complexity). Moreover, when the accumulated echoes are insufficient, a certain number of echoes can be reused by two different sub-apertures. In general, the selection of the number of sub-apertures requires thorough consideration.
In the following part of this subsection, the computational complexity is briefly numerically analyzed, with the detailed derivation shown in the Appendix. As mentioned above, the proposed algorithm contains three parts. In the first part, we use the CDA to solve the optimization in each sub-aperture. In each loop, the computational burden is devoted mainly to obtaining the entropy and its first and second derivatives. The computational complexity of these operations is denoted as
$$ \begin{aligned} N_{mul}^{OP} \sim \Theta \left({M \cdot N \cdot {{\log }_{2}}N} \right) \end{aligned} $$
(23)
$$ \begin{aligned} N_{add}^{OP} \sim \Theta \left({M \cdot N \cdot {{\log }_{2}}N} \right) \end{aligned} $$
(24)
where \(N_{mul}^{OP}\) and \(N_{add}^{OP}\) represent the numbers of multiplications and additions of the proposed algorithm’s optimization procedure, respectively, and M and N denote the numbers of Doppler and range cells of each sub-aperture, respectively. It also can be seen that each sub-aperture is independent; thus, parallel programming can be used to execute all optimizations concurrently.
The second step is the ACM in each loop, of which the computational burden arises mainly from obtaining the correlation function between the current profile and the reference. The computational complexity of this step is
$$ \begin{aligned} N_{mul}^{ACM} \sim \Theta \left({SN \cdot N \cdot {{\log }_{2}}N} \right) \end{aligned} $$
(25)
$$ \begin{aligned} N_{add}^{ACM} \sim \Theta \left({SN \cdot N \cdot {{\log }_{2}}N} \right) \end{aligned} $$
(26)
where \(N_{mul}^{ACM}\) and \(N_{add}^{ACM}\) represent the numbers of multiplications and additions of the proposed algorithm’s ACM procedure, respectively, N denotes the number of range cells of each sub-aperture, and SN is the number of sub-apertures.
With reference to [30], LOESS also decomposes the problem into independent pieces, with all operations being completely parallel. Therefore, with well-designed concurrent programming, the computational complexity of the LOESS step is equal to that of a weighted least squares operation with few points.